Testing cointegrating coefficients in vector autoregressive error correction models

Economics Letters 58 (1998) 1–5

Testing cointegrating coefficients in vector autoregressive error correction models Gerd Hansen, Jeong-Ryeol Kim, Stefan Mittnik* Institute of Statistics and Econometrics, Christian Albrechts University at Kiel, Olshausenstr. 40, D-24098 Kiel, Germany Received 7 November 1996; accepted 19 June 1997

Abstract Tests of cointegrating coefficients in vector autoregressive error correction models ignore the Cauchy-like behavior of the estimator’s finite-sample distribution. This causes excessive rejections of the null in standard x 2 tests. We propose a Cauchy-based x 2 test, and show, via simulation, that it yields adequate rejection rates.  1998 Elsevier Science S.A. Keywords: Cointegration; Error correction model; Granger causality; Chi-square test; Cauchy distribution JEL classification: C12; C22; C32

1. Introduction Testing for causality is a central issue in macroeconometrics. Since Granger (1969) introduced an operational concept of causality, the Wald test has been widely used for testing zero restrictions implying Granger-noncausality. It is especially important to find causal structures in vector autoregressive (VAR) analyses, because all variables are – in the sense of Sims (1980) – assumed to be endogenous. In view of the fact that most macroeconomic variables are nonstationary, Engle and Granger (1987) introduced the concepts of cointegration and error correction. Johansen (1988) proposed a vector autoregressive error correction model (VEC) based on canonical correlation and full information maximum likelihood (FIML) estimation. Ahn and Reinsel (1990) investigated the asymptotic distribution of the FIML estimator of VEC coefficients when variables are nonstationary. The efficiency of this estimator was analyzed in Saikkonen (1991). Toda and Phillips (1994) considered causality testing in a VEC and showed that both the Wald statistic for long-run and that for short-run Granger-noncausality are asymptotically x 2 distributed. Johansen and Juselius (1994) found that this also holds for the likelihood-ratio test statistic when testing overidentifying zero restrictions on cointegrating coefficients. Investigating the finite-sample distribution of the FIML estimator of cointegrating coefficients in a VEC Phillips (1994) found that the reduced rank regression estimator has a heavy-tailed distribution *Corresponding author. Tel.: 149 431 8802166; fax: 149 431 8802673; e-mail: [email protected] 0165-1765 / 98 / $19.00  1998 Elsevier Science S.A. All rights reserved. PII S0165-1765( 97 )00199-7

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in finite samples. Simulations in Hansen and Kim (1996) confirm this phenomenon by yielding an excessively large number of seemingly ‘‘outlying’’ estimates for cointegrating coefficients. In view of the non-normality of the finite-sample FIML estimator of VEC cointegrating coefficients, a standard x 2 test for testing cointegrating coefficients is inappropriate. The usual x 2 distribution is a poor approximation of the finite-sample distribution of the test statistic, inducing an excessive rejection tendency. Since the finite sample properties derived in Phillips (1994) also carry over to the likelihood-ratio test statistic, the use of the usual x 2 distribution is also inappropriate for this test. In this paper we consider a modified x 2 test of cointegrating coefficients which takes their heavy-tailed finite-sample distribution explicitly into account. The modification makes use of a x 2 -type distribution for heavy-tailed random variables investigated in Mittnik et al. (1996) which enables us to obtain more appropriate critical values for the x 2 statistic. The paper is organized as follows. Section 2 discusses the asymptotic and finite-sample distributions of the reduced rank regression estimator of the cointegrating coefficients in a VEC. The modified x 2 test and simulation results are presented in Section 3. Section 4 concludes with a brief empirical application using German macroeconomic data.

