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On the Use of InnovationCorrelations to Study CyclicalCo-Movements in GDP and ItsComponentsProfessor Abeysinghe Tilak a & Dr. Choy Keen Meng aa National University of SingaporePublished online: 28 Jul 2006.

To cite this article: Professor Abeysinghe Tilak & Dr. Choy Keen Meng (2002) On the Useof Innovation Correlations to Study Cyclical Co-Movements in GDP and Its Components,International Economic Journal, 16:2, 37-45, DOI: 10.1080/10168730200000012

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INTERNATIONAL ECONCbMIC JOURNAL Volume 16, Number 2, Summer 2002

ON THE USE OF INNOVATION CORRELATIONS TO STUDY

CYCLICAL CO-MOVEMENTS IN GDP AND ITS COMPONENTS

TILAK ABEYSINGHE AND KEEN MENG CHOY*

National University of Singapore

Innovation cross correlations are sometimes used as indicators of cyclical co- movements among economic variables. This note shows that care is needed in making inferences about business cycle co-movements between GDP and its components from an innovation cross correlation analysis because of the effect of the national income accounting identity. The point is illustrated with an empirical example from Singapore. [C22, E32]

1. INTRODUCTION

Numerous studies have appeared in the macroeconomic literature seeking to discover the "stylized facts" of macroeconomic fluctuations through an examination of the empirical co-movements between aggregate output and key macroeconomic variables (see for example Blanchard and Fischer, 1989: Ch. 1; Kydland and Prescott, 1990). One of the statistical methodologies employed for this purpose is cross correlation analysis, whereby bivariate innovation cross correlation coefficients are calculated between real GDPIGNP and selected variables to give an indication of the degree of co-movement in the fluctuations of economic aggregates.

Almost invariably, the evidence ftom these studies shows that the major expenditure components of GDP - in particular private consumption and investment expenditures - are strongly procyclical, as indicated by strong positive contemporaneous correlations. These are, in fact, well-established stylized facts of the business cycle (Lucas, 1977). Observing high contemporaneous (innovation) correlations, however, Blanchard and Fischer (1989: footnote 20, 3 1) note that "Since output is the sum of its components, it should come as no surprise that the average contemporaneous correlation is positive . . .".

It would appear, therefore, that empirical cross correlations calculated for output and any one expenditure category would have to be non-zero - almost by definition. The present note examines this proposition analytically by deriving

*The valuable comments of two anonymous referees are greatly appreciated. Of course, we alone remain responsible for any errors.

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theoretical cross correlation coefficients under an accounting constraint. These results are expounded in Section 2. In Section 3, we discuss the practical implications of the national income identity for the cyclical behaviour of GDP components and in Section 4, we present an empirical example to illustrate our points. In the final section, we conclude that inferences about business cycle co- movements among aggregate output and its components based on innovation cross correlation analysis may not be very informative.

2. THEORETICAL CROSS CORRELATIONS

To keep matters simple, consider the case where real GDP (Y) is constituted of only two components, denoted Xt and Z,. Then we can write

Abstracting from trends, assume that Xt and Zt are stationary and invertible ARMA(pl, ql) and ARMA(p2, q2) processes respectively. It has been shown that the aggregate series Yt is an ARM@, q) process - if Xt and Zt are uncorrelated, then p I p l + p~ and q I p + max(ql - p,, 92 - p2); when X, and Zt are correlated, p and q will also depend on the nature of the cross covariances between them (Engel, 1984).

To address the issue of this paper, we need more specific assumptions on the orders of the ARMA processes. For analytical convenience, let Xt and Zt be AR(1) processes with zero means (more general ARM@, q) processes will not affect the basic results below except for complicating the derivation). Then,

It follows from (1) and (2) that Y, can be expressed as:

[I- ( 4 + ry)L + ( @ r y ) ~ ~ l ~ t = (1 - ryL) ut + (1 - @)vt (3)

In this case, it is not difficult to show that Y, - ARMA(2, I ) regardless of whether Xt and Zt are correlated or not. However, a simpler model will result if common factors occur (Granger and Newbold, 1986: 28-31).

