Fitting survey expectations and uncertainty about trend inflation

Journal of Macroeconomics 35 (2013) 172–185

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Fitting survey expectations and uncertainty about trend inflation Steffen R. Henzel ⇑ Ifo Institute – Leibniz Institute for Economic Research at the University of Munich e.V., Poschingerstrasse 5, 81679 Munich, Germany

a r t i c l e

i n f o

Article history: Received 24 January 2012 Accepted 10 October 2012 Available online 29 November 2012 JEL classification: C32 E31 E37 Keywords: Survey expectations Trend learning Stochastic volatility

a b s t r a c t Many studies document that the inflation rate is governed by persistent trend shifts and time-varying uncertainty about trend inflation. As both these quantities are unobserved, a forecaster has to learn about changes in trend inflation by a signal extraction procedure. I suggest that the forecaster uses a simple IMA(1, 1) model because it is well suited to forecast inflation and it provides an efficient way to solve the signal extraction problem. I test whether this model provides a good fit for expectations from the Survey of Professional Forecasters. The model appears to be well suited to model observed inflation expectations if we allow for stochastic volatility. When I estimate the implied learning rule, results are supportive for the trend learning hypothesis. Moreover, stochastic volatility seems to influence the way agents learn over time. It appears that survey participants systematically adapt their learning behavior when inflation uncertainty changes. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Many studies document that the inflation rate is governed by persistent trend shifts.1 Moreover, it is acknowledged that inflation is characterized by stochastic volatility, i.e. time-varying uncertainty.2 However, trend inflation and inflation uncertainty are unobserved. Hence, a practical forecaster needs to solve a signal extraction problem to learn about trend shifts and transitory movements in order to arrive at a reasonable inflation forecast. Trend learning behavior on part of private agents is also put forward in a number of theoretical papers.3 Particularly, these studies argue that trend learning behavior enables us to replicate some of the main features of observed inflation expectations such as temporary bias and persistent forecast errors.4 The goal of this study is to fit a simple trend learning model to observed inflation expectations and to investigate whether and how shifts in trend inflation and time-varying uncertainty about trend inflation impact the expectation formation process. The contribution of the present paper is threefold. First, I analyze whether a simple trend learning model may be used to approximate the Survey of Professional Forecasters (SPFs). I do so by estimating a simultaneous model that consists of an inflation forecasting equation and SPF expectations. Second, the parameters of the learning model are estimated directly. To this end, I derive the learning rule associated with the forecast model. Moreover, I analyze how observed expectations adapt

⇑ Corresponding author. E-mail address: [email protected] See, among others, Cogley and Sargent (2005), Ireland (2007), Stock and Watson (2007), Stock and Watson (2010) and Cecchetti et al. (2007). See also Cogley et al. (2010) and Grassi and Proietti (2010). 3 Assuming that agents act in an environment of incomplete information, for instance, Cukierman and Meltzer (1986), Erceg and Levin (2003), Kozicki and Tinsley (2005), and Keen (2010) show how a lack of information gives rise to learning behavior. Commonly, learning behavior is implemented to better match the dynamics of the inflation rate. 4 See Thomas (1999) and the papers cited there. See also Andolfatto et al. (2008) who emphasize that persistent forecast errors may occur in small samples when agents apply an efficient signal extraction technique. 1 2

0164-0704/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmacro.2012.10.007

S.R. Henzel / Journal of Macroeconomics 35 (2013) 172–185

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when trend inflation shifts. Third, I test for time variation in the learning parameter and study whether the changes in the learning parameter of survey participants are consistent with the forecast model. Despite the effort to include trend learning into theoretical models, observed inflation expectations are rarely used formally when the model is taken to the data. Del Negro and Eusepi (2010) analyze whether the introduction of trend learning behavior into an otherwise standard New Keynesian DSGE model helps to predict observed inflation expectations and find that this is not the case. Their model involves numerous cross-equation restrictions which seem to be too restrictive to allow for a reasonable fit of observed expectations. Moreover, to ensure that the estimation of the model does not suffer from parameter instability, the authors use an estimation period that is characterized by stable trend inflation which hampers the identification of trend learning behavior. In contrast to Del Negro and Eusepi (2010), the present study starts from an empirical perspective and considers only univariate forecast models that account for trend shifts and, possibly, changing volatility. There is also evidence in favor of learning behavior in expectations. Branch and Evans (2006) calibrate a least square learning algorithm in the spirit of Evans and Honkapohja (2001) to SPF expectations. They find that a constant gain learning rule compares favorably to other learning schemes. A very similar approach is followed by Orphanides and Williams (2005) and Molnar and Reppa (2010). The latter emphasize that the learning parameter depends on the amount of parameter uncertainty in an economy. Note that this strand of literature assumes that the parameters of the (otherwise known) rational expectations solution of the model are unobserved and have to be estimated by OLS. That is, agents learn as more and more data becomes available. However, the recursive learning rule (Recursive least squares, RLSs) implies that the gain is a decreasing deterministic function of the sample size. In general, RLS assumes that the parameters of the economy are stable over time and can eventually be recovered by private households. Note that this is not in line with permanent shifts in the inflation process. Moreover, it is difficult to motivate a deterministically decreasing gain parameter when the inflation process is subject to changing uncertainty. In the present study, I put forward a (stochastic) time-varying gain which is rationalized by the time series properties of inflation. I find that a simple model where inflation varies around a random walk trend because of transitory disturbances can be used to approximate SPF expectations. Notably, accounting for stochastic volatility improves the fit. It turns out that survey expectations deviate from the model predictions only by a random error. This motivates a trend learning rule with a timevarying gain parameter. When we estimate such a learning rule, constant gain learning can be rejected for short-term forecasts. Hence, a structural interpretation for stochastic volatility of observed expectations seems to be that agents adapt their learning scheme to changes in the inflation process. In addition, changes in the gain parameter appear to be largely consistent with changes in the Kalman gain predicted by the forecast model. However, I also find that agents adjust their estimate of trend inflation slower than predicted by the model which can be explained by comparatively slow adjustment of the gain parameter. 2. Forecasting inflation in the presence of trend shifts To model the behavior of observed inflation expectations, I start from an empirical perspective and consider univariate forecast models that allow for different assumptions about trend inflation and deviations from the trend (see, among others, Clark and Doh, 2011).5 I consider a random walk trend, a constant trend, cyclical behavior around the trend, and a model where inflation enters in first differences. Trend plus noise, IMA The Trend plus noise model is motivated by Barsky (1987), Ball and Cecchetti (1990), and Ball and Croushore (2003) who opt for a model where inflation varies around its trend because of transitory monetary disturbances which themselves follow a white noise process. It can be shown that the model generates an autocovariance structure identical to an integrated moving average process of order one (IMA(1, 1)). It is used by Cecchetti et al. (2007) to analyze the inflation process in the G7, and it is argued by Stock and Watson (2007) that such a model – despite its simplicity – is well suited to predict US inflation. One reason is that it accounts for permanent shifts in the inflation rate. The model in state space form is given by observation Eq. (1) and state Eq. (2).

