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Testing short-run neutrality
Journal of Monetary Economics 17 (1986) 409-423. North-Holland
TESTING
SHORT-RUN
NEUTRALITY
Stephen G. CECCHE’ITI Graduate
School
of Business Administration, New York, NY 10006,
New USA
* York
Uniuersity,
This paper derives and implements a robust test of the short-run neutrality of anticipated policy that is valid when the information set used to form expectations is incorrectly specified and the effect of unanticipated policy on deviations of output from the natural rate varies over time. The test is based on the observation that if unanticipated policy affects output with a maximum lag of it quarters, then any information prior to n should be uninformative about output movements. The test is applied to quarterly data for the United States and the paper concludes that policy is not neutral.
1. Introduction
Typically when testing the proposition that output movements are unaffected by anticipated changes in aggregate demand policy variables such as the stock of money, econometricians are forced to specify an information set on which to base estimates of these anticipations. However, if the information set used by agents in the economy to form their expectations differs from the one specified by the econometrician in executing the test, it has been shown that standard procedures will yield inconsistent estimates of the parameters of interest invalidating the subsequent test of neutrality.’ The purpose of this paper is to derive and implement a test of short-run neutrality that is valid under very general conditions, including the possible misspecification of the information set. The test has been applied to quarterly data for the United States and the results indicate that anticipated aggregate demand policy variables help predict deviations of output from the natural rate. Barro (1977,1978), Barro and Rush (1980), and Misbkin (1983) all test for the neutrality of money by examining whether anticipated money is correlated *I owe a large debt to the work of Roman Frydman and Peter Rappoport as well as a suggestion by Larry Christiano. Conversations with Bob Cumby and Rick Mishkin led to some of the main results in the paper. John Boschen, George Sotianos. Paul Wachtel and an anonymous referee provided helpful comments on an earlier draft. As usual any remaining errors are my responsibility. ‘McCallum (1979). Germany and Srivastava (1979). Wickens (1982). Mishkin (1983). and Hoffman, Low and Schlagenhauf (1984) all discuss this problem. Frydman amd Rappoport (1984) provide a proof in the most general case. 0304-3923/86/$3.5001986,
Elsevier Science Publishers B.V. (North-Holland)
410
S. G. Cecchetti,
Testing
short-run
neutrality
with output fluctuations once the effect of unanticipated money has been taken into account. Mishkin finds evidence against the neutrality proposition, while the Barro and Barro and Rush papers support it. The test procedures used by these authors suffer from two flaws: They assume both that the econometrician uses the correct information set in estimatizg anticipated money and that the coefficients on unanticipated money in the output equation are constant over time. The first of these assumptions is uLkely to hold in practice, while the second is contradicted by the theory. With this in mind, it is useful to derive a test for the neutrality of money that is valid under more general conditions. The test is based on the following simple observat:on. If unanticipated policy affects output with a maximum lag of n quarters, then the mathematical expectation of the level of output conditional on any information prior to n should be the natural rate. If this is not true, then there is evidence against neutrality. The remainder of this paper first presents the Barro-Mishkin framework for testing the neutrality of anticipated money, describes the problems created by relaxing their restrictive assumptions, and derives two techniques that yield valid tests in the more general environment. Then the tests are implemented. Included are results for tests of the effect of anticipated money on output and anticipated inflation on unemployment. The paper concludes that there is very little evidence in favor of short-run neutrality. 2. A test of neutrality
This section begins with a description of the standard procedure for testing neutrality along with a brief discussion of how it is suspect on statistical grounds when either of two crucial assumptions are relaxed. This is followed by the derivation of two test procedures both of which remain valid when the effect of unanticipated money on output varies over time and the econometrician has only a portion of the information set actually used by agents in the economy to form expectations. The standard neutrality test. The standard approach for examining the effect of anticipated money on output deviations from the natural rate used by Barro (1977,1978) and Mishkin (1983) is based on
+ 5 SiE(AM,-iIfi,-i-1) + 5,* i-0
(1)
S.G. Cecchetti,
Testing
short-run
neutrality
411
where y, and J, are the logs of the actual level of output and the natural rate of output at time I, AM, is the first difference in the log of the money stock at time t, ai are parameters, and E is the expectation operator. /3,.(r) is the elasticity of output deviations with respect to unanticipated money and may change over time as the variance of the money stock process relative to the variance of the shocks to market specific supply and demand varies [see Lucas (1973)]. 9,-i is the information set used by agents in the economy to form expectations of variables at time t, and 5, is an additive error with expectation zero orthogonal to AM, and 9,-i. Finally, n *and n’ in eq. (1) are the maximum lag with which unanticipated money and anticipated money affect output deviations, respectively. For convenience assume n’ I n.2 De&ring 6 as the vector of Si’s, the standard test of neutrality is performed by examining H,: 6=0. Implementation of the neutrality test requires a proxy for expected money. To proceed, first assume that agents in the economy form expectations rationally such that E(AM,]B,-,)
= AM, - m,,
(2)
where the expectation error m, is serially uncorrelated with expectation zero and is orthogonal to 52,-i. Not knowing the true information set, the econometrician measures expectations from a linear projection of Z,-, on AM, yielding Z,- iy( L), where y(L) is a polynomial in the lag operator L. If either Z,- i does not mimic O,- i exactly or agents use a non-linear method for formulating their expectations, the true and measured expectations will not be equal. Define the error in measurement as u, = E(AM,P,-,)
- Z,-,Y(L)>
(3)
where u, is the projection of AM, on the portion of 62,-i orthogonal to Z,-,. Simple substitution yields the following two-equation system analogous to the one studied by Mishkin:
Y,=J,+5 SiZ,-i-Iv(L)+ i Bi(r)(AM,-i-Zt-i-ty(L)) i-0
i-0
i-0
i-0
and (4b) “As is emphasized below, in this context there is no theoretical basis for the choice of a particular n, but setting n to be too small leads to inconsistency.
412
S.G. Cecchetti,
Testing
short-run
tteutralip
If Pi(t) does not vary over time and 2,-i = a,-, so u, = 0, then under circumstances described by Mishkin alI of the parameters in (4) are identified. If these conditions are met, simultaneous estimation of the two-equation system provides a test of neutrality.3 But as is well known, if either &(t) is time varying or u, is not zero, the standard procedure yields inconsistent estimates of the parameters of interest. The difficulty emphasized in earlier work and discussed in Frydman and Rappoport (1984) is that the composite error including lagged uI’s is not orthogonal to the lagged AM,%. The problems just described can be resolved in a way analogous to the one used by Abel and Mishkin (1983) to derive a solution for the case with no lags (i.e., n = 0). First substitute (2) into (1) and impose the null hypothesis that all the 6’s are zero. This yields A general solution.
y,=Yt+
i
(5)
PitfJmt-i+5t.
i-0
Taking expectations of (5) conditional on information at (t - n - l), it immediately follows that if anticipated money is neutral, E(y, - jj,,,I52,-,-i) = 0. The orthogonality of output deviations to the lagged information set means that ( y, -J,) must be orthogonal to any elements of 9,-,,- i taken individually. So for Z, a subset of a,, E(y, - jj,,lZ,-,,-,) = 0 as well. Because D is unknown, the true expectation error m, in (5) cannot be identified and the p’s cannot be estimated. But consider the following regression: y, =J, + q-,-l+)
+ E,,
(6)
where &,= i
fli(r)m,-i+5,*
i-0
Under the null hypothesis, Z,-,,-i is orthogonal to the composite error E, and the coefficients in (Y(L) can be estimated by ordinary least squares. Once a consistent estimate of the covariance matrix of these coefficient estimates is obtained, a test of whether they are all simultaneously zero is a test of neutrality. This analysis .shows that a valid test of neutrality can be constructed by regressing any lagged elements of the information set on deviations of output from the natural rate. Unfortunately, doing this blindly is unlikely to yield robust or convincing results. A problem of statistical power arises for the ‘Pagan (1984) discusses many of the problems associated with efficient estimation and testing of a system like (4).
