COMMON SERIAL CORRELATION AND COMMON BUSINESS CYCLES: A CAUTIOUS NOTE
Gianluca Cubadda Dipartimento di Scienze Economiche Gestionali e Sociali Università degli Studi del Molise Via De Sanctis - 86100 Campobasso, Italy Tel. +39-0874404467; Fax. +39-0874311124 E-mail:
[email protected]
Abstract This paper examines the frequency-domain implications of the serial correlation common feature in order to evaluate its merits as an indicator of common business cycles among economic variables. It is shown that the presence of the serial correlation common feature in the first differences of a set of I(1) time series is not informative for the degree and the lead-lag structure of their comovements at the business cycle frequencies. Keywords: Business cycles, Serial correlation common feature, Frequencydomain methods. JEL classification: C22, E32
1. Introduction A long standing viewpoint in macroeconomics is that lower and higher frequency comovements in variables have different implications, with the former being informative for growth and the latter for the business cycle. Consequently, in business cycle empirics macro-variables are usually detrended prior to investigating the comovements at the business cycle frequencies [henceforth BCF]. A popular detrending method is low-frequency filtering, particularly by means of the symmetric linear filter suggested by Hodrick and Prescott (1997). However, this practice has recently been criticized on the grounds of a body of empirical evidence favorable to the presence of a unit-root in many aggregate variables. In fact, although the most widespread low-frequency filters are capable to stationarize I(1) time series (King and Rebelo, 1993), it has been well documented that low-frequency filtering of I(1) processes may generate spurious fluctuations at the BCF (see inter alia Cogley and Nason, 1995; Harvey and Jaeger, 1993).
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An alternative way of detrending I(1) time series is removing the randomwalk component from the series, as originally advocated by Beveridge and Nelson [henceforth BN] (1981). An important progress in this direction is the work by Vahid and Engle (1993). These authors showed that I(1) variables have common cycles in their multivariate BN decomposition when a linear combination of the first differences of these variables eliminates all correlation with the past. This measure of comovement among stationary series was defined by Engle and Kozicki (1993) as the serial correlation common feature [henceforth SCCF]. Moreover, Vahid and Engle (1993) indicated that statistical inference for common BN cycles can be easily implemented by canonical correlations methods. The purpose of this note is to clarify to business cycle researchers what they can learn on comovements at the BCF by testing for SCCF. Specifically, it is shown that the existence of SCCF between the first differences of I(1) time series neither implies nor is implied by high coherence at the BCF or absence of phase-shifts. Moreover, since it is argued that low-frequency symmetric linear filtering of I(1) series generate cycles having the same coherence and phase-spectrum as the first differences of these series, SCCF reveals to be not informative for the degree of comovement and the leadinglagging behavior of series detrended by means of filters with the aforementioned characteristics. Hence, the notion of common cycles based on SCCF appears to be inconsistent with the more traditional definition of common business cycles, namely "cyclical comovements between important macroeconomic variables with periods of around five year" (Englund et al., 1992). Section two, after briefly reviewing the definition of SCCF, derives the frequency-domain implications of this concept. Section three motivates the comparison of SCCF with coherence and phase-spectrum. Section four shows that SCCF is unrelated to coherence at BCF, Section five shows that SCCF is not connected to phase-shifts. Finally, Section six presents conclusions.
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2. SCCF and Common Cycles To simplify matters, we will focus on the bivariate case. Hence let ∆yt be the first differences of a 2-vector I(1) time series. The Wold representation of these series is ∆yt = B(L)ut
(2.1)
where B(L) is a polynomial matrix in the the lag operator L such that B(0) = I2,
< ∞ and ut is a 2-vector white noise series with a positive
j Bj
j≥1
definite variance matrix Σ. The bivariate BN trend-cycle representation of variables yt is yt = τt + ct
where ∆τt
=
B(1)ut and ct
=
(2.2)
B(L) - B(1) ut. 1 - L
Relevant characteristics of this decomposition are that the cycle ct is a stationary remainder after subtracting the random walk trend τt and that both components are driven by the vector of the Wold innovations. Following Engle and Kozicki (1993), we say that elements of ∆yt have a SCCF if there exists a 2-vector α =/ 0 such that α′∆yt is an innovation with respect to the available information up to time t. This is equivalent to requiring that α′B(L)ut = α′ut
(2.3)
i.e. all the coefficient matrices Bj for j > 1 must be orthogonal to the cofeature vector. Equation (2.3) implies α′ct = 0
3
(2.4)
and hence we can write ct = βηt
(2.5)
where β is a 2-vector and ηt is an univariate I(0) process. Equation (2.5) motivates Vahid and Engle (1993) to assert that variables having a SCCF share a common cycle.1 Let us now deal with the frequency-domain implications of the presence of SCCF. From equation (2.1), the spectral density matrix of ∆yt is equal to 1 F(ω) = 2π B(z-1)ΣB′(z)
(2.6)
