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The long-run Phillips curve: A structural VAR investigation
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The Long-Run Phillips Curve: A Structural VAR Investigation Luca Benati
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Journal of Monetary Economics
Received date: 17 March 2013 Revised date: 23 June 2015 Accepted date: 28 June 2015 Cite this article as: Luca Benati, The Long-Run Phillips Curve: A Structural VAR Investigation, Journal of Monetary Economics, http://dx.doi.org/10.1016/j.jmoneco.2015.06.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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The Long-Run Phillips Curve: A Structural VAR Investigation Luca Benatia,∗†
2
a
University of Bern
Received Date; Received in Revised Form Date; Accepted Date
3
Abstract
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Both cointegration methods, and non-cointegrated structural VARs identified based on either
5
long-run restrictions, or a combination of long-run and sign restrictions, are used in order to explore
6
the long-run trade-off between inflation and the unemployment rate in the post-WWII U.S., U.K.,
7
Euro area, Canada, and Australia. Overall, neither approach produces clear evidence of a non-
8
vertical trade-off. The extent of uncertainty surrounding the estimates is however substantial, thus
9
implying that a researcher holding alternative priors about what a reasonable slope of the long-run
10
trade-off might be will likely not see her views falsified.
11
Keywords: Inflation; unemployment; Phillips curve; unit roots; cointegration; Bayesian VARs; structural VARs; long-run
12
restrictions; sign restrictions.
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JEL classification: E30; E32 ∗
Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland. Email: [email protected]; Phone: (+41)-031-631-5598 † I wish to thank the Co-Editor (Ricardo Reis) and an anonymous referee for extremely helpful comments and suggestions. Thanks to Fabio Canova and Russell Cooper for helpful discussions; Julio Carrillo, Harald Uhlig and seminar participants at CREST, the European University Institute, Norges Bank, the Swiss National Bank’s 2012 research conference ‘Policy Challenges and Developments in Monetary Economics’, and the 2013 Workshop on Empirical Macroeconomics held at the University of Ghent for comments; and Juan Rubio-Ramirez for helpful suggestions. Usual disclaimers apply.
The Long-Run Phillips Curve: A Structural VAR Investigation
1
1.
2
Introduction
2
The long-run Phillips curve (henceforth, LRPC)—describing the alternative combinations
3
of inflation and the unemployment rate an economy can achieve in equilibrium—plays a
4
central role in monetary policy. If the LRPC is vertical there is no long-run trade-off between
5
the two variables, and the central bank ought therefore simply aim at keeping inflation low
6
and stable. If, on the other hand, the LRPC were negatively sloped, this would open up the
7
possibility of trading off a permanent increase in inflation for a permanent reduction in the
8
unemployment rate.
9
In spite of such a crucial role played by the slope of the LRPC in monetary policymaking,
10
surprisingly little econometric work has been devoted to investigating the nature of the long-
11
run trade-off. In particular, the only existing investigation of the LRPC based on structural
12
VAR methods is King and Watson (1994)’s ‘revisionist econometric history’ of the post-
13
WWII U.S. Phillips curve, which produced evidence of a negative, and statistically significant
14
long-run trade-off conditional on aggregate demand-side shocks.1
15
This paper uses both cointegration methods, and non-cointegrated structural VARs fea-
16
turing either one or more permanent inflation shocks, in order to investigate the nature of
17
the LRPC for the post-WWII U.S., U.K., Euro area, Canada, and Australia. Since, as ex-
18
tensively discussed by King and Watson (1994)—who, in turn, built on Sargent (1971) and
19
Lucas (1972a)—the very nature of the LRPC involves permanent variation in (i.e., perma-
20
nent shocks to) inflation and the unemployment rate, it starts by exploring a ‘strong-form 1
Some papers (see, e.g., Bruno and Easterly, 1998) have eschewed time-series single-country studies, focusing instead on the impact on output growth of ‘discrete high inflation crises’, defined as periods in which inflation is above some threshold. Taking 40 per cent to be such threshold, Bruno and Easterly found that ‘growth falls sharply during discrete high inflation crises, then recovers rapidly and strongly after inflation falls’. The key problem with this type of evidence (which is common to the entire literature on growth regressions) is the difficulty of establishing a clear direction of causality. Since high-inflation episodes cannot be regarded as random experiments, the very same factors causing high inflation (e.g., an incompetent government mismanaging the economy) may well be behind the fall in average output growth during the episode.
The Long-Run Phillips Curve: A Structural VAR Investigation
3
1
LRPC’, in which the two series are possibly cointegrated. Evidence that this may be the
2
case is overall weak, and we therefore move to investigating, via structural VAR methods,
3
a ‘medium-form LRPC’ in which permanent inflation shocks, which are not further disen-
4
tangled, are allowed to have a permanent impact on the unemployment rate. For neither
5
country it is possible to reject the null hypothesis of a vertical LRPC. The extent of uncer-
6
tainty is however substantial, thus implying that a researcher with an alternative prior on
7
what a reasonable slope of the LRPC may be would likely not see her views falsified. Finally,
8
since absence of a ‘medium-form LRPC’ is not incompatible with the notion that some of
9
the shocks having a permanent impact on inflation may generate a non-vertical long-run
10
trade-off, the paper proceeds to explore a ‘weak-form LRPC’, in which permanent inflation
11
shocks are disentangled into demand- and supply-side ones by imposing sign restrictions on
12
their impacts on the endogenous variables at t=0. For neither of the four identified shocks
13
there is any evidence that it may generate a non-vertical LRPC. This is true, in particular,
14
for the monetary policy shock, which in the light of the central role played by the slope of the
15
LRPC in monetary policy, is the single most important one among those identified herein.
16
The present work expands upon King and Watson (1994) under several respects. First,
17
King and Watson’s analysis was entirely based on a bivariate VAR for the first differences of
18
inflation and the unemployment rate. As shown by Evans (1994) in his comment, however,
19
even based on their identification strategy, evidence based on trivariate VARs was sometimes
20
significantly different, pointing in some cases towards a vertical long-run Phillips curve.
21
This paper therefore uses six-variables VARs featuring a broader informational content, in
22
particular about the stance of monetary policy and the state of the business cycle. Second, in
23
the years since 1994 structural VAR econometrics has seen important developments in terms
24
of identification. When King and Watson wrote, short-run restrictions were still either
25
of the inertial type (i.e., based on imposing zeros in the impact matrix of the structural
26
shocks at t=0), or they were based on the notion of calibrating some of the impacts based
The Long-Run Phillips Curve: A Structural VAR Investigation
4
1
on information extraneous to the VAR. As previously mentioned, in the present work the
2
non-cointegrated VARs are identified based on either long-run restrictions, or a combination
3
of long-run and sign restrictions. Finally, beyond the U.S. the paper consider four other
4
countries which have exhibited, over the sample periods considered herein, strong evidence
5
of permanent variation in inflation.
6
The fact that in order to be able to identify the slope of the LRPC inflation ought to
7
have a unit root (Sargent, 1971; Lucas, 1972a; King and Watson, 1994) immediately defines
8
the relationship between the present work and Svensson’s (2015) recent investigation of the
9
slope of the LRPC for a group of inflation-targeting countries. As we previously documented
10
(Benati, 2008), under inflation targeting evidence of a unit root in inflation has entirely
11
vanished, and inflation has instead been, in most cases, statistically indistiguishable from
12
white noise. This logically implies that Svensson’s estimates are uninformative about the
13
slope of the LRPC for these countries.
14
The paper is organized as follows. The next section discusses the three different notions
15
of long-run Phillips curve considered in this paper. Section 3 discusses the data, and reports
16
results from unit root tests. Section 4 reports and discusses the evidence, starting from the
17
results from cointegration analysis, and then moving to evidence based on non-cointegrated
18
structural VARs featuring one and, respectively, four permanent inflation shocks. Section 5
19
concludes.
20
2.
21
22
Three Alternative Notions of the Long-Run Phillips Curve In what follows we consider three alternative forms of the long-run Phillips curve (hence-
forth, LRPC).
The Long-Run Phillips Curve: A Structural VAR Investigation
1
2.1.
5
A strong-form LRPC
2
First, a ‘strong form of the LRPC’, in which inflation and the unemployment rate are
3
(possibly) cointegrated. Such a strong form implies that (i) all shocks having a permanent
4
impact on inflation also have a permanent impact on the unemployment rate (and vice
5
versa), and (ii ) for either of these shocks, the ratio between the long-run impacts on the
6
two variables is exactly the same (thus implying that inflation and unemployment share the
7
same stochastic trend across all possible shocks). Standard cointegration methods are used
8
in order to investigate whether the data are consistent with such a strong from of the LRPC.
9
2.2.
A medium-form LRPC
10
Since either (i ) or (ii) may well be violated, the paper also considers two weaker forms
11
of the LRPC, starting from a ‘medium-form LRPC’ conceptually in line with King and
12
Watson (1994), in which the unit root component of the unemployment rate is driven by
13
both idiosyncratic shocks, and permanent shocks to inflation. It is not difficult to think of
14
circumstances under which this may be the case. For example, permanent shocks to the
15
tax rate on labor should lead to increases in the natural rate of unemployment, but—being
16
shocks to the level of the tax rate, as opposed to its rate of change—they should only have
17
a transitory impact on inflation. By the same token, given the widely documented stylized
18
fact that unemployment rates differ systematically by age/sex/race groups, secular changes
19
in the composition of the labor force should lead to permanent changes in the overall natural
20
rate of unemployment, but at the same time they should not have any permanent impact on
21
inflation. So the bottom line is that it is not difficultfs to think of circumstances in which
22
the strong form of the LRPC does not hold, but the medium-form may hold.
23
In the spirit of King and Watson (1994), in what follows the paper will search for a
24
medium-form LRPC based on structural VARs featuring, beyond the first differences of
25
inflation and the unemployment rate, a measure of the output gap, the consumption/GDP
The Long-Run Phillips Curve: A Structural VAR Investigation
6
1
ratio, a long-short spread, and (the first difference of) the short-term monetary policy rate.
2
These variables expand the informational content of King and Watson’s original bivariate
3
VAR along several dimensions. The short rate and, to a lesser extent, the long-short spread
4
contain information about the monetary policy stance. As recently shown by Kurmann
5
and Otrok (2013), the long-short spread possesses a strong informational content for future
6
movements in technology. Finally, the consumption/GDP ratio and the output gap measure
7
can be regarded as two ‘noisy estimates’ of the state of the business cycle. There are two
8
reasons for considering these additional variables. First, King and Watson’s analysis was
9
entirely based on a bivariate VAR for the first differences of inflation and the unemployment
10
rate, but, as shown by Evans (1994) in his comment, even based on their identification
11
strategy, evidence based on trivariate VARs was sometimes significantly different, and in
12
some cases it pointed towards a vertical long-run Phillips curve. Second, as recently discussed
13
by Forni and Gambetti (2014), small VARs with a limited informational content may well
14
produce unreliable results. A single, ‘aggregate’ permanent inflation shock will be identified
15
via standard long-run restrictions, and we will then explore whether such a shock has a
16
non-zero long-run impact on the unemployment rate.
17
2.3.
A weak-form LRPC
18
Finally, the paper explores a ‘weak-form LRPC’, in which the just-mentioned ‘aggregate’
19
permanent inflation shock is disentangled into four different disturbances, which are allowed
20
to give rise to idiosyncratic (i.e., shock-specific) long-run trade-offs between inflation and
21
unemployment. This is achieved by imposing Canova and Paustian’s (2011) DSGE-based
22
‘robust sign restrictions’ on the impact of the shocks on the endogenous variables at t=0. The
23
paper then proceeds to explore whether any of these shocks has a statistically significant long-
24
run impact on the unemployment rate. Although four different shocks having a permanent
25
impact on inflation are identified, most of the focus is placed upon disturbances of a monetary
The Long-Run Phillips Curve: A Structural VAR Investigation
7
1
nature. The reason for doing so is that the motivation behind all explorations of the slope
2
of the LRPC has always been understanding whether the monetary authority might be
3
able to permanently lower the unemployment rate by engineering a permanently higher rate
4
of inflation. Under this respect, the long-run Phillips trade-offs which might originate from
5
permanent inflation shocks of a non-monetary nature are therefore irrelevant. The framework
6
used in order to explore the weak-form LRPC has an obvious connection to the one used
7
to explore the medium-form one. Whereas, by construction, the permanent inflation shocks
8
identified by the ‘medium-form LRPC’ VAR explain 100 per cent of the infinite long-run
9
variance of inflation, the four permanent inflation shocks identified by the ‘weak-form LRPC’
10
VAR jointly explain all of the long-run variance of inflation.
11
3.
The Data
12
As mentioned in the Introduction, the possibility of identifying the long-run impact on
13
the unemployment rate of permanent inflation shocks crucially hinges on the fact that both
14
series contain an I(1) component. As documented by Benati (2008), however, evidence of high
15
inflation persistence is weak-to-non-existent for all sample periods which are not dominated
16
by the Great Inflation episode (and especially so for monetary regimes such as inflation
17
targeting and European Monetary Union, henceforth, EMU). In what follows we therefore
18
consider the following sample periods: for the Euro area, 1970Q1-1998Q4; for the U.K.,
19
1972Q2-1992Q3; for Canada, 1961Q2-1990Q4; for Sweden 1970Q1-1992Q4; for Australia,
20
1969Q3-1994Q2; and for Japan the period following the collapse of Bretton Woods.2 Finally, 2
For the Euro area, EMU started in January 1999, whereas Euro area data are only available starting from 1970Q1. For the U.K., June 1972 marked the floating of the pound vis-` a-vis the U.S. dollar, whereas inflation-targeting was introduced in October 1992. As shown by Benati (2008), before the June 1972 floating of the pound U.K. inflation exhibited quite significantly lower persistence. In Canada, inflation targeting was introduced in February 1991, whereas 1961Q1 is when Canadian national account data first become available. In Sweden, inflation targeting was introduced in January 1993, whereas the unemployment rate is only available since 1970Q1. In Australia, the current inflation targeting regime was never clearly announced, and we therefore follow Bernanke et al., 1999, in marking its start in 1994Q3. On the other hand, the short rate is available since 1969Q3. As for Japan, since 1971 the Japanese government has not explicitly introduced any new monetary regime, and has instead relied on a generic committment to price
The Long-Run Phillips Curve: A Structural VAR Investigation
8
1
for the U.S., as discussed below, several alternative sample periods are considered. All of the
2
data and their sources are described in Appendix A.
3
4
[Table 1 here] 3.1.
Results from unit root tests
5
Table 1 reports, for either country, bootstrapped p-values for Elliot et al.’s (1996) unit
6
root tests for inflation, the unemployment rate, and the short-term interest rate.3 Based on
7
either the CPI or the GDP deflator, the null of a unit root in inflation cannot be rejected
8
for either the Euro area, the U.K., Canada, or Australia, with the p-values being uniformly
9
greater than 10 per cent, in most cases comfortably so. Evidence for Sweden and Japan,
10
on the other hand, is not clear-cut, and in what follows I therefore exclude these countries
11
from the analysis. Finally, for the U.S. evidence of a unit root based on the entire sample
12
period since January 1959 is strong based on the PCE deflator, and it is just slightly less
13
so based on the GDP deflator. Splitting the full sample in August 1979, however, clearly
14
shows how evidence of a unit root entirely originates from the pre-Volcker period, and it is
15
instead absent from the second sub-period, for which the p-values are uniformly very low.
16
This suggests that results for the U.S. based on the full sample period should be viewed with
17
suspicion, and in what follows I will therefore uniquely focus on the pre-Volcker period.
18
Evidence of a unit root in the unemployment rate is strong for either the Euro area, the
19
U.K., Canada, or Australia. For the U.S., evidence is compatible with the notion that the
20
unemployment rate contains a unit root in both sub-periods, whereas results for the entire
21
sample period are weak.
22
As for the output gap measure, the consumption/GDP ratio, and the spread, Elliot et stability. 3 Estimated models include an intercept, but no time trend. For either series, p-values have been computed by bootstrapping 10,000 times estimated ARIMA(p,1,0) processes. In all cases, the bootstrapped processes are of length equal to the series under investigation. As for the lag order, p, since, as it is well known, results from unit root tests may be sensitive to the specific lag order which is being used, for reasons of robustness we consider four alternative lag orders, 2, 4, 6 and 8 quarters (for the U.S., for which we use monthly data, the corresponding lag orders are 6, 12, 18, and 24).
The Long-Run Phillips Curve: A Structural VAR Investigation
9
1
al.’s (1996) tests clearly point towards these series being stationary (these results are not
2
reported for reasons of space, but they are available upon request). As for the short rate,
3
the evidence in Table 1 strongly points towards a unit root for all countries, and based on
4
either lag order, with the single exception of the U.K.. In what follows, the short rate will
5
therefore enter the VAR in levels for the U.K., and in first differences for all other countries.4 [Table 2 here]
6
7
4.
Evidence We start by exploring the possible presence of a ‘strong-form LRPC’ via standard coin-
8
9
10
tegration methods. 4.1.
Results based on cointegration analysis
11
Table 2 reports bootstrapped p-values for Johansen’s trace test of the null of no coin-
12
tegration, against the alternative of one or more cointegrating vectors, for either inflation
13
and unemployment, inflation and the short rate (with the single exception of the U.K., for
14
which results from unit root tests strongly reject the null of a unit root in the short rate),
15
or inflation, the unemployment rate, and the short rate. Following Cavaliere, Rahbek, and
16
Taylor (2012), p-values have been computed by bootstrapping the VAR estimated for the
17
first difference of the relevant vector of series.5 For the U.S. no evidence of cointegration 4
For reasons of robustness we have also considered tests based on Phillips’ Zρ and Zt statistics (for details, see Section 3 in the online appendix). These results are almost uniformly in line with those from Elliot et al.’s tests. The main difference— which is however irrelevant for our purposes—pertains to U.S. inflation for the full sample period since January 1959, for which the null of a unit root is strongly rejected (on the other hand, in line with the results from Elliot et al.’s tests, results for the pre-Volcker period clearly point towards a unit root in U.S. inflation based on either price index). 5 To be clear, the VAR which is being bootstrapped is not a cointegrated VAR, that is, it is equal to the VECM representation without the error-correction term. Given the vector of the relevant series Yt , we start by selecting the lag order for cointegration tests as the maximum between the lag orders selected based on the Schwartz and the Hannan-Quinn criteria (we eschew the AIC since, with I(1) variables, it is an inconsistent lag selection criterion, see Luetkepohl (1991)), and we perform Johansen’s trace test of the null of no cointegration. Then, we estimate the VAR for ΔYt ; we bootstrap it 10,000 times, thus generating bootstrapped artificial series ΔY˜tj ; based on each of them we compute corresponding bootstrapped artificial series Y˜tj (that is, those for the levels of the series); and finally, based on each of them we perform the same
The Long-Run Phillips Curve: A Structural VAR Investigation
10
1
between inflation and the unempoyment rate is detected. On the other hand, conceptually in
2
line with Fama (1975) and Barsky (1987), there is strong evidence of cointegration between
3
inflation and the Federal Funds rate. The median estimate of the second element of the
4
normalized cointegration vector is -1.018, with the p-value for rejecting the null hypothesis
5
that the cointegration vector is [1; -1] being equal to 0.423. For the Euro area, Canada,
6
and Australia on the other hand, bivariate cointegration tests do not detect any evidence of
7
cointegration between inflation and the short rate.6 Finally, for either the U.S., the Euro
8
area or, Canada there is strong evidence of cointegration between inflation, unemployment,
9
and the short rate, whereas for the U.K. evidence points, at the 10 per cent level, towards
10
cointegration between inflation and the unemployment rate.
11
For the U.S., the Euro area and Canada, for which evidence of cointegration between
12
the three series was detected, the table also reports bootstrapped p-values for testing the
13
null hypothesis of one single cointegrating vector, versus the alternative of two cointegrating
14
vectors.7 Whereas the p-value for Canada, at 0.2896, is very far from being significant at any
15
conventional level, those for the Euro area and the U.S., at 0.0531 and 0.0728, respectively,
16
point towards the presence of an additional cointegrating vector at the 10 per cent level.
17
[Figure 1 here]
18
For the Euro area and the U.K., for which evidence of cointegration in the bivariate
19
representation for Yt = [π t , Ut ] was detected at the 5 per cent level, Figure 1 reports
20
the bootstrapped distributions of the ratio between the permanent impacts of the common
21
shock on the unemployment rate and inflation, respectively (that is: the slope of the longtrace test we previously computed based on the actual data, thus building up the empirical distribution of the trace statistic under the null of no cointegration. Based on this distribution, we then compute critical values (not reported here) and p-values. 6 Although at first sight puzzling—taken at face value, these results imply a rejection of the Fisher hypothesis that permanent shifts in inflation should map one-to-one into corresponding shifts in interest rates—it ought to be stressed that empirical evidence on the violation of the Fisher hypothesis is widespread. 7 Details of the bootstrapping procedure are the same as before, with the only difference that, instead of bootstrapping the estimated VAR for ΔYt under the null of no cointegration, we bootstrap the VECM estimated conditional on there being one single cointegrating vector.
The Long-Run Phillips Curve: A Structural VAR Investigation
11
1
run Phillips curve implied by the estimated cointegrating vector); and of the elements of
2
the loading vector on the cointegration residual in the VECM representation. For either
3
country, the long-run impact on the unemployment rate induced by the common shock
4
(normalized in such a way as to induce a one per cent permanent increase in inflation) is
5
highly statistically significantly different from zero, as implied by the results from the trace
6
test, and as testified by the fact that mass of the bootstrapped distribution is pretty much
7
away from zero. Further, the estimated impact is also sizeable, with median estimates equal
8
to -1.074 for the Euro area and -1.537 for the U.K., and 90 per cent-coverage bootstrapped
9
confidence intervals equal to [-1.327; -0.866] and [-2.278; -0.759], respectively.
10
Taken at face value, these results point towards the presence of a negative and sizeable
11
long-run trade-off induced by the common permanent shock in both the Euro area and the
12
U.K.. The next two sub-sections provide some perspective on this, by exploring the extent
13
to which these results may just be statistical flukes, and then focusing on whether—even
14
assuming that the results are genuine—they are in fact compatible with the notion of a
15
trade-off which can be exploited by policymakers.
16
4.1.1.
17
The case of two independent random walks A first possibility is that the results for
18
the Euro area and the U.K. are statistical flukes, possibly due to the bad performance of
19
Johansen’s procedure in small-samples. In order to assess this possiblity we consider five sets
20
of 10,000 Monte Carlo simulations of lengths equal to the actual sample lengths for the five
21
countries. For each simulation, two independent random walks are randomly generated, and
22
the same procedure which has previously been applied to the actual data is used in order to
23
test for the null of no cointegration between them. Ideally, out of the 10,000 simulations, the
24
fraction of bootstrapped p-values below x per cent should be equal to x per cent. As the
25
results reported in Table A.2 in the online appendix8 show, the procedure used herein gets 8
Monte Carlo evidence on the performance of Johansen’s procedure
Supplementary material is available online at Science Direct.
The Long-Run Phillips Curve: A Structural VAR Investigation
12
1
quite remarkably close to this ideal: the fraction of simulations for which the bootstrapped
2
trace statistic incorrectly rejects the null of no cointegration at a given significance level
3
ranges between 11.3 and 11.9 per cent at the 10 per cent level; between 5.5 and 6.0 per cent
4
at the 5 per cent level; and between 1.1 and 1.3 per cent at the 1 per cent level.
5
[Table 3 here]
6
Taking the non-cointegrated structural VARs as data-generation processes Al-
7
though the procedure performs well conditional on a data-generation process (henceforth,
8
DGP) in which the series of interest are independent random walks, it is an open question
9
how well it might perform conditional on DGPs such as the non-cointegrated structural VARs
10
featuring permanent inflation shocks estimated in Sections 4.2 and 4.3, respectively. Since
11
these results will produce no evidence of a non-vertical long-run Phillips trade-off conditional
12
on any kind of permanent inflation shock, it is of interest to explore the performance of Jo-
13
hansen’s procedure conditional on these DGPs. Table 3 reports evidence on this, by showing
14
the fraction of simulations for which the bootstrapped trace statistic incorrectly rejects the
15
null of no cointegration at a given significance level. As the tables shows, Johansen’s proce-
16
dure tends to spuriously detect cointegration a non-negligible fraction of the times. Taking
17
the estimated Bayesian structural VARs featuring four permanent inflation shocks as DGPs,
18
for example, for the Euro area and the U.K., the fractions of bootstrapped p-values for the
19
trace statistic for testing the null of no cointegration between inflation and the unemploy-
20
ment rate which are smaller than 10 per cent are equal to 26.9 and 23.6 per cent respectively.
21
This means that, at the 10 per cent level, the trace test would incorrectly reject the null of no
22
cointegration between inflation and unemployment about one-fourth of the times. As for the
23
trace test of cointegration between inflation, unemployment, and the short rate, the fraction
24
of bootstrapped p-values smaller than 10 per cent range between 45.4 and 72.1 per cent,
25
thus essentially pointing towards the unreliability of such tests conditional on these DGPs.