S.G. Ceccherti,
Testing
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neutrality
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following reason. Poor choice of a set of Z’s may lead the estimates of the parameters of interest, the CY’Sin (6), to be inconsistent under the alternative hypothesis of nonneutrality. While estimates will always be consistent under the null hypothesis, it may be difficult to spot rejections even asymptotically. For this reason it is useful to construct test procedures that yield consistent estimates of parameters like the (Y’Seven when the 6’s are not zero. Two procedures will be described below. Both are based on the result just described. The first, based on a vector autoregression (and termed the VAR procedure), allows estimation of linear combinations of the 6’s and places mild restrictions on the information set used, the Z’s. The second and simpler of the two, is an instrumental variables (IV) procedure that allows direct estimation of the 6’s. Since the two procedures have different advantages and disadvantages, they will both be presented. The VAR procedure. To help understand how this test is constructed, consider a simple example where the econometrician assumes that AM, follows a first-order autoregressive process. 4 This means that Z, includes only AM, and y(L) is of order zero. The projection equation used to estimate expected money, the analog to (4b), is then
AM,=y,AM,-,+w,.
(7)
Taking n’ = n, the output equation is y,=Jt+
i SiylAM,-,-, i-o
+ Y,,
(8)
where
i-0
i-o
As was the case in (6), rn, and /3 become part of the composite error v,. For the purposes of the example assume w, is white noise. This is unlikely to be true in practice, but the assumption is made to simplify the example. The general form of the test uses an assumption of this form. The first-order AR process (7) can always be rewritten so that AM, is a function of money growth at some arbitrary lag, AM,-,, and the intervening errors, (w,, . . . , w,-,). This can be used to eliminate all terms in AM dated between (t - 1) and (t - n) in eq. (8). These are the terms in AM that are correlated with the error v. Successive substitution into the output equation (8) 4Both the example and the general case for the VAR method use a technique described in Flavin (1984). The example was suggested by Larry Christiano.
S.G.
414
Cecchetri,
Testing
short-run
ueurrality
yields
(9)
y,=j,+blAM,-,,-,+v:, where nb,
=
c
1 6,(y,)“-‘+a,,
i-0 n-i-l
n-l
V:=
C i-0
6i (
C j-0
+
(Yl)jw,-i-j-l
v,*
1
Eq. (9) is constructed so the error v: is orthogonal to the right-hand side variable AMt-,-l, E(AM,-,-,v:) = 0. To see this notice that v;” is the sum of white noise innovations in the money growth process, w,‘s, and serially uncorrelated expectation errors, m,‘s, dated (t - n) to t. By construction E(w,~AM,-~) = 0 for i > 0. Since E( m,la,- r) = 0 by assumption, E(m,lAM,-,) = 0 as well, and v: is orthogonal to AM,-,. This means that b, can be estimated by ordinary least squares. Since b, is a linear combination of the S’s, a test that rejects H,: b, = 0 rejects H,: 6 = 0 as well. Note that if w, is not white noise this estimate of b, is inconsistent under the alternative hypothesis that the 6’s differ from zero. The example can be extended to the case where the econometrician specifies a general information set 2, including AM,. Assume that Z, can be described by a vector autoregression (VAR)
W)z,=
e,,
00)
whose first row is given by the money growth equation (4b), so e,, = v, + m,. A(L) is a matrix of k&order lag polynomials, and e, is a vector of mutually independent and serially uncorrelated white noise innovations; E(ei,ejs) = 0 for all i +i and s # t. This condition implies that e, is orthogonal to all Z’s dated prior to t. To see why, assume A(L) is invertible and rewrite (10) as a moving average Z,=A-‘(L)e,. Then E(Z,+e,) = E(A-‘(L)e,-,e,). This is zero when all cross-correlations E(ei,ejs) are zero. The assumption imposes testable restrictions on the list of variables in Z and the length of their maximum lag k in the projection equation. A procedure for checking that the condition holds will be suggested below. Just as the univariate expression for money growth (9) in the example can be transformed, it is possible to rewrite the VAR eliminating an arbitrary number of lags in the Z’s and replacing them with linear combinations of the
S.G.
innovations.
Cecchetti,
Testing
short-run
neutrality
415
It is straight forward to show that (10) can be rewritten as
Z,=B(L)Z,-,-,+C(L)e,,
(11)
where B(L) is a matrix of (k - l)-order lag polynomials and C(L) is a matrix of s-order lag polynomials. Eq. (11) implies that the money growth equation (4b) can be rewritten as a linear function of money growth and the other Z’s all dated (t - s - 1) to (t - s - k) plus a linear function of all the e’s dated (t - s) to t. Following the example, use (11) to substitute for all the AM’s dated (t - 1) to (t - n) in the output equation. Again taking n’ = n, write the output equation as y,=J,+
i
Zr-n-idi+ii,S
(12)
i-l
where
fjr= 2 e,-ifi+ i (fii(t)-6i)m,-i+ i-0
i si”t-i+5,v i-0
i-0
In the expression (12) the di’s and fi’s are row vectors of coefficients conformable to Z, and e,. Each individual coefficient in the d’s is a complex linear combination of the 6’s such that if all the d’s are non-zero, then all the 6’s cannot be zero. As was the case in the example, (12) is constructed such that the error !j, is orthogonal to Z,-a-i for i > 0, even when the 6’s are not zero. This implies that consistent estimates of the d’s can be obtained from ordinary least squares, and a test of whether they are all simultaneously zero constitutes a test of neutrality. This is the VAR procedure. It is a test for the predictability of deviations in output from the natural rate. To derive the instrumental variables procedure consider estimation of the output equation as a simple errorsin-variables problem. First substitute the definition of money expectations (2) into (1) to obtain The instrumental
y,=J,+
variables procedure.
i
6iAM,-i+u,,
(13)
i-0
where U,=
i i-0
fii(t)m,-i+
2 i-0
6iM,-i+[r.
416
S.G. Ceccherri,
Testing
short-run
neufrali!v
Again m, and fl are part of the error. Clearly, AM,- i is correlated with the error u, since the latter is made up of expectation errors. But estimation of (13) can be performed directly if a set instruments can be found that is both correlated with AM but uncorrelated with u,. Z,-,,-i is the best candidate. Rational expectations imply that if Z is a subset of the full information set s2, then E(Z ,-“- im,-i) = 0 for all i I n, so E(Z ,-,,- ru,) = 0. This allows direct estimation of the 6’s and a test of whether they are all simultaneously zero follows immediately. This procedure is more general than the VAR procedure since it places no preconditions on the measurement errors. But this lack of structure along with the fact that the Z’s must be highly correlated with money growth at potentially long lags suggests that the test will have lower power.’ Inference. To enable inference in both the IV and the VAR procedures, an estimate of the covariance matrix of the coefficient estimates is needed. In both cases, the composite error term in the equation used to perform the test contains weighted sums of expectations errors and errors generated from using the wrong information set as well as the additive error 5,. If m, and u, are a joint stationary time series process with zero mean and zero contemporaneous covariance, it immediately follows that U, and ii, are at a minimum n&order moving averages. In fact the composite errors have non-zero autocovariances beyond n as long as E(u,m,-,) is non-zero for i > 0. Beyond the serial correlation, U, and 5, are conditionally heteroskedastic due both to the time variation in &(t ) and as a consequence of possible conditional heteroskedastio ity in m, and u,. Hansen (1982) has suggested an estimator for the covariance matrix of coefficient estimates obtained from a regression with errors that are conditionally heteroskedastic and serially correlated of unknown degree. The estimator is obtained by computing the spectral density matrix evaluated at the-zero frequency of the time series process formed by taking the product of the estimated regression residuals and the right-hand side variables. Once this estimate is obtained, a simple Wald test for whether the parameter estimates, either the 6’s for the IV test or the d’s for the VAR test, differ from zero constitutes a test of neutrality. The contribution of both the IV and the VAR tests is that they allow rejection of neutrality under the maintained hypothesis that the econometriSA simple variant of the IV procedure is to project AM, on Z,-,,-r and then use lags of the fitted values as measures of expected money in the output equation. This procedure yields an expression exactly like eq. (6) and results in a valid test. But the estimates of the 6’s in this case are consistent under the alternative hypothesis only in the unlikely event that the measurement errors are orthogonal to past Z’s. Among other things, this would rule out the possibility of serial correlation in the ok. Results using this procedure are qualitatively the same as those included and are omitted to conserve space.