where z = exp(-iω) and ω ∈ [0, 2π). Hence, by taking the spectrum at each side of (2.3), we derive the restrictions that SCCF imposes on the spectral density matrix: 1 α′F(ω)α = 2π α′Σα
(2.7)
for all ω. Interestingly enough, a first remark suggested by equation (2.7) is that SCCF requires that the spectral density matrix F(ω) is not null at any frequency ω. This should discourage the use of seasonally adjusted series in applied works on SCCF, since Maravall (1993) shows that, under fairly general conditions, the most widespread procedures of seasonal adjustment annihilate the spectrum at the seasonal frequencies.2
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As Lippi and Reichlin (1994) note, this result does not extend to trendcycle decompositions where the permanent component is different from a random-walk. 2 See Cubadda (1999) on the extension of common cycle analysis to time series having unit roots also at the seasonal frequencies. 4
3. Coherence, Phase-spectrum and Low-frequency Filtering In order to evaluate the merits of SCCF as indicator of cyclical comovements in elements of ∆yt, we are going to consider the implications of SCCF in terms of spectral coherence and phase-spectrum. Particularly, let flj(ω) (l,j = 1,2) be the generic element of the matrix F(ω). Hence, the spectral coherence is defined by C(ω) = f12(ω) 2 [f11(ω)f22(ω)]
(3.1)
and it measures the strength of linear relationship between the ω-frequency components of series ∆yt. Moreover, the phase spectrum is defined by
(ω) = tan-1[Im{f12(ω)}/Re{f12(ω)}]
(3.2)
and it represents the average phase-shifts between the ω-frequency components of series ∆yt. The comparison of SCCF with these spectral measures is interesting not only per se, but also because it is helpful to clarify to what extent SCCF is informative on comovements among low-frequency filtered series. In fact, in the appendix it is shown that cycles obtained by low-frequency symmetric linear filtering of the I(1) variables yt have the same coherence and phasespectrum as the series ∆yt.3 Moreover, since the Hodrick-Prescott filter emphasizes fluctuations at BCF when applied to I(1) variables (Cogley and Nason, 1995), examining cross-correlations at various lags and leads between series detrended by this filter is qualitatively similar to looking at their coherence and phase-spectrum at BCF. However, these spectral measures are not subject to the criticism of detecting spurious cyclical comovements induced by the filtering method.
3 As Cogley and Nason (1995) note, coherence is generally invariant to univariate linear filtering.
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4. SCCF and Coherence at Business Cycle Frequencies Let us start the analysis by assuming that C(λ) = 1
(4.1)
where λ belongs to BCF. Note that, being C(ω) uniformly continuous for all ω, equation (4.1) implies that coherence is high around frequency λ. Hence, if equation (4.1) holds we are strongly tempted to say that time series yt share a common business cycle.4 Would this be related to the presence of SCCF in elements of ∆yt? To answer this question, consider that equation (4.1) requires that there exists a complex-valued 2-vector δ =/ 0 such that δ*F(λ)δ = 0
(4.2)
where δ* is the complex conjugate transpose of vector δ. Evaluating equation (2.7) at ω = λ and comparing it with equation (4.2) we see that the vectors α and δ are eigenvectors corresponding to different eigenvalues of matrix F(λ). Hence these vectors must be linearly independent. It follows that in a set of differenced I(1) time series the existence of SCCF neither implies nor is implied by high coherence at BCF. This result has an intuitive explanation. In fact, note that the vector δ is associated in the time-domain to a polynomial vector δ(L) by the equation δ(L) = Re{δ} - Im{δ}[1/tan(λ) - L/sin(λ)]
(4.3)
Equation (4.2) tells us that the spectrum of the scalar process δ(L)′∆yt is null at frequency λ. Hence, we see that the polynomial vector δ(L) cancels out the common λ-frequency component in ∆yt. In contrast, SCCF requires the existence of a vector α which modulates the amplitude of this common λ-frequency component in order to ensure that the spectrum of the scalar process α′∆yt is constant over different frequencies.
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Note that when elements of yt are cointegrated it holds that C(0) = 1. 6
5. SCCF and Phase-shifts Once established that SCCF has little to do with coherence at BCF, we may wonder if SCCF is connected to the lead-lag properties of observed fluctuations. In order to explore this possibility, let us assume that elements of ∆yt are in phase. In this case, we have that (ω) = 0 for all ω, which is equivalent to Im{F(ω)} = 0
(5.1)
Consider now that, in view of equation (2.3), when elements of ∆yt have a SCCF we can write ∆yt
=
ut + βψ(L)′ut
(5.2)
where βψ(L)′ = [B(L) - I2]. At a first glance, equation (5.2) seems to suggest that SCCF implies that elements of ∆yt are synchronous, since the elements of each piece of ∆yt are clearly in phase. However, this intuition turns out to be misleading. This can immediately be understood by considering that the two pieces of ∆yt in equation (5.2) are contemporaneously uncorrelated but not linearly independent. Hence, if we take the spectrum at both sides of equation (5.2) we have 1 F(ω) = 2π [Σ + βψ(z)′Σψ(z-1)β′ + βψ(z)′Σ + Σψ(z-1)β′]
(5.3)
Equation (5.3) indicates that the spectral density matrix of ∆yt is generally complex-valued under SCCF’s restrictions. It follows that SCCF does not imply absence of phase shifts between elements of ∆yt. It is easy to see that the converse implication also fails.