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USE OF INNOVATION CORRELATIONS 3 9

Empirical cross correlations between output and the various expenditure components are computed using pre-whitened series, or the innovations from ARIMA time series models of the variables in question (see Haugh, 1976 or Pierce, 1977 for a description of the approach). This is the methodology adopted by Blanchard and Fischer (1989: Ch. l), and it has the advantage of not being subject to the criticism of spurious correlation that has been leveled against the use of the Hodrick-Prescott filter to detrend macroeconomic series (Cogley and Nason, 1995). With respect to the bivariate example considered here, the innovations in X, and Zt are u, and vt respectively, while the innovations in Y, are seen (from (3)) to be a function of u, and v,. To obtain an expression for the latter, we will proceed under the general assumption that X, and 2, are contemporaneously correlated, i.e. Cov(u, v,) = a,,, but uncorrelated at all other lags, i.e. Cov(ut,vFk) =

0 for k # 0. First, let the moving average part of Yt in (3) be denoted by cot:

Being an M ( 1 ) process, w, can be written as:

where c, is the innovation process of Y,. Next, equate (4) and (5) to yield the following expression for E,:

Using (6), the contemporaneous covariance between the innovations in Y, and the innovations in Xt is given by

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1

and the product of their variances by

The contemporaneous cross correlation coefficient between Yt and X, in terms of their innovations is defined as:

Corr (E , , u, ) = COV(&,, u, )

JVar(4)Varo

Moreover, the innovation cross correlation coefficient at lag 1 is

For lag k 2 2, it is easily verified that

COTT(E~, u ~ . ~ ) = e k-' Corr(Et, u~.I)

The symmetry between Xt and Zt implies that analogous expressions will be obtained for the innovation cross correlations of 2,; we therefore restrict attention to equations (7)-(9). Furthermore, the case where real GDP comprises of three or more components yields similar results except for the presence of additional covariance terms in (7) and (8). These additional terms may lead to cancellation of the effects coming from individual components.

3. DISCUSSION

The implications of the theoretical results of the foregoing section depend on the nature of the contemporaneous correlation between the innovations of the GDP

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USE OF INNOVATION CORRELATIONS 4 1

components. There are three possibilities: (i) o,, = 0; (ii) o,, > 0; and (iii) o,, < 0. In the first two cases the contemporaneous innovation correlation given in (7) will always be positive. Therefore, both components of real GDP will be interpreted as procyclical. Moreover, in the first case, their cross correlations at all lags are also non-zero unless 0= I,V in (8), although the correlations decay exponentially as k increases. In the second case, however, it is not possible to predict a priori the values of the lagged cross correlation coefficients.

The only situation where a non-positive contemporaneous correlation could occur is the third case when o,, < 0. Even in this case a zero or negative correlation appears only if 0; < lo,, 1. In other words, even when the innovations of the components are negatively correlated, the correlation of a given component with GDP could still be positive, which may lead to the conclusion that the component is procyclical.

Further insight into the problem can be gained by looking at a special case where 0= y/= 4 in which case E, = u, + v,. In this case, the contemporaneous correlation in (7) reduces to

Corr (E, , u, ) = 1 + &?,"

Jm where A= o, / o, > 0 and pU, = o,, / ouo%, . When p,," = 0 (components are uncorrelated), the above correlation depends on A (or the variance ratio). As A increases, the correlation decreases; when A = 1, C o r r ( ~ , u,) = 0.71. If A= 1 and the components are perfectly positively correlated (p,, = l), then Corr(&, u,) = +l . In this case the two components are nothing but the same.

When the components are perfectly negatively correlated (p,, = - l), Corr(&, u,) = (1 - A) I / A - 11 and the solution depends on A. For A = 1, this correlation is undefined. For 0 < i. < 1, Corr(&, u,) = + 1 and for A > 1, Corr(&, u,) = -1. Thus, even when the components are perfectly negatively correlated, their procyclicality or counter-cyclicality depends on the variance ratio. The size of the variance ratio, honrever, is unlikely to depend on the procyclical or counter- cyclical nature of the components. In short, the contemporaneous innovation cross correlation does not convey much information about the cyclical co- movements of the series with GDP because of the accounting identity problem.