S. G. Cecchetti,
Testing
short-run
neutrality
417
cian’s information set differs from that of the agents in the economy and that the affect of unanticipated money on output deviations from the natural rate varies over time in an unspecified way. In addition, the two tests procedures provide estimates of the parameters of interest that are ‘consistent under the alternative hypothesis of non-neutrality. 3. Applications
of the test
The two test procedures are employed to examine the short-run effects of money on output and inflation on unemployment. Before either can be implemented a number of choices must be made. First a vector of variables known to be predetermined but correlated with money growth, Z, must be chosen along with their maximum lag length k in the money growth equation. Then some proxy for 9, the natural rate of output, must be specified. Both tests call for choice of a value for n, the maximum lag with which unanticipated money influences output. In addition, the IV procedure requires choice of n’, the maximum lag of anticipated money in the output equation, and the VAR procedure necessitates testing that the Z’s meet preconditions described in the previous section. The choice of the Z’s is motivated by the earlier literature. For the VAR tests of the neutrality of money, three sets of Z’s are used: (1) the first difference in the log of MI (AM) alone with maximum lag of eight quarters (k = 8); (2) AM and the average of the three-month Treasury bill rate (TB), with k = 4; and (3) the information set used by Barr0 and Barr0 and Rush including AM, the total labor force unemployment rate that includes the military (UNM) and a constructed government expenditure variable FEDV [see Barro (1977) for details] with k = 2. In addition to these three, some of the IV tests of the neutrality of money use the Mishkin information set including AM, TB and the high employment surplus (HES) with k = 4. All tests of the short-run effect of inflation on output use an information set containing either inflation in the implicit GNP deflator (AP) alone with k = 8, or A P and TB with k = 4. For both tests the natural rate of output is proxied by a constant plus either a simple time trend (T) or the spline &led middle-expansion-trend GNP (YM) computed by de Leeuw and Holloway (1983). Following Mishkin, the natural rate of unemployment is specified as a constant plus a time trend. It is important to realize that the choice of a measure of J is not very crucial because of the way the test statistic is constructed. From the derivation of the test in the previous section it is clear the 4, the additive error in the original output equation, can be a moving average of order n. This may be a result of the misspecification of j. The structure of both tests requires assuming that unanticipated money does not affect output at a lag beyond n. There is no a priori reason to choose any
418
S.G. Cecchetti,
Testing
short-run
neutrality
particular n. In fact the test is invalid if the choice of n is smaller than the true value.” This suggests computation of the test statistic for different values of n. As the test rejects for successively larger n, it becomes increasingly dillicult to believe that only unanticipated money matters. Finally, the VAR procedure requires that an autoregressive system constructed from the information set yield errors that are mutually independent and serially uncorrelated. To ensure that these conditions are met, equationby-equation ord+ry least squares was used to obtain an estimate of A(L) in eq. (10). From A(L) an estimate of the innovations 2 follows. These are then tested to determine if the preconditions are met. The test has two stages. First a Box-Pierce test is used to examine if the residuals from each equation are white noise. Then a test suggested by Haugh (1976) is employed fo test if the innovations in different equations of the VAR are pairwise independent. Define B,.,(s) as the sample correlation between gi, and gj,,-,. The Haugh test for the independence of e, and ej is based on the sum from -M to +M of T{ &.j(~)2}. This statistic is distributed as x2 with (2M+ 1) degrees of freedom.’ Failure to reject by all of the Box-Pierce and Haugh tests simultaneously is taken as an indication that an information set can be employed in the VAR technique. Table 1 reports the results of the series of tests performed on the five information sets used. The null hypothesis is either an individual series is white noise or that two series are independent. The sample period for all tests is 1957 : 01 to 1984 : 03. Box-Pierce statistics, Q, use twenty-eight autocorrelations and Haugh statistics, H, use the contemporaneous plus twenty-eight iads and lags for a total of fifty-seven 6’s. The marginal significance of one of the Q statistics is near the 90% level and another is just below the 75% level, while all the others are near or below .the 50% level. These information sets are assumed to meet the conditions required for the VAR test. It is interesting to note that the information set including AM, TB and HES used by Mishkin fails to pass the required tests. For k = 4, the residuals of the HES equation are not white noise. The IV tests employ n’, the maximum lag of anticipated money or inflation, of three and seven quarters. Following Mishkin, tests are also reported for n’ = 20, where the S’s are assumed to follow a fourth-order polynomial distributed lag (PDL). For the VAR technique, n’ = n. Results are reported for every fourth value of n. All IV estimates employ the Cumby, Huizinga and ‘If the true n is n, and the investigator chooses x2 < n,, then the error in the output equation includes omitted m,‘s from (t - nt) to (t - n,). These are correlated with AM,, so the parameters of interest cannot be estimated consistently. This is true even if 2, = Q, and /l,(t) is timeinvariant. When n2 > n1 there is no problem. ‘Both the Box-Pierce and Haugh test require that the series in question, here the Z’s, meet certain regularity conditions, the most important is that they be weak covariance stationary and invertible.
S.G. Cecchetti, Testing short-run neutrality
419
Table 1 Testing information sets8 Information (1) (II) (III)
(Iv) 07
set
k=8 AM AM k=4 TB AM k=2 UNM FED V k=8 AP k=4 AP TB
Box-Pierce statistic Q (28) 14.3 22.9 27.4 36.0 27.3 27.6 13.8 20.7 32.3
Hat@ statistic H (57) AM, TB = 53.9 AM, l/NM = 38.1 A M, FED V = 32.1 UNM, FEDV = 29.6 A P, TB = 36.8
“All statistics are computed from the residuals of a VAR on the information set with maximum lag k. See the text for a complete description of the tests. All Q statistics are x&, all H statistics are &. A x& lies above 32.6 with 25% probability and above 37.9 with 10% probability.
Obstfeld (1983) two-step two-stage least squares procedure. Computation of the covariance matrix uses the frequency domain procedure in the Cumby and Huizinga (1984) ‘Two Step Two Stage Least Squares Program’. With the exception of the PDL tests, all estimates of the covariance matrix of the coefficient estimates are based on averaging six sample periodogram ordinates on either side of the zero frequency using an inverted ‘V’ window of the matrix formed by the product of the residuals and the right-hand side variables. The PDL tests average seven ordinates. The number of ordinates averaged was increased until the test statistics stabilized.* Each column of table 2 reports the results of a series of neutrality tests for various specifications of y, J, Z and k using both the IV and VAR procedures. For each case a Wald test to determine if the parameters of interest are all zero simultaneously is performed and the resulting statistic is reported. Tests examine both the effects of money on output and inflation on unemployment. g Output is measured as the log of real GNP and tmemployment is the log(UN/(l - UN)), w h ere UN is the civilian labor force unem‘In addition to choosing the number of ordinates to average, the frequency domain procedure allows the investigator to pre-whiten the cross-product series prior to computation of the sample periodogram. While this does not change the asymptotic consistency of the covariance matrix estimator, in finite samples it often changes the results. Substantial experimentation with both pre-whitening and changing the number of ordinates averaged did not change the general character of the results reported below. ‘While the IV procedure allows direct estimation of the S’s they are not reported since they have no economic interpretation under the alternative hypothesis. This is both because of the lack of a structural model for how anticipated money affects output and because of the fact that under rational expectations the 6’s are likely to change with changes in the monetary regime.