pt ¼ pt þ gt gt  Nð0; r2g Þ ptþ1 ¼ pt þ e1;t e1;t  Nð0; r2e1 Þ:

ð1Þ ð2Þ

Inflation pt is driven by two components, a time-varying trend pt that captures the low frequency movements of inflation and temporary shocks gt . The trend is subject to permanent shocks e1;t . It is modeled as a driftless random walk capturing a smooth transition from a high inflation state to a low inflation state. The random walk assumption can be justified by recognizing that it is difficult to a distinguish a random walk from a highly persistent mean reverting process by empirical testing. Most often, both modeling approaches lead to estimates that are observationally equivalent. Apparently, the IMA model is the simplest model that is flexible enough to allow for trend shifts. Most important, the IMA model entails trend learning behavior as the Kalman filter is used to disentangle permanent shocks e1;t and temporary shocks gt . Note that, under fairly general conditions, the Kalman filter allows for an efficient estimation of the unobserved components. 5 Ball (2000) argues that a simple univariate model can be seen as a ‘‘near rational’’ approximation to more sophisticated models. That is, when modeling inflation, unforeseeable structural breaks occur which disrupt the estimated relationships between inflation and other economic variables.

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Table 1 Deviation between model forecast and SPF. horizon

IMA

Const. tr.

Trend cycle

pt1 Trend

Root mean squared deviation h¼1 h¼4

0.7712 1.1297

0.8276 1.2723

0.7715 1.1301

0.8465 1.1711

Mean absolute deviation h¼1 h¼4

0.5985 0.8474

0.6118 0.8576

0.5991 0.8479

0.6637 0.8918

0.0117 0.0168

0.0998 0.2762

0.0120 0.0166

0.0014 0.0189

Mean deviation h¼1 h¼4

Constant trend plus cycle.

pt ¼ p þ gt gt ¼ a1 gt1 þ a2 gt2 þ ft ft  Nð0; r2f Þ

ð3Þ ð4Þ

The constant trend model is a variant of the IMA model. It assumes that inflation fluctuates around a constant value (r2e1 ¼ 0). Note that there are no trend shifts in inflation and, hence, forecasts are mean reverting. Compared to the IMA model, deviations from the trend may occur for a prolonged time span. Trend plus cycle.

pt ¼ pt þ gt pt ¼ pt1 þ f1;t f1;t  Nð0; r2f1 Þ gt ¼ a1 gt1 þ a2 gt2 þ f2;t f2;t  Nð0; r2f2 Þ:

ð5Þ ð6Þ ð7Þ

The trend cycle model is also a variant of the IMA model. It combines a random walk trend as in the IMA model with a cyclical component that follows an autoregressive process. Hence, it allows for a more sluggish adjustment towards the trend. pt1 trend.

ðpt  pt1 Þ ¼ a1 ðpt1  pt2 Þ þ a2 ðpt2  pt3 Þ þ ft

ft  Nð0; r2f Þ

ð8Þ

If inflation is characterized by a stochastic trend, an autoregressive process in first differences is also a valid forecast model. Note that there is no trend learning behavior. To rank these model according to their ability to replicate the dynamics of inflation expectations, I generate a sequence of forecasts from each of these models. These forecasts can then be compared to observed inflation expectations. The models are estimated using Maximum Likelihood techniques and the Kalman filter. Recursive pseudo out-of-sample forecasts for each model are obtained as follows. The model is estimated based on the first vintage of data for GDP inflation covering the time period 1960 Q1–1970 Q1. I then record the forecast for GDP inflation one quarter ahead (h ¼ 1) and one year ahead (h ¼ 4). As the GDP Deflator is sometimes heavily revised, I use real time data to ensure that the information set is the same for the forecaster at that time and the model forecast.6 I recursively iterate over the vintages expanding the sample until 2010 Q4. Finally, the root mean squared deviation and the mean absolute deviation between the generated forecasts and observed expectations from SPF can be calculated. Furthermore, to reveal possible bias in the approximation, the mean of the deviation is also reported. Table 1 summarizes the results: It turns out that the models with a random walk trend (IMA, Trend cycle) outperform the other models. Overall, the pt1 trend model performs worst for one-quarter-ahead expectations. Note that it is the only model without explicit trend learning. Moreover, the IMA model and the Trend plus cycle model give a comparable performance although the IMA specification is more parsimonious. Such a result may be explained by the fact that the estimated AR coefficients a1 and a2 are close to zero. This is in line with the arguments in Ball and Cecchetti (1990) who find that deviations from trend inflation are not persistent and last largely one quarter. Finally, it appears that the approximation is better for one-quarter-ahead expectations when compared to 1-year-ahead expectations as the average deviation is always smaller. Furthermore, it turns out that the approximation bias is small for all models but the Constant trend model. In this case, the 1-year-ahead forecasts from the model are somewhat higher than SPF expectations. The smallest bias is obtained for the pt1 trend model. That is, negative deviations largely offset the positive deviations during the sample period analyzed here. To see whether the results hinge on a particular monetary regime, I calculate the performance measures for four different sub-periods: the pre-Volcker era (1970 Q1–1979 Q2),7 the Volcker period (1979 Q3–1987 Q2), the Greenspan period (1987 6

A description and a plot of the data is given in Appendix A. During this time the Federal Reserve Board has witnessed three chairmen: William McChesney Martin, Jr. (1951–1970), Arthur F. Burns (1970–1978), G. William Miller (1978–1979). 7

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S.R. Henzel / Journal of Macroeconomics 35 (2013) 172–185 Table 2 Deviation between model forecast and SPF for different sub-samples. 1970–1979 Horizon

IMA

1979–1987 Const. tr.

Trend cycle

pt1 Trend

Trend cycle

pt1 Trend

0.6795 0.8841

0.6972 0.9899

0.7535 1.0000

0.5847 0.8559

0.5507 0.7464

0.5846 0.8563

0.6512 0.8768

0.1588 0.3619

0.0823 0.1641

0.1592 0.3624

0.1478 0.3908

IMA

Root mean squared deviation 1.1289 h¼1 1.8320 h¼4

1.3177 2.2885

1.1289 1.8320

1.2567 1.9038

0.6975 0.9895

Mean absolute deviation 0.9054 h¼1 1.4077 h¼4

1.0293 1.7800

0.9054 1.4077

1.0338 1.5123

Mean deviation 0.4359 h¼1 0.9845 h¼4

0.5719 1.1901

0.4359 0.9845

0.5082 1.0516

1987–2006

Const. tr.