26
Results based on taking Bayesian VARs featuring a single permanent inflation shock as the
The Long-Run Phillips Curve: A Structural VAR Investigation
13
1
DGP are in line with those discussed so far for the trace test of no cointegration between
2
inflation, unemployment, and the short rate, and are instead significantly better for the test
3
of no cointegration between inflation and unemployment.
4
Overall, these results suggest that evidence such as that reported in Table 2 should be
5
discounted, as these results might as well be due to the limitations of cointegration tests
6
conditional on these specific DGPs.
7
4.1.2.
The adjustment dynamics implied by the estimated loading vectors
8
Even ignoring the results reported in Table 3, the evidence shown in the last two columns
9
of Figure 1 is hardly compatible with the notion of a long-run trade-off a policymaker might
10
be able to purposefully exploit in order to permanently decrease the unemployment rate
11
by engineering a higher equilibrium inflation rate.9 The bootstrapped distributions of the
12
estimates of the loading on the first difference of the unemployment rate for the error-
13
correction term in the VECM reported in Figure 1, indeed, are extremely small for either
14
country. Further, in the case of the Euro area the estimated loading is insignificantly different
15
from zero at conventional levels (for the U.K., on the other hand, it is significant at the 10
16
per cent level, but not at the 5 per cent level). This implies that, following a shock to the
17
common stochastic trend, the bulk of the dynamic adjustment to disequilibrium takes place
18
through movements in inflation, rather than via movements in the unemployment rate, which
19
is difficult to reconcile with the notion of an exploitable Phillips trade-off. 9
As pointed out by Evans (1994, p. 222) in his comment on King and Watson (1994), ‘[...] for a trade-off to be viewed as exploitable, a decision-maker must have confidence that unemployment can be reduced by engineering a higher rate of inflation [...] It would be of little comfort to many politicians if the Phillips curve trade-off simply implied that if unemployment instead turned higher, at least inflation would be lower.’
The Long-Run Phillips Curve: A Structural VAR Investigation
1
2
3
4.2.
14
Evidence from non-cointegrated SVARs featuring a single permanent inflation shock
We start by considering VARs featuring a single permanent inflation shock. For either country the VAR(p) model Yt = B0 + B1 Yt−1 + ... + Bp Yt−p + ut ,
4
E[ut ut ] = Ω
(1)
5
is estimated, where Yt ≡ [yt , Δπ t , ΔUt , ΔRt , St ] for the Euro area, Canada, and Australia,
6
and Yt ≡ [yt , Δπ t , ΔUt , Rt , CYt , St ] for the U.K., with yt , π t , Ut , Rt , CYt , and St being the
7
output gap, inflation, the unemployment rate, the short rate, the consumption/GDP ratio,
8
and the long-short spread, respectively. The U.S. is the only country for which Johansen’s
9
tests produces evidence of cointegration between the short rate and inflation (see Table
10
2), with the bootstrapped, bias-corrected estimate of the second element of the normalized
11
cointegrating vector being equal to -1.018, and with a p-value of 0.423 for rejecting the null
12
hypothesis that it is equal to -1. For the U.S. Yt is therefore set to Yt ≡ [yt , Δπ t , ΔUt , CRt ,
13
Rt -π t , St ] , where Rt -π t is the ex post real rate (i.e., the theoretical cointegration residual).
14
Finally, for the Euro area, Canada, and the U.K. the lag order is set to p=4. For the U.S.,
15
for which the data are monthly, the lag order is set to p=12.
16
For reasons of robustness, the VARs are estimated based on both Classical and Bayesian
17
methods. Since the results produced by the two approaches are qualitatively the same, and
18
numerically very close, in what follows only those based on the Bayesian approach are re-
19
ported and discussed. The online appendix, however, contains a detailed description of the
20
Classical estimation methodology, and reports the full set of results (see Tables A.3 and A.8).
21
4.2.1.
Bayesian estimation
22
Working within a Bayesian context, the reduced-form VAR (1) is estimated as in Uhlig
23
(1998, 2005). Specifically, Uhlig’s approach is exactly followed in terms of both distribu-
24
tional assumptions—the distributions for the VAR’s coefficients and its covariance matrix
The Long-Run Phillips Curve: A Structural VAR Investigation
15
1
are postulated to belong to the Normal-Wishart family—and of priors. For estimation de-
2
tails the reader is therefore referred to either the Appendix of Uhlig (1998), or to Appendix
3
B of Uhlig (2005). Results are based on 10,000 draws from the posterior distribution of the
4
VAR’s reduced-form coefficients and the covariance matrix of its reduced-form innovations
5
(the draws are computed exactly as in Uhlig (1998, 2005)).
6
4.2.2.
Identification
7
A single permanent inflation shock is identified based on the restriction that it is the
8
only shock impacting upon inflation in the infinite long run. Since this shock is exactly
9
identified, drawing from the VAR’s reduced form parameters is sufficient in order to get the
10
impulse-response functions, because for each draw of the reduced form parameters there is
11
just one (normalized) impulse-response function.
12
[Table 4 here]
13
[Figure 2 here]
14
4.2.3.
Evidence
15
Figure 2 shows, for either country, the distribution of the long-run impact on the unem-
16
ployment rate of a one per cent permanent shock to inflation, together with the distribution
17
of the fraction of the permanent component of unemployment which is explained by the per-
18
manent inflation shock. Table 4 reports the median and the 90%-coverage percentiles of the
19
distribution of the long-run impact on the unemployment rate of a one per cent permanent
20
shock to inflation, together with the fraction of the mass of the distribution for which the
21
impact is estimated to be negative.
22
Results are consistent across countries, and for neither of them do they allow to reject the
23
‘natural’ null hypothesis of a vertical long-run Phillips curve. In particular, for either coun-
24
try the 90 per cent confidence interval for the estimated long-run impact on unemployment
25
of a permanent inflation shock contains zero, and the fraction of the mass of the posterior
The Long-Run Phillips Curve: A Structural VAR Investigation
16
1
distribution for which the impact is estimated to be negative is uniformly greater than stan-
2
dard levels of statistical significance. Further, for either country the posterior distribution
3
of the fraction of the permanent component of the unemployment rate which is explained by
4
permanent inflation shocks is clustered towards zero, thus implying that such shocks explain
5
very little of the unit root component of unemployment, which provides additional evidence
6
in support of the notion that the long-run Phillips curve is indeed vertical.10
7
A researcher looking for evidence of a negative long-run trade-off may find some limited
8
support from the results for the U.S. and Canada, for which the fraction of draws from the
9
posterior for which the long-run impact is estimated to be negative is slightly above 80 per
10
cent. Although far from standard levels of statistical significance, still, this provides some
11
support to the notion of a negative long-run trade-off. Further, the median estimate of the
12
long-run impact for the U.S., at -0.27, implies that a permanent increase in inflation by
13
10 percentage points would permanently decrease the unemployment rate by 2.7 per cent.
14
Overall, however, the picture emerging from Table 2 and Figure 2 (and from Table A.2 and
15
Figure A.2 in the online appendix) points towards no evidence of a long-run trade-off. For
16
the other four countries, for example, the median estimates of the permanent decrease in
17
the unemployment rate associated with a permanent increase in inflation by 10 percentage
18
points range between -0.4 and -1.3 per cent. Even if one were willing to believe that the
19
median estimates capture the authentic unemployment-inflation trade-offs out there, these
20
are hardly trade-offs which might induce policymakers to ‘try to play the Phillips curve’.
21
4.3.
Evidence on monetary policy shocks from non-cointegrated SVARs
22
Since, in principle, the results discussed in the previous section are not incompatible
23
with the notion that some of the shocks exerting a permanent impact on inflation may
24
induce a non-vertical long-run Phillips trade-off, we then proceed to disentangle permanent 10
By definition, a vertical long-run Phillips curve implies that the fraction of the permanent component of the unemployment rate which is explained by the permanent inflation shock is equal to zero.
The Long-Run Phillips Curve: A Structural VAR Investigation
17
1
inflation shocks into demand- and supply-side ones, by imposing Canova and Paustian’s
2
(2011) DSGE-based ‘robust sign restrictions’ on their impact on the endogenous variables
3
at t=0.11
4
4.3.1.
Identification
5
Our identification strategy combines long-run and sign restrictions. First, the VAR’s
6
structural shocks are separated into two sets, depending on the fact that they do, or they
7
do not have a permanent impact on inflation. Let the structural VAR(p) model be given
8
by Yt = B0 + B1 Yt−1 + ... + Bp Yt−p + A0 t , with Yt being defined as before; A0 being the
9
O K impact matrix of the structural shocks at t = 0; and t ≡ [Tt E , M , Tt A , M , Tt R ] being t t
10
the structural shocks, which, as standard practice, are assumed to be unit-variance and
11
O K orthogonal to one another, with Tt E , M , Tt A , M being Canova and Paustian’s (2011) t t
12
‘technology’, ‘monetary policy’, ‘taste’, and ‘markup’ shocks, which are here allowed to exert
13
a permanent impact on inflation, and Tt R being instead a 2×1 vector of shocks which, by
14
construction, have a transitory impact on inflation. The second row of the matrix of long-run
15
impacts of the structural shocks, [IN −B(1)]−1 A0 —i.e., the row corresponding to inflation—is
16
therefore postulated to have the following structure,
E M O T A M K T R T t t t t t
x
17
x
x
x
01×2
(2)
18
—where a ‘0’ means that the corresponding long-run impact has been restricted to zero,
19
O whereas an ‘x’ means that it has been left unrestricted—thus implying that Tt E , M , Tt A , t
20
K and M may have a permanent impact on inflation, whereas Tt R does not. The restriction t
21
that the two shocks in Tt R are the only shocks which do not have a permanent impact on 11
Since Canova and Paustian’s ‘robust sign restrictions’ have originally been derived based on a New Keynesian model log-linearized around a zero-inflation steady-state, and in which inflation is stationary, it is in principle an open question whether such restrictions would also hold in the case in which, within the same model, inflation has a unit root. In fact, this is indeed the case, with the results for the case in which inflation is I(1) being numerically very close to the ‘benchmark’ results one obtains based on the specification reported in Canova and Paustian’s Table 1 (all of these results are available upon request).
The Long-Run Phillips Curve: A Structural VAR Investigation
18
1
inflation is sufficient to disentangle them from the other four shocks. As for separating Tt E ,
2
O K M , Tt A , and M from one another, this is achieved by imposing on impact the set of sign t t
3
restrictions reported in Table 5, where ‘+’ means ‘greater than, or equal to zero’, and ‘-’
4
means ‘smaller than, or equal to zero’. These restrictions are the same as the ‘robust sign
5
restrictions’ reported by Canova and Paustian (2011) in their Table 2 for their benchmark
6
DSGE model featuring sticky prices, sticky wages, and several standard frictions (see the
7
column labelled there as ‘M’), with the only obvious difference that, since their model features
8
employment, instead of the unemployment rate, the signs which are here being imposed on
9
the unemployment rate are the opposite of those reported by Canova and Paustian for
10
employment.
11
[Table 5 here]
12
In what follows, the sign restrictions are imposed only on impact. The reason for do-
13
ing this is that, as stressed by Canova and Paustian (2011), whereas sign restrictions on
14
impact are, in general, robust—in the specific sense that they hold for the vast majority
15
of sub-classes within a specific class of DSGE models, and for the vast majority of plausi-
16
ble parameters’ configurations—restrictions at longer horizons are instead, as they put it,
17
‘whimsical’, meaning that they are hard to pin down, and in general, they are not robust
18
across sub-classes of models, and for alternative plausible parameters’ configurations. It is
19
to be noticed that Canova and Paustian (2011) reached this conclusion based on a quarterly
20
DSGE model. Since for the U.S. the data are monthly, for reasons of consistency the sign
21
restrictions are imposed both on impact, and for the two months after impact.
22
4.3.2.
Computing the structural impact matrix A0
23
For each draw from the posterior distribution of the VAR’s reduced-form parameters, the
24
structural impact matrix, A0 , is computed via the methodology proposed by Arias, Rubio-
25
Ramirez, and Waggoner (2014) for combining zero and sign restrictons (the methodology is
The Long-Run Phillips Curve: A Structural VAR Investigation
1
described in the online appendix).
2
[Figure 3 here]
3
[Figure 4 here]
4
[Figure 5 here]
5
19
4.3.3.
Evidence
6
Figures 3-5 and Table 6 show the results for monetary shocks, whereas Figures A.4-A.6
7
and Tables A.4-A.6 in the online appendix report the full set of results for all the four
8
shocks. The reason for narrowly focusing on monetary shocks is that the motivation behind
9
all explorations of the slope of the long-run Phillips trade-off has always been understanding
10
whether the monetary authority might be able to permanently lower the unemployment
11
rate by engineering a permanently higher rate of inflation. Under this respect, the long-run
12
Phillips trade-offs which might originate from permanent inflation shocks of a non-monetary
13
nature are therefore irrelevant. It is to be stressed, however, that results for the other three
14
permanent inflation shocks are qualitatively the same as those for monetary shocks, and
15
clearly suggest that, even ‘digging’ into the permanent inflation shock identified in Section
16
4.2, it is not possible to find any evidence that at least some shocks may generate a non-
17
vertical trade-off.
18
Figure 3 shows, for either country, the posterior distribution of the long-run impact on
19
the unemployment rate of a monetary shock having a one per cent permanent impact on
20
inflation, whereas Figures 4 and 5 show the posterior distributions of the fractions of the
21
permanent components of inflation and the unemployment rate, respectively, explained by
22
monetary shocks. Table 6 reports the the median and the 90%-coverage percentiles of the
23
posterior distribution of the long-run impact of monetary shocks on unemployment; the
24
fraction of draws for which the impact is estimated to be negative; and the medians and
25
the 90%-coverage percentiles of the posterior distributions of the fractions of the permanent
The Long-Run Phillips Curve: A Structural VAR Investigation
1
20
components of inflation and unemployment explained by monetary shocks.
2
Overall, results for monetary shocks are in line with those for the ‘aggregate’ permanent
3
inflation shocks discussed in Section 4.2, and in no way provide support to the notion that
4
the monetay authority might be able to permanently reduce the unemployment rate by
5
engineering a permanently higher inflation rate. Once again, the fractions of the mass of
6
the posterior distribution for which the impact is estimated to be negative, ranging between
7
0.480 for the Euro area and 0.836 for Canada, are uniformly greater than standard levels
8
of statistical significance, and for either country the 90 per cent confidence interval for the
9
estimated long-run impact of a monetary shock on unemployment contains zero. Finally, for
10
either country monetary shocks are estimated to have accounted for very small fractions of
11
the unit root component of the unemployment rate.
12
Once again, however, the uncertainty associated with the estimated long-run trade-offs
13
is uniformly substantial, so that these results are in fact compatible with a wide range of
14
possible slopes. This means that, although the null hypothesis of a vertical long-run trade-off
15
should be regarded as the natural one, a researcher with a different view of the world, and
16
therefore different priors about what the natural benchmark should be, might not see her
17
views about the slope of the long-run Phillips curve falsified.12 [Table 6 here]
18
19
5.
Conclusions
20
Overall, the evidence discussed in this paper provides essentially no support to the no-
21
tion of a non-vertical long-run Phillips trade-off—in particular of a trade-off which can be
22
actively exploited by a policymaker in order to permanently reduced the unemployment rate. 12
This originates from the fact that the feature of the data we are here estimating pertains to the infinite long-run, and, as it is well known—see, first and foremost, Faust and Leeper (1997)—this inevitably produces imprecise estimates, unless the researcher is willing to impose upon the data very strong informational assumptions. Within the present context, this problem is compounded by our use of sign restrictions, which, as stressed by Fry and Pagan (2011), are intrinsically ‘weak information’, and should therefore not be expected to produce strong inference.
The Long-Run Phillips Curve: A Structural VAR Investigation
21
1
Uncertainty, however, is uniformly substantial, so that a researcher who did not regard a ver-
2
tical long-run Phillips trade-off as the ‘natural null hypothesis’, and who entertained instead
3
alternative notions on what a reasonable slope of the long-run trade-off could be, might well
4
not see her priors falsified. For the United States, for example, it is possible to reject at the
5
10 per cent level all values of the long-run impact on the unemployment rate of a one per cent
6
permanent inflation shock smaller than -0.69, corresponding to slopes of the LRPC flatter
7
than -1.45. In particular, the p-value associated with a slope equal to -1 is equal to 0.0325,
8
thus implying that a one-for-one trade-off can be rejected with great confidence. Although
9
the extent of econometric uncertainty associated with our estimates is substantial, it still
10
therefore allows us to reject moderately-to-very flat LRPCs. At the same time, however, any
11
slope steeper than -1.4493 cannot be rejected at the 10 per cent level, thus implying that a
12
researcher holding the prior view that the slope of the LRPC is (e.g.) equal to -2 would not
13
see her view of the world falsified. Such a large extent of econometric uncertainty is nothing
14
but a manifestation of the intrinsic difficulty of reliably estimating the slope of the long-run
15
Phillips curve. There are several reasons for this. First of all, since the slope of the long-run
16
Phillips curve is a feature of the data pertaining, in principle, to the infinite long-run, Faust
17
and Leeper’s (1997) well-known point about the difficulty of capturing the long-run based on
18
finite samples automatically applies. On top of this, however, there is a key problem which
19
is specific to long-run Phillips curve itself. As first discussed by Sargent (1971) and Lucas
20
(1972a), and then reiterated by King and Watson (1994), the possibility of identifying the
21
slope of the long-run Phillips curve crucially hinges on the fact that inflation has exhibited,
22
over the sample period, permanent variation. But here is precisely the problem: as it has
23
been documented (Barsky, 1987; Benati, 2008) before WWI inflation was white noise, and,
24
if anything, it exhibited some mild negative serial correration; between 1914 and the onset of
25
the Great Inflation, inflation persistence increased, but a unit root can typically be rejected;
26
finally, after the Great Inflation, inflation persistence has disappeared in all countries having
The Long-Run Phillips Curve: A Structural VAR Investigation
22
1
an explicit inflation objective (Benati, 2008). So far, the Great Inflation episode is therefore
2
the only one during which inflation has clearly exhibited a unit-root-like behavior. This
3
logically implies that, unless, in the future, central banks will lose control of inflation for a
4
sufficiently long period of time—thus injecting a unit root into it—the only data researchers
5
will be able to use to explore the slope of the long-run Phillips curve will be those we ana-
6
lyzed herein. To put it differently, if central banks will continue being successful at keeping
7
inflation under control, economists will forever be unable to obtain precise estimates of the
8
slope of the long-run Phillips curve.
The Long-Run Phillips Curve: A Structural VAR Investigation
1
6.
23
References
2
Arias, J., Rubio-Ramirez, J., and Waggoner, D., 2014. Inference Based on SVARs Iden-
3
tified with Sign and Zero Restrictions: Theory and Applications. Federal Reserve Board,
4
Duke University, and Federal Reserve Bank of Atlanta, mimeo.
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10
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14
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18
19
20
21
22
23
24
25
26
Barsky, R., 1987. The Fisher hypothesis and the forecastability and persistence of inflation. Journal of Monetary Economics 19(1), 3-24. Benati, L., 2008. Investigating inflation persistence across monetary regimes. Quarterly Journal of Economics 123(3), 1005-1060. Bernanke, B. S., Laubach, T., Mishkin, F.S., Posen, A., 1999. Inflation targeting: Lessons from the international experience, Princeton University Press. Bruno, M., Easterly, W., 1998. Inflation crises and long-run growth. Journal of Monetary Economics 41, 3-26. Canova, F., Paustian, M., 2011. Measurement with some theory: Using sign restrictions to evaluate business cycle models. Journal of Monetary Economics 58, 345-361. Canova, F., Pina, J., 2005. What VAR tell us about DSGE models. In Diebolt, C. Kyrtsou, C., New trends in macroeconomics, Springer Verlag. Cavaliere, G, Rahbek, A., Taylor, R., 2012. Bootstrap determination of the co-integration rank in vector autoregressive models. Econometrica 80(4), 1721–1740 Elliot, G., Rothenberg, T., Stock, J.H., 1996. Efficient tests for an autoregressive unit root. Econometrica 64(4), 813-836 Evans, C., 1994. Comment on: The post-war U.S. phillips curve, A revisionist econometric history. Carnegie-Rochester Conference Series on Public Policy 41, 221-230. Fama, E., 1975. Short-term interest rates as predictors of inflation. American Economic Review 65(3), 269-282. Faust, J., 1998. The robustness of identified VAR conclusions about money. CarnegieRochester Conference Series on Public Policy, 49, 207-244.
The Long-Run Phillips Curve: A Structural VAR Investigation
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2
3
4
24
Forni, M., Gambetti, L., 2014. Testing for sufficient information in structural VARs. Journal of Monetary Economics 66, 124-136 Fry, R., Pagan, A., 2007. Sign restrictions in structural vector autoregressions: A critical review. Journal of Economic Literature 49(4), 938-60
5
Hamilton, J., 1994. Time series analysis, Princeton, N.J., Princeton University Press.
6
Kilian, L., 1998. Small-sample confidence intervals for impulse-response functions. Re-
7
8
9
10
11
view of Economics and Statistics, 218-230. King, R., Watson, M., 1994. The post-war U.S. phillips curve: A revisionist econometric history. Carnegie-Rochester Conference Series on Public Policy 41, 157-219. Kurmann, A., Otrok, C., 2013. News shocks and the slope of the term structure of interest rates. American Economic Review 103(6), 2612-32
12
Lucas Jr., R. E., 1972a. Econometric testing of the natural rate hypothesis. In Eckstein,
13
O., ed., The econometrics of price determination, Washington, D.C., Board of Governors of
14
the Federal Reserve System, 50-59.
15
16
17
18
19
20
21
22
23
24
25
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Lucas Jr., R. E., 1972b. Expectations and the neutrality of money. Journal of Economic Theory 4(April), 103-24. Lucas Jr., R. E., 1973. Some international evidence on output-inflation trade-offs. American Economic Review 63(3), 326-334. Sargent, T. J., 1971. A note on the accelerationist controversy. Journal of Money, Credit and Banking 3(3), 721-725. Svensson, L.E.O., 2015. The possible unemployment cost of average inflation below a credible target. American Economic Journal: Macroeconomics 7(1), 258-96. Uhlig, H., 1998. Comment on: The robustness of identified VAR conclusions about money. Carnegie-Rochester Conference Series on Public Policy 49, 245-263. Uhlig, H., 2005. What are the effects of monetary policy on output? Results from an agnostic identification procedure. Journal of Monetary Economics 52(2), 381-419.
Figure 1
Bootstrapped distributions of the slope of the long-run Phillips curve implied by the estimated cointegrating vector between inflation and unemployment, and of the elements of the loadings 22 vector
23
Figure 2 Results based on Bayesian VARs allowing for a single permanent inflation shock: posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to inflation; 24 and of the fractions of the permanent component of unemployment explained by the permanent inflation shock
Figure 3 Results based on Bayesian VARs allowing for four permanent inflation shocks: 25 posterior distribution of the long-run impact on the unemployment rate of a monetary shock having a one per cent permanent impact on inflation
26
Figure 4 Results based on Bayesian VARs allowing for four permanent inflation shocks: posterior distribution of the fraction of the permanent component of inflation explained by monetary shocks
27
Figure 5 Results based on Bayesian VARs allowing for four permanent inflation shocks: posterior distribution of the fraction28of the permanent component of the unemployment rate explained by monetary shocks
29
30
31
32
33
34
35
36
37
Figure 1 CPI inflation and the unemployment rate
38
39
1
Online appendix for:
2
‘The Long-Run Phillips Curve: A Structural VAR Investigation’ Luca Benatid>W
3
d
4
1.
The Data Here follows a detailed description of the dataset.
5
6
University of Bern
1.1. United States
7
A monthly seasonally adjusted series for the civilian unemployment rate (‘UNRATE:
8
Civilian Unemployment Rate, Seasonally Adjusted, Percent’) is from the U.S. the Depart-
9
ment of Labor, Bureau of Labor Statistics, and it has been converted to the quarterly
10
frequency by taking averages within the quarter. A monthly seasonally unadjusted series
11
for the 3-month Treasury Bill (‘TB3MS: 3-Month Treasury Bill: Secondary Market Rate’)
12
is from Board of Governors of the Federal Reserve System, and it has been converted to
13
the quarterly frequency by taking averages within the quarter. By the same token, monthly
14
seasonally unadjusted series for the 1-, 3-, 5-, and 10-year Treasury constant maturity rates
15
(acronyms are GS1, GS3, GS5, and GS10, respectively), and for the e ective Federal Funds
16
rate (acronyms is FEDFUNDS) are all from the Board of Governors of the Federal Reserve
17
System. A monthly seasonally adjusted series for the consumer price index (‘CPIAUCSL:
18
Consumer Price Index, Seasonally Adjusted, Monthly, Index 1982-84=100’) is from the U.S.