420
S.G. Cecchetti,
Testing Table
Tests of short-run n
(1)
(2)
4 12 16 20 24 28 32
177.2 122.3 16.4 10.2 12.5 19.8 82.2 -
98.8 45.3 27.2 23.5 26.1 21.0 204.9 -
d.f.
a
a
10.6 152.6 7.4 13.6 3.9 16.2 18.4 -
28.2 3.3 14.7 12.7 25.9 7.1 9.4 -
8 12 16 20 24 28 32
0.2 3.3 0.2 1.7 10.3 22.1 -
0.2 12.5 1.8 0.2 14.2 7.5 -
neutrality.a (5)
(6)
(7)
466.3 132.3 38.6 26.9 41.5 24.6 60.7 25.9
93.5 48.7 22.3 28.0 11.9 10.5 12.7 11.7
66.3 207.1 160.8 113.8 308.7 192.9 185.4 -
135.5 842.7 51.2 65.5 43.1 183.0 130.8 74.8
a
6
8
a
7.9 16.1 17.5 29.1 6.2 1.5 12.4 27.3
6.8 13.3 4.5 9.4 4.8 31.1 22.7 -
48.4 905.7 14.6 24.1 1.5 5.7 9.8 26.5
49.7 4.6 1.5 6.8 .2.5 12.3 34.0
-
0.0 0.6 0.0 2.6 19.0 43.4 -
114.2 12.9 0.1 1.2 2.3 5.2 5.7
GNP YM 4 M, TB
GNP T 2 M, FEDV, LUNM
UN T 8 P
UN T 4 P,TB
(4) ( n’ = n )
400.7 674.8 73.0 10.9 13.9 16.1 33.3 63.5 8 IV estimates
4 8 12 16 20 24 28 32
neutrality
2
(3) VA R esfimates
a
short-run
(n’ = 3) xi
32.3 13.1
40.5 18.8
1.5 7.4
8:;
7.7 6.3 10.9 6.2
379.: 19:1 58.6
IV estimates
(n’ = 7) xi
4.7 0.1 0.4 2.3 2.6 11.4 14.0 Test conditions
Y z Z
“Critical q = 8,15.5.
T 8 M
GNP YM a M
GNP T 4 M, TB
values at the 5% level for the x: distribution See text for description of procedures.
are for q = 4.9.5,
for q = 6,12.6,
and for
ployment rate. For the IV test with n’ = 7, Barr03 information set with k = 2 cannot be used since it has only six instruments and a minimum of eight are required. As is expected, the IV tests reject less frequently than the VAR tests. The data contain enough information to reject neutrality using the VAR test but not enough information to determine the mechanism as is required by the IV test. The pattern of the results is interesting. In all cases rejection is more likely when n is either small or large. For smaIl n, specification error is likely
S.G. Cecchetti,
Testing
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421
neutrality
Table 3 Tests of short-run neutrality? Barro Z’s
Mishkin Z’s (AM. TB. HES)
n IV estimates,
(A M. f/NM,
fourth-orderpolynomial
distributed
FED V)
lags (n’ = 20) xi
20 24 28 32
110.2 153.7 17.0 48.1
35.1 21.8 140.2 103.9
53.4 24.0 87.4 213.7
18.5 86.7 14.0 225.8
3
T
YIU
T
YM
BCritical value at the 5% level for xi is 11.1. See text for description of procedure.
to be causing the rejection. At n = 4, it is more likely that unanticipated money at a lag of five quarters affects output than that money is non-neutral. This argues for concentrating on high values of n. The most persuasive rejections are by the VAR tests for n above twenty. Table 3 contains results using the IV procedure with n’ = 20 and the fourth-order PDL. Both the full Mishkin and Barro information sets. with k = 4 are used. While the results using the Barro information set with k = 2 reported in column 5 of table 2 fail to reject especially for large n, with k = 4 and n’ = 20 rejection is common. In fact, in every case but one all the tests with the Barre and Mishkin information sets in table 3 reject neutrality at the 1% level.‘O For each neutrality test reported above, a test of conditional heteroskedasticity was carried out by regressing the squared residuals for each equation on a constant, J and the particular set of Z’s. A Wald test was performed to test if all the coefficients excluding the constant differed from zero simultaneously. These tests also use the covariance matrix estimator constructed using the frequency domain procedure. Of the total of 162 tests, a total of 24 failed to reject at the 1% level, 13 of which also failed to reject at the 5% level. This is very strong evidence against homoskedasticity and suggests the need for the robust method of computing the covariance matrix estimates. Taken together these results imply that unless unanticipated movements in aggregate demand variables affect output deviations at a lag of eight years or more, anticipated policy matters. Given how implausible it is that unantic‘“Stulz and Wasserfallen (1985) suggest that output is a stochastic, not a deterministic, trend. This implies that the natural rate of output should be specified as J, = a +Y,-~ + E,. Then the appropriate dependent variable in the regressions to test neutrality is the first difference in the log of output. Experimentation with this specification yielded results nearly identical to those reported in tables 2 and 3.
422
S.G.
Cecchetti,
Testing
short-run
neutrality
ipated events have such lasting effects, this is strong evidence against the neutrality postulate of the new classical macroeconomics.” In light of these results, it is likely that Barro’s failure to reject neutrality can be traced to a combination of three factors: (1) improper specification of the information set on which to base forecasts of policy,‘* (2) failure to account for the time-varying nature of the elasticity of output with respect to unanticipated policy, and (3) choice of too small a maximum lag length for either anticipated or unanticipated money or both. The results in table 3 for n’ = 20 and n 2 20 suggest that Barro’s information set is not primarily at fault, instead some combination of the second and third possibilities is responsible. 4. Conclusion This paper has derived and implemented a test of the short-run neutrality of anticipated aggregate demand policy. The test is valid even when the effect of unanticipated policy on output varies over time in an unknown way and the econometrician does not correctly specify the information set relevant for forecasting policy. The test is used to examine whether either anticipated money or anticipated inflation matter. The conclusion is that they do. The generality of the economic and statistical model underlying the test makes clear that there is no credible empirical evidence for policy neutrality. This rejection must lead one to question the basis of new classical macroeconomic models. References Abel, Andrew B. and Frederic S. Mist&in, 1983, Ao integrated view of tests of rationality, market efficiency and short-run neutrality of monetary policy, Journal of Monetary Economics 11, 3-24. Barre, Robert J., 1977, Unanticipated money growth and unemployment in the United States, American Economic Review 67,101-115. Barre, Robert J., 1978, Unanticipated money, output and the price level, Journal of Political Economy 86,549-580. Barre, Robert J. and Mark Rush, 1980, Unanticipated money and economic activity, in: S. Fischer, ed., Rational expectations and economic policy (University of Chicago Press, cbieago, IL). Cumby, Robert E., John Huizinga and Maurice Obstfeld, 1983, Two-step two-step least squares estimation in models with rational expectations, Journal of Econometrics 21, 333-355. “It remains possible that these results are the consequence of misspeciftcation of the natural rate of output. If, for example, anticipated policy were to affect j, then it would not be surprising that these tests would reject neutrality. But without a structural model for the determination of .F there is very little that can be done about this. 12Germany and Srivastava show that use of the wrong information set can bias the estimates of the relevant parameters toward zero, increasing the probability of false acceptance.