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6. Conclusions This note is not meant to be a disapproval of the use of SCCF. Its purpose is rather to serve empirical researchers by carefully examining the frequencydomain properties of SCCF. This analysis has revealed that the presence of SCCF between the first differences of I(1) variables is unrelated to the degree (coherence) and the lead-lag structure (phase-shifts) of their comovements at the BCF. Moreover, since the considered spectral functions are invariant to symmetric low-frequency filtering of I(1) variables, SCCF is not informative for comovements among series detrended by this method. These results suggest that caution should be taken in interpreting SCCF as an indicator of common business cycles among economic time series. For instance, the evidence reported by Engle and Issler (1995) that US sectoral GNP growth rates have very similar and well synchronized cyclical movements turns out to be irrelevant for their finding of SCCF among these growth rates.
Appendix Let the detrended series be xt = D(L)yt, where D(L) is such that: (i) (ii) (iii)
D(L) is a diagonal filter matrix D(L) contains the factor (1 - L), i.e., D(L) = (1 - L)G(L) D(L) is symmetric, i.e., D(L) = D(L-1)
Note that this general class of filtering includes as special cases the most widespread detrending methods in business cycle empirics (see e.g. King and Rebelo, 1993). Let Fx(ω), Cx(ω), and ! x(ω) respectively be the spectral density matrix, the coherence, and the phase-spectrum of the detrended series xt. Assumption (ii) implies that Fx(ω) = G(z)F(ω)G(z-1)′
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(A.1)
Hence, in view of assumption (i) and equation (A.1), we see that Cx(ω) = C(ω) and "
# $ &
(
x(ω)
(z-1)}
%
(z-1)} '
Re{f (ω)}Im{g1(z)g2 + Im{f12(ω)}Re{g1(z)g2 = tan-1 .) * * * * * * * * * * * * * * * * * * * 12 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * +-, 2 Re{f (ω)}Re{g (z)g (z-1)} / / / / 0 Im{f (ω)}Im{g (z)g (z-1)} 31 12 1 2 12 1 2 4
(A.2)
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where gj(z) (j = 1,2) is the generic element of the diagonal matrix G(z). Moreover, combining assumptions (ii) and (iii) we can write G(L) = -L-1G(L-1)
(A.3)
which highlights the asymmetric character of the filter matrix G(L). However, from (A.3) it follows that g1(z)g2(z-1) = g1(z-1)g2(z)
(A.4)
Since equation (A.4) implies that Im{g1(z)g2(z-1)} = 0 and in view of equation (A.2), we finally see that 6 x(ω) = 7 (ω).
Acknowledgments Financial support from The Italian Ministry of the Universities and Scientific Research (MURST) is gratefully acknowledged. Thanks are due to M. Lippi, C. Lupi, as well as an anonymous referee for useful comments. The usual disclaimers apply.
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References Beveridge, S. and C.R. Nelson (1981) "A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle", Journal of Monetary Economics, 7, 151-174. Cogley, T. and J.M. Nason (1995) "Effects of the Hodrick-Prescott filter on trend and difference stationary time series. Implications for business cycle research", Journal of Economic Dynamics and Control, 19, 253-278. Cubadda, G. (1999) "Common cycles in seasonal non-stationary time series", Journal of Applied Econometrics, 14, 273-291. Engle, R.F. and S. Kozicki (1993) "Testing for common features", Journal of Business and Economic Statistics, 11, 369-395 with discussions. Engle, R.F. and V. Issler. (1995) "Estimating common sectoral cycles", Journal of Monetary Economics, 35, 83-113. Englund, P., Persson, T. and L.E.O. Svensson (1992) "Swedish business cycles: 1861-1988", Journal of Monetary Economics, 30, 347-371. Harvey, A.C. and A. Jaeger (1993) "Detrending, stylized facts and the business cycle", Journal of Applied Econometrics, 8, 231-247. Hodrick, R. and C. Prescott (1997) "Postwar US business cycles: An empirical investigation", Journal of Money, Credit and Banking, 29,1-16. King, R.G. and S.T. Rebelo (1993) "Low frequency filtering and real business cycles", Journal of Economic Dynamics and Control, 17, 207-231.
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Lippi, M. and L. Reichlin (1994) "Common and uncommon trends and cycles", European Economic Review, 38, 624-635. Maravall, A. (1993) "Stochastic linear trends", Journal of Econometrics, 56, 5-37. Vahid, F. and R.F. Engle (1993) "Common trends and common cycles", Journal of Applied Econometrics, 8, 341-360.
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