19
the Department of Labor, Bureau of Labor Statistics, and it has been converted to the
20
quarterly frequency by taking averages within the quarter. Quarterly seasonally adjusted
21
series for real GDP (‘GDPC96: Real Gross Domestic Product, 3 Decimal, Seasonally Ad
Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland. Email: [email protected]
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 2
1
justed Annual Rate, Billions of Chained 2005 Dollars’) and the GDP de ator (‘GDPCTPI:
2
Gross Domestic Product: Chain-type Price Index, Seasonally Adjusted’) are from the U.S.
3
Department of Commerce: Bureau of Economic Analysis. The quarterly series for potential
4
GDP (‘GDPPOT: Real Potential Gross Domestic Product, Quarterly, Billions of Chained
5
2005 Dollars’) is from the U.S. Congressional Budget O!ce, Budget and Economic Outlook.
6
Real consumption expenditure of non-durable goods and services has been computed based
7
on Table 1.1.6 of the U.S. National Income and Product Accounts. The output gap has been
8
computed as the percentage deviation of real GDP from the Congressional Budget O!ce
9
estimate of potential real GDP. It is important to stress that, since both series are expressed
10
in the same unit (billions of chained 2005 dollars), such a computation is indeed perfectly
11
legitimate. Monthly interpolated seasonally adjusted series for real GDP, the GDP de ator,
12
and real consumption expenditure are from Mark Watson’s website, and are available from
13
January 1959 to June 2010. The quarterly estimate of the output gap implied by the Con-
14
gressional Budget O!ce estimate of potential output has been interpolated to the monthly
15
frequency as in Bernanke, Gertler, and Watson (1997), using, as interpolators, the monthly
16
ratios between investment in non-residential structures and GDP, between investment in
17
residential structures and GDP, and between the change in inventories and GDP. In turn,
18
these ratios have been constructed based on the monthly interpolated national accounts data
19
found at Mark Watson’s website.
20
1.2. Euro area
21
Quarterly, seasonally adjusted series for real GDP, the GDP de ator, the CPI, real con-
22
sumption expenditure, real gross xed capital formation, the unemployment rate, the output
23
gap, and a short and a long rate are from the European Central Bank’s database.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 3
1
1.3. United Kingdom
2
The output gap estimate is from Pybus (2011). I wish to thank Tom Pybus, of the U.K.
3
O!ce for Budget Responsibility, for kindly sharing his output gap series with me. Quar-
4
terly, seasonally adjusted series for real GDP (acronym is ‘ABMI’), real nal consumption
5
by households (‘ABJR’), real total gross xed capital formation (‘NPQT’), nominal GDP
6
(‘YBHA’), and real change in inventories (‘CAEX’) are from the U.K. O!ce for National
7
Statistics. A quarterly, seasonally adjusted series for the GDP de ator has been computed as
8
the ratio between YBHA and ABMI. Monthly, seasonally adjusted series for the unemploy-
9
ment rate based on the claimant count (acronym is ‘BCJE’) and the retail price index are
10
from the U.K. O!ce for National Statistics, and they have been converted to the quarterly
11
frequency by taking averages within the quarter. Monthly, seasonally unadjusted series for
12
the discount rate and a long-term rate (‘Interest Rates, Government Securities, Government
13
Bonds’) are from the International Monetary Fund’s International Financial Statistics, and
14
they have been converted to the quarterly frequency by taking averages within the quarter.
15
Since, over the sample period, the consumption/GDP ratio exhibits some trend (which, ac-
16
cording to the permanent income hypothesis should obviously not be there), before entering
17
the ratio into the VARs I quadratically detrend it.
18
1.4. Canada
19
Quarterly, seasonally adjusted series for real GDP (acronym is ‘CANRGDPQDSNAQ’),
20
the harmonized unemployment rate for all persons (‘CANURHARMQDSMEI’), real private
21
nal consumption expenditure (‘CANPFCEQDSNAQ’), real gross xed capital formation
22
(‘CANGFCFQDSNAQ’), the GDP implicit price de ator (‘CANGDPDEFQISMEI’), and
23
the CPI are from the Organisation for Economic Co-operation and Development (either
24
from the Quarterly National Accounts, or from the Main Economic Indicators). Since, over
25
the sample period, the consumption/GDP ratio exhibits some trend (which, according to the
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 4
1
permanent income hypothesis should obviously not be there), before entering the ratio into
2
the VARs I quadratically detrend it. A monthly, seasonally unadjusted series for the short
3
rate, treasury bill auction, average yields, is from the Bank of Canada. A monthly, seasonally
4
unadjusted series for a long rate, is from the International Monetary Fund’s International
5
Financial Statistics. Both monthly series have been converted to the quarterly frequency
6
by taking averages within the quarter. As for the output gap, an ‘o!cial’ series computed
7
by the Bank of Canada, and available from its website, starts in 1981Q1. Since our sample
8
period starts in 1961Q1, in our analysis we use instead HP- ltered log real GDP, which, since
9
1981Q1, has exhibited a remarkably close co-movement witrh the Bank of Canada’s output
10
gap estimate. Speci cally, (i) the contemporaneous correlation between the two series has
11
been equal to 0.949, whereas (ii) the fraction of variance explained by the rst principal
12
component of the panel composed of the two standardized series has been equal to 0.974.
13
1.5. Japan
14
Monthly, seasonally adjusted series for the unemployment rate and the consumer price
15
indices are from the International Monetary Fund’s International Financial Statistics. Both
16
monthly series have been converted to the quarterly frequency by taking averages within the
17
quarter.
18
1.6. Australia
19
Monthly, seasonally adjusted series for the unemployment rate and the consumer price
20
indices are from the International Monetary Fund’s International Financial Statistics. Both
21
monthly series have been converted to the quarterly frequency by taking averages within the
22
quarter.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 5
1
1.7. Sweden
2
Monthly, seasonally adjusted series for the unemployment rate and the consumer price
3
indices are from the International Monetary Fund’s International Financial Statistics. Both
4
monthly series have been converted to the quarterly frequency by taking averages within the
5
quarter.
6
2.
Methodological Issues
7
How should we estimate the slope of the long-run Phillips curve? Before Sargent (1971)
8
and Lucas (1972a), the standard approach, due to Solow (1968) and Tobin (1968), was to
9
estimate the Phillips curve regression w = 0 + 1 w31 + 2 w32 + === + s w3s + 1 Xw31 + 2 Xw32 + === + s Xw3s + w
10
(1)
11
–with w and Xw being in ation and the unemployment rate, respectively–and to test
12
whether 1 + 2 + === + s = (1) = 1. Failure to reject the null hypothesis that (1)=1
13
was regarded as evidence in favor of the notion of a vertical long-run Phillips curve, whereas
14
an estimate of (1) signi cantly smaller than 1 was regarded as pointing towards a negative
15
long-run trade-o .
16
2.1. Why single-equation methods are misleading Sargent and Lucas highlighted the fallacy intrinsic to such an approach. As pointed out
17
18
by Lucas (1972a),
19
‘[...] the natural rate hypothesis, correctly formulated, has no implications for
20
the coe!cients of distributed lags Phillips curves, or for any other single-equation
21
expression of the empirical in ation-real output trade-o .’1 He went on to further stress that
22
1
Emphasis in the original.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 6
1
‘[...] a valid test of the natural rate hypothesis involves a test of a restriction
2
on the parameters across equations of a complete simultaneous equations model.
3
The esistence of a natural rate is thus a systems property, like stability or iden-
4
ti ability.’2
5
But if (1) cannot capture the slope of the long-run Phillips curve, what does it capture,
6
in fact? As stressed by Sargent (1971), it captures in ation persistence.3 As he pointed out
7
(see Sargent, 1971, p. 724),
8
‘[...] even the most casual glance at the price history of the United States makes
9
it clear that the in ation rate has not been a strongly drifting variable. [...] It
10
is not surprising, therefore, that most empirical studies have estimated to be
11
markedly less than unity. Unfortunately, as usually interpreted, these estimates
12
tell us virtually nothing about the validity of the accelerationist thesis.’
13
2.2. An empirical illustration based on an estimated DSGE model
14
How relevant is this problem is practice? Since, as I have previously documented (see Be-
15
nati (2008)), under in ation targeting regimes in ation persistence has essentially disappeared–
16
to the point that, in most cases, in ation has been, so far, statistically indistinguishable from 2
Emphasis added. This can be illustrated via the following simple example. Let aggregate supply be given by a Lucas (1972b)-type supply curve incorporating long-run neutrality (that is, a vertical long-run Phillips curve), |w = [ w w|w1 ] + xw , with w being in ation, w|w1 being its rational expectation based on information at time w-1, |w being the deviation of output from its natural level, and being the slope of the short-run Phillips curve. In ation is postulated to be equal to money growth, w . Finally, money growth is set by the policymaker according to the following AR(1) process, w = w1 + }w , and all of the shocks are postulated to be white noise, with innovation variances respectively equal to 2x and 2} . The model’s solutions for w and |w are given by w = w1 + }w and |w = }w + xw , from which the theoretical value of the coe!cient on w1 in the Phillips curve regression of in ation on lags of itself and of the deviation of output from its natural level turns out to be equal to ˆ ROV = . This implies that if $ 1, so that money growth and in ation tend to unit root processes, ˆ ROV $ 1, and the Solow-Tobin approach therefore ‘uncovers’ the truth of a vertical long-run Phillips curve. However, if in ation is weakly persistent the Solow-Tobin approach will turn out to be misleading. In the extreme case in which = 0, so that money growth and in ation are white noise processes, ˆ ROV = 0, thus pointing towards a perfectly at long-run Phillips curve. 3
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 7
1
white noise4 –by Sargent and Lucas’ argument we should logically expect that, by applying
2
the Solow-Tobin approach to data generated under such regimes, we would ‘uncover’ very
3
at long-run Phillips curves. In this sub-section I therefore estimate a simple New Keyne-
4
sian DSGE model featuring a vertical long-run Phillips curve based on data from Sweden,
5
the U.K., and Canada under in ation-targeting, and I show that the application of the
6
Solow-Tobin approach to this data-generating processes (henceforth, DGPs) would lead a
7
researcher to uncover at long-run Phillips curves for either country, in spite of the fact that,
8
by construction, the model encodes a vertical long-run Phillips curve. The New Keynesian model I am using is near-identical to the one I estimated in Benati
9
10
(2008), and is described by the following equations,
11
|w = |w+1|w + (1 )|w31 31 (Uw w+1|w ) + |>w
(2)
12
w =
w+1|w + w31 + |w + >w 1 + 1 +
(3)
Uw = Uw31 + (1 )[! w + !| |w ] + U>w
13
(4)
14
where w , |w and Uw are in ation, the output gap, and the nominal rate; is the forward-
15
looking component in the intertemporal IS curve; is price setters’ extent of indexation to
16
past in ation, and >w , |>w and U>w are white noise disturbances.5 All of the variables in
17
(2)-(4) are expressed as log-deviations from a non-stochastic steady-state. I set =1, thus
18
imposing a vertical long-run Phillips curve, and I estimate the model based on Swedish, U.K.,
19
and Canadian data from the in ation-targeting regimes. Both the Bayesian methodology and
20
the priors are identical to those I used in Benati (2008), to which the reader is referred to. The 4
Updating Benati’s (2008) estimates to 2011Q4, Hansen (1999) ‘grid bootstrap’ estimates of the sum of the autoregressive coe!cients in univariate AR(p) representations for GDP de ator in ation are equal to -0.15 in the U.K., to 0.06 in Sweden, and to 0.13 in Canada, with 90%-coverage bootstrapped con dence intervals equal to [-0.43 0.15], [-0.26 0.38], and [-0.09 0.34] respectively. A unit root in in ation can therefore be strongly rejected, and evidence points instead towards in ation being white noise. 5 The assumption that all of the structural disturbances are white noise is made uniquely for the sake of simplicity, it can be easily relaxed, and it plays no role in my results.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 8
1
VAR representation of the estimated model for Sweden conditional on the median estimates
4
is given by6 5 6 5 65 9 Uw : 9 0.788 0.007 0.208 : 9 Uw31 9 : 9 :9 9 : = 9 -0.230 0.037 0.288 : 9 9 w : 9 : 9 w31 7 8 7 87 -0.147 -0.001 1.004 |w |w31
5
in ation-targeting, the coe!cients on w31 in the equation for w is equal to 0.037. Interpret-
6
ing the second equation of the VAR as an estimated short-run Phillips curve, by applying the
7
Solow-Tobin approach we can immediately back out the estimate of the slope of the long-run
8
Phillips curve, which is equal to 0.299, in spite of the fact that the underlying DGP features
9
a vertical Phillips curve by construction. Results for the other two countries are qualitatively
2
3
10
6
5
: 9 xU>w : 9 :+9 x : 9 >w 8 7 x|>w
6 : : : : 8
(5)
Consistent with my previous results on the disappearance of in ation persistence under
the same, being equal to 0.306 for the United Kingdom, and to 0.245 for Canada.
11
These results provide a straighforward explanation for Svensson’s (2015) nding of a
12
comparatively at long-run Phillips trade-o in Sweden under the in ation targeting regime
13
based on the Solow-Tobin approach. Speci cally, Svensson (2015) estimates a version of (1)
14
for the period 1997Q4-2011Q4,7 ‘backs out’ the estimate of the long-run Phillips trade-o
15
from his short-run estimates, and performs the Solow-Tobin test that (1) = 1. He thus
16
summarizes his results, which point towards a negatively-sloped long-run Phillips curve:
17
‘From the estimates in table 2 it follows that the slope of the long-run Phillips curve is
18
about 0.8. With a standard error of 0.19, it is fairly precisely estimated. A 95-percent
19
con dence interval of the slope is the interval from 0.43 to 1.18. Clearly, the hypothesis
20
of a vertical Phillips curve, an in nite slope, can be rejected.’
21
In the light of the previous discussion of Sargent and Lucas’ criticism of the Solow-Tobin
22
approach, and of the fact that, under in ation-targeting, in ation in Sweden is, today, very 6
The covariance matrix of the VAR’s reduced-form innovations is not reported for the sake of simplicity. By the same token, I do not report the corresponding VAR representations of the estimated DSGE models for the other three countries, but they are available upon request. 7 See Svensson (2013, Table 2).
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 9
1
close to white noise8 , Svensson’s rejection of a vertical long-run Phillips curve based on the
2
test that (1) = 1 is to be expected. At the same time, however, as pointed out by Sargent
3
(1971) in the previous quotation, this evidence is uninformative about the authentic slope
4
of Sweden’s long-run Phillips curve.
5
2.3. King and Watson’s (1994) alternative approach
6
Since the results produced by single-equation methods–especially when applied to data
7
generated by monetary regimes which cause in ation to be strongly mean-reverting, such
8
as in ation targeting, European Monetary Union (henceforth, EMU), and the Swiss ‘new
9
monetary policy concept’–are unreliable, in this paper I follow King and Watson (1994),
10
and I use structural VAR methods to estimate the permanent impact (if any) on the unem-
11
ployment rate of permanent shocks to in ation. As stressed by King and Watson (1994, p.
12
177), indeed, if in ation has a unit root the data are in fact informative about the slope of
13
the long-run Phillips curve, since they do contain the relevant ‘experiment’:
14
‘When in ation is stationary, [...] as stressed by Lucas and Sargent, the relevant
15
experiment–permanent changes in the rate of in ation–are absent from the in ation
16
data and so the long-run Phillips trade-o could not be estimated [...]. In the unit
17
root case, by contrast, [...] the relevant experiments are present in the data: variation
18
in ¯ w [i.e.: the permanent component of in ation] allows the long-run slope to be
19
determined.’
As a simple illustration of King and Watson’s point, suppose that the economy is de-
20
21
scribed by a Lucas (1972b)-type supply curve possibly incorporating a long-run non-neutrality
22
(that is, a non vertical long-run Phillips curve), |w = [ w w|w31 ] + xw >
23
8
See footnote 11.
(6)
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 10
1
with w being in ation, w|w31 being its rational expectation based on information at time
2
w-1, |w being the deviation of output from its natural level, being the slope of the short-
3
run Phillips curve, and being the parameter upon which the slope of the long-run Phillips
4
curve crucially depends. Expression (6) implies that the long-run impact on |w of a permanent
5
change in w is equal to (1-). Finally, let’s assume that in ation is a pure random walk,
6
7
w = w31 + }w . The VAR representation of the model is given by 65 65 5 6 5 6 5 6 9 |w : 9 0 (1-) : 9 |w31 : 9 1 : 9 xw : 87 87 7 8=7 8+7 8 0 1 0 1 w w31 }w
(7)
8
from which the long-run impact on |w of a permanent shock to w at t = 0 can be immediately
9
computed as
10
C|w = (1 ), w<" C}0 lim
11
which is precisely the slope of the long-run Phillips curve.
12
2.4. Why a DSGE-based approach is not a viable option
(8)
13
In principle, an alternative approach would be to estimate a (DSGE) structural macro-
14
economic model in which the slope of the long-run trade-o is encoded in one or more of the
15
model’s structural parameters, and then to test whether, conditional on the model’s esti-
16
mates, it is possible to reject the null hypothesis that the long-run Phillips curve is vertical.
17
There are two key drawbacks to this approach.
18
First, and least importantly, as a matter of logic results would be, in general, model-
19
dependent (that is: they would be conditional on the speci c structural model which is
20
being estimated), and they could not be regarded as possessing a general validity.
21
Second–and this is key–this approach crucially rests on the assumption that the pa-
22
rameters of the estimated model are truly structural in the sense of Lucas (1976), and,
23
in particular, that they are invariant to changes in the characteristics of the in ationary
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 11
1
environment– rst and foremost, equilibrium in ation. This is key because if and only if
2
this assumption is satis ed we can be con dent that the long-run Phillips trade-o extracted
3
from a structural macroeconomic model estimated based on a sample period during which
4
average in ation has been equal to, say, two per cent, will hold at higher equilibrium in ation
5
rates. In recent years, however, evidence has been accumulating that several DSGE mod-
6
els’ supposedly structural parameters are in fact, most likely, not structural in the sense of
7
Lucas (1976). Benati (2008), for example, has shown that the indexation parameter in New
8
Keynesian backward- and forward-looking Phillips curves changes systematically with the
9
monetary regime, whereas the recent work of Fernandez-Villaverde and Rubio-Ramirez (and
10
their co-authors) has documented instability in several structural parameters. Fernández-
11
Villaverde and Rubio-Ramírez (2008), in particular, have shown that the extent of price
12
stickiness negatively and systematically co-moves with trend in ation. All of this implies
13
that the notion of extracting, from an estimated DSGE model, a long-run Phillips trade-o
14
which can safely be regarded as invariant to changes in equilibrium in ation rests, most
15
likely, on a leap of faith, unless we will be able, in the future, to e ectively model the way
16
in which those speci c DSGE models’ features which crucially determine the slope of the
17
long-run trade-o ( rst and foremost, price stickiness and the extent of backward-looking
18
indexation) co-move systematically with equilibrium in ation.
19
The attractiveness of King and Watson’s approach, on the other hand, is that the only
20
assumption it crucially hinges upon is that in ation has a unit root, and, conditional on this
21
assumption, the slope of the long-run trade-o can be investigated without possessing any
22
detailed knowledge of the underlying structural model of the economy.9 9
This point is also made by Fisher and Seater (1993): ‘[...] absent knowledge of the underlying structure, the consequences of an event cannot be inferred if the event has not occurred. In order for inferences regarding [long-run neutrality (long-run superneutrality)] to be drawn from a reduced form, the data must contain permanent stochastic changes in the level (growth rate) of the money supply.’.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 12
1
2
2.5. Why it is so di!cult to nd the ‘right’ data to estimate the slope of the long-run Phillips curve
3
The fact that, for a researcher to be able to econometrically identify the slope of the
4
long-run Phillips curve, in ation must contain a unit root, highlights why it is so di!cult
5
to nd the ‘right’ data to explore this issue. As it has been extensively documented, e.g.,
6
in Barsky (1987) and Benati (2008), under metallic standards (that is, before WWI) in a-
7
tion had consistently been indistinguishable from white noise, and, if anything, it had been
8
weakly negatively serially correlated. By the same token, during the period between the
9
outbreak of WWI and the early part of the post-WWII period in ation had exhibited some
10
persistence, but it had still been most likely stationary. Further, although for sample periods
11
which are dominated by the Great In ation episode evidence of a unit root in in ation is
12
sometimes (but, as I discuss in Section 4 below, not always) strong, following the disin a-
13
tions of the early 1980s in ation persistence has dramatically decreased, so that for sample
14
periods starting in mid-1980s the null of a unit root can once again be strongly rejected at
15
conventional signi cance levels. Even more ominously for the possibility of identifying the
16
slope of the long-run Phillips curve, under in ation targeting regimes, EMU, and the Swiss
17
‘new monetary policy concept’ in ation is, today, close to white noise. This means that, his-
18
torically, only sample periods dominated by the Great In ation episode can legitimately be
19
used to investigate the slope of the long-run Phillips curve, simply because such sample are
20
the only ones containing the relevant ‘experiment’ mentioned by King and Watson (1994),
21
whereas data generated under either metallic standards or in ation targeting regimes are
22
exactly the kind of data that cannot be used for this purpose.
23
2.6. The long-run Phillips trade-o in a low-in ation environment
24
A conceptually related issue pertains to the slope of the long-run Phillips curve in a
25
low-in ation environment. Several papers–from Akerlof, Dickens, and Perry (1996), to
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 13
1
Benigno and Ricci (2011)–have argued that, due to the presence of downward nominal
2
wage rigidity, the long-run Phillips trade-o may become quite signi cantly at within a
3
low-in ation environment, even if it is very steep, or even vertical, at higher in ation rates.
4
Unfortunately, a proper econometric investigation of this hypothesis is, and will most likely
5
remain, impossible.
6
First, for the reasons I discussed in section 2.4, extracting the long-run Phillips trade-o
7
from an estimated structural (DSGE) macroeconomic model is (at least, currently) not a
8
feasible option.
9
Second, for a pure time-series approach to be feasible, in ation should exhibit, over the
10
sample period, two characteristics which, historically, have been mutually exclusive: it should
11
be ‘low’ (which, in practice, means that it should be bounded between, say, minus two and
12
two per cent), and it should contain a unit root. The problem is that although it is easy to
13
nd periods during which either (i) in ation has been low, stable, and the null of a unit
14
root can be strongly rejected (e.g., in ation targeting regimes), or (ii) in ation has been
15
high, volatile, and the null of a unit root cannot be rejected (for several countries, the Great
16
In ation episode and the surrounding years), historically there has not been a single period
17
during which in ation has been low, stable, and it has exhibited a unit root behaviour.
18
As a result, the notion that the long-run Phillips curve may become at within a low-
19
in ation environment will most likely remain a theoretically compelling, but empirically
20
unproven conjecture.
21
3.
Results from Unit Root Tests Based on Phillips and Perron’s Test Statistics
22
For reasons of robustness we have also considered tests based on Phillips’ ] and ]w
23
statistics. The tests have been performed based on estimated models including an inter-
24
cept, but no time trend, and bootstrapped p-values have been computed as before. Again
25
for reasons of robustness, we have considered four values for T, the maximum number of
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 14
1
autocovariances to be considered for Phillips’ tests, that is, 2, 4, 6 and 8 quarters (for the
2
U.S. the corresponding values of T are 6, 12, 18, and 24). Table A.1 in the online appendix
3
reports results for the ] statistic, whereas results for the ]w tests are not reported because
4
they are uniformly in line with those from the ] tests. These results are almost uniformly in
5
line with those from Elliot et al.’s tests. The main di erence– which is however irrelevant
6
for our purposes–pertains to U.S. in ation for the full sample period since January 1959,
7
for which the null of a unit root is strongly rejected (on the other hand, in line with the
8
results from Elliot et al.’s tests, results for the pre-Volcker period clearly point towards a
9
unit root in U.S. in ation based on either price index).
10
11
4.
Classical Estimation of the Non-Cointegrated VARs Featuring a Single Permanent In ation Shock
12
Tables A.3 and A.8 in the online appendix report–based on the CPI and on the GDP
13
de ator, respectively–the full set of results for the non-cointegrated VARs featuring a single
14
permanent in ation shock. As mentioned in the text, Bayesian estimation of the reduced-
15
form VARs has been performed based on the estimator discussed by Uhlig (1998, 2005).
16
Working within a Classical context, we estimate equation (??) in the text via OLS based on
17
the standard formulas found in Luetkepohl (1991). Concerning the estimation of the impact
18
matrix of the structural shocks, as extensively discussed by Christiano et al. (2006), reliably
19
estimating F" requires a good estimate of the spectral density of \w at the frequency $ = 0,
20
0 V\ (0) = F" F" , an object which VARs, given their focus on tting the short-run dynamics
21
of the data, should not necessarily be expected to capture well. Following Christiano et
22
al. (2006) we therefore consider, beyond the standard estimator of V\ (0) produced by
23
31 ˆ 31 0 ˆ ˆ the VAR–that is, Vˆ\Y DU (0) = [LQ E(1)] } , where Yˆ is the estimated Y {[LQ E(1)]
24
covariance matrix of the VAR’s reduced-form innovations–the estimator produced by the
25
Bartlett estimate of the spectral density matrix (see Hamilton (1994)).
