- Email: [email protected]
On persistence of shocks to economic variables
Journal of Monetary Econo
(1992) 87-93. Nort
arm Lippi Dipartimento di Science Economiche, Rome. It&
Okserratoise Fransais des Conjonctures Economiques, Paris. France Received June 1990, final version received September
1991
Empirical results on U.S. GNP have provided estimates of the Beveridge and Nelson’s persistence measure (almost) always greater than unity when using ARIMA models and always smaller than unity when using UCARIMA models. This paper shows that a measure of persistence less than unity is a mathematical consequence of the definition of UCARlMA models and not an independent estimation result. Therefore, ARIMA and UCARIMA estimates of persistence cannot be compared on the same ground.
Following the works by Beveridge and Nelson (1981) and Nelson and Plosser (1982), many studies have provided estimates of the measure of persistence of shocks to the GNP or closely related measures. The importance that has been attributed to such measures in recen macroeconomic debate may be briefly su marized as foil c persistence of traditional representation as a shocks, the further the G P process from deterministic trend plus a stationary fluct ly a temporal e *The authors wish to thank Larry Christiano. Danny Quah, and an anonymous referee very useful suggestions. Financial support by the Consiglio Nazionale delle Ricerc Qbservatoire Franfais des Conjonctures Economiques is gratefully acknowledged.
O384-3932/92/$05.00
X?1992-Elsevier
Science
ublishers B.V. All rights reserved
for
M. Lippi and L. Reichlin, On petsistence of shocks
88
Starting with Nelson and Plosser’s estimate, which was higher than unity, subsequent works found values both smaller and greater than unity. In particular, unconstrained AR1 A models usual1 persistence well above one [see .g., Campbell standard unobserved components models (U which (1) the trend is a random walk and (2) trend and cycle orthogonal - always gave a measure less than Watson (1986)]. has been rhis divergence of results between ARIMA and UCA commented on and analysed in some important papers in this research area: .Watson ., (19G61, Cochrane ClOSS), Evans (1989). In particular, Watson has shown that, although ARIMA and UCARIMA models for the real GNP provide different estimates of the persistence measure, they do not imply a substantial difference in the likelihood. This has left the empirical issue of comparison of such estimates without a conclusive answer. Our paper does not deal with the issue of estimation. Rather, it addresses a purely analytical point which explains why UCARIMA models have always given measures of persistence less than unity. We show that a measure of persistence less than unity is a mathematical consequence of the definition of UCARIMA models or, equivalently, that no UCARIMA representation exists for an ARIMA with a measure of persistence greater or equal to unity. Thus, comparisons between persistence measures estimated with ARIMA or UCARIMA models should contain the proviso that UC-AR1ML4 constrain a priori the range of persistence in the interval [0, 1). It should be noted that this asymmetry property had been firstly pointed out in Nelson and Plosser’s paper for the MA(l) case, but neglected in the subsequent debate. Our result generalizes it to any difference-stationary stochastic process admitting a Wold representation. In section 2 we recall the two measures of persistence which have been proposed in the literature. In section 3 we give an intuitive-geometric proof of our theorem. A formal proof is left to the appendix.
Let us briefly recall definition of a difference-stationary process (IX henceforth). Assume t nonstationary, whereas Ay, = y, - y,_ , is stationary and admits a Wo
J$=b+
AiEr_i=b
+A( L)&f,
ise
be
(1)
i=O
i-mess
,
6.x
uncorrelated), while (I) (L) has no zeros (1
Definition.
The process y, is difference-stationary
if
?c A(1) = i=O
The crucial implication of this definition is that a DS process y, does not admit a representation like the followin -stationary, or TS): y,=a+bt+s
(2)
I’
where s, is stationary with Wold representation:
In fact, if representation
(2143) held, then
by,=b+(l-L)B(L)rJ,. Comparison with (1) and uniqueness of the Weld representation would imply A(L) = (1 - L)B(L), so that A(1) = 0, contrary to the assumption. Beveridge and Nelson showed that
lim [E,(Y,-d -
E,-,(Y&]
k+x
=A(!)&,,
which gives the revision of the long-run prediction due to shock E, as E, times A(1). This motivates their definition of A(1) as the measure of persistence for the shocks to y,. As A( 1) vanishes for TS processes, the size of A(1) tells also how far we are from the traditional decomposition (2)~(3). The spectral density of ily, is given by g(A)
=
1A(e-"')
I’oz,
which implies the following equation linking A(1) to the spectra frequency zero: g(O)/u; = A( 1)‘.