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Testing
short-run
neutrality
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Cumby. Robert E. and John Huizinga, 1984, Two-step two-stage least squares user’s guide, Version 2.1. Mimeo. (New York Universitv. Graduate School of Business. New York). de Leeuw, Frank and Thomas M. Holloway, i983. Cyclical adjustments of the Federal budget and Federal debt, Survey of Current Business 63, 25-40. FIavin, Marjorie, 1984, Excess sensitivity of consumption to current income: Liquidity constraints or myopia?, Canadian Journal of Economics 18,117-136. Frydman. Roman and Peter Rappoport, 1984, The effects of mismeasurement of rational expectations on empirical evidence concerning the central propositions of the new classical macroeconomics, Mimeo. (New York University, Department of Economics, New York). Germany, J. David and Sanjay Srivastava, 1979, Empirical estimates of unanticipated pohcy: Issues in stability and identification, Mimeo. (M.I.T., Cambridge, MA). Hansen, Lam Peter, 1982, Large sample properties of generalized method of moments estimators, Econometrica 50,1029-1054. Hauah. Larry D.. 1976, Checking the independence of stationary time series: A univariate residual c&s-correlation approach, J&maI of ihe American Statisti&.l Association 71, 378-385. Hoffman. Dennis L.. Stuart A. Low and Don E. Schlaeenhauf. 1984. Tests of rationalitv.<. neutrahtv and market efficiency, Journal of Monetary Econo&ics 14; 339-363. Lucas, Robert E. Jr., 1973, Some international evidence on output inflation tradeoffs, American Economic Review 63,326-334. McCaIlum, Bennett T., 1979, Topics concerning the formulation, estimation and use of macroeconometric models with rational expectations, American Statistical Association Proceedings of the Business and Economic Statistics Section, 65-72. Mishkin. Frederic S., 1983, A rational expectations approach to macroeconometrics (University of Chicago Press, Chicago, IL). Pagan. Adrian, 1984, Econometric issues in the analysis of regressions with generated regressors, International Economic Review 25,221-247. Stub, Rene M. and Walter Wasserfallen, 1985, Macroeconomic time-series, business cycles and macroeconomic policies, Carnegie-Rochester Conference Series on Public Policy 22,9-53. White, Halbert, 1980, A heteroskedasticity-consistent covariance matrix estimator and a direct test of heteroskedasticity, Econometrica 48, 817-839. Wickens, M.R.. 1982, The efficient estimation of econometric models with rational expectations, Review of Economic Studies 49,55-67.
TESTING
SHORT-RUN
NEUTRALITY
Stephen G. CECCHE’ITI Graduate
School
of Business Administration, New York, NY 10006,
New USA
* York
Uniuersity,
This paper derives and implements a robust test of the short-run neutrality of anticipated policy that is valid when the information set used to form expectations is incorrectly specified and the effect of unanticipated policy on deviations of output from the natural rate varies over time. The test is based on the observation that if unanticipated policy affects output with a maximum lag of it quarters, then any information prior to n should be uninformative about output movements. The test is applied to quarterly data for the United States and the paper concludes that policy is not neutral.
1. Introduction
Typically when testing the proposition that output movements are unaffected by anticipated changes in aggregate demand policy variables such as the stock of money, econometricians are forced to specify an information set on which to base estimates of these anticipations. However, if the information set used by agents in the economy to form their expectations differs from the one specified by the econometrician in executing the test, it has been shown that standard procedures will yield inconsistent estimates of the parameters of interest invalidating the subsequent test of neutrality.’ The purpose of this paper is to derive and implement a test of short-run neutrality that is valid under very general conditions, including the possible misspecification of the information set. The test has been applied to quarterly data for the United States and the results indicate that anticipated aggregate demand policy variables help predict deviations of output from the natural rate. Barro (1977,1978), Barro and Rush (1980), and Misbkin (1983) all test for the neutrality of money by examining whether anticipated money is correlated *I owe a large debt to the work of Roman Frydman and Peter Rappoport as well as a suggestion by Larry Christiano. Conversations with Bob Cumby and Rick Mishkin led to some of the main results in the paper. John Boschen, George Sotianos. Paul Wachtel and an anonymous referee provided helpful comments on an earlier draft. As usual any remaining errors are my responsibility. ‘McCallum (1979). Germany and Srivastava (1979). Wickens (1982). Mishkin (1983). and Hoffman, Low and Schlagenhauf (1984) all discuss this problem. Frydman amd Rappoport (1984) provide a proof in the most general case. 0304-3923/86/$3.5001986,
Elsevier Science Publishers B.V. (North-Holland)
410
S. G. Cecchetti,
Testing
short-run
neutrality
with output fluctuations once the effect of unanticipated money has been taken into account. Mishkin finds evidence against the neutrality proposition, while the Barro and Barro and Rush papers support it. The test procedures used by these authors suffer from two flaws: They assume both that the econometrician uses the correct information set in estimatizg anticipated money and that the coefficients on unanticipated money in the output equation are constant over time. The first of these assumptions is uLkely to hold in practice, while the second is contradicted by the theory. With this in mind, it is useful to derive a test for the neutrality of money that is valid under more general conditions. The test is based on the following simple observat:on. If unanticipated policy affects output with a maximum lag of n quarters, then the mathematical expectation of the level of output conditional on any information prior to n should be the natural rate. If this is not true, then there is evidence against neutrality. The remainder of this paper first presents the Barro-Mishkin framework for testing the neutrality of anticipated money, describes the problems created by relaxing their restrictive assumptions, and derives two techniques that yield valid tests in the more general environment. Then the tests are implemented. Included are results for tests of the effect of anticipated money on output and anticipated inflation on unemployment. The paper concludes that there is very little evidence in favor of short-run neutrality. 2. A test of neutrality
This section begins with a description of the standard procedure for testing neutrality along with a brief discussion of how it is suspect on statistical grounds when either of two crucial assumptions are relaxed. This is followed by the derivation of two test procedures both of which remain valid when the effect of unanticipated money on output varies over time and the econometrician has only a portion of the information set actually used by agents in the economy to form expectations. The standard neutrality test. The standard approach for examining the effect of anticipated money on output deviations from the natural rate used by Barro (1977,1978) and Mishkin (1983) is based on
+ 5 SiE(AM,-iIfi,-i-1) + 5,* i-0
(1)
S.G. Cecchetti,
Testing
short-run
neutrality
411
where y, and J, are the logs of the actual level of output and the natural rate of output at time I, AM, is the first difference in the log of the money stock at time t, ai are parameters, and E is the expectation operator. /3,.(r) is the elasticity of output deviations with respect to unanticipated money and may change over time as the variance of the money stock process relative to the variance of the shocks to market specific supply and demand varies [see Lucas (1973)]. 9,-i is the information set used by agents in the economy to form expectations of variables at time t, and 5, is an additive error with expectation zero orthogonal to AM, and 9,-i. Finally, n *and n’ in eq. (1) are the maximum lag with which unanticipated money and anticipated money affect output deviations, respectively. For convenience assume n’ I n.2 De&ring 6 as the vector of Si’s, the standard test of neutrality is performed by examining H,: 6=0. Implementation of the neutrality test requires a proxy for expected money. To proceed, first assume that agents in the economy form expectations rationally such that E(AM,]B,-,)
= AM, - m,,
(2)
where the expectation error m, is serially uncorrelated with expectation zero and is orthogonal to 52,-i. Not knowing the true information set, the econometrician measures expectations from a linear projection of Z,-, on AM, yielding Z,- iy( L), where y(L) is a polynomial in the lag operator L. If either Z,- i does not mimic O,- i exactly or agents use a non-linear method for formulating their expectations, the true and measured expectations will not be equal. Define the error in measurement as u, = E(AM,P,-,)
- Z,-,Y(L)>
(3)
where u, is the projection of AM, on the portion of 62,-i orthogonal to Z,-,. Simple substitution yields the following two-equation system analogous to the one studied by Mishkin:
Y,=J,+5 SiZ,-i-Iv(L)+ i Bi(r)(AM,-i-Zt-i-ty(L)) i-0
i-0
i-0
i-0
and (4b) “As is emphasized below, in this context there is no theoretical basis for the choice of a particular n, but setting n to be too small leads to inconsistency.