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 15
1
We use standard bootstrapping techniques in order to both bias-correct the estimated
2
long-run impacts of the permanent in ation shock as in Kilian (1998), and characterise the
3
extent of uncertainty around the bias-corrected estimates (so, to be clear, the only di erence
4
between Kilian’s (1998) paper and the present work is that he dealt with the bias-correction
5
of IRFs, whereas we are here bias-correcting the long-run impacts of the structural shocks).
6
Speci cally, we bootstrap the estimated reduced-form VAR 10,000 times, and based on each
7
bootstrap replication we estimate a VAR(p); we impose the same identi cation scheme we
8
imposed on the VAR estimated based on the actual data; and we compute the implied
9
long-run impact on the unemployment rate of the permanent in ation shock. Then, we
10
use such bootstrapped distributions, rst, to bias-correct the simple estimate of the long-run
11
impact; and second, to characterise the extent of uncertainty surrounding such bias-corrected
12
estimates, by simply rescaling the original bootstrapped distribution in such a way that its
13
median be equal to the bias-corrected estimate (in general, however, the extent of the bias
14
is very small, so that, in the end, bias-correcting does not make a material di erence to the
15
results).
16
4.0.1.
Computing the structural impact matrix D0
17
For each draw from the posterior distribution of the VAR’s reduced-form parameters we
18
compute the structural impact matrix, D0 , via the methodology proposed by Arias, Rubio-
19
Ramirez, and Waggoner (2014) for combining zero and sign restrictons. Speci cally, let E0q ,
20
E1q , ..., Esq , and lq be the q-th draw from the posterior distribution for the intercept, the
21
VAR matrices, and the covariance matrix of reduced-form innovations of the VAR (??), for
22
q = 1, 2, 3, ..., Q . Let Sq Gq Sq0 be the eigenvalue-eigenvector decomposition of lq . We
23
start by computing an initial estimate of Dq0 –let’s call it D˜q0 –as D˜q0 = Sq Gq2 with the
24
˜ q = [LN E q (1)]31 D˜q0 , corresponding matrix of long-run impacts of the structural shocks O
25
where N is the number of series in the VAR, and E q (1) = E1q + E2q + ... + Esq . Based
1
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 16
1
2
on the Gibbs-sampling algorithm described in section 3.6.3 of Arias et al. (2014), we then draw ] random orthonormal matrices of dimension N × N from the uniform distribution,
3
conditional on the zero restrictions on the long-run impacts of the structural shocks on
4
in ation shown in (??)–that is: four shocks are allowed to have a non-zero long-run impact
5
on in ation, whereas the remaining two shocks are forced to have no long-run impact. Let
6
Tq} , } = 1, 2, 3, ..., ], be the }-th random orthonormal matrix, with Tq} (Tq} )0 = LN . We then
7
combine each of the ] random orthonormal matrices with the initial estimate of the long-run
8
˜ q , in order to obtain a randomly rotated long-run impact impact of the structural shocks, O
9
˜ q Tq} . By construction, each O}q , } = 1, 2, 3, ..., ], satis es the zero long-run matrix, O}q = O
10
restrictions in (??). From O}q we then obtain the corresponding candidate estimate of the
11
structural impact matrix, D}0>q = [LN E q (1)]O}q . Finally, we check whether D}0>q satis es the
12
sign restrictions reported in the table in Section 4.3.1, and out of the ] candidate structural
13
impact matrices we only keep, for draw q, those satisfying the sign restrictions. For each
14
draw from the posterior we consider 10,000 random rotation matrices. Finally, we set the
15
number of Gibbs-sampling iterations in the algorithm to O = 5.10
10
As pointed out by Arias et al. (2014, p. 18), ‘[t]here is also the question of how large O should be to obtain convergence. Experiments show that for the starting value given below, even O = 1 gives a good approximation of the desired distribution. In practice, increasing values of O can be used to determine when convergence has occurred.’ Our own experience with the algorithm con rms this. Although we set O = 5, convergence was typically achieved at most at the second iteration.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 17
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Structural Parameters?”, NBER Macroeconomics Annuals 2007, Cambridge, The MIT Press,
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Gregory, A.W., and B.E. Hansen (1996): “Tests for Cointegration in Models with Regime and Trend Shifts”, Oxford Bulletin of Economics and Statistics, 58(3), 555-560" }
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Kurmann, A., and C. Otrok (2013): “News Shocks and the Slope of the Term Structure
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Lucas, R. E. (1972b): “Expectations and the Neutrality of Money”, Journal of Economic Theory, 4(April 1972), 103-24. Lucas, R. E. (1973): “Some International Evidence on Output-In ation Trade-O s”, American Economic Review, 63(3), 326-334. Lucas, R. E. (1976): “Econometric Policy Evaluation: A Critique”, Carnegie-Rochester Conference Series on Public Policy, 1, 19-46. Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis, II edition. SpringerVerlag. Pybus, T. (2011): “Estimating the UK’s Historical Output Gap”, O!ce for Budget Responsibility Working paper No. 1, November 2011. Roberts, J. M. (1993): “The Sources of Business Cycles: A Monetarist Interpretation”, International Economic Review, 34(4), 923-934.
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Rubio-Ramirez, J., D. Waggoner, and T. Zha (2005): “Structural Vector Autoregressions:
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Theory of Identi cation and Algorithms for Inference”, Review of Economic Studies, 77(2),
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Sargent, T. J. (1987): Macroeconomic Theory, II Edition. Orlando, Academic Press.
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Solow, R. (1968): “Recent Controversy on the Theory of In ation: An Eclectic View”,
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Rousseaus (ed.), New York: New York University.
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Stock, J. H., and M.W. Watson (2007): “Why Has U.S. In ation Become Harder to Forecast?”, Journal of Money, Credit, and Banking, 39(1), 3-33. Svensson, Lars E O. (2015): “The Possible Unemployment Cost of Average In ation below a Credible Target”, American Economic Journal: Macroeconomics, 7(1): 258-96. Tobin, J. (1968): “Discussion”, in Proceedings of a Symposium on In ation: Its Causes, Consequences, and Control, S. Rousseaus (ed.), New York: New York University. Uhlig, H. (1998): “Comment On: The Robustness of Identi ed VAR Conclusions About Money”, Carnegie-Rochester Conference Series on Public Policy, 49, 245-263. Uhlig, H. (2005): “What are the E ects of Monetary Policy on Output? Results from an Agnostic Identi cation Procedure”, Journal of Monetary Economics, 52(2), 381-419.
Table A.1 Bootstrapped p-valuesd for Phillips ] testsf without trend GDP de ator in ation Q=2 Q=4 Q=6 Q=8
Q=2
CPI in ation Q=4 Q=6
Q=8
0.002 0.150
0.001 0.130
0.001 0.200
0.000 0.196
0.000 0.189
0.000 0.183
0.001 0.000 0.000 0.000 0.453 0.432 0.441 0.425 0.100 0.097 0.094 0.093 0.131 0.121 0.113 0.109 0.157 0.161 0.164 0.160 — — — — 0.159 0.148 0.148 0.144 Unemployment rate Q=2 Q=4 Q=6 Q=8
0.001 0.570 0.033 0.209 0.302 0.005 0.110
0.000 0.531 0.032 0.205 0.308 0.006 0.108 Short Q=4
0.000 0.541 0.031 0.194 0.310 0.007 0.107 rate Q=6
0.000 0.550 0.034 0.175 0.315 0.013 0.109
0.066 0.132
0.070 0.165
0.179 0.216 0.194 0.017 0.296 0.186 0.200
0.189 0.262 0.240 0.015 0.317 0.238 0.221
e
United States full sample (Jan. 1959-Jun. 2010) pre-Volcker (Jan. 1959-Jul. 1979) Volcker, Greenspan, and Bernanke (Aug. 1979-Jun. 2010) Euro area (1970Q1-1998Q4) Japan (1966Q4-2011Q4) United Kingdom (1972Q1-1992Q3) Canada (1961Q1-1990Q4) Sweden (1970Q1-1992Q4) Australia (1969Q3-1994Q2)
0.001 0.153
0.001 0.136
Q=2
Q=8
e
United States full sample (Jan. 1959-Jun. 2010) 0.059 0.029 0.025 0.026 0.094 0.090 pre-Volcker (Jan. 1959-Jul. 1979) 0.286 0.225 0.221 0.221 0.180 0.168 Volcker, Greenspan, and Bernanke (Aug. 1979-Jun. 2010) 0.363 0.216 0.172 0.153 0.220 0.197 Euro area (1970Q1-1998Q4) 0.367 0.381 0.386 0.387 0.198 0.186 Japan (1966Q4-2011Q4) 0.576 0.548 0.524 0.521 0.192 0.174 United Kingdom (1972Q1-1992Q3) 0.479 0.453 0.431 0.422 0.035 0.025 Canada (1961Q1-1990Q4) 0.421 0.409 0.424 0.431 0.315 0.303 Sweden (1970Q1-1992Q4) 0.993 0.874 0.542 0.305 0.187 0.168 Australia (1969Q3-1994Q2) 0.411 0.396 0.405 0.431 0.248 0.205 d Based on 10,000 bootstrap replications of estimated ARIMA processes. e Based on monthly data, the corresponding values of Q are 6, 12, 18, and 24. f Q is the maximum number of autocovariances considered for the Phillips test.
Table A.2 Monte Carlo evidence on the performance of the Johansen procedure: fraction of simulations for which the bootstrapped trace statistic incorrectly rejects the null of no cointegration between two independent random walks at a given signi cance leveld
Sample size (in quarters) T = 82 (U.S., pre-Volcker: 1959Q1-1979Q3) T = 115 (Euro area: 1970Q1-1998Q4) T = 81 (United Kingdom: 1972Q1-1992Q3) T = 118 (Canada: 1961Q1-1990Q4) T = 98 (Australia: 1970Q1-1994Q2) d Based on 10,000 simulations.
Fractions of bootstrapped p-values for Johansen’s trace statistic below: 0.1 0.05 0.01 0.119 0.059 0.013 0.118 0.060 0.012 0.118 0.059 0.011 0.113 0.055 0.011 0.115 0.060 0.011
ă ă ă ă ă ă ă ă ă ă ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăIxooăvhwăriăuesulwvăeasegărqăwkhăFSL
Table A.3 Results based on VARs allowing for a single permanent in ation shock: bootstrapped and posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of the mass of the distribution for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Euro area (1970Q1-1998Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian United Kingdom (1972Q1-1992Q3) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Canada (1961Q1-1990Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Australia (1970Q1-1994Q2) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian
Fraction of the mass of the distribution for which the impact is negative
matrix matrix
-0.226 -0.256 -0.276
-0.257 -0.264 -0.274
[-0.688; 0.134] [-0.677; 0.126] [-0.950; 0.324]
0.869 0.867 0.818
matrix matrix
-0.056 -0.024 -0.040
-0.043 -0.041 -0.036
[-0.474; 0.402] [-0.472; 0.399] [-0.784; 0.763]
0.576 0.573 0.547
matrix matrix
-0.088 -0.088 -0.088
-0.084 -0.0865 -0.0865
[-0.357; 0.180] [-0.359; 0.180] [-0.503; 0.330]
0.730 0.733 0.688
matrix matrix
-0.153 -0.121 -0.121
-0.131 -0.130 -0.131
[-0.337; 0.063] [-0.336; 0.063] [-0.467; 0.167]
0.872 0.870 0.805
matrix matrix
-0.056 -0.056 -0.056
-0.044 -0.044 -0.045
[-0.175; 0.074] [-0.176; 0.076] [-0.240; 0.135]
0.734 0.734 0.702
Table A.4 Results based on Bayesian VARs allowing for four permanent in ation shocks: posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of draws for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of draws for which impact is negative
-0.414 -0.489 -0.263 -0.113
-0.292 -0.472 -0.161 -0.237
[-3.827; [-3.592; [-1.370; [-3.615;
3.247] 3.426] 1.123] 3.714]
0.642 0.738 0.659 0.637
-0.551 -0.050 -0.050 -0.050
-0.5090 0.0233 0.0214 0.0201
[-4.525; [-3.034; [-2.168; [-3.256;
2.410] 2.518] 2.486] 2.542]
0.809 0.480 0.481 0.476
-0.141 -0.181 -0.100 0.100
-0.101 -0.214 -0.089 0.109
[-1.856; [-1.148; [-2.085; [-2.137;
1.629] 0.678] 1.893] 1.850]
0.600 0.819 0.584 0.371
-0.140 -0.180 -0.020 0.180
-0.241 -0.183 -0.067 -0.113
[-1.670; [-0.733; [-1.552; [-4.131;
0.952] 0.174] 1.696] 2.858]
0.767 0.836 0.578 0.561
-0.169 -0.024 -0.169 -0.024
-0.181 -0.042 -0.096 0.028
[-0.938; [-0.979; [-1.389; [-0.571;
0.451] 0.732] 1.056] 0.636]
0.840 0.586 0.619 0.437
Table A.5 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of in ation explained by individual shocks
Median, and 90%coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.109 0.194 0.276 0.131
[0.001; [0.002; [0.005; [0.001;
0.633] 0.791] 0.834] 0.695]
0.481 0.366 0.269 0.445
0.346 0.259 0.181 0.325
0.152 0.121 0.079 0.159
0.227 0.119 0.214 0.156
[0.002; [0.001; [0.003; [0.002;
0.823] 0.614] 0.746] 0.720]
0.341 0.462 0.315 0.401
0.238 0.331 0.235 0.283
0.111 0.151 0.091 0.130
0.110 0.316 0.073 0.205
[0.001; [0.005; [0.001; [0.002;
0.728] 0.864] 0.529] 0.805]
0.472 0.247 0.563 0.359
0.335 0.161 0.421 0.258
0.156 0.073 0.195 0.119
0.145 0.433 0.152 0.064
[0.002; [0.017; [0.001; [0.001;
0.639] 0.887] 0.727] 0.435]
0.414 0.143 0.404 0.605
0.300 0.086 0.288 0.444
0.139 0.034 0.136 0.204
0.245 0.151 0.064 0.282
[0.004; [0.002; [0.001; [0.005;
0.797] 0.754] 0.483] 0.819]
0.297 0.396 0.595 0.251
0.204 0.281 0.446 0.159
0.084 0.129 0.199 0.067
Table A.6 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of the unemployment rate explained by individual shocks
United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock s mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Median, and 90%coverage percentiles
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.122 0.236 0.057 0.120
[0.002; [0.004; [0.000; [0.001;
0.634] 0.727] 0.400] 0.627]
0.454 0.293 0.631 0.458
0.317 0.198 0.477 0.333
0.143 0.085 0.220 0.155
0.282 0.056 0.080 0.066
[0.007; [0.000; [0.001; [0.001;
0.717] 0.376] 0.479] 0.408]
0.239 0.650 0.561 0.602
0.159 0.472 0.389 0.436
0.062 0.229 0.176 0.215
0.085 0.161 0.077 0.1496
[0.001; [0.002; [0.001; [0.002;
0.561] 0.631] 0.500] 0.657]
0.542 0.389 0.557 0.402
0.392 0.265 0.410 0.283
0.177 0.125 0.189 0.122
0.091 0.113 0.091 0.139
[0.001; [0.001; [0.001; [0.001;
0.515] 0.559] 0.557] 0.628]
0.524 0.477 0.519 0.423
0.373 0.342 0.387 0.272
0.164 0.152 0.178 0.122
0.210 0.063 0.066 0.076
[0.002; [0.000; [0.001; [0.000;
0.695] 0.517] 0.497] 0.529]
0.321 0.607 0.587 0.558
0.215 0.459 0.439 0.408
0.094 0.224 0.205 0.193
ă ă ă ă ă ă ăă ă ă ă ă ă ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăIxooăvhwăriăuhvxowvăedvhgărqăwkhăJGSăghiodwru
Table A.7 Bootstrapped p-valuesd for Johansen’s cointegration tests Trace test of the null of no cointegration between: w and Xw w and Uw w , Xw and Uw United States, pre-Volcker (January 1959-July 1979) 0.141 1.0e-4 0.000 Euro area (1970Q1-1998Q4) 0.023 0.568 0.005 United Kingdom (1972Q1-1992Q3) 0.094 NAe NAe Canada (1961Q1-1990Q4) 0.509 0.493 0.010 Australia (1970Q1-1994Q2) 0.143 0.063 0.149 Test of the null of one cointegrating vector, versus the alternative of two, for w , Xw and Uw : United States, pre-Volcker (January 1959-July 1979) 0.074 Euro area (1970Q1-1998Q4) 0.012 Canada (1961Q1-1990Q4) 0.668 d e Based on 10,000 bootstrap replications. For the United Kingdom, results from the ADF tests reported in Table 1 strongly reject the null of a unit root in the short rate.
Table A.8 Results based on VARs allowing for a single permanent in ation shock: bootstrapped and posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of the mass of the distribution for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Euro area (1970Q1-1998Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian United Kingdom (1972Q1-1992Q3) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Canada (1961Q1-1990Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Australia (1970Q1-1994Q2) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian
Fraction of the mass of the distribution for which the impact is negative
matrix matrix
-0.105 -0.216 -0.200
-0.200 -0.191 -0.200
[-0.620; 0.185] [-0.613; 0.185] [-0.869; 0.404]
0.820 0.811 0.750
matrix matrix
0.056 -0.040 -0.008
-0.011 0.003 0.012
[-0.426; 0.453] [-0.432; 0.452] [-0.767; 0.758]
0.518 0.495 0.485
matrix matrix
-0.088 -0.104 -0.072
-0.081 -0.080 -0.065
[-0.308; 0.158] [-0.302; 0.157] [-0.411; 0.287]
0.734 0.736 0.680
matrix matrix
-0.121 -0.121 -0.088
-0.096 -0.096 -0.100
[-0.271; 0.070] [-0.270; 0.070] [-0.373; 0.134]
0.838 0.837 0.796
matrix matrix
-0.137 -0.121 -0.121
-0.116 -0.116 -0.116
[-0.241; 0.004] [-0.237; 0.003] [-0.267; 0.034]
0.945 0.946 0.914
Table A.9 Results based on Bayesian VARs allowing for four permanent in ation shocks: posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of draws for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of draws for which impact is negative
-0.338 -0.338 -0.038 -0.564
-0.332 -0.401 -0.086 -0.262
[-3.773; [-3.164; [-1.052; [-3.743;
3.084] 2.029] 1.148] 2.871]
0.679 0.745 0.594 0.627
-0.26 0.02 0.14 -0.18
-0.38 0.04 0.12 0.10
[-6.16; [-2.59; [-0.98; [-2.68;
4.40] 2.89] 1.68] 2.76]
0.655 0.475 0.374 0.427
-0.09 -0.17 -0.05 0.03
-0.08 -0.16 -0.04 0.04
[-1.47; [-0.81; [-1.51; [-1.66;
1.08] 0.45] 1.48] 1.49]
0.616 0.818 0.556 0.446
-0.156 -0.180 -0.012 0.036
-0.149 -0.183 -0.084 0.048
[-1.559; [-0.894; [-1.553; [-1.444;
1.095] 0.198] 1.161] 1.536]
0.724 0.850 0.606 0.433
-0.169 -0.121 -0.169 -0.072
-0.214 -0.135 -0.143 -0.039
[-0.720; [-1.274; [-1.066; [-0.485;
0.244] 0.698] 0.675] 0.590]
0.897 0.727 0.739 0.612
Table A.10 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of in ation explained by individual shocks
Median, and 90%coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.085 0.181 0.332 0.129
[0.001; [0.002; [0.006; [0.001;
0.565] 0.779] 0.844] 0.714]
0.543 0.363 0.231 0.447
0.386 0.260 0.153 0.321
0.176 0.109 0.065 0.159
0.154 0.109 0.369 0.118
[0.001; [0.001; [0.014; [0.001;
0.726] 0.557] 0.872] 0.640]
0.406 0.484 0.172 0.454
0.294 0.336 0.106 0.320
0.135 0.163 0.041 0.128
0.124 0.367 0.096 0.145
[0.001; [0.007; [0.001; [0.002;
0.675] 0.870] 0.624] 0.714]
0.451 0.198 0.507 0.425
0.331 0.134 0.364 0.291
0.161 0.059 0.167 0.129
0.132 0.311 0.133 0.133
[0.001; [0.007; [0.001; [0.001;
0.730] 0.834] 0.736] 0.660]
0.442 0.235 0.435 0.438
0.322 0.155 0.315 0.308
0.148 0.064 0.142 0.143
0.300 0.086 0.082 0.308
[0.006; [0.001; [0.001; [0.005;
0.842] 0.534] 0.516] 0.851]
0.264 0.532 0.553 0.235
0.171 0.396 0.385 0.158
0.062 0.182 0.166 0.069
Table A.11 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of the unemployment rate explained by individual shocks
Median, and 90%coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.105 0.187 0.059 0.147
[0.001; 0.576] [0.002; 0.711] [0.000 0.427] [0.002; 0.680]
0.488 0.356 0.626 0.410
0.355 0.256 0.480 0.281
0.166 0.110 0.239 0.125
0.316 0.057 0.078 0.059
[0.005; [0.001; [0.001; [0.000;
0.778] 0.439] 0.509] 0.436]
0.221 0.619 0.573 0.630
0.153 0.469 0.422 0.470
0.069 0.220 0.195 0.235
0.088 0.176 0.077 0.133
[0.001; [0.001; [0.001; [0.002;
0.570] 0.656] 0.492] 0.637]
0.539 0.364 0.555 0.438
0.396 0.254 0.411 0.301
0.181 0.124 0.187 0.130
0.094 0.142 0.082 0.095
[0.001; [0.002; [0.001; [0.001;
0.545] 0.655] 0.579] 0.536]
0.515 0.410 0.544 0.506
0.375 0.290 0.390 0.374
0.175 0.121 0.179 0.176
0.304 0.068 0.064 0.074
[0.007; [0.001; [0.001; [0.001;
0.786] 0.508] 0.512] 0.512]
0.233 0.582 0.598 0.572
0.148 0.436 0.443 0.415
0.058 0.206 0.192 0.172
ă ă ă ă ă ă ă ă ă ă ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăIxooăvhwăriăuesulwvăeasegărqăwkhăFSL
FigureăD11 CPI inflation and the unemployment rate 22
Figure D12 Bootstrapped distributions of the slope of the long-run Phillips curve implied ăăă by the estimated cointegrating vector between inflation and unemployment, ăăă and of the elements of the loadings vector (based on the CPI) 23
Figure D16ăăResults based on VARs allowing for a single permanent inflation shock: bootstrapped or ăăă posterior distributions of (i) the long-run impacts on the unemployment rate of a one ăăă per cent permanent shock to inflation, and of (ii) the fractions of the permanent compoăăă nent of unemployment explained by the permanent inflation shock (based on the CPI) 24
Figure D14 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the long-run impacts on the unemployment rate of a ăăă one per cent permanent shock to inflation (based on the CPI) 25
Figure D15 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of inflation ăăă explained by individual shocks (based on the CPI) 26
Figure D16 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of the unemăăă ployment rate explained by individual shocks (based on the CPI) 27
ă ă ă ă ă ă ăă ă ă ă ă ă ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăIxooăvhwăriăuhvxowvăedvhgărqăwkhăJGSăghiodwru
Figure D17 GDP deflator inflation and the unemployment rate 29
FigureăD18 ăăă ăăă
Bootstrapped distributions of the slope of the long-run Phillips curve implied by the estimated cointegrating vector between inflation and unemployment, and of the elements of the loadings vector (based on the GDP deflator) 30
Figure D19 ăăă ăăă ăăă ăăă
Results based on VARs allowing for a single permanent inflation shock: bootstrapped or posterior distributions of (i) the long-run impacts on the unemployment rate of a one per cent permanent shock to inflation, and of (ii) the fractions of the permanent component of unemployment explained by the permanent inflation shock (based on the GDP deflator) 31
Figure D110 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the long-run impacts on the unemployment rate of a ăăă one per cent permanent shock to inflation (based on the GDP deflator) 32
Figure D111 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of inflation ăăă explained by individual shocks (based on the GDP deflator) 33
Figure D112 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of the unemăăă ployment rate explained by individual shocks (based on the GDP deflator) 34
The Long-Run Phillips Curve: A Structural VAR Investigation Luca Benati
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PII: DOI: Reference:
S0304-3932(15)00079-3 http://dx.doi.org/10.1016/j.jmoneco.2015.06.007 MONEC2788
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Journal of Monetary Economics
Received date: 17 March 2013 Revised date: 23 June 2015 Accepted date: 28 June 2015 Cite this article as: Luca Benati, The Long-Run Phillips Curve: A Structural VAR Investigation, Journal of Monetary Economics, http://dx.doi.org/10.1016/j.jmoneco.2015.06.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
The Long-Run Phillips Curve: A Structural VAR Investigation Luca Benatia,∗†
2
a
University of Bern
Received Date; Received in Revised Form Date; Accepted Date
3
Abstract
4
Both cointegration methods, and non-cointegrated structural VARs identified based on either
5
long-run restrictions, or a combination of long-run and sign restrictions, are used in order to explore
6
the long-run trade-off between inflation and the unemployment rate in the post-WWII U.S., U.K.,
7
Euro area, Canada, and Australia. Overall, neither approach produces clear evidence of a non-
8
vertical trade-off. The extent of uncertainty surrounding the estimates is however substantial, thus
9
implying that a researcher holding alternative priors about what a reasonable slope of the long-run
10
trade-off might be will likely not see her views falsified.