that the above formul
ivale
itio
M. Lippi and L. Reichlin, On persistence of shocks
90
Consider now a decomposition of y, into a random walk and a stationary process: Y, = r* + s,,
T,=b+
(5)
T,-, + u,,
with u, white noise (Beveridge and Nelson showed that at least one such decomposition exists). As noted by Cochrane (1988, p. 9051, the variance of u,, i.e., the absolute measure of the random walk component in yt, is equal to g(O), irrespective of the particular decomposition (5) under consideration. Thus, as shown by (41, A(1j2 measures the size of the innovation of the random w+ component in y, relative to the variance of the aggregate shock to Y,As an altexstive measure of persistence Cochrane (1988) proposed the ratio:
u2 g(O) =A( 1)‘~, 2
-
%Y
%p
i.e., the size of the innovation of the random walk component relative to the variance of dy, itself. In the next section we will concentrate on measure (4), i.e., Beveridge and Nelson’s definition, which is the one estimated by Clark, Watson, Campbell and Mankiw. The result obtained can obviously be employed in the analysis of the second measure. 3.
efore formally stating our result let us recall that the variance of E, may be recovered from the spectral density of A y, by means of Kolmogorov’s formula:
log g(A) dh. ) has a very sj e and suggestive interpretation: oF2 is nothing e geometric mean of the spectral density graph. three cases are considered (corresponding to very simple stochastic processes): ,
i.e., o,Z,is
Fig. 1. Spectral density of Jy,.
Case B.
three cases.
Here, on the contrary, g(O) is a global maximum: hence A(1) > 1.
Case C. In this case we cannot resort to visual inspection so straightforwardly. Piotting the graph of log g(h) could help, because the geometric mean of g(A) is greater or smaller than g(O) according to whether the ordinary mean of log g(A) is greater or smaller than log g(O), and this may be decided by visual inspection if such a mean is sufficiently far from log g(O). We are now ready to state and discuss our result. Theurem. Let y, be DS and (1) be its Woid representation. Assume further that yI admits the following decomposition (standard unobserr*ed component 1: yr = T, + s I’
(7)
where (a) s, is stationary and admits a Wold representation, var (s, 1 > 0; (b) q is a random walk, i.e,. TI = b + u, with u, white noise, varted,) > 0; (c) cov(AT,, s-,_~) = 0, for all integers k. Then A(1) < 1.
The shape of the spectral density of Jay, for unobserve models is perfectly exemplified by case A: the dashed horizo the spectral density of the random walk (differencedl, to whi density of the orthogonal stationary co above analysis of case A can represent respect to the formal proof in the appendix. &mark
that
1.
eco
An
nt formulation of the t oes not exist for F!.
92
M. Lippi and L. Reichlin, On persistence of shocks
be pointed out that A(1) < 1 does not imply tbQ+ a decomposition like (7) exists. For instance, take Ay, = (1 + O.tx)( 1 - OSL)&,.
In this case (which is case B in fig. 1) we obtain immediately ,4( 1) = 0.9, g(0) = 0.81~:, but g(‘~) = (1 - 0.8)2(1 + 0.5)‘~~~= O.O9a~, so that g(O) is not a global minimum. Remark 2. In Nelson and Plosser’s paper an MA(l), i.e., Ay, = (1 + aLb,, was estimated and analyzed. In that case, a > 0 implies both A(1) > 1 and that the random walk component in y, has a larger variance than Ay, itself: = (1 + a)*a,2 > (1 + a*b: = CT&,.Such inequalities hold more in general ever the spectral density of Ay, is monotonically decreasing. However, g(O) is neither a minimum nor a maximum for the s ectral density, we may have A(1) > 1 but g(O) =A(~)“u~’ < u:,,~ as the following example shows: Ay, = (1 + 0.9L)( 1 - 0.4L2)+ Here A(1) = 1.14, so that 2.0996~~~.