4. EMPIRICAL EXAMPLE

To illustrate the above ideas, we examine the innovation cross correlations between real GDP (Y) and its expenditure components in Singapore over the period 1975Q1-199943. For the purpose of comparison with the previous theoretical results, we first examine the results for the identity y, = c, + c, where

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c is real consumption expenditure and 'F is real non-consumption expenditure. All three series show seasonal variation. Since it is not very clear whether to use seasonal differencing or seasonal dummies, we estimated ARIMA models using both approaches (see Abeysinghe, 1994 and references therein). The best fitting models have the following general forms:

AA, In z, = (1 - 8L)(l- OL')~, (10)

where d's are seasonal dummies. We do not report the estimated models for brevity. For the real GDP series, the M(1) parameter is insignificant in both models and the presence or absence of this parameter in the model does not change the results much. The estimated innovation cross correlations are reported in the first two panels of Table 1. Although the results based on seasonal differencing and seasonal dummies are comparable, we observe that seasonal differencing results in over-differencing at least in the case of c. For this reason, the innovation variances and the covariance reported below the table pertain only to the seasonal dummy models.

Table 1. Innovation Cross Correlations Between y, and Components x,k

Lag k (Quarters) -3 -2 -1 0 1 2 3

Based on seasonal differencing

y and c 0.09 0.11 0.24 0.41 0.22 0.12 0.13 y and F 0.10 0.11 0.25 0.75 0.05 0.05 0.08 c and F 0.10 0.19 0.36 -0.20 0.33 0.16 0.01

Based on seasonal dummies

y and c 0.16 0.06 0.18 0.43 0.26 0.05 0.01 y and F 0.04 0.05 0.26 0.75 -0.03 -0.04 0.07 c and F 0.02 0.14 0.40 -0.1 8 0.25 0.08 0.10

y and i- j x .I4 .27 -25 .13 -.03 .07 -.05 y and i-inv . I8 .O 1 .22 .41 .03 -.09 .08 y and net-x -.25 -.04 -.02 .04 .04 .20 .20 Y and g -.01 -.02 -.23 -.I5 -.01 -.I9 -.I4

Notes: 2 0 limits for the correlation estimates are approximately k0.21. The highlighted figures are the statistically significant ones. The innovation variances based on seasonal dummy models are: Var(e,) = 2.76x10-', Var(e,.) = 5 . 5 6 ~ 1 0 - ~ , Var( e, ) = 1.0x10-~. The contemporaneous Cov( e,, e, ) = -1.25~10.~.

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USE OF INNOVATION CORRELATIONS 43

The empirical results are largely in line with our theoretical findings. Although the significant correlations at lags f 1 have implications with regard to Granger causality, here we concentrate only on the contemporaneous correlations (at k = 0). The contemporaneous correlations between the innovations of y and c and y and F are both strongly positive, implying procyclical co-movement of both components. The contemporaneous correlation between the innovations of c and F seems to be either zero or negative. Even if it is negative, the covariance between the two is much smaller than their variances (see notes to the table). The net result, therefore, is the positive contemporaneous correlations between real GDP and its components. However, if the components are negatively correlated, it is rather inappropriate to conclude that both components are procyclical.

It would be instructive to examine the innovation cross correlations between y and the individual cotnponents of C namely gross fixed capital formation (i-fix), changes in stocks or inventory investment (i-inv), net exports (net-x) and government consumption expenditure (g). The models fitted to i-fix and g are of the form given by (1 1). Since i-inv and net-x series contain negative and positive values we modeled them in their levels. Both series show heteroscedastic behavior and net-x has been trending upward since 1987. For i-inv we fitted a seasonal dummy model with an AR(1)-ARCH(1) error term. For net-x we fitted a seasonal dummy model with t (time) and t2 as additional regressors with an AR(4)-ARCH(1) error tenn. We then used the standardized residuals as the innovations. The resulting cross correlation estimates are given in the third panel of Table 1.

These results show that the disaggregation of Z has reduced the magnitude of the contemporaneous correlation coefficients, leaving only one of them statistically significant - that for inventory investment. Government consumption expenditure shows insignificant negative estimates except for the correlation at lag -1. Just based on these cross correlation estimates alone, it would be difficult to conclude that g is counter-cyclical. In contrast to the Singapore case, Blanchard and Fischer (1989: 16) find a statistically significant positive contemporaneous correlation between y and g for the US economy.