11
Keywords: Inflation; unemployment; Phillips curve; unit roots; cointegration; Bayesian VARs; structural VARs; long-run
12
restrictions; sign restrictions.
13
JEL classification: E30; E32 ∗
Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland. Email: [email protected]; Phone: (+41)-031-631-5598 † I wish to thank the Co-Editor (Ricardo Reis) and an anonymous referee for extremely helpful comments and suggestions. Thanks to Fabio Canova and Russell Cooper for helpful discussions; Julio Carrillo, Harald Uhlig and seminar participants at CREST, the European University Institute, Norges Bank, the Swiss National Bank’s 2012 research conference ‘Policy Challenges and Developments in Monetary Economics’, and the 2013 Workshop on Empirical Macroeconomics held at the University of Ghent for comments; and Juan Rubio-Ramirez for helpful suggestions. Usual disclaimers apply.
The Long-Run Phillips Curve: A Structural VAR Investigation
1
1.
2
Introduction
2
The long-run Phillips curve (henceforth, LRPC)—describing the alternative combinations
3
of inflation and the unemployment rate an economy can achieve in equilibrium—plays a
4
central role in monetary policy. If the LRPC is vertical there is no long-run trade-off between
5
the two variables, and the central bank ought therefore simply aim at keeping inflation low
6
and stable. If, on the other hand, the LRPC were negatively sloped, this would open up the
7
possibility of trading off a permanent increase in inflation for a permanent reduction in the
8
unemployment rate.
9
In spite of such a crucial role played by the slope of the LRPC in monetary policymaking,
10
surprisingly little econometric work has been devoted to investigating the nature of the long-
11
run trade-off. In particular, the only existing investigation of the LRPC based on structural
12
VAR methods is King and Watson (1994)’s ‘revisionist econometric history’ of the post-
13
WWII U.S. Phillips curve, which produced evidence of a negative, and statistically significant
14
long-run trade-off conditional on aggregate demand-side shocks.1
15
This paper uses both cointegration methods, and non-cointegrated structural VARs fea-
16
turing either one or more permanent inflation shocks, in order to investigate the nature of
17
the LRPC for the post-WWII U.S., U.K., Euro area, Canada, and Australia. Since, as ex-
18
tensively discussed by King and Watson (1994)—who, in turn, built on Sargent (1971) and
19
Lucas (1972a)—the very nature of the LRPC involves permanent variation in (i.e., perma-
20
nent shocks to) inflation and the unemployment rate, it starts by exploring a ‘strong-form 1
Some papers (see, e.g., Bruno and Easterly, 1998) have eschewed time-series single-country studies, focusing instead on the impact on output growth of ‘discrete high inflation crises’, defined as periods in which inflation is above some threshold. Taking 40 per cent to be such threshold, Bruno and Easterly found that ‘growth falls sharply during discrete high inflation crises, then recovers rapidly and strongly after inflation falls’. The key problem with this type of evidence (which is common to the entire literature on growth regressions) is the difficulty of establishing a clear direction of causality. Since high-inflation episodes cannot be regarded as random experiments, the very same factors causing high inflation (e.g., an incompetent government mismanaging the economy) may well be behind the fall in average output growth during the episode.
The Long-Run Phillips Curve: A Structural VAR Investigation
3
1
LRPC’, in which the two series are possibly cointegrated. Evidence that this may be the
2
case is overall weak, and we therefore move to investigating, via structural VAR methods,
3
a ‘medium-form LRPC’ in which permanent inflation shocks, which are not further disen-
4
tangled, are allowed to have a permanent impact on the unemployment rate. For neither
5
country it is possible to reject the null hypothesis of a vertical LRPC. The extent of uncer-
6
tainty is however substantial, thus implying that a researcher with an alternative prior on
7
what a reasonable slope of the LRPC may be would likely not see her views falsified. Finally,
8
since absence of a ‘medium-form LRPC’ is not incompatible with the notion that some of
9
the shocks having a permanent impact on inflation may generate a non-vertical long-run
10
trade-off, the paper proceeds to explore a ‘weak-form LRPC’, in which permanent inflation
11
shocks are disentangled into demand- and supply-side ones by imposing sign restrictions on
12
their impacts on the endogenous variables at t=0. For neither of the four identified shocks
13
there is any evidence that it may generate a non-vertical LRPC. This is true, in particular,
14
for the monetary policy shock, which in the light of the central role played by the slope of the
15
LRPC in monetary policy, is the single most important one among those identified herein.
16
The present work expands upon King and Watson (1994) under several respects. First,
17
King and Watson’s analysis was entirely based on a bivariate VAR for the first differences of
18
inflation and the unemployment rate. As shown by Evans (1994) in his comment, however,
19
even based on their identification strategy, evidence based on trivariate VARs was sometimes
20
significantly different, pointing in some cases towards a vertical long-run Phillips curve.
21
This paper therefore uses six-variables VARs featuring a broader informational content, in
22
particular about the stance of monetary policy and the state of the business cycle. Second, in
23
the years since 1994 structural VAR econometrics has seen important developments in terms
24
of identification. When King and Watson wrote, short-run restrictions were still either
25
of the inertial type (i.e., based on imposing zeros in the impact matrix of the structural
26
shocks at t=0), or they were based on the notion of calibrating some of the impacts based
The Long-Run Phillips Curve: A Structural VAR Investigation
4
1
on information extraneous to the VAR. As previously mentioned, in the present work the
2
non-cointegrated VARs are identified based on either long-run restrictions, or a combination
3
of long-run and sign restrictions. Finally, beyond the U.S. the paper consider four other
4
countries which have exhibited, over the sample periods considered herein, strong evidence
5
of permanent variation in inflation.
6
The fact that in order to be able to identify the slope of the LRPC inflation ought to
7
have a unit root (Sargent, 1971; Lucas, 1972a; King and Watson, 1994) immediately defines
8
the relationship between the present work and Svensson’s (2015) recent investigation of the
9
slope of the LRPC for a group of inflation-targeting countries. As we previously documented
10
(Benati, 2008), under inflation targeting evidence of a unit root in inflation has entirely
11
vanished, and inflation has instead been, in most cases, statistically indistiguishable from
12
white noise. This logically implies that Svensson’s estimates are uninformative about the
13
slope of the LRPC for these countries.
14
The paper is organized as follows. The next section discusses the three different notions
15
of long-run Phillips curve considered in this paper. Section 3 discusses the data, and reports
16
results from unit root tests. Section 4 reports and discusses the evidence, starting from the
17
results from cointegration analysis, and then moving to evidence based on non-cointegrated
18
structural VARs featuring one and, respectively, four permanent inflation shocks. Section 5
19
concludes.
20
2.
21
22
Three Alternative Notions of the Long-Run Phillips Curve In what follows we consider three alternative forms of the long-run Phillips curve (hence-
forth, LRPC).
The Long-Run Phillips Curve: A Structural VAR Investigation
1
2.1.
5
A strong-form LRPC
2
First, a ‘strong form of the LRPC’, in which inflation and the unemployment rate are
3
(possibly) cointegrated. Such a strong form implies that (i) all shocks having a permanent
4
impact on inflation also have a permanent impact on the unemployment rate (and vice
5
versa), and (ii ) for either of these shocks, the ratio between the long-run impacts on the
6
two variables is exactly the same (thus implying that inflation and unemployment share the
7
same stochastic trend across all possible shocks). Standard cointegration methods are used
8
in order to investigate whether the data are consistent with such a strong from of the LRPC.
9
2.2.
A medium-form LRPC
10
Since either (i ) or (ii) may well be violated, the paper also considers two weaker forms
11
of the LRPC, starting from a ‘medium-form LRPC’ conceptually in line with King and
12
Watson (1994), in which the unit root component of the unemployment rate is driven by
13
both idiosyncratic shocks, and permanent shocks to inflation. It is not difficult to think of
14
circumstances under which this may be the case. For example, permanent shocks to the
15
tax rate on labor should lead to increases in the natural rate of unemployment, but—being
16
shocks to the level of the tax rate, as opposed to its rate of change—they should only have
17
a transitory impact on inflation. By the same token, given the widely documented stylized
18
fact that unemployment rates differ systematically by age/sex/race groups, secular changes
19
in the composition of the labor force should lead to permanent changes in the overall natural
20
rate of unemployment, but at the same time they should not have any permanent impact on
21
inflation. So the bottom line is that it is not difficultfs to think of circumstances in which
22
the strong form of the LRPC does not hold, but the medium-form may hold.
23
In the spirit of King and Watson (1994), in what follows the paper will search for a
24
medium-form LRPC based on structural VARs featuring, beyond the first differences of
25
inflation and the unemployment rate, a measure of the output gap, the consumption/GDP
The Long-Run Phillips Curve: A Structural VAR Investigation
6
1
ratio, a long-short spread, and (the first difference of) the short-term monetary policy rate.
2
These variables expand the informational content of King and Watson’s original bivariate
3
VAR along several dimensions. The short rate and, to a lesser extent, the long-short spread
4
contain information about the monetary policy stance. As recently shown by Kurmann
5
and Otrok (2013), the long-short spread possesses a strong informational content for future
6
movements in technology. Finally, the consumption/GDP ratio and the output gap measure
7
can be regarded as two ‘noisy estimates’ of the state of the business cycle. There are two
8
reasons for considering these additional variables. First, King and Watson’s analysis was
9
entirely based on a bivariate VAR for the first differences of inflation and the unemployment
10
rate, but, as shown by Evans (1994) in his comment, even based on their identification
11
strategy, evidence based on trivariate VARs was sometimes significantly different, and in
12
some cases it pointed towards a vertical long-run Phillips curve. Second, as recently discussed
13
by Forni and Gambetti (2014), small VARs with a limited informational content may well
14
produce unreliable results. A single, ‘aggregate’ permanent inflation shock will be identified
15
via standard long-run restrictions, and we will then explore whether such a shock has a
16
non-zero long-run impact on the unemployment rate.
17
2.3.
A weak-form LRPC
18
Finally, the paper explores a ‘weak-form LRPC’, in which the just-mentioned ‘aggregate’
19
permanent inflation shock is disentangled into four different disturbances, which are allowed
20
to give rise to idiosyncratic (i.e., shock-specific) long-run trade-offs between inflation and
21
unemployment. This is achieved by imposing Canova and Paustian’s (2011) DSGE-based
22
‘robust sign restrictions’ on the impact of the shocks on the endogenous variables at t=0. The
23
paper then proceeds to explore whether any of these shocks has a statistically significant long-
24
run impact on the unemployment rate. Although four different shocks having a permanent
25
impact on inflation are identified, most of the focus is placed upon disturbances of a monetary
The Long-Run Phillips Curve: A Structural VAR Investigation
7
1
nature. The reason for doing so is that the motivation behind all explorations of the slope
2
of the LRPC has always been understanding whether the monetary authority might be
3
able to permanently lower the unemployment rate by engineering a permanently higher rate
4
of inflation. Under this respect, the long-run Phillips trade-offs which might originate from
5
permanent inflation shocks of a non-monetary nature are therefore irrelevant. The framework
6
used in order to explore the weak-form LRPC has an obvious connection to the one used
7
to explore the medium-form one. Whereas, by construction, the permanent inflation shocks
8
identified by the ‘medium-form LRPC’ VAR explain 100 per cent of the infinite long-run
9
variance of inflation, the four permanent inflation shocks identified by the ‘weak-form LRPC’
10
VAR jointly explain all of the long-run variance of inflation.
11
3.
The Data
12
As mentioned in the Introduction, the possibility of identifying the long-run impact on
13
the unemployment rate of permanent inflation shocks crucially hinges on the fact that both
14
series contain an I(1) component. As documented by Benati (2008), however, evidence of high
15
inflation persistence is weak-to-non-existent for all sample periods which are not dominated
16
by the Great Inflation episode (and especially so for monetary regimes such as inflation
17
targeting and European Monetary Union, henceforth, EMU). In what follows we therefore
18
consider the following sample periods: for the Euro area, 1970Q1-1998Q4; for the U.K.,
19
1972Q2-1992Q3; for Canada, 1961Q2-1990Q4; for Sweden 1970Q1-1992Q4; for Australia,
20
1969Q3-1994Q2; and for Japan the period following the collapse of Bretton Woods.2 Finally, 2
For the Euro area, EMU started in January 1999, whereas Euro area data are only available starting from 1970Q1. For the U.K., June 1972 marked the floating of the pound vis-` a-vis the U.S. dollar, whereas inflation-targeting was introduced in October 1992. As shown by Benati (2008), before the June 1972 floating of the pound U.K. inflation exhibited quite significantly lower persistence. In Canada, inflation targeting was introduced in February 1991, whereas 1961Q1 is when Canadian national account data first become available. In Sweden, inflation targeting was introduced in January 1993, whereas the unemployment rate is only available since 1970Q1. In Australia, the current inflation targeting regime was never clearly announced, and we therefore follow Bernanke et al., 1999, in marking its start in 1994Q3. On the other hand, the short rate is available since 1969Q3. As for Japan, since 1971 the Japanese government has not explicitly introduced any new monetary regime, and has instead relied on a generic committment to price
The Long-Run Phillips Curve: A Structural VAR Investigation
8
1
for the U.S., as discussed below, several alternative sample periods are considered. All of the
2
data and their sources are described in Appendix A.
3
4
[Table 1 here] 3.1.
Results from unit root tests
5
Table 1 reports, for either country, bootstrapped p-values for Elliot et al.’s (1996) unit
6
root tests for inflation, the unemployment rate, and the short-term interest rate.3 Based on
7
either the CPI or the GDP deflator, the null of a unit root in inflation cannot be rejected
8
for either the Euro area, the U.K., Canada, or Australia, with the p-values being uniformly
9
greater than 10 per cent, in most cases comfortably so. Evidence for Sweden and Japan,
10
on the other hand, is not clear-cut, and in what follows I therefore exclude these countries
11
from the analysis. Finally, for the U.S. evidence of a unit root based on the entire sample
12
period since January 1959 is strong based on the PCE deflator, and it is just slightly less
13
so based on the GDP deflator. Splitting the full sample in August 1979, however, clearly
14
shows how evidence of a unit root entirely originates from the pre-Volcker period, and it is
15
instead absent from the second sub-period, for which the p-values are uniformly very low.
16
This suggests that results for the U.S. based on the full sample period should be viewed with
17
suspicion, and in what follows I will therefore uniquely focus on the pre-Volcker period.
18
Evidence of a unit root in the unemployment rate is strong for either the Euro area, the
19
U.K., Canada, or Australia. For the U.S., evidence is compatible with the notion that the
20
unemployment rate contains a unit root in both sub-periods, whereas results for the entire
21
sample period are weak.
22
As for the output gap measure, the consumption/GDP ratio, and the spread, Elliot et stability. 3 Estimated models include an intercept, but no time trend. For either series, p-values have been computed by bootstrapping 10,000 times estimated ARIMA(p,1,0) processes. In all cases, the bootstrapped processes are of length equal to the series under investigation. As for the lag order, p, since, as it is well known, results from unit root tests may be sensitive to the specific lag order which is being used, for reasons of robustness we consider four alternative lag orders, 2, 4, 6 and 8 quarters (for the U.S., for which we use monthly data, the corresponding lag orders are 6, 12, 18, and 24).
The Long-Run Phillips Curve: A Structural VAR Investigation
9
1
al.’s (1996) tests clearly point towards these series being stationary (these results are not
2
reported for reasons of space, but they are available upon request). As for the short rate,
3
the evidence in Table 1 strongly points towards a unit root for all countries, and based on
4
either lag order, with the single exception of the U.K.. In what follows, the short rate will
5
therefore enter the VAR in levels for the U.K., and in first differences for all other countries.4 [Table 2 here]
6
7
4.
Evidence We start by exploring the possible presence of a ‘strong-form LRPC’ via standard coin-
8
9
10
tegration methods. 4.1.
Results based on cointegration analysis
11
Table 2 reports bootstrapped p-values for Johansen’s trace test of the null of no coin-
12
tegration, against the alternative of one or more cointegrating vectors, for either inflation
13
and unemployment, inflation and the short rate (with the single exception of the U.K., for
14
which results from unit root tests strongly reject the null of a unit root in the short rate),
15
or inflation, the unemployment rate, and the short rate. Following Cavaliere, Rahbek, and
16
Taylor (2012), p-values have been computed by bootstrapping the VAR estimated for the
17
first difference of the relevant vector of series.5 For the U.S. no evidence of cointegration 4
For reasons of robustness we have also considered tests based on Phillips’ Zρ and Zt statistics (for details, see Section 3 in the online appendix). These results are almost uniformly in line with those from Elliot et al.’s tests. The main difference— which is however irrelevant for our purposes—pertains to U.S. inflation for the full sample period since January 1959, for which the null of a unit root is strongly rejected (on the other hand, in line with the results from Elliot et al.’s tests, results for the pre-Volcker period clearly point towards a unit root in U.S. inflation based on either price index). 5 To be clear, the VAR which is being bootstrapped is not a cointegrated VAR, that is, it is equal to the VECM representation without the error-correction term. Given the vector of the relevant series Yt , we start by selecting the lag order for cointegration tests as the maximum between the lag orders selected based on the Schwartz and the Hannan-Quinn criteria (we eschew the AIC since, with I(1) variables, it is an inconsistent lag selection criterion, see Luetkepohl (1991)), and we perform Johansen’s trace test of the null of no cointegration. Then, we estimate the VAR for ΔYt ; we bootstrap it 10,000 times, thus generating bootstrapped artificial series ΔY˜tj ; based on each of them we compute corresponding bootstrapped artificial series Y˜tj (that is, those for the levels of the series); and finally, based on each of them we perform the same
The Long-Run Phillips Curve: A Structural VAR Investigation
10
1
between inflation and the unempoyment rate is detected. On the other hand, conceptually in
2
line with Fama (1975) and Barsky (1987), there is strong evidence of cointegration between
3
inflation and the Federal Funds rate. The median estimate of the second element of the
4
normalized cointegration vector is -1.018, with the p-value for rejecting the null hypothesis
5
that the cointegration vector is [1; -1] being equal to 0.423. For the Euro area, Canada,
6
and Australia on the other hand, bivariate cointegration tests do not detect any evidence of
7
cointegration between inflation and the short rate.6 Finally, for either the U.S., the Euro
8
area or, Canada there is strong evidence of cointegration between inflation, unemployment,
9
and the short rate, whereas for the U.K. evidence points, at the 10 per cent level, towards
10
cointegration between inflation and the unemployment rate.
11
For the U.S., the Euro area and Canada, for which evidence of cointegration between
12
the three series was detected, the table also reports bootstrapped p-values for testing the
13
null hypothesis of one single cointegrating vector, versus the alternative of two cointegrating
14
vectors.7 Whereas the p-value for Canada, at 0.2896, is very far from being significant at any
15
conventional level, those for the Euro area and the U.S., at 0.0531 and 0.0728, respectively,
16
point towards the presence of an additional cointegrating vector at the 10 per cent level.
17
[Figure 1 here]
18
For the Euro area and the U.K., for which evidence of cointegration in the bivariate
19
representation for Yt = [π t , Ut ] was detected at the 5 per cent level, Figure 1 reports
20
the bootstrapped distributions of the ratio between the permanent impacts of the common
21
shock on the unemployment rate and inflation, respectively (that is: the slope of the longtrace test we previously computed based on the actual data, thus building up the empirical distribution of the trace statistic under the null of no cointegration. Based on this distribution, we then compute critical values (not reported here) and p-values. 6 Although at first sight puzzling—taken at face value, these results imply a rejection of the Fisher hypothesis that permanent shifts in inflation should map one-to-one into corresponding shifts in interest rates—it ought to be stressed that empirical evidence on the violation of the Fisher hypothesis is widespread. 7 Details of the bootstrapping procedure are the same as before, with the only difference that, instead of bootstrapping the estimated VAR for ΔYt under the null of no cointegration, we bootstrap the VECM estimated conditional on there being one single cointegrating vector.
The Long-Run Phillips Curve: A Structural VAR Investigation
11
1
run Phillips curve implied by the estimated cointegrating vector); and of the elements of
2
the loading vector on the cointegration residual in the VECM representation. For either
3
country, the long-run impact on the unemployment rate induced by the common shock
4
(normalized in such a way as to induce a one per cent permanent increase in inflation) is
5
highly statistically significantly different from zero, as implied by the results from the trace
6
test, and as testified by the fact that mass of the bootstrapped distribution is pretty much
7
away from zero. Further, the estimated impact is also sizeable, with median estimates equal
8
to -1.074 for the Euro area and -1.537 for the U.K., and 90 per cent-coverage bootstrapped
9
confidence intervals equal to [-1.327; -0.866] and [-2.278; -0.759], respectively.
10
Taken at face value, these results point towards the presence of a negative and sizeable
11
long-run trade-off induced by the common permanent shock in both the Euro area and the
12
U.K.. The next two sub-sections provide some perspective on this, by exploring the extent
13
to which these results may just be statistical flukes, and then focusing on whether—even
14
assuming that the results are genuine—they are in fact compatible with the notion of a
15
trade-off which can be exploited by policymakers.
16
4.1.1.
17
The case of two independent random walks A first possibility is that the results for
18
the Euro area and the U.K. are statistical flukes, possibly due to the bad performance of
19
Johansen’s procedure in small-samples. In order to assess this possiblity we consider five sets
20
of 10,000 Monte Carlo simulations of lengths equal to the actual sample lengths for the five
21
countries. For each simulation, two independent random walks are randomly generated, and
22
the same procedure which has previously been applied to the actual data is used in order to
23
test for the null of no cointegration between them. Ideally, out of the 10,000 simulations, the
24
fraction of bootstrapped p-values below x per cent should be equal to x per cent. As the
25
results reported in Table A.2 in the online appendix8 show, the procedure used herein gets 8
Monte Carlo evidence on the performance of Johansen’s procedure
Supplementary material is available online at Science Direct.
The Long-Run Phillips Curve: A Structural VAR Investigation
12
1
quite remarkably close to this ideal: the fraction of simulations for which the bootstrapped
2
trace statistic incorrectly rejects the null of no cointegration at a given significance level
3
ranges between 11.3 and 11.9 per cent at the 10 per cent level; between 5.5 and 6.0 per cent
4
at the 5 per cent level; and between 1.1 and 1.3 per cent at the 1 per cent level.
5
[Table 3 here]
6
Taking the non-cointegrated structural VARs as data-generation processes Al-
7
though the procedure performs well conditional on a data-generation process (henceforth,
8
DGP) in which the series of interest are independent random walks, it is an open question
9
how well it might perform conditional on DGPs such as the non-cointegrated structural VARs
10
featuring permanent inflation shocks estimated in Sections 4.2 and 4.3, respectively. Since
11
these results will produce no evidence of a non-vertical long-run Phillips trade-off conditional
12
on any kind of permanent inflation shock, it is of interest to explore the performance of Jo-
13
hansen’s procedure conditional on these DGPs. Table 3 reports evidence on this, by showing
14
the fraction of simulations for which the bootstrapped trace statistic incorrectly rejects the
15
null of no cointegration at a given significance level. As the tables shows, Johansen’s proce-
16
dure tends to spuriously detect cointegration a non-negligible fraction of the times. Taking
17
the estimated Bayesian structural VARs featuring four permanent inflation shocks as DGPs,
18
for example, for the Euro area and the U.K., the fractions of bootstrapped p-values for the
19
trace statistic for testing the null of no cointegration between inflation and the unemploy-
20
ment rate which are smaller than 10 per cent are equal to 26.9 and 23.6 per cent respectively.
21
This means that, at the 10 per cent level, the trace test would incorrectly reject the null of no
22
cointegration between inflation and unemployment about one-fourth of the times. As for the
23
trace test of cointegration between inflation, unemployment, and the short rate, the fraction
24
of bootstrapped p-values smaller than 10 per cent range between 45.4 and 72.1 per cent,
25
thus essentially pointing towards the unreliability of such tests conditional on these DGPs.
26
Results based on taking Bayesian VARs featuring a single permanent inflation shock as the
The Long-Run Phillips Curve: A Structural VAR Investigation
13
1
DGP are in line with those discussed so far for the trace test of no cointegration between
2
inflation, unemployment, and the short rate, and are instead significantly better for the test
3
of no cointegration between inflation and unemployment.