2006–2010

Root mean squared deviation 0.5356 h¼1 0.6961 h¼4

0.5045 0.5886

0.5367 0.6971

0.5640 0.7273

0.7459 0.7493

0.7437 0.7243

0.7459 0.7493

0.8447 0.7897

Mean absolute deviation 0.4324 h¼1 0.5867 h¼4

0.4106 0.4819

0.4337 0.5875

0.4642 0.6106

0.6211 0.6794

0.6225 0.5948

0.6211 0.6794

0.6813 0.7191

Mean deviation h¼1 h¼4

0.0607 0.0068

0.1805 0.3221

0.1911 0.3369

0.0362 0.0258

0.0575 0.2398

0.0362 0.0258

0.0481 0.0381

0.1800 0.3219

Q3–2005 Q4), and the period beginning with the chairmanship of Ben Bernanke (2006 Q1–2010 Q4). The results are given in Table 2. First, it appears that the pt1 trend model is outperformed in all sub-samples; this is consistent with the result in Table 1. Second, the models with a random walk trend (IMA, Trend cycle) both perform well on average and, as before, the values are almost identical. Finally, the Constant trend model seems to work remarkably well for all sub-samples but the pre-Volcker period (1970–1979) where SPF expectations experience a swift increase. Not surprisingly, it outperforms the other approaches during the Greenspan period (1987–2006) where trend inflation was rather stable; recall that the Constant trend model is a restricted version of the Trend plus cycle model where shocks to trend inflation are absent. Also note that the present out-of-sample simulation exercise is based on recursive estimation and, hence, the parameters of all forecast models adapt over time. 3. Fitting observed inflation expectations 3.1. The IMA model The IMA model seems to be a good candidate model when it comes to fitting observed inflation expectations to a simple trend learning rule. In the following, I fit observed inflation expectations to this model. That is, I estimate a simultaneous structural time series model for inflation and inflation expectations. The aim is to analyze whether the model obtains an approximation to observed expectations that is reasonably close. The IMA model for inflation consists of Eqs. (1) and (2). Moreover, the following observation equation is added to model the movements of inflation expectations:

petþhjt ¼ p etþhjt þ getþhjt

ð9Þ

Here, t þ hjt denotes a forecast for period t þ h given information up to time t and superscript e is attached to all variables  etþhjt is the that relate to survey expectations. Eq. 9 states that survey forecasts stem from a trend learning model, where p trend forecast and getþhjt is the forecast of the transitory component. The equality of model forecasts and survey forecasts is introduced by the following relations:

p etþhjt ¼ p tþhjt þ e2;t getþhjt ¼ gtþhjt :

e2;t  Nð0; r2e2 Þ

ð10Þ ð11Þ

We add one irregular component e2;t capturing the differences between SPF expectations and the forecast of the trend emerging from the model. If forecasts are derived from the IMA model the following restrictions have to hold: the trend fore tþhjt ¼ p  t . Moreover, the deviations from trend are white noise and gtþhjt ¼ 0. Hence, cast is equal to the last trend estimate p SPF expectations are equal to the last trend estimate if the IMA model holds. Further, if the IMA model yields a good approximation then e2;t will be a (unsystematic) random error. Hence, e2;t can be used to test the appropriateness of the IMA model

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S.R. Henzel / Journal of Macroeconomics 35 (2013) 172–185

Table 3 Estimation results. 1970–2010

1970–1979

Param.

p-Value

1.067

0.000

0.323

0.000

0.165

Param.

1979–1987

1987–2006

2006–2010

p-Value

Param.

p-Value

Param.

p-Value

Param.