g(0) =A(l)*o,’ = 1.2996~:,
whereas
c$, =
The debate on the unit root issue has brought about many important advances, both theoretical and empirical, on the time-series analysis of GNP as well as other nonstationary macroeconomic variables. Yet, the question whether the measure of persistence of shocks to GNP is greater or smaller t unity has remained controversial in spite of the considerable amount of work spent in modeling and estimation. This paper clarifies one issue relative to this debate, namely that a measure of persistence less than one for A models is a mathematical consequence of the structure of these els. Therefore, estimation cannot but confirm such inequality.
representation possess a spectral here [see Rozanov (1967, pp. 52-64, in (6) in section 3 holds [see
that g(O) is equal to crU2and is a o$=exp$j:
f g(A) in E-sr,n$
logg(h)dh ?r
recalled, f;(A) is positive almost everyw ere in OF2> O[f = g(O). Q.E.D. As
just
[ - 7z-+273;
thus we have
eferences Beveridge, S. and C.R. Nelson, 1981, A new approach to the decomposition of economic time series into permanent and transitory component with particular attention to measurement of the business cycle, Journal of Monetary Economics 7. 15I- 154. Campbell, J-W. and N.G. Mankiw, 1987, Are output fluctuations transitory?, Quarterly Jxxnal of Economics 102.857-880. Clark, P.K., 1987, The cyclical component of US economic activity, Quarterly Journal of Economics 102, 797- 814. Cochrane, J.H., 1988, How big is the random walk component in the GNP’?. Journal of Political Economy 96, 893-920. Evans, G.W.. 1989, Output and unemployment dynamics in the United States: 1950-1985, Journal of Applied Econometrics 4, 213-237. Hannan, E.J., 1970, Multiple time series (Wiley, New York. NY). Nelson, C.R. and C.I. Plosser. 1982. Trends and random walks in macroeconomic time series: Some evidence and implication, Journal of Monetary Economics 10, 139-162. Rozanov. Y.A., 1967, Stationary random processes (Holden Day. San Francisco, CA). Watson, M.W., 1986. Univariate detrending methods with stochastic trends, Journal of Monetary omits 18. 49-75.
(1992) 87-93. Nort
arm Lippi Dipartimento di Science Economiche, Rome. It&
Okserratoise Fransais des Conjonctures Economiques, Paris. France Received June 1990, final version received September
1991
Empirical results on U.S. GNP have provided estimates of the Beveridge and Nelson’s persistence measure (almost) always greater than unity when using ARIMA models and always smaller than unity when using UCARIMA models. This paper shows that a measure of persistence less than unity is a mathematical consequence of the definition of UCARlMA models and not an independent estimation result. Therefore, ARIMA and UCARIMA estimates of persistence cannot be compared on the same ground.
Following the works by Beveridge and Nelson (1981) and Nelson and Plosser (1982), many studies have provided estimates of the measure of persistence of shocks to the GNP or closely related measures. The importance that has been attributed to such measures in recen macroeconomic debate may be briefly su marized as foil c persistence of traditional representation as a shocks, the further the G P process from deterministic trend plus a stationary fluct ly a temporal e *The authors wish to thank Larry Christiano. Danny Quah, and an anonymous referee very useful suggestions. Financial support by the Consiglio Nazionale delle Ricerc Qbservatoire Franfais des Conjonctures Economiques is gratefully acknowledged.