4
Overall, these results suggest that evidence such as that reported in Table 2 should be
5
discounted, as these results might as well be due to the limitations of cointegration tests
6
conditional on these specific DGPs.
7
4.1.2.
The adjustment dynamics implied by the estimated loading vectors
8
Even ignoring the results reported in Table 3, the evidence shown in the last two columns
9
of Figure 1 is hardly compatible with the notion of a long-run trade-off a policymaker might
10
be able to purposefully exploit in order to permanently decrease the unemployment rate
11
by engineering a higher equilibrium inflation rate.9 The bootstrapped distributions of the
12
estimates of the loading on the first difference of the unemployment rate for the error-
13
correction term in the VECM reported in Figure 1, indeed, are extremely small for either
14
country. Further, in the case of the Euro area the estimated loading is insignificantly different
15
from zero at conventional levels (for the U.K., on the other hand, it is significant at the 10
16
per cent level, but not at the 5 per cent level). This implies that, following a shock to the
17
common stochastic trend, the bulk of the dynamic adjustment to disequilibrium takes place
18
through movements in inflation, rather than via movements in the unemployment rate, which
19
is difficult to reconcile with the notion of an exploitable Phillips trade-off. 9
As pointed out by Evans (1994, p. 222) in his comment on King and Watson (1994), ‘[...] for a trade-off to be viewed as exploitable, a decision-maker must have confidence that unemployment can be reduced by engineering a higher rate of inflation [...] It would be of little comfort to many politicians if the Phillips curve trade-off simply implied that if unemployment instead turned higher, at least inflation would be lower.’
The Long-Run Phillips Curve: A Structural VAR Investigation
1
2
3
4.2.
14
Evidence from non-cointegrated SVARs featuring a single permanent inflation shock
We start by considering VARs featuring a single permanent inflation shock. For either country the VAR(p) model Yt = B0 + B1 Yt−1 + ... + Bp Yt−p + ut ,
4
E[ut ut ] = Ω
(1)
5
is estimated, where Yt ≡ [yt , Δπ t , ΔUt , ΔRt , St ] for the Euro area, Canada, and Australia,
6
and Yt ≡ [yt , Δπ t , ΔUt , Rt , CYt , St ] for the U.K., with yt , π t , Ut , Rt , CYt , and St being the
7
output gap, inflation, the unemployment rate, the short rate, the consumption/GDP ratio,
8
and the long-short spread, respectively. The U.S. is the only country for which Johansen’s
9
tests produces evidence of cointegration between the short rate and inflation (see Table
10
2), with the bootstrapped, bias-corrected estimate of the second element of the normalized
11
cointegrating vector being equal to -1.018, and with a p-value of 0.423 for rejecting the null
12
hypothesis that it is equal to -1. For the U.S. Yt is therefore set to Yt ≡ [yt , Δπ t , ΔUt , CRt ,
13
Rt -π t , St ] , where Rt -π t is the ex post real rate (i.e., the theoretical cointegration residual).
14
Finally, for the Euro area, Canada, and the U.K. the lag order is set to p=4. For the U.S.,
15
for which the data are monthly, the lag order is set to p=12.
16
For reasons of robustness, the VARs are estimated based on both Classical and Bayesian
17
methods. Since the results produced by the two approaches are qualitatively the same, and
18
numerically very close, in what follows only those based on the Bayesian approach are re-
19
ported and discussed. The online appendix, however, contains a detailed description of the
20
Classical estimation methodology, and reports the full set of results (see Tables A.3 and A.8).
21
4.2.1.
Bayesian estimation
22
Working within a Bayesian context, the reduced-form VAR (1) is estimated as in Uhlig
23
(1998, 2005). Specifically, Uhlig’s approach is exactly followed in terms of both distribu-
24
tional assumptions—the distributions for the VAR’s coefficients and its covariance matrix
The Long-Run Phillips Curve: A Structural VAR Investigation
15
1
are postulated to belong to the Normal-Wishart family—and of priors. For estimation de-
2
tails the reader is therefore referred to either the Appendix of Uhlig (1998), or to Appendix
3
B of Uhlig (2005). Results are based on 10,000 draws from the posterior distribution of the
4
VAR’s reduced-form coefficients and the covariance matrix of its reduced-form innovations
5
(the draws are computed exactly as in Uhlig (1998, 2005)).
6
4.2.2.
Identification
7
A single permanent inflation shock is identified based on the restriction that it is the
8
only shock impacting upon inflation in the infinite long run. Since this shock is exactly
9
identified, drawing from the VAR’s reduced form parameters is sufficient in order to get the
10
impulse-response functions, because for each draw of the reduced form parameters there is
11
just one (normalized) impulse-response function.
12
[Table 4 here]
13
[Figure 2 here]
14
4.2.3.
Evidence
15
Figure 2 shows, for either country, the distribution of the long-run impact on the unem-
16
ployment rate of a one per cent permanent shock to inflation, together with the distribution
17
of the fraction of the permanent component of unemployment which is explained by the per-
18
manent inflation shock. Table 4 reports the median and the 90%-coverage percentiles of the
19
distribution of the long-run impact on the unemployment rate of a one per cent permanent
20
shock to inflation, together with the fraction of the mass of the distribution for which the
21
impact is estimated to be negative.
22
Results are consistent across countries, and for neither of them do they allow to reject the
23
‘natural’ null hypothesis of a vertical long-run Phillips curve. In particular, for either coun-
24
try the 90 per cent confidence interval for the estimated long-run impact on unemployment
25
of a permanent inflation shock contains zero, and the fraction of the mass of the posterior
The Long-Run Phillips Curve: A Structural VAR Investigation
16
1
distribution for which the impact is estimated to be negative is uniformly greater than stan-
2
dard levels of statistical significance. Further, for either country the posterior distribution
3
of the fraction of the permanent component of the unemployment rate which is explained by
4
permanent inflation shocks is clustered towards zero, thus implying that such shocks explain
5
very little of the unit root component of unemployment, which provides additional evidence
6
in support of the notion that the long-run Phillips curve is indeed vertical.10
7
A researcher looking for evidence of a negative long-run trade-off may find some limited
8
support from the results for the U.S. and Canada, for which the fraction of draws from the
9
posterior for which the long-run impact is estimated to be negative is slightly above 80 per
10
cent. Although far from standard levels of statistical significance, still, this provides some
11
support to the notion of a negative long-run trade-off. Further, the median estimate of the
12
long-run impact for the U.S., at -0.27, implies that a permanent increase in inflation by
13
10 percentage points would permanently decrease the unemployment rate by 2.7 per cent.
14
Overall, however, the picture emerging from Table 2 and Figure 2 (and from Table A.2 and
15
Figure A.2 in the online appendix) points towards no evidence of a long-run trade-off. For
16
the other four countries, for example, the median estimates of the permanent decrease in
17
the unemployment rate associated with a permanent increase in inflation by 10 percentage
18
points range between -0.4 and -1.3 per cent. Even if one were willing to believe that the
19
median estimates capture the authentic unemployment-inflation trade-offs out there, these
20
are hardly trade-offs which might induce policymakers to ‘try to play the Phillips curve’.
21
4.3.
Evidence on monetary policy shocks from non-cointegrated SVARs
22
Since, in principle, the results discussed in the previous section are not incompatible
23
with the notion that some of the shocks exerting a permanent impact on inflation may
24
induce a non-vertical long-run Phillips trade-off, we then proceed to disentangle permanent 10
By definition, a vertical long-run Phillips curve implies that the fraction of the permanent component of the unemployment rate which is explained by the permanent inflation shock is equal to zero.
The Long-Run Phillips Curve: A Structural VAR Investigation
17
1
inflation shocks into demand- and supply-side ones, by imposing Canova and Paustian’s
2
(2011) DSGE-based ‘robust sign restrictions’ on their impact on the endogenous variables
3
at t=0.11
4
4.3.1.
Identification
5
Our identification strategy combines long-run and sign restrictions. First, the VAR’s
6
structural shocks are separated into two sets, depending on the fact that they do, or they
7
do not have a permanent impact on inflation. Let the structural VAR(p) model be given
8
by Yt = B0 + B1 Yt−1 + ... + Bp Yt−p + A0 t , with Yt being defined as before; A0 being the
9
O K impact matrix of the structural shocks at t = 0; and t ≡ [Tt E , M , Tt A , M , Tt R ] being t t
10
the structural shocks, which, as standard practice, are assumed to be unit-variance and
11
O K orthogonal to one another, with Tt E , M , Tt A , M being Canova and Paustian’s (2011) t t
12
‘technology’, ‘monetary policy’, ‘taste’, and ‘markup’ shocks, which are here allowed to exert
13
a permanent impact on inflation, and Tt R being instead a 2×1 vector of shocks which, by
14
construction, have a transitory impact on inflation. The second row of the matrix of long-run
15
impacts of the structural shocks, [IN −B(1)]−1 A0 —i.e., the row corresponding to inflation—is
16
therefore postulated to have the following structure,
E M O T A M K T R T t t t t t
x
17
x
x
x
01×2
(2)
18
—where a ‘0’ means that the corresponding long-run impact has been restricted to zero,
19
O whereas an ‘x’ means that it has been left unrestricted—thus implying that Tt E , M , Tt A , t
20
K and M may have a permanent impact on inflation, whereas Tt R does not. The restriction t
21
that the two shocks in Tt R are the only shocks which do not have a permanent impact on 11
Since Canova and Paustian’s ‘robust sign restrictions’ have originally been derived based on a New Keynesian model log-linearized around a zero-inflation steady-state, and in which inflation is stationary, it is in principle an open question whether such restrictions would also hold in the case in which, within the same model, inflation has a unit root. In fact, this is indeed the case, with the results for the case in which inflation is I(1) being numerically very close to the ‘benchmark’ results one obtains based on the specification reported in Canova and Paustian’s Table 1 (all of these results are available upon request).
The Long-Run Phillips Curve: A Structural VAR Investigation
18
1
inflation is sufficient to disentangle them from the other four shocks. As for separating Tt E ,
2
O K M , Tt A , and M from one another, this is achieved by imposing on impact the set of sign t t
3
restrictions reported in Table 5, where ‘+’ means ‘greater than, or equal to zero’, and ‘-’
4
means ‘smaller than, or equal to zero’. These restrictions are the same as the ‘robust sign
5
restrictions’ reported by Canova and Paustian (2011) in their Table 2 for their benchmark
6
DSGE model featuring sticky prices, sticky wages, and several standard frictions (see the
7
column labelled there as ‘M’), with the only obvious difference that, since their model features
8
employment, instead of the unemployment rate, the signs which are here being imposed on
9
the unemployment rate are the opposite of those reported by Canova and Paustian for
10
employment.
11
[Table 5 here]
12
In what follows, the sign restrictions are imposed only on impact. The reason for do-
13
ing this is that, as stressed by Canova and Paustian (2011), whereas sign restrictions on
14
impact are, in general, robust—in the specific sense that they hold for the vast majority
15
of sub-classes within a specific class of DSGE models, and for the vast majority of plausi-
16
ble parameters’ configurations—restrictions at longer horizons are instead, as they put it,
17
‘whimsical’, meaning that they are hard to pin down, and in general, they are not robust
18
across sub-classes of models, and for alternative plausible parameters’ configurations. It is
19
to be noticed that Canova and Paustian (2011) reached this conclusion based on a quarterly
20
DSGE model. Since for the U.S. the data are monthly, for reasons of consistency the sign
21
restrictions are imposed both on impact, and for the two months after impact.
22
4.3.2.
Computing the structural impact matrix A0
23
For each draw from the posterior distribution of the VAR’s reduced-form parameters, the
24
structural impact matrix, A0 , is computed via the methodology proposed by Arias, Rubio-
25
Ramirez, and Waggoner (2014) for combining zero and sign restrictons (the methodology is
The Long-Run Phillips Curve: A Structural VAR Investigation
1
described in the online appendix).
2
[Figure 3 here]
3
[Figure 4 here]
4
[Figure 5 here]
5
19
4.3.3.
Evidence
6
Figures 3-5 and Table 6 show the results for monetary shocks, whereas Figures A.4-A.6
7
and Tables A.4-A.6 in the online appendix report the full set of results for all the four
8
shocks. The reason for narrowly focusing on monetary shocks is that the motivation behind
9
all explorations of the slope of the long-run Phillips trade-off has always been understanding
10
whether the monetary authority might be able to permanently lower the unemployment
11
rate by engineering a permanently higher rate of inflation. Under this respect, the long-run
12
Phillips trade-offs which might originate from permanent inflation shocks of a non-monetary
13
nature are therefore irrelevant. It is to be stressed, however, that results for the other three
14
permanent inflation shocks are qualitatively the same as those for monetary shocks, and
15
clearly suggest that, even ‘digging’ into the permanent inflation shock identified in Section
16
4.2, it is not possible to find any evidence that at least some shocks may generate a non-
17
vertical trade-off.
18
Figure 3 shows, for either country, the posterior distribution of the long-run impact on
19
the unemployment rate of a monetary shock having a one per cent permanent impact on
20
inflation, whereas Figures 4 and 5 show the posterior distributions of the fractions of the
21
permanent components of inflation and the unemployment rate, respectively, explained by
22
monetary shocks. Table 6 reports the the median and the 90%-coverage percentiles of the
23
posterior distribution of the long-run impact of monetary shocks on unemployment; the
24
fraction of draws for which the impact is estimated to be negative; and the medians and
25
the 90%-coverage percentiles of the posterior distributions of the fractions of the permanent
The Long-Run Phillips Curve: A Structural VAR Investigation
1
20
components of inflation and unemployment explained by monetary shocks.
2
Overall, results for monetary shocks are in line with those for the ‘aggregate’ permanent
3
inflation shocks discussed in Section 4.2, and in no way provide support to the notion that
4
the monetay authority might be able to permanently reduce the unemployment rate by
5
engineering a permanently higher inflation rate. Once again, the fractions of the mass of
6
the posterior distribution for which the impact is estimated to be negative, ranging between
7
0.480 for the Euro area and 0.836 for Canada, are uniformly greater than standard levels
8
of statistical significance, and for either country the 90 per cent confidence interval for the
9
estimated long-run impact of a monetary shock on unemployment contains zero. Finally, for
10
either country monetary shocks are estimated to have accounted for very small fractions of
11
the unit root component of the unemployment rate.
12
Once again, however, the uncertainty associated with the estimated long-run trade-offs
13
is uniformly substantial, so that these results are in fact compatible with a wide range of
14
possible slopes. This means that, although the null hypothesis of a vertical long-run trade-off
15
should be regarded as the natural one, a researcher with a different view of the world, and
16
therefore different priors about what the natural benchmark should be, might not see her
17
views about the slope of the long-run Phillips curve falsified.12 [Table 6 here]
18
19
5.
Conclusions
20
Overall, the evidence discussed in this paper provides essentially no support to the no-
21
tion of a non-vertical long-run Phillips trade-off—in particular of a trade-off which can be
22
actively exploited by a policymaker in order to permanently reduced the unemployment rate. 12
This originates from the fact that the feature of the data we are here estimating pertains to the infinite long-run, and, as it is well known—see, first and foremost, Faust and Leeper (1997)—this inevitably produces imprecise estimates, unless the researcher is willing to impose upon the data very strong informational assumptions. Within the present context, this problem is compounded by our use of sign restrictions, which, as stressed by Fry and Pagan (2011), are intrinsically ‘weak information’, and should therefore not be expected to produce strong inference.
The Long-Run Phillips Curve: A Structural VAR Investigation
21
1
Uncertainty, however, is uniformly substantial, so that a researcher who did not regard a ver-
2
tical long-run Phillips trade-off as the ‘natural null hypothesis’, and who entertained instead
3
alternative notions on what a reasonable slope of the long-run trade-off could be, might well
4
not see her priors falsified. For the United States, for example, it is possible to reject at the
5
10 per cent level all values of the long-run impact on the unemployment rate of a one per cent
6
permanent inflation shock smaller than -0.69, corresponding to slopes of the LRPC flatter
7
than -1.45. In particular, the p-value associated with a slope equal to -1 is equal to 0.0325,
8
thus implying that a one-for-one trade-off can be rejected with great confidence. Although
9
the extent of econometric uncertainty associated with our estimates is substantial, it still
10
therefore allows us to reject moderately-to-very flat LRPCs. At the same time, however, any
11
slope steeper than -1.4493 cannot be rejected at the 10 per cent level, thus implying that a
12
researcher holding the prior view that the slope of the LRPC is (e.g.) equal to -2 would not
13
see her view of the world falsified. Such a large extent of econometric uncertainty is nothing
14
but a manifestation of the intrinsic difficulty of reliably estimating the slope of the long-run
15
Phillips curve. There are several reasons for this. First of all, since the slope of the long-run
16
Phillips curve is a feature of the data pertaining, in principle, to the infinite long-run, Faust
17
and Leeper’s (1997) well-known point about the difficulty of capturing the long-run based on
18
finite samples automatically applies. On top of this, however, there is a key problem which
19
is specific to long-run Phillips curve itself. As first discussed by Sargent (1971) and Lucas
20
(1972a), and then reiterated by King and Watson (1994), the possibility of identifying the
21
slope of the long-run Phillips curve crucially hinges on the fact that inflation has exhibited,
22
over the sample period, permanent variation. But here is precisely the problem: as it has
23
been documented (Barsky, 1987; Benati, 2008) before WWI inflation was white noise, and,
24
if anything, it exhibited some mild negative serial correration; between 1914 and the onset of
25
the Great Inflation, inflation persistence increased, but a unit root can typically be rejected;
26
finally, after the Great Inflation, inflation persistence has disappeared in all countries having
The Long-Run Phillips Curve: A Structural VAR Investigation
22
1
an explicit inflation objective (Benati, 2008). So far, the Great Inflation episode is therefore
2
the only one during which inflation has clearly exhibited a unit-root-like behavior. This
3
logically implies that, unless, in the future, central banks will lose control of inflation for a
4
sufficiently long period of time—thus injecting a unit root into it—the only data researchers
5
will be able to use to explore the slope of the long-run Phillips curve will be those we ana-
6
lyzed herein. To put it differently, if central banks will continue being successful at keeping
7
inflation under control, economists will forever be unable to obtain precise estimates of the
8
slope of the long-run Phillips curve.
The Long-Run Phillips Curve: A Structural VAR Investigation
1
6.
23
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Arias, J., Rubio-Ramirez, J., and Waggoner, D., 2014. Inference Based on SVARs Iden-
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Lucas Jr., R. E., 1972a. Econometric testing of the natural rate hypothesis. In Eckstein,
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25
26
Lucas Jr., R. E., 1972b. Expectations and the neutrality of money. Journal of Economic Theory 4(April), 103-24. Lucas Jr., R. E., 1973. Some international evidence on output-inflation trade-offs. American Economic Review 63(3), 326-334. Sargent, T. J., 1971. A note on the accelerationist controversy. Journal of Money, Credit and Banking 3(3), 721-725. Svensson, L.E.O., 2015. The possible unemployment cost of average inflation below a credible target. American Economic Journal: Macroeconomics 7(1), 258-96. Uhlig, H., 1998. Comment on: The robustness of identified VAR conclusions about money. Carnegie-Rochester Conference Series on Public Policy 49, 245-263. Uhlig, H., 2005. What are the effects of monetary policy on output? Results from an agnostic identification procedure. Journal of Monetary Economics 52(2), 381-419.
Figure 1
Bootstrapped distributions of the slope of the long-run Phillips curve implied by the estimated cointegrating vector between inflation and unemployment, and of the elements of the loadings 22 vector
23
Figure 2 Results based on Bayesian VARs allowing for a single permanent inflation shock: posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to inflation; 24 and of the fractions of the permanent component of unemployment explained by the permanent inflation shock
Figure 3 Results based on Bayesian VARs allowing for four permanent inflation shocks: 25 posterior distribution of the long-run impact on the unemployment rate of a monetary shock having a one per cent permanent impact on inflation
26
Figure 4 Results based on Bayesian VARs allowing for four permanent inflation shocks: posterior distribution of the fraction of the permanent component of inflation explained by monetary shocks
27
Figure 5 Results based on Bayesian VARs allowing for four permanent inflation shocks: posterior distribution of the fraction28of the permanent component of the unemployment rate explained by monetary shocks
29
30
31
32
33
34
35
36
37
Figure 1 CPI inflation and the unemployment rate
38
39
1
Online appendix for:
2
‘The Long-Run Phillips Curve: A Structural VAR Investigation’ Luca Benatid>W
3
d
4
1.
The Data Here follows a detailed description of the dataset.
5
6
University of Bern
1.1. United States
7
A monthly seasonally adjusted series for the civilian unemployment rate (‘UNRATE:
8
Civilian Unemployment Rate, Seasonally Adjusted, Percent’) is from the U.S. the Depart-
9
ment of Labor, Bureau of Labor Statistics, and it has been converted to the quarterly
10
frequency by taking averages within the quarter. A monthly seasonally unadjusted series
11
for the 3-month Treasury Bill (‘TB3MS: 3-Month Treasury Bill: Secondary Market Rate’)
12
is from Board of Governors of the Federal Reserve System, and it has been converted to
13
the quarterly frequency by taking averages within the quarter. By the same token, monthly
14
seasonally unadjusted series for the 1-, 3-, 5-, and 10-year Treasury constant maturity rates
15
(acronyms are GS1, GS3, GS5, and GS10, respectively), and for the e ective Federal Funds
16
rate (acronyms is FEDFUNDS) are all from the Board of Governors of the Federal Reserve
17
System. A monthly seasonally adjusted series for the consumer price index (‘CPIAUCSL:
18
Consumer Price Index, Seasonally Adjusted, Monthly, Index 1982-84=100’) is from the U.S.
19
the Department of Labor, Bureau of Labor Statistics, and it has been converted to the
20
quarterly frequency by taking averages within the quarter. Quarterly seasonally adjusted
21
series for real GDP (‘GDPC96: Real Gross Domestic Product, 3 Decimal, Seasonally Ad
Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland. Email: [email protected]
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 2
1
justed Annual Rate, Billions of Chained 2005 Dollars’) and the GDP de ator (‘GDPCTPI:
2
Gross Domestic Product: Chain-type Price Index, Seasonally Adjusted’) are from the U.S.
3
Department of Commerce: Bureau of Economic Analysis. The quarterly series for potential
4
GDP (‘GDPPOT: Real Potential Gross Domestic Product, Quarterly, Billions of Chained
5
2005 Dollars’) is from the U.S. Congressional Budget O!ce, Budget and Economic Outlook.
6
Real consumption expenditure of non-durable goods and services has been computed based
7
on Table 1.1.6 of the U.S. National Income and Product Accounts. The output gap has been
8
computed as the percentage deviation of real GDP from the Congressional Budget O!ce
9
estimate of potential real GDP. It is important to stress that, since both series are expressed
10
in the same unit (billions of chained 2005 dollars), such a computation is indeed perfectly
11
legitimate. Monthly interpolated seasonally adjusted series for real GDP, the GDP de ator,
12
and real consumption expenditure are from Mark Watson’s website, and are available from
13
January 1959 to June 2010. The quarterly estimate of the output gap implied by the Con-
14
gressional Budget O!ce estimate of potential output has been interpolated to the monthly
15
frequency as in Bernanke, Gertler, and Watson (1997), using, as interpolators, the monthly
16
ratios between investment in non-residential structures and GDP, between investment in
17
residential structures and GDP, and between the change in inventories and GDP. In turn,
18
these ratios have been constructed based on the monthly interpolated national accounts data
19
found at Mark Watson’s website.
20
1.2. Euro area
21
Quarterly, seasonally adjusted series for real GDP, the GDP de ator, the CPI, real con-
22
sumption expenditure, real gross xed capital formation, the unemployment rate, the output
23
gap, and a short and a long rate are from the European Central Bank’s database.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 3
1
1.3. United Kingdom
2
The output gap estimate is from Pybus (2011). I wish to thank Tom Pybus, of the U.K.
3
O!ce for Budget Responsibility, for kindly sharing his output gap series with me. Quar-
4
terly, seasonally adjusted series for real GDP (acronym is ‘ABMI’), real nal consumption
5
by households (‘ABJR’), real total gross xed capital formation (‘NPQT’), nominal GDP
6
(‘YBHA’), and real change in inventories (‘CAEX’) are from the U.K. O!ce for National
7
Statistics. A quarterly, seasonally adjusted series for the GDP de ator has been computed as
8
the ratio between YBHA and ABMI. Monthly, seasonally adjusted series for the unemploy-
9
ment rate based on the claimant count (acronym is ‘BCJE’) and the retail price index are
10
from the U.K. O!ce for National Statistics, and they have been converted to the quarterly
11
frequency by taking averages within the quarter. Monthly, seasonally unadjusted series for
12
the discount rate and a long-term rate (‘Interest Rates, Government Securities, Government
13
Bonds’) are from the International Monetary Fund’s International Financial Statistics, and
14
they have been converted to the quarterly frequency by taking averages within the quarter.