412
S.G. Cecchetti,
Testing
short-run
tteutralip
If Pi(t) does not vary over time and 2,-i = a,-, so u, = 0, then under circumstances described by Mishkin alI of the parameters in (4) are identified. If these conditions are met, simultaneous estimation of the two-equation system provides a test of neutrality.3 But as is well known, if either &(t) is time varying or u, is not zero, the standard procedure yields inconsistent estimates of the parameters of interest. The difficulty emphasized in earlier work and discussed in Frydman and Rappoport (1984) is that the composite error including lagged uI’s is not orthogonal to the lagged AM,%. The problems just described can be resolved in a way analogous to the one used by Abel and Mishkin (1983) to derive a solution for the case with no lags (i.e., n = 0). First substitute (2) into (1) and impose the null hypothesis that all the 6’s are zero. This yields A general solution.
y,=Yt+
i
(5)
PitfJmt-i+5t.
i-0
Taking expectations of (5) conditional on information at (t - n - l), it immediately follows that if anticipated money is neutral, E(y, - jj,,,I52,-,-i) = 0. The orthogonality of output deviations to the lagged information set means that ( y, -J,) must be orthogonal to any elements of 9,-,,- i taken individually. So for Z, a subset of a,, E(y, - jj,,lZ,-,,-,) = 0 as well. Because D is unknown, the true expectation error m, in (5) cannot be identified and the p’s cannot be estimated. But consider the following regression: y, =J, + q-,-l+)
+ E,,
(6)
where &,= i
fli(r)m,-i+5,*
i-0
Under the null hypothesis, Z,-,,-i is orthogonal to the composite error E, and the coefficients in (Y(L) can be estimated by ordinary least squares. Once a consistent estimate of the covariance matrix of these coefficient estimates is obtained, a test of whether they are all simultaneously zero is a test of neutrality. This analysis .shows that a valid test of neutrality can be constructed by regressing any lagged elements of the information set on deviations of output from the natural rate. Unfortunately, doing this blindly is unlikely to yield robust or convincing results. A problem of statistical power arises for the ‘Pagan (1984) discusses many of the problems associated with efficient estimation and testing of a system like (4).
S.G. Ceccherti,
Testing
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neutrality
413
following reason. Poor choice of a set of Z’s may lead the estimates of the parameters of interest, the CY’Sin (6), to be inconsistent under the alternative hypothesis of nonneutrality. While estimates will always be consistent under the null hypothesis, it may be difficult to spot rejections even asymptotically. For this reason it is useful to construct test procedures that yield consistent estimates of parameters like the (Y’Seven when the 6’s are not zero. Two procedures will be described below. Both are based on the result just described. The first, based on a vector autoregression (and termed the VAR procedure), allows estimation of linear combinations of the 6’s and places mild restrictions on the information set used, the Z’s. The second and simpler of the two, is an instrumental variables (IV) procedure that allows direct estimation of the 6’s. Since the two procedures have different advantages and disadvantages, they will both be presented. The VAR procedure. To help understand how this test is constructed, consider a simple example where the econometrician assumes that AM, follows a first-order autoregressive process. 4 This means that Z, includes only AM, and y(L) is of order zero. The projection equation used to estimate expected money, the analog to (4b), is then
AM,=y,AM,-,+w,.
(7)
Taking n’ = n, the output equation is y,=Jt+
i SiylAM,-,-, i-o
+ Y,,
(8)
where
i-0
i-o
As was the case in (6), rn, and /3 become part of the composite error v,. For the purposes of the example assume w, is white noise. This is unlikely to be true in practice, but the assumption is made to simplify the example. The general form of the test uses an assumption of this form. The first-order AR process (7) can always be rewritten so that AM, is a function of money growth at some arbitrary lag, AM,-,, and the intervening errors, (w,, . . . , w,-,). This can be used to eliminate all terms in AM dated between (t - 1) and (t - n) in eq. (8). These are the terms in AM that are correlated with the error v. Successive substitution into the output equation (8) 4Both the example and the general case for the VAR method use a technique described in Flavin (1984). The example was suggested by Larry Christiano.
S.G.
414
Cecchetri,
Testing
short-run
ueurrality
yields
(9)
y,=j,+blAM,-,,-,+v:, where nb,
=
c
1 6,(y,)“-‘+a,,
i-0 n-i-l
n-l
V:=
C i-0
6i (
C j-0
+
(Yl)jw,-i-j-l
v,*
1
Eq. (9) is constructed so the error v: is orthogonal to the right-hand side variable AMt-,-l, E(AM,-,-,v:) = 0. To see this notice that v;” is the sum of white noise innovations in the money growth process, w,‘s, and serially uncorrelated expectation errors, m,‘s, dated (t - n) to t. By construction E(w,~AM,-~) = 0 for i > 0. Since E( m,la,- r) = 0 by assumption, E(m,lAM,-,) = 0 as well, and v: is orthogonal to AM,-,. This means that b, can be estimated by ordinary least squares. Since b, is a linear combination of the S’s, a test that rejects H,: b, = 0 rejects H,: 6 = 0 as well. Note that if w, is not white noise this estimate of b, is inconsistent under the alternative hypothesis that the 6’s differ from zero. The example can be extended to the case where the econometrician specifies a general information set 2, including AM,. Assume that Z, can be described by a vector autoregression (VAR)
W)z,=
e,,
00)
whose first row is given by the money growth equation (4b), so e,, = v, + m,. A(L) is a matrix of k&order lag polynomials, and e, is a vector of mutually independent and serially uncorrelated white noise innovations; E(ei,ejs) = 0 for all i +i and s # t. This condition implies that e, is orthogonal to all Z’s dated prior to t. To see why, assume A(L) is invertible and rewrite (10) as a moving average Z,=A-‘(L)e,. Then E(Z,+e,) = E(A-‘(L)e,-,e,). This is zero when all cross-correlations E(ei,ejs) are zero. The assumption imposes testable restrictions on the list of variables in Z and the length of their maximum lag k in the projection equation. A procedure for checking that the condition holds will be suggested below. Just as the univariate expression for money growth (9) in the example can be transformed, it is possible to rewrite the VAR eliminating an arbitrary number of lags in the Z’s and replacing them with linear combinations of the
S.G.
innovations.
Cecchetti,
Testing
short-run
neutrality
415
It is straight forward to show that (10) can be rewritten as
Z,=B(L)Z,-,-,+C(L)e,,
(11)
where B(L) is a matrix of (k - l)-order lag polynomials and C(L) is a matrix of s-order lag polynomials. Eq. (11) implies that the money growth equation (4b) can be rewritten as a linear function of money growth and the other Z’s all dated (t - s - 1) to (t - s - k) plus a linear function of all the e’s dated (t - s) to t. Following the example, use (11) to substitute for all the AM’s dated (t - 1) to (t - n) in the output equation. Again taking n’ = n, write the output equation as y,=J,+
i
Zr-n-idi+ii,S
(12)
i-l
where
fjr= 2 e,-ifi+ i (fii(t)-6i)m,-i+ i-0
i si”t-i+5,v i-0
i-0
In the expression (12) the di’s and fi’s are row vectors of coefficients conformable to Z, and e,. Each individual coefficient in the d’s is a complex linear combination of the 6’s such that if all the d’s are non-zero, then all the 6’s cannot be zero. As was the case in the example, (12) is constructed such that the error !j, is orthogonal to Z,-a-i for i > 0, even when the 6’s are not zero. This implies that consistent estimates of the d’s can be obtained from ordinary least squares, and a test of whether they are all simultaneously zero constitutes a test of neutrality. This is the VAR procedure. It is a test for the predictability of deviations in output from the natural rate. To derive the instrumental variables procedure consider estimation of the output equation as a simple errorsin-variables problem. First substitute the definition of money expectations (2) into (1) to obtain The instrumental
y,=J,+
variables procedure.
i
6iAM,-i+u,,
(13)
i-0
where U,=
i i-0
fii(t)m,-i+
2 i-0
6iM,-i+[r.