If one were interested in Granger causality between GDP and its components based on statistically significant estimates at positive and negative lags, the findings would be that y and c form a feedback loop and y causes all the components of iZ Although the positive effects of GDP on fixed and inventory investments may be acceptable, the negative coefficients for net-x and g are puzzling. This brings us to the next important question on the usehlness of sample estimates of the innovation cross correlations. Haugh (1976) has shown that under the null hypothesis of series independence, the cross correlation estimates based on estimated innovations and those based on true innovations are both consistent and asymptotically normally distributed with zero means and variance N', where N is the number of observations in the sample. In a Granger

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causality test based on the above approach, Pierce (1977) found rather weak relationships among a number of monetary variables. In response to these findings, Schwert (1979) has shown that even if the true innovations were known, the distributed lag structure between innovations can be substantially different from that of the original series. Nelson and Schwert (1982) have further pointed out that ARMA residual-based tests of Granger causality are less powerful compared to some alternatives. These findings add further caution to inference based on innovation cross correlations.

5. CONCLUDING REMARKS

As part of an exercise to establish the stylized facts of business cycles in Singapore, we have analyzed the co-movements of a large number of macroeconomic variables with output (Choy, 2001). Statistical tools used in the research include innovation cross correlation analysis, cross spectral analysis and impulse response analysis. The theoretical results in this paper, however, demonstrate that the information contained in the cross correlation method is not easily interpreted.

Because of the implicit accounting identity problem, we would advise business cycle practitioners to exercise caution in drawing conclusions from this technique when dealing with variables involved in an accounting identity. A further weakness of the cross correlation approach is that non-zero cross correlations between series that do not possess cycles cannot be ruled out. In this regard, spectral analysis is much more informative with respect to establishing the cyclical properties of macroeconomic series and their co-movements with output.

REFERENCES

Abeysinghe, Tilak, "Deterministic Seasonal Models and Spurious Regressions," Journal of Econometrics, April 1994, 259-272.

Blanchard, Olivier J. and Fischer, Stanley, Lectures on Macroeconomics, Cambridge, Massachusetts: MIT Press, 1989.

Choy, Keen Meng, "Macroeconomic Fluctuations In Singapore: An Empirical Study," PhD, Dissertation, Department of Economics, National University of Singapore, 200 1.

Cogley, Timothy and Nason, James M., "Effects of the Hodrick-Prescott Filter on Trend and Difference Stationary Time Series: Implications for Business Cycle Research," Journal of Economic Dynamics and Control, January-February 1995,253-278.

Engel, Eduardo M. R. A., "A Unified Approach to the Study of Sums, Products, Time-Aggregation and Other Functions of ARMA Processes," Journal of Time Series Analysis, Vol. 5, No. 3, 1984, 159-171.

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USE OF INNOVATION CORRELATIONS 45

Granger, Clive W. J. and Newbold, Paul, Forecasting Economic Time Series, 2nd Edition, San Diego: Academic Press, 1986.

Haugh, Larry D., "Checking the Independence of Two Covariance-Stationary Time Series: A Univariate Residual Cross-Correlation Approach," Journal of the American Statistical Association, June 1976, 378-385.

Kydland, Finn E. and Prescott, Edward C., "Business Cycles: Real Facts and a Monetary Myth," Federal Reserve Bank of Minneapolis Quarterly Review, Spring 1990,3-18.

Lucas, Robert E., "Understanding Business Cycles," in Karl Brunner and Allan H. Meltzer, eds., Stabilization of the Domestic and International Economy, Camegie-Rochester Conference Series on Public Policy, 1977.

Nelson, Charles R. and Schwert, G. William, "Tests for Predictive Relationships Between Time Series Variables: A Monte Carlo Investigation," Journal of the American Statistical Association, March 1982, 11 - 18.

Pierce, David A., "Relationships - and the Lack Thereof - Between Economic Time Series, with Special Reference to Money and Interest Rates," Journal of the American Statistical Association, March 1977, 11 -22.

Schwert, G. William, "Tests of Causality: The Message in the Innovations," in Karl Brunner and Allan H. Meltzer, eds., Three Aspects of Policymaking, Amsterdam: North Holland, 1979.

Mailing Address: Professor Tilak Abeysinghe, Department of Economics, National University gf Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore. Tel: 65-68746116, Fax: 65-7794065, e-mail: TilakAbey@nus. edu.sg Mailing Address: Dr. Choy Keen Meng, Department of Economics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore. Tel: 65-68744874, Fax: 65-7794065, E-mail: ecsckm@nus.edu.sg

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