p-Value

1.740

0.000

0.898

0.000

0.580

0.000

1.466

0.046

0.860

0.232

0.381

0.000

0.081

0.002

0.121

0.732

0.000

0.481

0.057

0.260

0.000

0.018

0.010

0.089

0.809

0.216

0.020

0.024

0.987

0.130

0.736

0.424

0.001

0.070

0.945

1.965

0.000

4.796

0.031

1.597

0.000

0.658

0.000

1.750

0.000

0.168

0.000

0.372

0.061

0.241

0.000

0.043

0.000

0.049

0.103

0.046

0.008

0.016

0.533

0.124

0.000

0.033

0.000

0.024

0.343

0.338

0.000

0.396

0.027

0.212

0.444

0.191

0.245

0.325

0.295

h¼1

r2g r2e1 r2e2 qð1Þ h¼4

r2g r2e1 r2e2 qð1Þ

as an approximating model for inflation expectations. The complete model is given by Eqs. 1,2 and 9,10,11 with hyperparameters r2g ; r2e1 , and r2e2 . It is fitted to both, inflation and inflation expectations contained in the Survey of Professional Forecasters, simultaneously.8 The Likelihood function is evaluated using the state space representation of the system and the Kalman filter. Maximum Likelihood estimates for the full sample period (1970 Q1–2010 Q4) along with parameter significance levels are given in columns two and three of Table 3. Overall, the estimated parameters are significant at conventional levels. In particular, r2e1 is different from zero which confirms that trend inflation changes over the sample. As a comparatively large value is estimated for r2g , transitory disturbances appear to be important as well. It is now possible to test whether e2;t is a random error. To this end, I report the autocorrelation coefficient of e2;t (qð1Þ) along with the p-value of the associated Q-test. It turns out that a small amount of autocorrelation remains as qð1Þ ¼ 0:216 for the one-quarter-ahead forecast and qð1Þ ¼ 0:338 for the 1-year-ahead forecast. However, autocorrelation is significant at conventional levels which suggests that the deviations between the IMA model and observed expectations are predictable. Such a result may obtain because there have been changes to the parameters – i.e. the variance – of the inflation process. To account for possible changes in the inflation process, I re-estimate the model for different sub-periods. As before, the subperiods consist of the pre-Volcker era, the Volcker period, the Greenspan period, and the period beginning with the chairmanship of Ben Bernanke. Estimation results are given in Table 3. Note that, because of the shorter sample, the parameters are estimated with less precision than before. In particular, Table 3 reveals that there have been changes to the estimated variances of inflation over time. Considering one-quarter-ahead expectations, r2e1 decreases from 0.86 during the pre-Volcker era to roughly 0.10 during the presidency of Greenspan and Bernanke. Note that r2e1 is a measure of uncertainty about trend inflation because it determines the size of the shocks that hit trend inflation. If these shocks are large, it is harder to forecast the level of trend inflation. It appears that trend uncertainty seems to decrease over the monetary regimes. A similar tendency is observed for 1-year-ahead expectations. Note that, after the ‘‘Great Moderation’’ has taken place, trend shifts are of little importance; with a value of 0.08 the Greenspan period obtains the lowest trend uncertainty. Considering the pre-Volcker era, the accommodative policy of the FED following the oil price hikes and soaring wages during the early seventies leads to a persistent rise of inflation (see, among others, Clarida et al., 2000). Notably, rising trend inflation has to be acknowledged by private forecasters. For one-quarter-ahead expectations, it appears that the autocorrelation of e2;t is virtually zero and insignificant which indicates that the irregular component is a random error. Hence, the IMA model seems to be a valid description of the expectation formation process during the pre-Volcker era. The subsequent Volcker period is known as a period of disinflationary monetary policy and decreasing trend inflation. In conjunction with the fact that the Federal Reserve does not announce a target inflation rate, private agents need to learn about trend inflation. Therefore, the model should yield a good approximation to the expectation formation process during this sub-period. Table 3 reveals that there is no persistent deviation of survey expectations from the IMA model. A somewhat different picture emerges for the Greenspan era. Although all parameters are significant at conventional levels, a perfect fit is not obtained probably due to large oil price shocks.9 Turning to the most recent period, autocorrelation of e2;t is, again, low and insignificant. Table 3 also shows the results for 1-year-ahead expectations. Considering the sub-samples, it turns out that almost all model parameters remain significant if we use a shorter sample. However, autocorrelation of e2;t decreases only slightly when the sub-samples are compared to the full sample period, although it is insignificant for all periods but the pre-Volcker era. However, during the pre-Volcker era, 1-year-ahead expectations do not reflect the hike in inflation during the mid8 As in Section 2, inflation is measured as the annualized quarterly growth rate of the GDP deflator. Expectations are taken from the Survey of Professional Forecasters one quarter ahead (h ¼ 1) and 1 year ahead (h ¼ 4). 9 However, the autocorrelation of e2;t traces back to the last few quarters of the sample. This is probably due to the fact that large oil price shocks hit the economy when the US engaged in war in Iraq in 2003. The soaring oil prices apparently have led SPF expectations and GDP deflator inflation to drift apart temporarily. Hence, I re-estimate the model with GDP inflation replaced by CPI core inflation. As CPI inflation is consistently lower than GDP inflation for the sample considered a constant is included in Eq. (10). It turns out that autocorrelation disappears once we abstract from the effect of sudden oil price shocks.

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S.R. Henzel / Journal of Macroeconomics 35 (2013) 172–185 12

12

10

10

8

8

6

6

4

4

2 0

2 IMA model trend (95% interval, h=1) SPF (h=1)

70

74

78

82

86

90

94

98

02

06

10

0 70

IMA model trend (95% interval, h=4) SPF (h=4)

74

78

82

86

90

94

98

02

06

10

Fig. 1. SV-IMA model prediction and fitted SPF expectations.

seventies (see Fig. A.1 in Appendix A).10 Overall, results suggest that survey expectations do not deviate systematically from the model predictions when we account for shifts in the variance parameters. However, the IMA model seems to be better suited to approximate one-quarter-ahead expectations compared to 1-year-ahead expectations. 3.2. Stochastic volatility It appears that we need to account for changes in the variance when modeling expectations. Instead of concentrating on different regimes, I estimate the model of the previous Section 3.2 assuming stochastic volatility. The reason is that a number of studies emphasize that inflation is subject to stochastic volatility (see, for instance, Ball and Cecchetti, 1990; Stock and Watson, 2007; Cecchetti et al., 2007). Particularly, the forecast performance of the IMA model improves if we introduce stochastic volatility (Stock and Watson, 2007; Clark and Doh, 2011). If survey participants acknowledge that trend uncertainty changes, the volatility of observed expectations can be rationalized by changing uncertainty about trend inflation.11 The stochastic volatility version of the IMA model (SV-IMA) is given by Eqs. 1,2, and 9,10,11 and the following two equations capturing the dynamics of the variances:

log r2g;t ¼ log r2g;t1 þ fg;t log r

2 e1;t

2 e1;t1

¼ log r þ fe;t  0 ft ¼ fg;t fe;t  Nð0; c2 I2 Þ:

ð12Þ ð13Þ

Now variances r2g;t and r2e1;t obtain an additional time index t. The logs of the variances are assumed to follow two independent driftless random walks. Again, r2e1;t provides a time-varying measure of uncertainty about trend inflation, whereas r2g;t denotes uncertainty associated with the transitory component. Particularly, the SV-IMA model allows for the Kalman gain to be time-varying. That is, the Kalman gain adapts in an efficient way when the relative importance of both types of shocks changes over time.12 If, for instance, the variance of permanent shocks steadily decreases over time, trend shifts become less and less important for the forecast. This flexibility brings about the advantage over the simpler IMA model which assumes that trend shifts are equally important over the whole sample. Note that the introduction of stochastic volatility leaves us with a non-linear, non-gaussian state space model. Hence, the usual maximum likelihood estimator is inappropriate (see, for instance, Shephard, 1994). I use the Gibbs sampler to estimate  t , together with the model.13 Results of the estimation are depicted in Fig. 1. It shows the 95% region of the distribution of p  etþ1jt from the SV-IMA model. It appears that the deviations are small because SPF expectations fitted inflation expectations p lie within the estimated trend interval for the whole sample.  t implied by the SV-IMA model for one-quarterNote: The shaded area represents the 95% interval of the distribution of p ahead expectations (left) and 1-year-ahead expectations (right). The bold line depicts SPF expectations. Finally, I analyze whether the deviation e2;t is white noise. The estimated first order autocorrelation coefficient of e2;t for one-quarter-ahead expectations appears to be close to zero; the median value is qð1Þ ¼ 0:13. That is, the 95% interval covers values from 0.04 to 0.30. For 1-year-ahead expectations the median autocorrelation is 0.08 (0.04 to 0.21). Hence, it ap10 One explanation might be that SPF participants seem to judge the hike during the seventies as a largely temporary phenomenon and adapted their estimate of trend inflation slower than predicted by the IMA model. Note that for this sub-period, the estimated variance of the transitory component is more than twice as high as the estimate obtained for one-quarter-ahead expectations which means that transitory movements get a relatively large weight during the preVolcker era. Hence, the approximation obtained with the IMA model is worse than for the other sub-periods. To check whether this is the case, I re-estimate the model for this sub-period using the Trend cycle model and it turns out that a1 ¼ 0:56; a2 ¼ 0:07, and the autocorrelation of e2;t disappears. 11 For long-term inflation expectations, Clark and Davig (2009) document that the volatility decreases over time. 12 The steady state Kalman gain K tjT can be derived from r2g;tjT and the steady state covariance of pt , Pt : K tjT ¼ P t =ðPt þ r2g;tjT Þ (see Harvey, 1989). The steady qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi state covariance Pt is given by Pt ¼ 0:5ðr2e1;tjT r2g;tjT þ ðr2e1;tjT r2g;tjT Þ2 þ 4ðr2e1;tjT r2g;tjT ÞÞ (see also Grassi and Proietti, 2010). 13

The model equations and a description of the estimation routine are given in Appendix B.