15
Since, over the sample period, the consumption/GDP ratio exhibits some trend (which, ac-
16
cording to the permanent income hypothesis should obviously not be there), before entering
17
the ratio into the VARs I quadratically detrend it.
18
1.4. Canada
19
Quarterly, seasonally adjusted series for real GDP (acronym is ‘CANRGDPQDSNAQ’),
20
the harmonized unemployment rate for all persons (‘CANURHARMQDSMEI’), real private
21
nal consumption expenditure (‘CANPFCEQDSNAQ’), real gross xed capital formation
22
(‘CANGFCFQDSNAQ’), the GDP implicit price de ator (‘CANGDPDEFQISMEI’), and
23
the CPI are from the Organisation for Economic Co-operation and Development (either
24
from the Quarterly National Accounts, or from the Main Economic Indicators). Since, over
25
the sample period, the consumption/GDP ratio exhibits some trend (which, according to the
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 4
1
permanent income hypothesis should obviously not be there), before entering the ratio into
2
the VARs I quadratically detrend it. A monthly, seasonally unadjusted series for the short
3
rate, treasury bill auction, average yields, is from the Bank of Canada. A monthly, seasonally
4
unadjusted series for a long rate, is from the International Monetary Fund’s International
5
Financial Statistics. Both monthly series have been converted to the quarterly frequency
6
by taking averages within the quarter. As for the output gap, an ‘o!cial’ series computed
7
by the Bank of Canada, and available from its website, starts in 1981Q1. Since our sample
8
period starts in 1961Q1, in our analysis we use instead HP- ltered log real GDP, which, since
9
1981Q1, has exhibited a remarkably close co-movement witrh the Bank of Canada’s output
10
gap estimate. Speci cally, (i) the contemporaneous correlation between the two series has
11
been equal to 0.949, whereas (ii) the fraction of variance explained by the rst principal
12
component of the panel composed of the two standardized series has been equal to 0.974.
13
1.5. Japan
14
Monthly, seasonally adjusted series for the unemployment rate and the consumer price
15
indices are from the International Monetary Fund’s International Financial Statistics. Both
16
monthly series have been converted to the quarterly frequency by taking averages within the
17
quarter.
18
1.6. Australia
19
Monthly, seasonally adjusted series for the unemployment rate and the consumer price
20
indices are from the International Monetary Fund’s International Financial Statistics. Both
21
monthly series have been converted to the quarterly frequency by taking averages within the
22
quarter.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 5
1
1.7. Sweden
2
Monthly, seasonally adjusted series for the unemployment rate and the consumer price
3
indices are from the International Monetary Fund’s International Financial Statistics. Both
4
monthly series have been converted to the quarterly frequency by taking averages within the
5
quarter.
6
2.
Methodological Issues
7
How should we estimate the slope of the long-run Phillips curve? Before Sargent (1971)
8
and Lucas (1972a), the standard approach, due to Solow (1968) and Tobin (1968), was to
9
estimate the Phillips curve regression w = 0 + 1 w31 + 2 w32 + === + s w3s + 1 Xw31 + 2 Xw32 + === + s Xw3s + w
10
(1)
11
–with w and Xw being in ation and the unemployment rate, respectively–and to test
12
whether 1 + 2 + === + s = (1) = 1. Failure to reject the null hypothesis that (1)=1
13
was regarded as evidence in favor of the notion of a vertical long-run Phillips curve, whereas
14
an estimate of (1) signi cantly smaller than 1 was regarded as pointing towards a negative
15
long-run trade-o .
16
2.1. Why single-equation methods are misleading Sargent and Lucas highlighted the fallacy intrinsic to such an approach. As pointed out
17
18
by Lucas (1972a),
19
‘[...] the natural rate hypothesis, correctly formulated, has no implications for
20
the coe!cients of distributed lags Phillips curves, or for any other single-equation
21
expression of the empirical in ation-real output trade-o .’1 He went on to further stress that
22
1
Emphasis in the original.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 6
1
‘[...] a valid test of the natural rate hypothesis involves a test of a restriction
2
on the parameters across equations of a complete simultaneous equations model.
3
The esistence of a natural rate is thus a systems property, like stability or iden-
4
ti ability.’2
5
But if (1) cannot capture the slope of the long-run Phillips curve, what does it capture,
6
in fact? As stressed by Sargent (1971), it captures in ation persistence.3 As he pointed out
7
(see Sargent, 1971, p. 724),
8
‘[...] even the most casual glance at the price history of the United States makes
9
it clear that the in ation rate has not been a strongly drifting variable. [...] It
10
is not surprising, therefore, that most empirical studies have estimated to be
11
markedly less than unity. Unfortunately, as usually interpreted, these estimates
12
tell us virtually nothing about the validity of the accelerationist thesis.’
13
2.2. An empirical illustration based on an estimated DSGE model
14
How relevant is this problem is practice? Since, as I have previously documented (see Be-
15
nati (2008)), under in ation targeting regimes in ation persistence has essentially disappeared–
16
to the point that, in most cases, in ation has been, so far, statistically indistinguishable from 2
Emphasis added. This can be illustrated via the following simple example. Let aggregate supply be given by a Lucas (1972b)-type supply curve incorporating long-run neutrality (that is, a vertical long-run Phillips curve), |w = [ w w|w1 ] + xw , with w being in ation, w|w1 being its rational expectation based on information at time w-1, |w being the deviation of output from its natural level, and being the slope of the short-run Phillips curve. In ation is postulated to be equal to money growth, w . Finally, money growth is set by the policymaker according to the following AR(1) process, w = w1 + }w , and all of the shocks are postulated to be white noise, with innovation variances respectively equal to 2x and 2} . The model’s solutions for w and |w are given by w = w1 + }w and |w = }w + xw , from which the theoretical value of the coe!cient on w1 in the Phillips curve regression of in ation on lags of itself and of the deviation of output from its natural level turns out to be equal to ˆ ROV = . This implies that if $ 1, so that money growth and in ation tend to unit root processes, ˆ ROV $ 1, and the Solow-Tobin approach therefore ‘uncovers’ the truth of a vertical long-run Phillips curve. However, if in ation is weakly persistent the Solow-Tobin approach will turn out to be misleading. In the extreme case in which = 0, so that money growth and in ation are white noise processes, ˆ ROV = 0, thus pointing towards a perfectly at long-run Phillips curve. 3
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 7
1
white noise4 –by Sargent and Lucas’ argument we should logically expect that, by applying
2
the Solow-Tobin approach to data generated under such regimes, we would ‘uncover’ very
3
at long-run Phillips curves. In this sub-section I therefore estimate a simple New Keyne-
4
sian DSGE model featuring a vertical long-run Phillips curve based on data from Sweden,
5
the U.K., and Canada under in ation-targeting, and I show that the application of the
6
Solow-Tobin approach to this data-generating processes (henceforth, DGPs) would lead a
7
researcher to uncover at long-run Phillips curves for either country, in spite of the fact that,
8
by construction, the model encodes a vertical long-run Phillips curve. The New Keynesian model I am using is near-identical to the one I estimated in Benati
9
10
(2008), and is described by the following equations,
11
|w = |w+1|w + (1 )|w31 31 (Uw w+1|w ) + |>w
(2)
12
w =
w+1|w + w31 + |w + >w 1 + 1 +
(3)
Uw = Uw31 + (1 )[! w + !| |w ] + U>w
13
(4)
14
where w , |w and Uw are in ation, the output gap, and the nominal rate; is the forward-
15
looking component in the intertemporal IS curve; is price setters’ extent of indexation to
16
past in ation, and >w , |>w and U>w are white noise disturbances.5 All of the variables in
17
(2)-(4) are expressed as log-deviations from a non-stochastic steady-state. I set =1, thus
18
imposing a vertical long-run Phillips curve, and I estimate the model based on Swedish, U.K.,
19
and Canadian data from the in ation-targeting regimes. Both the Bayesian methodology and
20
the priors are identical to those I used in Benati (2008), to which the reader is referred to. The 4
Updating Benati’s (2008) estimates to 2011Q4, Hansen (1999) ‘grid bootstrap’ estimates of the sum of the autoregressive coe!cients in univariate AR(p) representations for GDP de ator in ation are equal to -0.15 in the U.K., to 0.06 in Sweden, and to 0.13 in Canada, with 90%-coverage bootstrapped con dence intervals equal to [-0.43 0.15], [-0.26 0.38], and [-0.09 0.34] respectively. A unit root in in ation can therefore be strongly rejected, and evidence points instead towards in ation being white noise. 5 The assumption that all of the structural disturbances are white noise is made uniquely for the sake of simplicity, it can be easily relaxed, and it plays no role in my results.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 8
1
VAR representation of the estimated model for Sweden conditional on the median estimates
4
is given by6 5 6 5 65 9 Uw : 9 0.788 0.007 0.208 : 9 Uw31 9 : 9 :9 9 : = 9 -0.230 0.037 0.288 : 9 9 w : 9 : 9 w31 7 8 7 87 -0.147 -0.001 1.004 |w |w31
5
in ation-targeting, the coe!cients on w31 in the equation for w is equal to 0.037. Interpret-
6
ing the second equation of the VAR as an estimated short-run Phillips curve, by applying the
7
Solow-Tobin approach we can immediately back out the estimate of the slope of the long-run
8
Phillips curve, which is equal to 0.299, in spite of the fact that the underlying DGP features
9
a vertical Phillips curve by construction. Results for the other two countries are qualitatively
2
3
10
6
5
: 9 xU>w : 9 :+9 x : 9 >w 8 7 x|>w
6 : : : : 8
(5)
Consistent with my previous results on the disappearance of in ation persistence under
the same, being equal to 0.306 for the United Kingdom, and to 0.245 for Canada.
11
These results provide a straighforward explanation for Svensson’s (2015) nding of a
12
comparatively at long-run Phillips trade-o in Sweden under the in ation targeting regime
13
based on the Solow-Tobin approach. Speci cally, Svensson (2015) estimates a version of (1)
14
for the period 1997Q4-2011Q4,7 ‘backs out’ the estimate of the long-run Phillips trade-o
15
from his short-run estimates, and performs the Solow-Tobin test that (1) = 1. He thus
16
summarizes his results, which point towards a negatively-sloped long-run Phillips curve:
17
‘From the estimates in table 2 it follows that the slope of the long-run Phillips curve is
18
about 0.8. With a standard error of 0.19, it is fairly precisely estimated. A 95-percent
19
con dence interval of the slope is the interval from 0.43 to 1.18. Clearly, the hypothesis
20
of a vertical Phillips curve, an in nite slope, can be rejected.’
21
In the light of the previous discussion of Sargent and Lucas’ criticism of the Solow-Tobin
22
approach, and of the fact that, under in ation-targeting, in ation in Sweden is, today, very 6
The covariance matrix of the VAR’s reduced-form innovations is not reported for the sake of simplicity. By the same token, I do not report the corresponding VAR representations of the estimated DSGE models for the other three countries, but they are available upon request. 7 See Svensson (2013, Table 2).
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 9
1
close to white noise8 , Svensson’s rejection of a vertical long-run Phillips curve based on the
2
test that (1) = 1 is to be expected. At the same time, however, as pointed out by Sargent
3
(1971) in the previous quotation, this evidence is uninformative about the authentic slope
4
of Sweden’s long-run Phillips curve.
5
2.3. King and Watson’s (1994) alternative approach
6
Since the results produced by single-equation methods–especially when applied to data
7
generated by monetary regimes which cause in ation to be strongly mean-reverting, such
8
as in ation targeting, European Monetary Union (henceforth, EMU), and the Swiss ‘new
9
monetary policy concept’–are unreliable, in this paper I follow King and Watson (1994),
10
and I use structural VAR methods to estimate the permanent impact (if any) on the unem-
11
ployment rate of permanent shocks to in ation. As stressed by King and Watson (1994, p.
12
177), indeed, if in ation has a unit root the data are in fact informative about the slope of
13
the long-run Phillips curve, since they do contain the relevant ‘experiment’:
14
‘When in ation is stationary, [...] as stressed by Lucas and Sargent, the relevant
15
experiment–permanent changes in the rate of in ation–are absent from the in ation
16
data and so the long-run Phillips trade-o could not be estimated [...]. In the unit
17
root case, by contrast, [...] the relevant experiments are present in the data: variation
18
in ¯ w [i.e.: the permanent component of in ation] allows the long-run slope to be
19
determined.’
As a simple illustration of King and Watson’s point, suppose that the economy is de-
20
21
scribed by a Lucas (1972b)-type supply curve possibly incorporating a long-run non-neutrality
22
(that is, a non vertical long-run Phillips curve), |w = [ w w|w31 ] + xw >
23
8
See footnote 11.
(6)
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 10
1
with w being in ation, w|w31 being its rational expectation based on information at time
2
w-1, |w being the deviation of output from its natural level, being the slope of the short-
3
run Phillips curve, and being the parameter upon which the slope of the long-run Phillips
4
curve crucially depends. Expression (6) implies that the long-run impact on |w of a permanent
5
change in w is equal to (1-). Finally, let’s assume that in ation is a pure random walk,
6
7
w = w31 + }w . The VAR representation of the model is given by 65 65 5 6 5 6 5 6 9 |w : 9 0 (1-) : 9 |w31 : 9 1 : 9 xw : 87 87 7 8=7 8+7 8 0 1 0 1 w w31 }w
(7)
8
from which the long-run impact on |w of a permanent shock to w at t = 0 can be immediately
9
computed as
10
C|w = (1 ), w<" C}0 lim
11
which is precisely the slope of the long-run Phillips curve.
12
2.4. Why a DSGE-based approach is not a viable option
(8)
13
In principle, an alternative approach would be to estimate a (DSGE) structural macro-
14
economic model in which the slope of the long-run trade-o is encoded in one or more of the
15
model’s structural parameters, and then to test whether, conditional on the model’s esti-
16
mates, it is possible to reject the null hypothesis that the long-run Phillips curve is vertical.
17
There are two key drawbacks to this approach.
18
First, and least importantly, as a matter of logic results would be, in general, model-
19
dependent (that is: they would be conditional on the speci c structural model which is
20
being estimated), and they could not be regarded as possessing a general validity.
21
Second–and this is key–this approach crucially rests on the assumption that the pa-
22
rameters of the estimated model are truly structural in the sense of Lucas (1976), and,
23
in particular, that they are invariant to changes in the characteristics of the in ationary
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 11
1
environment– rst and foremost, equilibrium in ation. This is key because if and only if
2
this assumption is satis ed we can be con dent that the long-run Phillips trade-o extracted
3
from a structural macroeconomic model estimated based on a sample period during which
4
average in ation has been equal to, say, two per cent, will hold at higher equilibrium in ation
5
rates. In recent years, however, evidence has been accumulating that several DSGE mod-
6
els’ supposedly structural parameters are in fact, most likely, not structural in the sense of
7
Lucas (1976). Benati (2008), for example, has shown that the indexation parameter in New
8
Keynesian backward- and forward-looking Phillips curves changes systematically with the
9
monetary regime, whereas the recent work of Fernandez-Villaverde and Rubio-Ramirez (and
10
their co-authors) has documented instability in several structural parameters. Fernández-
11
Villaverde and Rubio-Ramírez (2008), in particular, have shown that the extent of price
12
stickiness negatively and systematically co-moves with trend in ation. All of this implies
13
that the notion of extracting, from an estimated DSGE model, a long-run Phillips trade-o
14
which can safely be regarded as invariant to changes in equilibrium in ation rests, most
15
likely, on a leap of faith, unless we will be able, in the future, to e ectively model the way
16
in which those speci c DSGE models’ features which crucially determine the slope of the
17
long-run trade-o ( rst and foremost, price stickiness and the extent of backward-looking
18
indexation) co-move systematically with equilibrium in ation.
19
The attractiveness of King and Watson’s approach, on the other hand, is that the only
20
assumption it crucially hinges upon is that in ation has a unit root, and, conditional on this
21
assumption, the slope of the long-run trade-o can be investigated without possessing any
22
detailed knowledge of the underlying structural model of the economy.9 9
This point is also made by Fisher and Seater (1993): ‘[...] absent knowledge of the underlying structure, the consequences of an event cannot be inferred if the event has not occurred. In order for inferences regarding [long-run neutrality (long-run superneutrality)] to be drawn from a reduced form, the data must contain permanent stochastic changes in the level (growth rate) of the money supply.’.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 12
1
2
2.5. Why it is so di!cult to nd the ‘right’ data to estimate the slope of the long-run Phillips curve
3
The fact that, for a researcher to be able to econometrically identify the slope of the
4
long-run Phillips curve, in ation must contain a unit root, highlights why it is so di!cult
5
to nd the ‘right’ data to explore this issue. As it has been extensively documented, e.g.,
6
in Barsky (1987) and Benati (2008), under metallic standards (that is, before WWI) in a-
7
tion had consistently been indistinguishable from white noise, and, if anything, it had been
8
weakly negatively serially correlated. By the same token, during the period between the
9
outbreak of WWI and the early part of the post-WWII period in ation had exhibited some
10
persistence, but it had still been most likely stationary. Further, although for sample periods
11
which are dominated by the Great In ation episode evidence of a unit root in in ation is
12
sometimes (but, as I discuss in Section 4 below, not always) strong, following the disin a-
13
tions of the early 1980s in ation persistence has dramatically decreased, so that for sample
14
periods starting in mid-1980s the null of a unit root can once again be strongly rejected at
15
conventional signi cance levels. Even more ominously for the possibility of identifying the
16
slope of the long-run Phillips curve, under in ation targeting regimes, EMU, and the Swiss
17
‘new monetary policy concept’ in ation is, today, close to white noise. This means that, his-
18
torically, only sample periods dominated by the Great In ation episode can legitimately be
19
used to investigate the slope of the long-run Phillips curve, simply because such sample are
20
the only ones containing the relevant ‘experiment’ mentioned by King and Watson (1994),
21
whereas data generated under either metallic standards or in ation targeting regimes are
22
exactly the kind of data that cannot be used for this purpose.
23
2.6. The long-run Phillips trade-o in a low-in ation environment
24
A conceptually related issue pertains to the slope of the long-run Phillips curve in a
25
low-in ation environment. Several papers–from Akerlof, Dickens, and Perry (1996), to
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 13
1
Benigno and Ricci (2011)–have argued that, due to the presence of downward nominal
2
wage rigidity, the long-run Phillips trade-o may become quite signi cantly at within a
3
low-in ation environment, even if it is very steep, or even vertical, at higher in ation rates.
4
Unfortunately, a proper econometric investigation of this hypothesis is, and will most likely
5
remain, impossible.
6
First, for the reasons I discussed in section 2.4, extracting the long-run Phillips trade-o
7
from an estimated structural (DSGE) macroeconomic model is (at least, currently) not a
8
feasible option.
9
Second, for a pure time-series approach to be feasible, in ation should exhibit, over the
10
sample period, two characteristics which, historically, have been mutually exclusive: it should
11
be ‘low’ (which, in practice, means that it should be bounded between, say, minus two and
12
two per cent), and it should contain a unit root. The problem is that although it is easy to
13
nd periods during which either (i) in ation has been low, stable, and the null of a unit
14
root can be strongly rejected (e.g., in ation targeting regimes), or (ii) in ation has been
15
high, volatile, and the null of a unit root cannot be rejected (for several countries, the Great
16
In ation episode and the surrounding years), historically there has not been a single period
17
during which in ation has been low, stable, and it has exhibited a unit root behaviour.
18
As a result, the notion that the long-run Phillips curve may become at within a low-
19
in ation environment will most likely remain a theoretically compelling, but empirically
20
unproven conjecture.
21
3.
Results from Unit Root Tests Based on Phillips and Perron’s Test Statistics
22
For reasons of robustness we have also considered tests based on Phillips’ ] and ]w
23
statistics. The tests have been performed based on estimated models including an inter-
24
cept, but no time trend, and bootstrapped p-values have been computed as before. Again
25
for reasons of robustness, we have considered four values for T, the maximum number of
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 14
1
autocovariances to be considered for Phillips’ tests, that is, 2, 4, 6 and 8 quarters (for the
2
U.S. the corresponding values of T are 6, 12, 18, and 24). Table A.1 in the online appendix
3
reports results for the ] statistic, whereas results for the ]w tests are not reported because
4
they are uniformly in line with those from the ] tests. These results are almost uniformly in
5
line with those from Elliot et al.’s tests. The main di erence– which is however irrelevant
6
for our purposes–pertains to U.S. in ation for the full sample period since January 1959,
7
for which the null of a unit root is strongly rejected (on the other hand, in line with the
8
results from Elliot et al.’s tests, results for the pre-Volcker period clearly point towards a
9
unit root in U.S. in ation based on either price index).
10
11
4.
Classical Estimation of the Non-Cointegrated VARs Featuring a Single Permanent In ation Shock
12
Tables A.3 and A.8 in the online appendix report–based on the CPI and on the GDP
13
de ator, respectively–the full set of results for the non-cointegrated VARs featuring a single
14
permanent in ation shock. As mentioned in the text, Bayesian estimation of the reduced-
15
form VARs has been performed based on the estimator discussed by Uhlig (1998, 2005).
16
Working within a Classical context, we estimate equation (??) in the text via OLS based on
17
the standard formulas found in Luetkepohl (1991). Concerning the estimation of the impact
18
matrix of the structural shocks, as extensively discussed by Christiano et al. (2006), reliably
19
estimating F" requires a good estimate of the spectral density of \w at the frequency $ = 0,
20
0 V\ (0) = F" F" , an object which VARs, given their focus on tting the short-run dynamics
21
of the data, should not necessarily be expected to capture well. Following Christiano et
22
al. (2006) we therefore consider, beyond the standard estimator of V\ (0) produced by
23
31 ˆ 31 0 ˆ ˆ the VAR–that is, Vˆ\Y DU (0) = [LQ E(1)] } , where Yˆ is the estimated Y {[LQ E(1)]
24
covariance matrix of the VAR’s reduced-form innovations–the estimator produced by the
25
Bartlett estimate of the spectral density matrix (see Hamilton (1994)).
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 15
1
We use standard bootstrapping techniques in order to both bias-correct the estimated
2
long-run impacts of the permanent in ation shock as in Kilian (1998), and characterise the
3
extent of uncertainty around the bias-corrected estimates (so, to be clear, the only di erence
4
between Kilian’s (1998) paper and the present work is that he dealt with the bias-correction
5
of IRFs, whereas we are here bias-correcting the long-run impacts of the structural shocks).
6
Speci cally, we bootstrap the estimated reduced-form VAR 10,000 times, and based on each
7
bootstrap replication we estimate a VAR(p); we impose the same identi cation scheme we
8
imposed on the VAR estimated based on the actual data; and we compute the implied
9
long-run impact on the unemployment rate of the permanent in ation shock. Then, we
10
use such bootstrapped distributions, rst, to bias-correct the simple estimate of the long-run
11
impact; and second, to characterise the extent of uncertainty surrounding such bias-corrected
12
estimates, by simply rescaling the original bootstrapped distribution in such a way that its
13
median be equal to the bias-corrected estimate (in general, however, the extent of the bias
14
is very small, so that, in the end, bias-correcting does not make a material di erence to the
15
results).
16
4.0.1.
Computing the structural impact matrix D0
17
For each draw from the posterior distribution of the VAR’s reduced-form parameters we
18
compute the structural impact matrix, D0 , via the methodology proposed by Arias, Rubio-
19
Ramirez, and Waggoner (2014) for combining zero and sign restrictons. Speci cally, let E0q ,
20
E1q , ..., Esq , and lq be the q-th draw from the posterior distribution for the intercept, the
21
VAR matrices, and the covariance matrix of reduced-form innovations of the VAR (??), for
22
q = 1, 2, 3, ..., Q . Let Sq Gq Sq0 be the eigenvalue-eigenvector decomposition of lq . We
23
start by computing an initial estimate of Dq0 –let’s call it D˜q0 –as D˜q0 = Sq Gq2 with the
24
˜ q = [LN E q (1)]31 D˜q0 , corresponding matrix of long-run impacts of the structural shocks O
25
where N is the number of series in the VAR, and E q (1) = E1q + E2q + ... + Esq . Based
1
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 16
1
2
on the Gibbs-sampling algorithm described in section 3.6.3 of Arias et al. (2014), we then draw ] random orthonormal matrices of dimension N × N from the uniform distribution,
3
conditional on the zero restrictions on the long-run impacts of the structural shocks on
4
in ation shown in (??)–that is: four shocks are allowed to have a non-zero long-run impact
5
on in ation, whereas the remaining two shocks are forced to have no long-run impact. Let
6
Tq} , } = 1, 2, 3, ..., ], be the }-th random orthonormal matrix, with Tq} (Tq} )0 = LN . We then
7
combine each of the ] random orthonormal matrices with the initial estimate of the long-run
8
˜ q , in order to obtain a randomly rotated long-run impact impact of the structural shocks, O
9
˜ q Tq} . By construction, each O}q , } = 1, 2, 3, ..., ], satis es the zero long-run matrix, O}q = O
10
restrictions in (??). From O}q we then obtain the corresponding candidate estimate of the
11
structural impact matrix, D}0>q = [LN E q (1)]O}q . Finally, we check whether D}0>q satis es the
12
sign restrictions reported in the table in Section 4.3.1, and out of the ] candidate structural
13
impact matrices we only keep, for draw q, those satisfying the sign restrictions. For each
14
draw from the posterior we consider 10,000 random rotation matrices. Finally, we set the
15
number of Gibbs-sampling iterations in the algorithm to O = 5.10
10
As pointed out by Arias et al. (2014, p. 18), ‘[t]here is also the question of how large O should be to obtain convergence. Experiments show that for the starting value given below, even O = 1 gives a good approximation of the desired distribution. In practice, increasing values of O can be used to determine when convergence has occurred.’ Our own experience with the algorithm con rms this. Although we set O = 5, convergence was typically achieved at most at the second iteration.