416
S.G. Ceccherri,
Testing
short-run
neufrali!v
Again m, and fl are part of the error. Clearly, AM,- i is correlated with the error u, since the latter is made up of expectation errors. But estimation of (13) can be performed directly if a set instruments can be found that is both correlated with AM but uncorrelated with u,. Z,-,,-i is the best candidate. Rational expectations imply that if Z is a subset of the full information set s2, then E(Z ,-“- im,-i) = 0 for all i I n, so E(Z ,-,,- ru,) = 0. This allows direct estimation of the 6’s and a test of whether they are all simultaneously zero follows immediately. This procedure is more general than the VAR procedure since it places no preconditions on the measurement errors. But this lack of structure along with the fact that the Z’s must be highly correlated with money growth at potentially long lags suggests that the test will have lower power.’ Inference. To enable inference in both the IV and the VAR procedures, an estimate of the covariance matrix of the coefficient estimates is needed. In both cases, the composite error term in the equation used to perform the test contains weighted sums of expectations errors and errors generated from using the wrong information set as well as the additive error 5,. If m, and u, are a joint stationary time series process with zero mean and zero contemporaneous covariance, it immediately follows that U, and ii, are at a minimum n&order moving averages. In fact the composite errors have non-zero autocovariances beyond n as long as E(u,m,-,) is non-zero for i > 0. Beyond the serial correlation, U, and 5, are conditionally heteroskedastic due both to the time variation in &(t ) and as a consequence of possible conditional heteroskedastio ity in m, and u,. Hansen (1982) has suggested an estimator for the covariance matrix of coefficient estimates obtained from a regression with errors that are conditionally heteroskedastic and serially correlated of unknown degree. The estimator is obtained by computing the spectral density matrix evaluated at the-zero frequency of the time series process formed by taking the product of the estimated regression residuals and the right-hand side variables. Once this estimate is obtained, a simple Wald test for whether the parameter estimates, either the 6’s for the IV test or the d’s for the VAR test, differ from zero constitutes a test of neutrality. The contribution of both the IV and the VAR tests is that they allow rejection of neutrality under the maintained hypothesis that the econometriSA simple variant of the IV procedure is to project AM, on Z,-,,-r and then use lags of the fitted values as measures of expected money in the output equation. This procedure yields an expression exactly like eq. (6) and results in a valid test. But the estimates of the 6’s in this case are consistent under the alternative hypothesis only in the unlikely event that the measurement errors are orthogonal to past Z’s. Among other things, this would rule out the possibility of serial correlation in the ok. Results using this procedure are qualitatively the same as those included and are omitted to conserve space.
S. G. Cecchetti,
Testing
short-run
neutrality
417
cian’s information set differs from that of the agents in the economy and that the affect of unanticipated money on output deviations from the natural rate varies over time in an unspecified way. In addition, the two tests procedures provide estimates of the parameters of interest that are ‘consistent under the alternative hypothesis of non-neutrality. 3. Applications
of the test
The two test procedures are employed to examine the short-run effects of money on output and inflation on unemployment. Before either can be implemented a number of choices must be made. First a vector of variables known to be predetermined but correlated with money growth, Z, must be chosen along with their maximum lag length k in the money growth equation. Then some proxy for 9, the natural rate of output, must be specified. Both tests call for choice of a value for n, the maximum lag with which unanticipated money influences output. In addition, the IV procedure requires choice of n’, the maximum lag of anticipated money in the output equation, and the VAR procedure necessitates testing that the Z’s meet preconditions described in the previous section. The choice of the Z’s is motivated by the earlier literature. For the VAR tests of the neutrality of money, three sets of Z’s are used: (1) the first difference in the log of MI (AM) alone with maximum lag of eight quarters (k = 8); (2) AM and the average of the three-month Treasury bill rate (TB), with k = 4; and (3) the information set used by Barr0 and Barr0 and Rush including AM, the total labor force unemployment rate that includes the military (UNM) and a constructed government expenditure variable FEDV [see Barro (1977) for details] with k = 2. In addition to these three, some of the IV tests of the neutrality of money use the Mishkin information set including AM, TB and the high employment surplus (HES) with k = 4. All tests of the short-run effect of inflation on output use an information set containing either inflation in the implicit GNP deflator (AP) alone with k = 8, or A P and TB with k = 4. For both tests the natural rate of output is proxied by a constant plus either a simple time trend (T) or the spline &led middle-expansion-trend GNP (YM) computed by de Leeuw and Holloway (1983). Following Mishkin, the natural rate of unemployment is specified as a constant plus a time trend. It is important to realize that the choice of a measure of J is not very crucial because of the way the test statistic is constructed. From the derivation of the test in the previous section it is clear the 4, the additive error in the original output equation, can be a moving average of order n. This may be a result of the misspecification of j. The structure of both tests requires assuming that unanticipated money does not affect output at a lag beyond n. There is no a priori reason to choose any
418
S.G. Cecchetti,
Testing
short-run
neutrality
particular n. In fact the test is invalid if the choice of n is smaller than the true value.” This suggests computation of the test statistic for different values of n. As the test rejects for successively larger n, it becomes increasingly dillicult to believe that only unanticipated money matters. Finally, the VAR procedure requires that an autoregressive system constructed from the information set yield errors that are mutually independent and serially uncorrelated. To ensure that these conditions are met, equationby-equation ord+ry least squares was used to obtain an estimate of A(L) in eq. (10). From A(L) an estimate of the innovations 2 follows. These are then tested to determine if the preconditions are met. The test has two stages. First a Box-Pierce test is used to examine if the residuals from each equation are white noise. Then a test suggested by Haugh (1976) is employed fo test if the innovations in different equations of the VAR are pairwise independent. Define B,.,(s) as the sample correlation between gi, and gj,,-,. The Haugh test for the independence of e, and ej is based on the sum from -M to +M of T{ &.j(~)2}. This statistic is distributed as x2 with (2M+ 1) degrees of freedom.’ Failure to reject by all of the Box-Pierce and Haugh tests simultaneously is taken as an indication that an information set can be employed in the VAR technique. Table 1 reports the results of the series of tests performed on the five information sets used. The null hypothesis is either an individual series is white noise or that two series are independent. The sample period for all tests is 1957 : 01 to 1984 : 03. Box-Pierce statistics, Q, use twenty-eight autocorrelations and Haugh statistics, H, use the contemporaneous plus twenty-eight iads and lags for a total of fifty-seven 6’s. The marginal significance of one of the Q statistics is near the 90% level and another is just below the 75% level, while all the others are near or below .the 50% level. These information sets are assumed to meet the conditions required for the VAR test. It is interesting to note that the information set including AM, TB and HES used by Mishkin fails to pass the required tests. For k = 4, the residuals of the HES equation are not white noise. The IV tests employ n’, the maximum lag of anticipated money or inflation, of three and seven quarters. Following Mishkin, tests are also reported for n’ = 20, where the S’s are assumed to follow a fourth-order polynomial distributed lag (PDL). For the VAR technique, n’ = n. Results are reported for every fourth value of n. All IV estimates employ the Cumby, Huizinga and ‘If the true n is n, and the investigator chooses x2 < n,, then the error in the output equation includes omitted m,‘s from (t - nt) to (t - n,). These are correlated with AM,, so the parameters of interest cannot be estimated consistently. This is true even if 2, = Q, and /l,(t) is timeinvariant. When n2 > n1 there is no problem. ‘Both the Box-Pierce and Haugh test require that the series in question, here the Z’s, meet certain regularity conditions, the most important is that they be weak covariance stationary and invertible.
S.G. Cecchetti, Testing short-run neutrality
419
Table 1 Testing information sets8 Information (1) (II) (III)
(Iv) 07
set
k=8 AM AM k=4 TB AM k=2 UNM FED V k=8 AP k=4 AP TB
Box-Pierce statistic Q (28) 14.3 22.9 27.4 36.0 27.3 27.6 13.8 20.7 32.3
Hat@ statistic H (57) AM, TB = 53.9 AM, l/NM = 38.1 A M, FED V = 32.1 UNM, FEDV = 29.6 A P, TB = 36.8
“All statistics are computed from the residuals of a VAR on the information set with maximum lag k. See the text for a complete description of the tests. All Q statistics are x&, all H statistics are &. A x& lies above 32.6 with 25% probability and above 37.9 with 10% probability.
Obstfeld (1983) two-step two-stage least squares procedure. Computation of the covariance matrix uses the frequency domain procedure in the Cumby and Huizinga (1984) ‘Two Step Two Stage Least Squares Program’. With the exception of the PDL tests, all estimates of the covariance matrix of the coefficient estimates are based on averaging six sample periodogram ordinates on either side of the zero frequency using an inverted ‘V’ window of the matrix formed by the product of the residuals and the right-hand side variables. The PDL tests average seven ordinates. The number of ordinates averaged was increased until the test statistics stabilized.* Each column of table 2 reports the results of a series of neutrality tests for various specifications of y, J, Z and k using both the IV and VAR procedures. For each case a Wald test to determine if the parameters of interest are all zero simultaneously is performed and the resulting statistic is reported. Tests examine both the effects of money on output and inflation on unemployment. g Output is measured as the log of real GNP and tmemployment is the log(UN/(l - UN)), w h ere UN is the civilian labor force unem‘In addition to choosing the number of ordinates to average, the frequency domain procedure allows the investigator to pre-whiten the cross-product series prior to computation of the sample periodogram. While this does not change the asymptotic consistency of the covariance matrix estimator, in finite samples it often changes the results. Substantial experimentation with both pre-whitening and changing the number of ordinates averaged did not change the general character of the results reported below. ‘While the IV procedure allows direct estimation of the S’s they are not reported since they have no economic interpretation under the alternative hypothesis. This is both because of the lack of a structural model for how anticipated money affects output and because of the fact that under rational expectations the 6’s are likely to change with changes in the monetary regime.