O384-3932/92/$05.00
X?1992-Elsevier
Science
ublishers B.V. All rights reserved
for
M. Lippi and L. Reichlin, On petsistence of shocks
88
Starting with Nelson and Plosser’s estimate, which was higher than unity, subsequent works found values both smaller and greater than unity. In particular, unconstrained AR1 A models usual1 persistence well above one [see .g., Campbell standard unobserved components models (U which (1) the trend is a random walk and (2) trend and cycle orthogonal - always gave a measure less than Watson (1986)]. has been rhis divergence of results between ARIMA and UCA commented on and analysed in some important papers in this research area: .Watson ., (19G61, Cochrane ClOSS), Evans (1989). In particular, Watson has shown that, although ARIMA and UCARIMA models for the real GNP provide different estimates of the persistence measure, they do not imply a substantial difference in the likelihood. This has left the empirical issue of comparison of such estimates without a conclusive answer. Our paper does not deal with the issue of estimation. Rather, it addresses a purely analytical point which explains why UCARIMA models have always given measures of persistence less than unity. We show that a measure of persistence less than unity is a mathematical consequence of the definition of UCARIMA models or, equivalently, that no UCARIMA representation exists for an ARIMA with a measure of persistence greater or equal to unity. Thus, comparisons between persistence measures estimated with ARIMA or UCARIMA models should contain the proviso that UC-AR1ML4 constrain a priori the range of persistence in the interval [0, 1). It should be noted that this asymmetry property had been firstly pointed out in Nelson and Plosser’s paper for the MA(l) case, but neglected in the subsequent debate. Our result generalizes it to any difference-stationary stochastic process admitting a Wold representation. In section 2 we recall the two measures of persistence which have been proposed in the literature. In section 3 we give an intuitive-geometric proof of our theorem. A formal proof is left to the appendix.
Let us briefly recall definition of a difference-stationary process (IX henceforth). Assume t nonstationary, whereas Ay, = y, - y,_ , is stationary and admits a Wo
J$=b+
AiEr_i=b
+A( L)&f,
ise
be
(1)
i=O
i-mess
,
6.x
uncorrelated), while (I) (L) has no zeros (1
Definition.
The process y, is difference-stationary
if
?c A(1) = i=O
The crucial implication of this definition is that a DS process y, does not admit a representation like the followin -stationary, or TS): y,=a+bt+s
(2)
I’
where s, is stationary with Wold representation:
In fact, if representation
(2143) held, then
by,=b+(l-L)B(L)rJ,. Comparison with (1) and uniqueness of the Weld representation would imply A(L) = (1 - L)B(L), so that A(1) = 0, contrary to the assumption. Beveridge and Nelson showed that
lim [E,(Y,-d -
E,-,(Y&]
k+x
=A(!)&,,
which gives the revision of the long-run prediction due to shock E, as E, times A(1). This motivates their definition of A(1) as the measure of persistence for the shocks to y,. As A( 1) vanishes for TS processes, the size of A(1) tells also how far we are from the traditional decomposition (2)~(3). The spectral density of ily, is given by g(A)
=
1A(e-"')
I’oz,
which implies the following equation linking A(1) to the spectra frequency zero: g(O)/u; = A( 1)‘.
that the above formul
ivale
itio
M. Lippi and L. Reichlin, On persistence of shocks
90
Consider now a decomposition of y, into a random walk and a stationary process: Y, = r* + s,,
T,=b+
(5)
T,-, + u,,
with u, white noise (Beveridge and Nelson showed that at least one such decomposition exists). As noted by Cochrane (1988, p. 9051, the variance of u,, i.e., the absolute measure of the random walk component in yt, is equal to g(O), irrespective of the particular decomposition (5) under consideration. Thus, as shown by (41, A(1j2 measures the size of the innovation of the random w+ component in y, relative to the variance of the aggregate shock to Y,As an altexstive measure of persistence Cochrane (1988) proposed the ratio:
u2 g(O) =A( 1)‘~, 2
-
%Y
%p
i.e., the size of the innovation of the random walk component relative to the variance of dy, itself. In the next section we will concentrate on measure (4), i.e., Beveridge and Nelson’s definition, which is the one estimated by Clark, Watson, Campbell and Mankiw. The result obtained can obviously be employed in the analysis of the second measure. 3.
efore formally stating our result let us recall that the variance of E, may be recovered from the spectral density of A y, by means of Kolmogorov’s formula:
log g(A) dh. ) has a very sj e and suggestive interpretation: oF2 is nothing e geometric mean of the spectral density graph. three cases are considered (corresponding to very simple stochastic processes): ,
i.e., o,Z,is
Fig. 1. Spectral density of Jy,.
Case B.
three cases.
Here, on the contrary, g(O) is a global maximum: hence A(1) > 1.