Online appendix for:‘The Long-Run Phillips Curve: A Structural VAR Investigation’ 17
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
5.
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Canova, F., and J. Pina (2005): “What VAR Tell Us About DSGE Models”, in Diebolt, C. and Kyrtsou, C., New Trends In Macroeconomics, Springer Verlag.
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Carlstrom, C., T. Fuerst, and M. Paustian (2009): “Monetary Policy Shocks, Choleski Identi cation, and DNK Models”, Journal of Monetary Economics, 56(7), 1014-1021.
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Christiano, L. J., M. Eichenbaum, and R. Vigfusson (2006): “Assessing Structural
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Cochrane, J. H. (1994): “Permanent and Transitory Components of GNP and Stock Prices”, Quarterly Journal of Economics, 109(1), 241-265. Elliot, G., Rothenberg, T., and Stock, J.H. (1996): “E!cient Tests for an Autoregressive Unit Root”, Econometrica, 64(4), 813-836
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Evans, C. (1994): “Comment on: The Post-War U.S. Phillips Curve, A Revisionist
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Econometric History”, Carnegie-Rochester Conference Series on Public Policy, 41, 221-230.
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Fama, E. (1975): “Short-Term Interest Rates as Predictors of In ation”, American Eco-
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Fernández-Villaverde, J., and J. F. Rubio-Ramírez (2008): “How Structural Are the
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Structural Parameters?”, NBER Macroeconomics Annuals 2007, Cambridge, The MIT Press,
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83-137.
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Fisher, M. E., and J. J. Seater (1993): “Long Run Neutrality and Superneutrality in an ARIMA Framework”, American Economic Review, 83(3), 402-415. Forni, M., and L. Gambetti (2014): “Testing for Su!cient Information in Structural VARs”, Journal of Monetary Economics, 66(September), 124-136 Fry, R., and A. Pagan (2007): “Sign Restrictions in Structural Vector Autoregressions: A Critical Review”, Journal of Economic Literature, 49(4), 938-60
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Gregory, A.W., and B.E. Hansen (1996): “Tests for Cointegration in Models with Regime and Trend Shifts”, Oxford Bulletin of Economics and Statistics, 58(3), 555-560" }
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Hamilton, J. (1994): Time Series Analysis, Princeton, N.J., Princeton University Press.
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Hansen, B. (1999): “The Grid Bootstrap and the Autoregressive Model”, Review of
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Economics and Statistics, 81(4), 594-607. Kilian, L. (1998): “Small-Sample Con dence Intervals for Impulse-Response Functions”, Review of Economics and Statistics, pp. 218-230.
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King, R., and M. Watson (1994): “The Post-War U.S. Phillips Curve: A Revisionist
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Econometric History”, Carnegie-Rochester Conference Series on Public Policy, 41, 157-219.
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Kurmann, A., and C. Otrok (2013): “News Shocks and the Slope of the Term Structure
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of Interest Rates”, American Economic Review, 103(6), 2612-32
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Lucas, R. E. (1972a): “Econometric Testing of the Natural Rate Hypothesis”, in Eckstein,
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O., ed., The Econometrics of Price Determination, Washington, D.C., Board of Governors
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of the Federal Reserve System, pp. 50-59.
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Lucas, R. E. (1972b): “Expectations and the Neutrality of Money”, Journal of Economic Theory, 4(April 1972), 103-24. Lucas, R. E. (1973): “Some International Evidence on Output-In ation Trade-O s”, American Economic Review, 63(3), 326-334. Lucas, R. E. (1976): “Econometric Policy Evaluation: A Critique”, Carnegie-Rochester Conference Series on Public Policy, 1, 19-46. Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis, II edition. SpringerVerlag. Pybus, T. (2011): “Estimating the UK’s Historical Output Gap”, O!ce for Budget Responsibility Working paper No. 1, November 2011. Roberts, J. M. (1993): “The Sources of Business Cycles: A Monetarist Interpretation”, International Economic Review, 34(4), 923-934.
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Rubio-Ramirez, J., D. Waggoner, and T. Zha (2005): “Structural Vector Autoregressions:
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Theory of Identi cation and Algorithms for Inference”, Review of Economic Studies, 77(2),
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665-696.
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Sargent, T. J. (1971): “A Note on the Accelerationist Controversy”, Journal of Money, Credit and Banking, 3(3), 721-725.
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Sargent, T. J. (1987): Macroeconomic Theory, II Edition. Orlando, Academic Press.
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Solow, R. (1968): “Recent Controversy on the Theory of In ation: An Eclectic View”,
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in Proceedings of a Symposium on In ation: Its Causes, Consequences, and Control, S.
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Rousseaus (ed.), New York: New York University.
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Stock, J. H., and M.W. Watson (2007): “Why Has U.S. In ation Become Harder to Forecast?”, Journal of Money, Credit, and Banking, 39(1), 3-33. Svensson, Lars E O. (2015): “The Possible Unemployment Cost of Average In ation below a Credible Target”, American Economic Journal: Macroeconomics, 7(1): 258-96. Tobin, J. (1968): “Discussion”, in Proceedings of a Symposium on In ation: Its Causes, Consequences, and Control, S. Rousseaus (ed.), New York: New York University. Uhlig, H. (1998): “Comment On: The Robustness of Identi ed VAR Conclusions About Money”, Carnegie-Rochester Conference Series on Public Policy, 49, 245-263. Uhlig, H. (2005): “What are the E ects of Monetary Policy on Output? Results from an Agnostic Identi cation Procedure”, Journal of Monetary Economics, 52(2), 381-419.
Table A.1 Bootstrapped p-valuesd for Phillips ] testsf without trend GDP de ator in ation Q=2 Q=4 Q=6 Q=8
Q=2
CPI in ation Q=4 Q=6
Q=8
0.002 0.150
0.001 0.130
0.001 0.200
0.000 0.196
0.000 0.189
0.000 0.183
0.001 0.000 0.000 0.000 0.453 0.432 0.441 0.425 0.100 0.097 0.094 0.093 0.131 0.121 0.113 0.109 0.157 0.161 0.164 0.160 — — — — 0.159 0.148 0.148 0.144 Unemployment rate Q=2 Q=4 Q=6 Q=8
0.001 0.570 0.033 0.209 0.302 0.005 0.110
0.000 0.531 0.032 0.205 0.308 0.006 0.108 Short Q=4
0.000 0.541 0.031 0.194 0.310 0.007 0.107 rate Q=6
0.000 0.550 0.034 0.175 0.315 0.013 0.109
0.066 0.132
0.070 0.165
0.179 0.216 0.194 0.017 0.296 0.186 0.200
0.189 0.262 0.240 0.015 0.317 0.238 0.221
e
United States full sample (Jan. 1959-Jun. 2010) pre-Volcker (Jan. 1959-Jul. 1979) Volcker, Greenspan, and Bernanke (Aug. 1979-Jun. 2010) Euro area (1970Q1-1998Q4) Japan (1966Q4-2011Q4) United Kingdom (1972Q1-1992Q3) Canada (1961Q1-1990Q4) Sweden (1970Q1-1992Q4) Australia (1969Q3-1994Q2)
0.001 0.153
0.001 0.136
Q=2
Q=8
e
United States full sample (Jan. 1959-Jun. 2010) 0.059 0.029 0.025 0.026 0.094 0.090 pre-Volcker (Jan. 1959-Jul. 1979) 0.286 0.225 0.221 0.221 0.180 0.168 Volcker, Greenspan, and Bernanke (Aug. 1979-Jun. 2010) 0.363 0.216 0.172 0.153 0.220 0.197 Euro area (1970Q1-1998Q4) 0.367 0.381 0.386 0.387 0.198 0.186 Japan (1966Q4-2011Q4) 0.576 0.548 0.524 0.521 0.192 0.174 United Kingdom (1972Q1-1992Q3) 0.479 0.453 0.431 0.422 0.035 0.025 Canada (1961Q1-1990Q4) 0.421 0.409 0.424 0.431 0.315 0.303 Sweden (1970Q1-1992Q4) 0.993 0.874 0.542 0.305 0.187 0.168 Australia (1969Q3-1994Q2) 0.411 0.396 0.405 0.431 0.248 0.205 d Based on 10,000 bootstrap replications of estimated ARIMA processes. e Based on monthly data, the corresponding values of Q are 6, 12, 18, and 24. f Q is the maximum number of autocovariances considered for the Phillips test.
Table A.2 Monte Carlo evidence on the performance of the Johansen procedure: fraction of simulations for which the bootstrapped trace statistic incorrectly rejects the null of no cointegration between two independent random walks at a given signi cance leveld
Sample size (in quarters) T = 82 (U.S., pre-Volcker: 1959Q1-1979Q3) T = 115 (Euro area: 1970Q1-1998Q4) T = 81 (United Kingdom: 1972Q1-1992Q3) T = 118 (Canada: 1961Q1-1990Q4) T = 98 (Australia: 1970Q1-1994Q2) d Based on 10,000 simulations.
Fractions of bootstrapped p-values for Johansen’s trace statistic below: 0.1 0.05 0.01 0.119 0.059 0.013 0.118 0.060 0.012 0.118 0.059 0.011 0.113 0.055 0.011 0.115 0.060 0.011
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Table A.3 Results based on VARs allowing for a single permanent in ation shock: bootstrapped and posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of the mass of the distribution for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Euro area (1970Q1-1998Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian United Kingdom (1972Q1-1992Q3) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Canada (1961Q1-1990Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Australia (1970Q1-1994Q2) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian
Fraction of the mass of the distribution for which the impact is negative
matrix matrix
-0.226 -0.256 -0.276
-0.257 -0.264 -0.274
[-0.688; 0.134] [-0.677; 0.126] [-0.950; 0.324]
0.869 0.867 0.818
matrix matrix
-0.056 -0.024 -0.040
-0.043 -0.041 -0.036
[-0.474; 0.402] [-0.472; 0.399] [-0.784; 0.763]
0.576 0.573 0.547
matrix matrix
-0.088 -0.088 -0.088
-0.084 -0.0865 -0.0865
[-0.357; 0.180] [-0.359; 0.180] [-0.503; 0.330]
0.730 0.733 0.688
matrix matrix
-0.153 -0.121 -0.121
-0.131 -0.130 -0.131
[-0.337; 0.063] [-0.336; 0.063] [-0.467; 0.167]
0.872 0.870 0.805
matrix matrix
-0.056 -0.056 -0.056
-0.044 -0.044 -0.045
[-0.175; 0.074] [-0.176; 0.076] [-0.240; 0.135]
0.734 0.734 0.702
Table A.4 Results based on Bayesian VARs allowing for four permanent in ation shocks: posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of draws for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of draws for which impact is negative
-0.414 -0.489 -0.263 -0.113
-0.292 -0.472 -0.161 -0.237
[-3.827; [-3.592; [-1.370; [-3.615;
3.247] 3.426] 1.123] 3.714]
0.642 0.738 0.659 0.637
-0.551 -0.050 -0.050 -0.050
-0.5090 0.0233 0.0214 0.0201
[-4.525; [-3.034; [-2.168; [-3.256;
2.410] 2.518] 2.486] 2.542]
0.809 0.480 0.481 0.476
-0.141 -0.181 -0.100 0.100
-0.101 -0.214 -0.089 0.109
[-1.856; [-1.148; [-2.085; [-2.137;
1.629] 0.678] 1.893] 1.850]
0.600 0.819 0.584 0.371
-0.140 -0.180 -0.020 0.180
-0.241 -0.183 -0.067 -0.113
[-1.670; [-0.733; [-1.552; [-4.131;
0.952] 0.174] 1.696] 2.858]
0.767 0.836 0.578 0.561
-0.169 -0.024 -0.169 -0.024
-0.181 -0.042 -0.096 0.028
[-0.938; [-0.979; [-1.389; [-0.571;
0.451] 0.732] 1.056] 0.636]
0.840 0.586 0.619 0.437
Table A.5 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of in ation explained by individual shocks
Median, and 90%coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.109 0.194 0.276 0.131
[0.001; [0.002; [0.005; [0.001;
0.633] 0.791] 0.834] 0.695]
0.481 0.366 0.269 0.445
0.346 0.259 0.181 0.325
0.152 0.121 0.079 0.159
0.227 0.119 0.214 0.156
[0.002; [0.001; [0.003; [0.002;
0.823] 0.614] 0.746] 0.720]
0.341 0.462 0.315 0.401
0.238 0.331 0.235 0.283
0.111 0.151 0.091 0.130
0.110 0.316 0.073 0.205
[0.001; [0.005; [0.001; [0.002;
0.728] 0.864] 0.529] 0.805]
0.472 0.247 0.563 0.359
0.335 0.161 0.421 0.258
0.156 0.073 0.195 0.119
0.145 0.433 0.152 0.064
[0.002; [0.017; [0.001; [0.001;
0.639] 0.887] 0.727] 0.435]
0.414 0.143 0.404 0.605
0.300 0.086 0.288 0.444
0.139 0.034 0.136 0.204
0.245 0.151 0.064 0.282
[0.004; [0.002; [0.001; [0.005;
0.797] 0.754] 0.483] 0.819]
0.297 0.396 0.595 0.251
0.204 0.281 0.446 0.159
0.084 0.129 0.199 0.067
Table A.6 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of the unemployment rate explained by individual shocks
United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock s mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Median, and 90%coverage percentiles
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.122 0.236 0.057 0.120
[0.002; [0.004; [0.000; [0.001;
0.634] 0.727] 0.400] 0.627]
0.454 0.293 0.631 0.458
0.317 0.198 0.477 0.333
0.143 0.085 0.220 0.155
0.282 0.056 0.080 0.066
[0.007; [0.000; [0.001; [0.001;
0.717] 0.376] 0.479] 0.408]
0.239 0.650 0.561 0.602
0.159 0.472 0.389 0.436
0.062 0.229 0.176 0.215
0.085 0.161 0.077 0.1496
[0.001; [0.002; [0.001; [0.002;
0.561] 0.631] 0.500] 0.657]
0.542 0.389 0.557 0.402
0.392 0.265 0.410 0.283
0.177 0.125 0.189 0.122
0.091 0.113 0.091 0.139
[0.001; [0.001; [0.001; [0.001;
0.515] 0.559] 0.557] 0.628]
0.524 0.477 0.519 0.423
0.373 0.342 0.387 0.272
0.164 0.152 0.178 0.122
0.210 0.063 0.066 0.076
[0.002; [0.000; [0.001; [0.000;
0.695] 0.517] 0.497] 0.529]
0.321 0.607 0.587 0.558
0.215 0.459 0.439 0.408
0.094 0.224 0.205 0.193
ă ă ă ă ă ă ăă ă ă ă ă ă ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăIxooăvhwăriăuhvxowvăedvhgărqăwkhăJGSăghiodwru
Table A.7 Bootstrapped p-valuesd for Johansen’s cointegration tests Trace test of the null of no cointegration between: w and Xw w and Uw w , Xw and Uw United States, pre-Volcker (January 1959-July 1979) 0.141 1.0e-4 0.000 Euro area (1970Q1-1998Q4) 0.023 0.568 0.005 United Kingdom (1972Q1-1992Q3) 0.094 NAe NAe Canada (1961Q1-1990Q4) 0.509 0.493 0.010 Australia (1970Q1-1994Q2) 0.143 0.063 0.149 Test of the null of one cointegrating vector, versus the alternative of two, for w , Xw and Uw : United States, pre-Volcker (January 1959-July 1979) 0.074 Euro area (1970Q1-1998Q4) 0.012 Canada (1961Q1-1990Q4) 0.668 d e Based on 10,000 bootstrap replications. For the United Kingdom, results from the ADF tests reported in Table 1 strongly reject the null of a unit root in the short rate.
Table A.8 Results based on VARs allowing for a single permanent in ation shock: bootstrapped and posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of the mass of the distribution for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Euro area (1970Q1-1998Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian United Kingdom (1972Q1-1992Q3) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Canada (1961Q1-1990Q4) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian Australia (1970Q1-1994Q2) Classical Bartlett estimator of the spectral density VAR-based estimator of the spectral density Bayesian
Fraction of the mass of the distribution for which the impact is negative
matrix matrix
-0.105 -0.216 -0.200
-0.200 -0.191 -0.200
[-0.620; 0.185] [-0.613; 0.185] [-0.869; 0.404]
0.820 0.811 0.750
matrix matrix
0.056 -0.040 -0.008
-0.011 0.003 0.012
[-0.426; 0.453] [-0.432; 0.452] [-0.767; 0.758]
0.518 0.495 0.485
matrix matrix
-0.088 -0.104 -0.072
-0.081 -0.080 -0.065
[-0.308; 0.158] [-0.302; 0.157] [-0.411; 0.287]
0.734 0.736 0.680
matrix matrix
-0.121 -0.121 -0.088
-0.096 -0.096 -0.100
[-0.271; 0.070] [-0.270; 0.070] [-0.373; 0.134]
0.838 0.837 0.796
matrix matrix
-0.137 -0.121 -0.121
-0.116 -0.116 -0.116
[-0.241; 0.004] [-0.237; 0.003] [-0.267; 0.034]
0.945 0.946 0.914
Table A.9 Results based on Bayesian VARs allowing for four permanent in ation shocks: posterior distributions of the long-run impact on the unemployment rate of a one per cent permanent shock to in ation, and fractions of draws for which the impact is negative
Mode, median, and 90%-coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of draws for which impact is negative
-0.338 -0.338 -0.038 -0.564
-0.332 -0.401 -0.086 -0.262
[-3.773; [-3.164; [-1.052; [-3.743;
3.084] 2.029] 1.148] 2.871]
0.679 0.745 0.594 0.627
-0.26 0.02 0.14 -0.18
-0.38 0.04 0.12 0.10
[-6.16; [-2.59; [-0.98; [-2.68;
4.40] 2.89] 1.68] 2.76]
0.655 0.475 0.374 0.427
-0.09 -0.17 -0.05 0.03
-0.08 -0.16 -0.04 0.04
[-1.47; [-0.81; [-1.51; [-1.66;
1.08] 0.45] 1.48] 1.49]
0.616 0.818 0.556 0.446
-0.156 -0.180 -0.012 0.036
-0.149 -0.183 -0.084 0.048
[-1.559; [-0.894; [-1.553; [-1.444;
1.095] 0.198] 1.161] 1.536]
0.724 0.850 0.606 0.433
-0.169 -0.121 -0.169 -0.072
-0.214 -0.135 -0.143 -0.039
[-0.720; [-1.274; [-1.066; [-0.485;
0.244] 0.698] 0.675] 0.590]
0.897 0.727 0.739 0.612
Table A.10 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of in ation explained by individual shocks
Median, and 90%coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.085 0.181 0.332 0.129
[0.001; [0.002; [0.006; [0.001;
0.565] 0.779] 0.844] 0.714]
0.543 0.363 0.231 0.447
0.386 0.260 0.153 0.321
0.176 0.109 0.065 0.159
0.154 0.109 0.369 0.118
[0.001; [0.001; [0.014; [0.001;
0.726] 0.557] 0.872] 0.640]
0.406 0.484 0.172 0.454
0.294 0.336 0.106 0.320
0.135 0.163 0.041 0.128
0.124 0.367 0.096 0.145
[0.001; [0.007; [0.001; [0.002;
0.675] 0.870] 0.624] 0.714]
0.451 0.198 0.507 0.425
0.331 0.134 0.364 0.291
0.161 0.059 0.167 0.129
0.132 0.311 0.133 0.133
[0.001; [0.007; [0.001; [0.001;
0.730] 0.834] 0.736] 0.660]
0.442 0.235 0.435 0.438
0.322 0.155 0.315 0.308
0.148 0.064 0.142 0.143
0.300 0.086 0.082 0.308
[0.006; [0.001; [0.001; [0.005;
0.842] 0.534] 0.516] 0.851]
0.264 0.532 0.553 0.235
0.171 0.396 0.385 0.158
0.062 0.182 0.166 0.069
Table A.11 Results based on Bayesian VARs allowing for four permanent in ation shocks: fractions of the permanent component of the unemployment rate explained by individual shocks
Median, and 90%coverage percentiles United States, pre-Volcker (January 1959-July 1979) technology shock monetary shock taste shock mark-up shock Euro area (1970Q1-1998Q4) technology shock monetary shock taste shock mark-up shock United Kingdom (1972Q1-1992Q3) technology shock monetary shock taste shock mark-up shock Canada (1961Q1-1990Q4) technology shock monetary shock taste shock mark-up shock Australia (1970Q1-1994Q2) technology shock monetary shock taste shock mark-up shock
Fraction of the mass of the posterior distribution below: 0.1 0.05 0.01
0.105 0.187 0.059 0.147
[0.001; 0.576] [0.002; 0.711] [0.000 0.427] [0.002; 0.680]
0.488 0.356 0.626 0.410
0.355 0.256 0.480 0.281
0.166 0.110 0.239 0.125
0.316 0.057 0.078 0.059
[0.005; [0.001; [0.001; [0.000;
0.778] 0.439] 0.509] 0.436]
0.221 0.619 0.573 0.630
0.153 0.469 0.422 0.470
0.069 0.220 0.195 0.235
0.088 0.176 0.077 0.133
[0.001; [0.001; [0.001; [0.002;
0.570] 0.656] 0.492] 0.637]
0.539 0.364 0.555 0.438
0.396 0.254 0.411 0.301
0.181 0.124 0.187 0.130
0.094 0.142 0.082 0.095
[0.001; [0.002; [0.001; [0.001;
0.545] 0.655] 0.579] 0.536]
0.515 0.410 0.544 0.506
0.375 0.290 0.390 0.374
0.175 0.121 0.179 0.176
0.304 0.068 0.064 0.074
[0.007; [0.001; [0.001; [0.001;
0.786] 0.508] 0.512] 0.512]
0.233 0.582 0.598 0.572
0.148 0.436 0.443 0.415
0.058 0.206 0.192 0.172
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FigureăD11 CPI inflation and the unemployment rate 22
Figure D12 Bootstrapped distributions of the slope of the long-run Phillips curve implied ăăă by the estimated cointegrating vector between inflation and unemployment, ăăă and of the elements of the loadings vector (based on the CPI) 23
Figure D16ăăResults based on VARs allowing for a single permanent inflation shock: bootstrapped or ăăă posterior distributions of (i) the long-run impacts on the unemployment rate of a one ăăă per cent permanent shock to inflation, and of (ii) the fractions of the permanent compoăăă nent of unemployment explained by the permanent inflation shock (based on the CPI) 24
Figure D14 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the long-run impacts on the unemployment rate of a ăăă one per cent permanent shock to inflation (based on the CPI) 25
Figure D15 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of inflation ăăă explained by individual shocks (based on the CPI) 26
Figure D16 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of the unemăăă ployment rate explained by individual shocks (based on the CPI) 27
ă ă ă ă ă ă ăă ă ă ă ă ă ăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăăIxooăvhwăriăuhvxowvăedvhgărqăwkhăJGSăghiodwru
Figure D17 GDP deflator inflation and the unemployment rate 29
FigureăD18 ăăă ăăă
Bootstrapped distributions of the slope of the long-run Phillips curve implied by the estimated cointegrating vector between inflation and unemployment, and of the elements of the loadings vector (based on the GDP deflator) 30
Figure D19 ăăă ăăă ăăă ăăă
Results based on VARs allowing for a single permanent inflation shock: bootstrapped or posterior distributions of (i) the long-run impacts on the unemployment rate of a one per cent permanent shock to inflation, and of (ii) the fractions of the permanent component of unemployment explained by the permanent inflation shock (based on the GDP deflator) 31
Figure D110 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the long-run impacts on the unemployment rate of a ăăă one per cent permanent shock to inflation (based on the GDP deflator) 32
Figure D111 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of inflation ăăă explained by individual shocks (based on the GDP deflator) 33
Figure D112 Results based on Bayesian VARs allowing for four permanent inflation shocks: ăăă posterior distributions of the fractions of the permanent component of the unemăăă ployment rate explained by individual shocks (based on the GDP deflator) 34