420
S.G. Cecchetti,
Testing Table
Tests of short-run n
(1)
(2)
4 12 16 20 24 28 32
177.2 122.3 16.4 10.2 12.5 19.8 82.2 -
98.8 45.3 27.2 23.5 26.1 21.0 204.9 -
d.f.
a
a
10.6 152.6 7.4 13.6 3.9 16.2 18.4 -
28.2 3.3 14.7 12.7 25.9 7.1 9.4 -
8 12 16 20 24 28 32
0.2 3.3 0.2 1.7 10.3 22.1 -
0.2 12.5 1.8 0.2 14.2 7.5 -
neutrality.a (5)
(6)
(7)
466.3 132.3 38.6 26.9 41.5 24.6 60.7 25.9
93.5 48.7 22.3 28.0 11.9 10.5 12.7 11.7
66.3 207.1 160.8 113.8 308.7 192.9 185.4 -
135.5 842.7 51.2 65.5 43.1 183.0 130.8 74.8
a
6
8
a
7.9 16.1 17.5 29.1 6.2 1.5 12.4 27.3
6.8 13.3 4.5 9.4 4.8 31.1 22.7 -
48.4 905.7 14.6 24.1 1.5 5.7 9.8 26.5
49.7 4.6 1.5 6.8 .2.5 12.3 34.0
-
0.0 0.6 0.0 2.6 19.0 43.4 -
114.2 12.9 0.1 1.2 2.3 5.2 5.7
GNP YM 4 M, TB
GNP T 2 M, FEDV, LUNM
UN T 8 P
UN T 4 P,TB
(4) ( n’ = n )
400.7 674.8 73.0 10.9 13.9 16.1 33.3 63.5 8 IV estimates
4 8 12 16 20 24 28 32
neutrality
2
(3) VA R esfimates
a
short-run
(n’ = 3) xi
32.3 13.1
40.5 18.8
1.5 7.4
8:;
7.7 6.3 10.9 6.2
379.: 19:1 58.6
IV estimates
(n’ = 7) xi
4.7 0.1 0.4 2.3 2.6 11.4 14.0 Test conditions
Y z Z
“Critical q = 8,15.5.
T 8 M
GNP YM a M
GNP T 4 M, TB
values at the 5% level for the x: distribution See text for description of procedures.
are for q = 4.9.5,
for q = 6,12.6,
and for
ployment rate. For the IV test with n’ = 7, Barr03 information set with k = 2 cannot be used since it has only six instruments and a minimum of eight are required. As is expected, the IV tests reject less frequently than the VAR tests. The data contain enough information to reject neutrality using the VAR test but not enough information to determine the mechanism as is required by the IV test. The pattern of the results is interesting. In all cases rejection is more likely when n is either small or large. For smaIl n, specification error is likely
S.G. Cecchetti,
Testing
short-run
421
neutrality
Table 3 Tests of short-run neutrality? Barro Z’s
Mishkin Z’s (AM. TB. HES)
n IV estimates,
(A M. f/NM,
fourth-orderpolynomial
distributed
FED V)
lags (n’ = 20) xi
20 24 28 32
110.2 153.7 17.0 48.1
35.1 21.8 140.2 103.9
53.4 24.0 87.4 213.7
18.5 86.7 14.0 225.8
3
T
YIU
T
YM
BCritical value at the 5% level for xi is 11.1. See text for description of procedure.
to be causing the rejection. At n = 4, it is more likely that unanticipated money at a lag of five quarters affects output than that money is non-neutral. This argues for concentrating on high values of n. The most persuasive rejections are by the VAR tests for n above twenty. Table 3 contains results using the IV procedure with n’ = 20 and the fourth-order PDL. Both the full Mishkin and Barro information sets. with k = 4 are used. While the results using the Barro information set with k = 2 reported in column 5 of table 2 fail to reject especially for large n, with k = 4 and n’ = 20 rejection is common. In fact, in every case but one all the tests with the Barre and Mishkin information sets in table 3 reject neutrality at the 1% level.‘O For each neutrality test reported above, a test of conditional heteroskedasticity was carried out by regressing the squared residuals for each equation on a constant, J and the particular set of Z’s. A Wald test was performed to test if all the coefficients excluding the constant differed from zero simultaneously. These tests also use the covariance matrix estimator constructed using the frequency domain procedure. Of the total of 162 tests, a total of 24 failed to reject at the 1% level, 13 of which also failed to reject at the 5% level. This is very strong evidence against homoskedasticity and suggests the need for the robust method of computing the covariance matrix estimates. Taken together these results imply that unless unanticipated movements in aggregate demand variables affect output deviations at a lag of eight years or more, anticipated policy matters. Given how implausible it is that unantic‘“Stulz and Wasserfallen (1985) suggest that output is a stochastic, not a deterministic, trend. This implies that the natural rate of output should be specified as J, = a +Y,-~ + E,. Then the appropriate dependent variable in the regressions to test neutrality is the first difference in the log of output. Experimentation with this specification yielded results nearly identical to those reported in tables 2 and 3.
422
S.G.
Cecchetti,
Testing
short-run
neutrality
ipated events have such lasting effects, this is strong evidence against the neutrality postulate of the new classical macroeconomics.” In light of these results, it is likely that Barro’s failure to reject neutrality can be traced to a combination of three factors: (1) improper specification of the information set on which to base forecasts of policy,‘* (2) failure to account for the time-varying nature of the elasticity of output with respect to unanticipated policy, and (3) choice of too small a maximum lag length for either anticipated or unanticipated money or both. The results in table 3 for n’ = 20 and n 2 20 suggest that Barro’s information set is not primarily at fault, instead some combination of the second and third possibilities is responsible. 4. Conclusion This paper has derived and implemented a test of the short-run neutrality of anticipated aggregate demand policy. The test is valid even when the effect of unanticipated policy on output varies over time in an unknown way and the econometrician does not correctly specify the information set relevant for forecasting policy. The test is used to examine whether either anticipated money or anticipated inflation matter. The conclusion is that they do. The generality of the economic and statistical model underlying the test makes clear that there is no credible empirical evidence for policy neutrality. This rejection must lead one to question the basis of new classical macroeconomic models. References Abel, Andrew B. and Frederic S. Mist&in, 1983, Ao integrated view of tests of rationality, market efficiency and short-run neutrality of monetary policy, Journal of Monetary Economics 11, 3-24. Barre, Robert J., 1977, Unanticipated money growth and unemployment in the United States, American Economic Review 67,101-115. Barre, Robert J., 1978, Unanticipated money, output and the price level, Journal of Political Economy 86,549-580. Barre, Robert J. and Mark Rush, 1980, Unanticipated money and economic activity, in: S. Fischer, ed., Rational expectations and economic policy (University of Chicago Press, cbieago, IL). Cumby, Robert E., John Huizinga and Maurice Obstfeld, 1983, Two-step two-step least squares estimation in models with rational expectations, Journal of Econometrics 21, 333-355. “It remains possible that these results are the consequence of misspeciftcation of the natural rate of output. If, for example, anticipated policy were to affect j, then it would not be surprising that these tests would reject neutrality. But without a structural model for the determination of .F there is very little that can be done about this. 12Germany and Srivastava show that use of the wrong information set can bias the estimates of the relevant parameters toward zero, increasing the probability of false acceptance.
S.G. Cecchetti,
Testing
short-run
neutrality
423
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