Case C. In this case we cannot resort to visual inspection so straightforwardly. Piotting the graph of log g(h) could help, because the geometric mean of g(A) is greater or smaller than g(O) according to whether the ordinary mean of log g(A) is greater or smaller than log g(O), and this may be decided by visual inspection if such a mean is sufficiently far from log g(O). We are now ready to state and discuss our result. Theurem. Let y, be DS and (1) be its Woid representation. Assume further that yI admits the following decomposition (standard unobserr*ed component 1: yr = T, + s I’
(7)
where (a) s, is stationary and admits a Wold representation, var (s, 1 > 0; (b) q is a random walk, i.e,. TI = b + u, with u, white noise, varted,) > 0; (c) cov(AT,, s-,_~) = 0, for all integers k. Then A(1) < 1.
The shape of the spectral density of Jay, for unobserve models is perfectly exemplified by case A: the dashed horizo the spectral density of the random walk (differencedl, to whi density of the orthogonal stationary co above analysis of case A can represent respect to the formal proof in the appendix. &mark
that
1.
eco
An
nt formulation of the t oes not exist for F!.
92
M. Lippi and L. Reichlin, On persistence of shocks
be pointed out that A(1) < 1 does not imply tbQ+ a decomposition like (7) exists. For instance, take Ay, = (1 + O.tx)( 1 - OSL)&,.
In this case (which is case B in fig. 1) we obtain immediately ,4( 1) = 0.9, g(0) = 0.81~:, but g(‘~) = (1 - 0.8)2(1 + 0.5)‘~~~= O.O9a~, so that g(O) is not a global minimum. Remark 2. In Nelson and Plosser’s paper an MA(l), i.e., Ay, = (1 + aLb,, was estimated and analyzed. In that case, a > 0 implies both A(1) > 1 and that the random walk component in y, has a larger variance than Ay, itself: = (1 + a)*a,2 > (1 + a*b: = CT&,.Such inequalities hold more in general ever the spectral density of Ay, is monotonically decreasing. However, g(O) is neither a minimum nor a maximum for the s ectral density, we may have A(1) > 1 but g(O) =A(~)“u~’ < u:,,~ as the following example shows: Ay, = (1 + 0.9L)( 1 - 0.4L2)+ Here A(1) = 1.14, so that 2.0996~~~.
g(0) =A(l)*o,’ = 1.2996~:,
whereas
c$, =
The debate on the unit root issue has brought about many important advances, both theoretical and empirical, on the time-series analysis of GNP as well as other nonstationary macroeconomic variables. Yet, the question whether the measure of persistence of shocks to GNP is greater or smaller t unity has remained controversial in spite of the considerable amount of work spent in modeling and estimation. This paper clarifies one issue relative to this debate, namely that a measure of persistence less than one for A models is a mathematical consequence of the structure of these els. Therefore, estimation cannot but confirm such inequality.
representation possess a spectral here [see Rozanov (1967, pp. 52-64, in (6) in section 3 holds [see
that g(O) is equal to crU2and is a o$=exp$j:
f g(A) in E-sr,n$
logg(h)dh ?r
recalled, f;(A) is positive almost everyw ere in OF2> O[f = g(O). Q.E.D. As
just
[ - 7z-+273;
thus we have
eferences Beveridge, S. and C.R. Nelson, 1981, A new approach to the decomposition of economic time series into permanent and transitory component with particular attention to measurement of the business cycle, Journal of Monetary Economics 7. 15I- 154. Campbell, J-W. and N.G. Mankiw, 1987, Are output fluctuations transitory?, Quarterly Jxxnal of Economics 102.857-880. Clark, P.K., 1987, The cyclical component of US economic activity, Quarterly Journal of Economics 102, 797- 814. Cochrane, J.H., 1988, How big is the random walk component in the GNP’?. Journal of Political Economy 96, 893-920. Evans, G.W.. 1989, Output and unemployment dynamics in the United States: 1950-1985, Journal of Applied Econometrics 4, 213-237. Hannan, E.J., 1970, Multiple time series (Wiley, New York. NY). Nelson, C.R. and C.I. Plosser. 1982. Trends and random walks in macroeconomic time series: Some evidence and implication, Journal of Monetary Economics 10, 139-162. Rozanov. Y.A., 1967, Stationary random processes (Holden Day. San Francisco, CA). Watson, M.W., 1986. Univariate detrending methods with stochastic trends, Journal of Monetary omits 18. 49-75.