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S.R. Henzel / Journal of Macroeconomics 35 (2013) 172–185

Table 4 Estimation results for the learning rule. Param.

s.e.

p-Value

r2g~;1

0.218

0.036

0.000

/1 /2

0.564 0.228 4.3e4

0.087 0.034 2.2e4

0.000 0.000 0.051

Param.

s.e.

p-Value

0.269

0.041

0.000

0.679 0.321

0.078 0.079

0.000 0.000

0.306 137.432 pðv21 Þ ¼ 0:000

0.026

0.000

Param.

s.e.

p-Value

0.197

0.041

0.000

0.022

0.016

0.165

0.170

0.023

0.000

0.004

0.002

0.087

h¼1

r2g~;2 K logL LRtest

129.684

162.838 pðv22 Þ ¼ 0:000

h¼4

r2g~;1

0.191

0.024

0.000

0.204

0.023

0.000

/1 /2

0.615 0.105 2.2e4

0.104 0.073 4.0e4

0.000 0.149 0.594

0.698 0.142

0.076 0.072

0.000 0.049

0.135 117.905 pðv21 Þ ¼ 0:158

0.014

0.032

r2g~;2 K logL LRtest

116.909

130.545 pðv22 Þ ¼ 0:000

pears that the SV-IMA model deviates from the survey only by an unsystematic random error. Overall, stochastic volatility seems to be a key feature of SPF expectations we need to consider if we want to fit the series to the trend learning model. 4. Estimating a simple forecast rule 4.1. The trend learning rule We now estimate the learning rule implied by the SV-IMA model explicitly. A forecast function can be derived from the Kalman filter recursions. If agents use the SV-IMA model inflation expectations emerge from the process given by Eqs. (14) and (15).

ptþhjt ¼ p tjt þ nt p tjt ¼ p t1jt1 þ Kt ðpt  ptjt1 Þ  t1jt1 þ Kt mt ¼p

ð14Þ ð15Þ

Eq. (14) states that the forecast of future inflation is given by a straight line emerging from the actual trend estimate. Eq. (15) states how agents update their belief about trend inflation in the light of past forecast errors, i.e. how signal extraction is accomplished. When estimating the unobserved components, agents learn from noisy information contained in the onestep-ahead forecast error mt ¼ pt  Et1 pt . If Kt is larger than zero, a trend learning mechanism is present in the data. Notably, the perceived trend is driven solely by past forecast errors. For estimation purpose, an irregular component nt is added to the forecast function. To obtain a well-specified model, the disturbance is allowed to be autocorrelated:

~ 1;t nt ¼ /1 nt1 þ /2 nt2 þ g

g~ 1  Nð0; r2g~;1 Þ

ð16Þ

Note that nt can be interpret as the portion of SPF expectations which cannot be explained by observed forecast errors. To operationalize estimation, ptþhjt is replaced by its observed counterpart petþhjt from SPF. In addition, mt is also replaced by its observed counterpart met . It is given by the one-quarter-ahead expectation error from SPF met ¼ pt  petjt1 . Note that I use real time data to calculate the expectation error, as the GDP deflator is sometimes heavily revised. That is pt is the (first) vintage that has been published at time t. Hence, the estimations are based only on information available to professional  t1jt1 is the trend perceived by survey participants when expectations forecasters at the time the forecast is provided. p are formed from past forecast errors. To distinguish between trend inflation as predicted by SV-IMA model and perceived  et1jt1 ). trend inflation, the latter obtains a superscript e (p Kt yields an estimate of the gain parameter underlying SPF expectations. Note that Kt has a structural interpretation because it determines by how much expectations adjust in the presence of trend shifts (see, for instance, Erceg and Levin, 2003). Further, a time-varying gain parameter may account for stochastic volatility of inflation expectations. As the variance of shocks changes over time agents account for this fact by adjusting their reaction to observed forecast errors. That is, stochastic volatility of observed expectations may be the outcome of changing uncertainty about trend inflation. To test whether there is evidence for time variation, Kt is modeled as a time-varying parameter:

~ 2;t Kt ¼ Kt1 þ g

g~ 2  Nð0; r2g~;2 Þ

ð17Þ

I estimate the system given by Eqs. (14)–(17). The model contains four hyperparameters (r2g~;1 ; /1 ; /2 ; r2g~;2 ). As stochastic volatility of expectations is reflected by time-varying Kt , there is no need to model additional time-varying shock variances.

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π

14

t

SPF trend inflation (h=1)

12

SPF trend inflation (h=4)

12

10

10

8

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4

4

2

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t

0 74

78

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70

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Fig. 2. Inflation and trend implied by SPF. Note: The thick black line indicates the trend associated with SPF one-quarter-ahead expectations (left) and 1year-ahead expectations (right). The thin black line depicts GDP deflator inflation. Grey shaded areas indicate a 90% confidence region.

Hence, the model is linear and can be put into state space form. It is estimated by Maximum likelihood where the likelihood is evaluated by the Kalman filter. 4.2. Estimation results The upper part of Table 4 presents estimated parameters based on one-quarter-ahead expectations of annualized quarterly GDP deflator growth. The left panel contains results for an unrestricted version of the model.14 It appears that r2g~ ;2 is significantly positive which implies that the gain parameter implied by SPF seems to vary over time. Table 4 also presents parameter estimates for a restricted estimation where Kt is assumed to be constant over time (r2g~ ;2 ¼ 0). Note that the estimate for the constant gain parameter K amounts to 0.31 which entails that roughly one third of the most recent forecast error is attributed to permanent shocks. To test whether a constant gain model is preferred over the unrestricted model, I conduct a likelihood ratio (LR) test. The LR statistic is significant at conventional levels. Hence, the constant gain is rejected in favor of the time-varying gain model. This suggests that SPF participants adjust their learning behavior according to changes in trend uncertainty. Furthermore, the parameters governing the dynamics of nt (r2g~ ;1 ; /1 ; /2 ) are individually significant at conventional levels. Alternatively, we impose the restriction /1 ¼ /2 ¼ 0 and assume that Kt is the only source of variation. Results are given in the right panel of Table 4. The variance of the gain parameter is somewhat higher but estimated with less precision. As the restricted model is rejected by the data the following analysis is based on the unrestricted version of the model. The lower part of Table 4 displays estimation results for 1-year-ahead expectations. It turns out that the model provides a reasonable fit for 1-year-ahead expectations as well. The estimated hyperparameters are roughly comparable to onequarter-ahead expectations. However, it turns out that r2g~;2 is only half as big as the estimate for one-quarter-ahead expectations. As a result, the constant gain version of the model in the middle panel of Table 4 is not rejected at conventional levels. This is probably due to the fact that 1-year-ahead expectations are somewhat less volatile than one-quarter-ahead expectations.15 However, K is significantly different from zero, yet it is roughly half the size obtained for one-quarter-ahead expectations (0.14). Similar to one-quarter-ahead expectations, the restricted model /1 ¼ /2 ¼ 0 is rejected. From Eq. (15) we obtain an estimate of the unobserved trend (petjt ) that SPF participants derive from past forecast errors. For both forecast horizons, Fig. 2 depicts trend inflation perceived by SPF participants obtained from the time-varying gain model. It turns out that the implied permanent component tracks the inflation rate quite closely. This is remarkable because perceived trend inflation is solely driven by past forecast errors. The trend for 1-year-ahead expectations seems to adapt at a slower rate, due to a smaller estimated gain parameter. Finally, it appears that perceived trend inflation changes rather slowly. It takes about 10 years beginning in the late seventies until inflation expectations stabilize around 3% at the beginning of the nineties. Overall, past forecast errors have a significant impact on observed inflation expectations in a way consistent with trend learning behavior in the presence of stochastic volatility. 4.3. Comparison of learning dynamics Now I turn to the question whether the gain parameter implied by SPF expectations is consistent with the (efficient) Kalman gain implied by the SV-IMA model. I use the Gibbs sampler to estimate the non-linear, non-gaussian SV-IMA model.16 14

Estimation diagnostics in Appendix C indicate that the model is well specified. It appears that h ¼ 4 expectations are somewhat less volatile than one-quarter-ahead expectations (the average standard deviation is 2.30 for h ¼ 1 and 1.96 for h ¼ 4). Hence, it might be more difficult to identify time-variation in the gain parameter. To see if this is the case, I assume that both expectation series have a common gain parameter which can be estimated by setting up a simultaneous model for both expectation series. It turns out that, for the simultaneous model, the restriction r2g~ ;2 ¼ 0 is rejected by the LR test, the test statistic amounts to 28.24. Detailed estimation results are available upon request. 16 As in Section 2, I use annualized quarterly change of the GDP deflator to measure inflation. A sketch of the estimation routine and estimates for timevarying variances are given in Appendix D. 15

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0.7

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Fig. 3. SPF and SV-IMA gain parameter. Note: The crossed line depicts the final estimate of the Kalman gain K t . The bold black line represents the estimated gain parameter underlying SPF expectations Kt along with the 90% confidence region for one-quarter-ahead expectations (left) and 1-year-ahead expectations (right).

Table 5 SPF and SV-IMA gain parameter. h ¼ 1 Level Kt ¼ 0:350 þ 1:045 K t 0:027

lags l corrðKt ; K tþl Þ

0:045

3 0:87

2 0:88

1 0:89

0 0:90

1 0:90

2 0:90

3 0:90

4 0:89

5 0:89

6 0:88

6 5 4 3 0:27 0:25 0:27 0:31 dðKt Þ ¼ 0:001 þ 0:251 dðK t6 Þ

2 0:24

1 0:20

0 0:17

1 0.13

2 0.06

3 0.01

4 0.02

5 0.02

6 0.01

3 0:90

2 0:89

1 0:88

0 0:87

1 0:86

2 0:84

3 0:82

4 0:80

5 0:78

6 0:77

6 5 4 3 0:53 0:54 0:57 0:51 dðKt Þ ¼ 0:0001 þ 0:353 dðK t2 Þ

2 0:59

1 0:57

0 0:57

1 0:53

2 0:48

3 0:47

4 0:44

5 0:38

6 0:35

6 0:83

5 0:84

4 0:85

Difference dðKt Þ ¼ 0:001 þ 0:162 dðK t Þ 0:000

Lags l CorrðdðKt Þ; dðK tþl ÞÞ

0:000

0:089

0:094

h ¼ 4 Level Kt ¼ 0:141 þ 0:504 K t 0:014

Lags l CorrðKt ; K tþl Þ

6 0:91

5 0:91

0:029

4 0:90

Difference dðKt Þ ¼ 0:0003 þ 0:380 dðK t Þ 0:0002

Lags l CorrðdðKt Þ; dðK tþl ÞÞ

0:0003

0:076

0:078

Note: A asterisk denotes significance at the 5% level. Newey-West adjusted standard errors are given in parenthesis

Estimation results are depicted in Fig. 3. It appears that there is considerable time variation in the Kalman gain due to stochastic volatility of inflation. The model attaches a large weight to trend shifts at the beginning of the ‘‘Great Moderation’’ of inflation. In subsequent years, permanent shocks become less and less important. To be precise, in the beginning of the sample, about 70% of the last forecast error is due to misperceptions of the permanent component, whereas the remaining 30% are due to transitory shocks. Towards the end of the sample, roughly only 40% of the forecast error are due to trend misperceptions according to the SV-IMA model. Next, I compare the gain parameters underlying the survey forecasts to the gain of the simple forecast model. Fig. 3 depicts the gain parameter underlying SPF expectations Kt . Note that Kt lies in a plausible range. For both forecast horizons it is significantly different from zero and below one in the first half of the sample. This finding emphasizes that, during the early periods, past forecast errors significantly drive observed inflation expectations. Notably, Kt is not restricted during estimation. Towards the end of the sample, the gain parameter approaches zero. However, this does not mean that the learning model is rejected. To begin with, SPF participants do not update their estimate of trend inflation during this specific time period. However, they may put more weight on trend shifts and start to revise their trend estimate again as soon as trend uncertainty starts to rise. Second, Kt is considerably lower than the gain parameter of the SV-IMA model. Hence, SPF participants give less weight to permanent shocks which leads perceived trend inflation to change relatively slowly (compare also Fig. 2). This holds for 1year-ahead expectations in particular. Hence, SPF participants seem to underestimate the role of permanent shocks in comparison to a simple forecast model. Third, it turns out that in the beginning of the sample Kt starts out relatively high and falls in subsequent years. Note that Kt broadly retraces the movements of K t . Hence, it appears that the ‘‘Great Moderation’’

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1

0.8

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Fig. 4. Impulse response of a shock to K t . Note: The bold black line shows Cholesky impulse responses of Kt for one-quarter-ahead expectations (left) and 1year-ahead expectations (right) to a shock to the Kalman gain K t . The dashed line is the response of K t to the same shock. The shaded areas are 68% and 95% confidence regions obtained from a bias adjusted bootstrap (Kilian, 1998).

is somehow reflected in the learning rule of survey participants. This point is reinforced by the significance of K t when it is regressed on Kt (see Table 5). This holds also true if variables are considered in first differences. Notably, the constant turns insignificant which implies that there is no systematical bias in the change of the gain parameter. To study the dynamic relation between the Kalman gain of the SV-IMA model K t and the gain parameter estimated for SPF Kt , Table 5 shows cross-correlations among both variables. It turns out that lags of K t are important. Hence, values of Kt are even closer associated with lagged values of the efficient gain parameter. The highest correlation is found for a lead of about six quarters. To analyze the effect of an exogenous increase of the Kalman gain, I estimate a bivariate VAR containing four lags of Kt and K t , where the lag length is chosen by BIC. The shocks are orthogonalized using a standard Cholesky decomposition. Kt is ordered first and K t is ordered last. To analyze the response of the gain parameter estimated for SPF when a shock to K t occurs, Fig. 4 presents impulse responses to a one standard deviation shock to the Kalman gain derived from the bivariate VAR. We observe that a sudden increase of the Kalman gain leads to an increase of the learning parameter of one-quarterahead observed expectations. Kt is significantly and persistently above the zero line. Thus, survey participants appear to adjust their gain parameter into the correct direction, once a change of the Kalman gain has taken place. However, the learning parameter takes some time to react to the shock. The delay seems to be roughly seven quarters which is in line with the cross-correlation in Table 5. A similar result is obtained for 1-year-ahead expectations. However, the response appears to be not only delayed but also dampened. It becomes significant after roughly 3 years.17 Overall, the gain parameter of 1year-ahead expectations adapts inefficiently slow. 5. Conclusion Policy advice is often derived from models assuming rationality and the fit of survey expectations is usually disregarded. Given that many economic decisions are governed by inflation expectations, we need a better understanding of how these quantities evolve over time. Further, many studies argue that inflation is governed by unobserved trend shifts and time-varying volatility, i.e. trend uncertainty. As trend inflation and trend uncertainty are unobserved, a forecaster has to solve a signal extraction problem to come up with a forecast for inflation. The analysis in the present paper is based on a forecast model where inflation varies around a random walk trend due to temporary shocks which themselves follow a white noise process. The model entails trend learning by signal extraction with the help of the Kalman filter. It can be shown that this simple trend learning model yields a closer approximation to observed expectations than other univariate candidate models that account for trend shifts. The model can be modified to account for stochastic volatility which leads to time variation in the learning rule. I fit the trend learning model to observed inflation expectations and investigate whether and how shifts in trend inflation and time-varying uncertainty about trend inflation impact the expectation formation process. First, I find that SPF expectations deviate from the learning model only by a random error if we account for time-varying volatility of inflation and inflation expectations. Second, I derive the trend learning rule associated with the forecast model. Moreover, stochastic volatility of expectations motivates a trend learning rule with a time-varying Kalman gain. When I estimate the learning rule explicitly it turns out that trend learning parameters are significant. That is, past forecast errors have a significant impact on observed inflation expectations in a way consistent with the trend learning model. However, I also find that the SPF gain parameter is lower than the Kalman gain implied by the forecast model. Hence, survey participants recognize a shift of the trend inflation rate comparatively slowly. 17

Results for the alternative ordering are presented in Appendix E. It turns out that the ordering does not affect the results.

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Furthermore, for short-term expectations, there is evidence that the gain parameter varies over time which provides a rationale for stochastic volatility in survey expectations. It turns out that the gain parameter decreases which means that the importance of permanent shifts in the inflation rate perceived by survey participants decreases. Note that uncertainty about trend inflation has also come down to very low levels over the past 40 years. Hence, agents appear to adjust their trend learning behavior in accordance with changes in trend uncertainty. However, I find that a change in the gain parameter is followed by a change in the SPF gain parameter with a lag of about seven quarters. Overall, if we want to model expectations in line with survey expectations, trend learning with a time-varying learning rule is a viable alternative. Furthermore there is a link between trend inflation uncertainty and the way inflation expectations are formed. Hence, the present study highlights how a reduction of uncertainty about trend inflation may contribute to low and stable inflation expectations. Acknowledgements I would like to thank Kai Carstensen, Michael Ehrmann, Gebhard Flaig, and Thomas Lubik as well as the editor and two anonymous referees for valuable comments and suggestions. I am also grateful to participants at the meeting of the European Economic Association and the German Economic Association, and participants at research seminars at the Universities of Munich, Goettingen, and Kiel. Financial support from the German Research Foundation (Grant No. CA 833/2) is gratefully acknowledged. Appendix A. Data Post-war real time data for the US GDP deflator are obtained from the real time database maintained at the Federal Reserve Bank of Philadelphia (Croushore and Stark, 2001). Inflation is calculated as the annualized quarterly growth rate of the GDP deflator. Observed inflation expectations are taken from The Survey of Professional Forecasters which is also obtained from the Federal Reserve Bank of Philadelphia.18 Here, I use the median forecast of the GDP deflator which is surveyed since 1968 Q4. From the forecast of the GDP deflator, the expected annualized quarterly inflation one quarter ahead (h ¼ 1) can be calculated. Due to the publication lag, the first forecast is always obtained for the current quarter. For instance, the survey in 2001 Q1 is based on observations up to time t which is 2000 Q4. The one-quarter-ahead forecast t þ 1jt is then the forecast for 2001 Q1. I also calculate 1-year-ahead expectations (h ¼ 4) which is the annualized quarterly growth rate predicted for the quarter 1 year from the last observation for inflation. For instance, for the survey conducted in 2001 Q1, the 1-year-ahead forecast t þ 4jt denotes expected quarterly inflation in 2001 Q4. Fig. A.1 Appendix B. Section 3.2: Gibbs sampling algorithm This appendix sketches the estimation algorithm used for the model in Section 3.2. The non-linear, non-Gaussian statespace model is given by the following equations:

pt ¼ pt þ gt gt  Nð0; r2g;t Þ e p ¼ ptþhjt ptþ1 ¼ pt þ e1;t e1;t  Nð0; r2e1;t Þ e ptþhjt ¼ pt þ e2;t e2;t  Nð0; r2e2 Þ log r2g;t ¼ log r2g;t1 þ fg;t log r2e1;t ¼ log r2e1;t1 þ fe;t  0 ft ¼ fg;t fe;t  Nð0; c2 I2 Þ: e tþhjt

The model is estimated using Markov-chain Monte Carlo methods. The quantities pt ; petþhjt ; ln r2g;t ; log r2e1;t , and tained from a multimove Gibbs sampler consisting of the following steps (see Shephard, 1994):

r2e2 are ob-

2;T  0; p  e0 ; log r2;T 1. Initialize p g and log re1 . 2;T 2;T  0 and p  e0 are initialized using a diffuse Here, log rg and log re1 denote the time series of log volatility up to period T. p 2;T prior, and log r2;T and log r are initialized using a flat prior. g e1 2;T 2 T; p  e;T  from p(½p T; p  e;T jr2;T 2. Sample the states ½p g ; re1 ; re2 ). Draws from the distribution are obtained using standard Kalman filtering techniques and the simulation smoother of Durbin and Koopman (2002). 2;T 2;T  T  e;T 2 3. Sample r2;T g from p(rg jp ; p ; re1 ; re2 ).  t ¼ rg;t gt , where To sample from the conditional distribution, we rewrite the non-linear signal equation pt  p rg;t  Nð0; 1Þ. Taking squares and calculating logs we arrive at a linear signal equation:

18

The data is obtained from http://www.philadelphiafed.org/research-and-data/real-time-center/survey-of-professional-forecasters/data-files/PGDP/.

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12 10 8 6 4 2 0 −2

69

73

77

81

85

89

93

97

01

05

09

Fig. A.1. Inflation and observed expectations (SPF). Note: The bold line represents one-quarter-ahead inflation expectations from SPF at time t þ 1jt, the dashed line represents 1-year-ahead inflation expectations t þ 4jt. The thin line is the first vintage of real-time data for GDP deflator inflation available at time t.

5

5

innovationt

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02

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10

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1

1 ACF(innovation)

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innovationt

−1

1

2

3

4

5

6

7

8

9

10 11 12

Fig. C.1. Innovations and ACF of innovations of the learning model. Note: Innovations and autocorrelation for the (unrestricted) learning model for onequarter-ahead expectations (left) and 1-year-ahead expectations (right).

 t Þ2 ¼ log r2g;t þ log g2t : logðpt  p 2 2 In principle, standard techniques may be used to sample log r2;T g . However, given the model assumptions log gt is a log v1 2 distributed variable and a gaussian smoother does not apply. As suggested by Shephard (1994), the log v1 distribution is approximated by a mixture of two normal distributions where the parameters and the mixture probabilities have been chosen to match the first four moments of the log v21 distribution (see also Stock and Watson (2002)).19 2;T  T  e;T 2;T 2 4. Sample r2;T e1 from p(re1 jp ; p ; rg ; re2 ). t  p  t1 ¼ re1;t e1;t may again be rewritten: To sample from this density, the non-linear signal equation p

t  p  t1 Þ2 ¼ log r2e1;t þ log e21;t : logðp A draw is obtained by standard simulation smoothing methods using the same techniques as in Step 3. 2;T T; p  e;T ; r2;T 5. Sample r2e2 from p(r2e2 jp g ; re1 Þ. I use a conjugate prior distribution which is IG(m0 =2; S0 =2) where S0 ¼ 0:4. Hence, the posterior is IG(m1 =2; S1 =2) with parameters given by m1 ¼ m0 þ T and S1 ¼ S0 þ e02;t e2;t . 2;T T; p  e;T ; p  T ; log r2;T 6. Write p g ; log re1 and go to Step 2.

m0 ¼ 40 and

 t ) and the variance of the After a burn-in phase, these steps are repeated 5000 times and the point estimate of the trend (p shocks (r2e1;t ; r2g;t ) are given by the median of the simulated distributions and intervals are obtained as the quantiles of the distribution. Appendix C. Section 4.1: Diagnostics Fig. C.1. 19

Estimations are based on the replication files of Stock and Watson (2007) which are available from: http://www.princeton.edu/mwatson/publi.html.

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1.6

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Fig. D.1. The SV-IMA model. Note: The left part depicts trend uncertainty obtained from estimation of the SV-IMA model. The right part depicts the variance of the shock to the transitory component.

Appendix D. Section 4.1: SV-IMA model estimates This appendix illustrates the implementation of the SV-IMA model and provides some results. The SV-IMA model is given by the following equations:

pt ¼ pt þ gt gt  Nð0; r2g;t Þ ptþ1 ¼ pt þ e1;t e1;t  Nð0; r2e1;t Þ log r2g;t ¼ log r2g;t1 þ fg;t log r2e1;t ¼ log r2e1;t1 þ fe;t  ft ¼ fg;t

fe;t

0

ð18Þ ð19Þ ð20Þ ð21Þ

 Nð0; c2 I2 Þ:

To estimate the SV-IMA model, I apply the Gibbs sampler described in Appendix B leaving out Step 5 and with appropriate modifications to the state space. The left panel in Fig. D.1 depicts the variance of the permanent component r2g;t . There are persistent shocks to inflation because r2g;t is different from zero. Moreover, it can be shown that trend uncertainty represented by the variance of permanent shocks is time-varying. Trend uncertainty decreases since the late seventies in accordance with the ‘‘Great Moderation’’ in inflation and slowly picks up towards the end of the sample. The variance of the temporary component r2e1;t is depicted in the right part of Fig. D.1. Overall, trend shocks gt become less and less important relative to shocks to the temporary component e1;t .

Appendix E. Section 4.3: Alternative ordering of variables The following impulse response functions are derived from a bivariate VAR where K t is ordered first and Kt is ordered last. (see Fig. E.1)

1

1

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32

Fig. E.1. Impulse response of a shock to K t (inverse ordering). Note: The bold black line shows Cholesky impulse responses of Kt for one-quarter-ahead expectations (left) and 1-year-ahead expectations (right) to a shock to the Kalman gain K t . The dashed line is the response of K t to the same shock. The shaded areas are 68% and 95% confidence regions obtained from a bias adjusted bootstrap (Kilian, 1998).

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