2. The FIML estimator for cointegrating coefficients in a VEC Johansen (1988) introduced the FIML estimation of the cointegrating vectors in a VAR process. The error correction form of a K-dimensional VAR process of order p with r cointegrating restrictions can be written as A(L)Dy t 5 BCy t2p 1 u t , t 5 1, . . . ,T, p 21

(1)

i

where A(L)5o i 50 A i L with A 0 5IK ; D denotes difference operator; and u t |iid N(0,S ). In (1), B is a K 3r matrix of loading coefficients and C an r3K matrix of cointegrating vectors. The reduced rank ˆ is obtained by minimizing determinant, uS00 2 regression estimator of C, denoted by C, 21 S0p C9(CSpp C9 ) CSp 0 u with respect to C, where Sij (i, j50, p) are the product moment matrices of the residuals from regressing Dy t and y t21 on Dy t 21 , . . . , Dy t2p 11 (see Johansen, 1988, p. 234). Johansen (1988) showed that this minimization is equivalent to solving the eigenvalue problem u lSpp 2 21 21 Sp 0 S 00 S0p u50. Letting l1 $ ? ? ? $ lK be the eigenvalues obtained by solving u lSpp 2Sp 0 S 00 S0p u50 and v1 , . . . , vK the corresponding orthonormal eigenvectors, the estimator of the cointegrating vectors ˆ To ensure the empirical associated with the r greatest eigenvectors is given by [v1 , . . . , vr ]9C. ˆ each of the unrestricted cointegrating vectors is to be normalized such that uniqueness of C, ˆ pp Cˆ 9 5 Ir . CS

(2)

These cointegrating vectors are generally not identified. In the rank-r case each cointegrating vector has to satisfy r21 linear identifying restrictions. Its asymptotic distribution is a mixture of normal distributions, namely two independent Brownian motions, implying that hypothesis tests can be conducted asymptotically with standard x 2 tests (see Johansen, 1988). Phillips (1994) analyzed the tail behavior of the finite sample distribution of the identified (normalized) cointegrating estimator (2) of the reduced rank regression and found it to be Cauchy-like ˆ distributed. Partitioning C5[C1 C2 ] the reduced rank estimator in (2) can be re-written as C5[I r

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21 21 Cˆ 1 Cˆ 2 ]. The distribution of the reduced rank regression estimate Cˆ 1 Cˆ 2 amounts to a quotient of independent normal variables which has a Cauchy-like distribution (see Phillips, 1994, Theorem 2.1 and Corollary 2.4).

3. A modified x 2 test To deal with problems arising from the poor approximation of the usual x 2 distribution for the Wald and likelihood ratio test statistics, arising from the heavy-tailedness of the reduced rank regression estimator for VEC cointegrating coefficients in finite samples, we suggest the use of a modified x 2 distribution, which is based on sums of squared standard Cauchy-distributed random variates. Mittnik et al. (1996) consider a more general class of x 2 -type distributions based on sums of squared stable Paretian random variates, which encompasses normal and Cauchy variates as special cases. Relying on simulations and response surface analysis they approximate the logarithm of the g -quantile of the Cauchy-based x 2 -type distribution with n degrees of freedom, denoted by lnC x g2 (n ), by ln C x 2g (n ) ¯ 1.5725 ln x 2g (n ) 2 ln(1 2 g )[0.6317 ln x 2g (n ) 2 0.0434(ln x 2g (n ))2 ],

(3)

where ln x 2g (n ) denotes the logarithm of the corresponding quantile of the usual (normal-based) x 2 distribution (see Mittnik et al., 1996, for details). Response surface (3) provides a rather close fit for g [[0.9, 0.99] and n 51, . . . , 30, allowing us to specify approximate critical values relevant in empirical work. In a small simulation study we illustrate the tail behavior of the finite-sample FIML estimator of VEC cointegrating coefficients and compare the performance of the standard and modified x 2 tests. For this purpose, we specify the following five-variable cointegrated model, which is in the triangular ECM-format, of Phillips (1991), as data generating process (DGP):

F

1 0.5

GF G F

2 0.5 1

y 1t u11 5 y 2t u21

u12 u13 u22 u23

x 1t u x 2t 1 1t , u 2t x 3t

G3 4 F G

(4)

where Dx 1t 5e 1t , Dx 2t 5e 2t , Dx 3t 5e 3t ; [u 1t , u 2t , e 1t , e 2t , e 3t ]95 et 1 Fet21 with et |iidN(0, S ); 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 F 5 0.5 0.5 0.5 0.5 0.5 ; 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

3

1 0.5 5 0.5 0.5 0

4O3

0.5 1 0 0 0.5

0.5 0 1 0 0

0.5 0 0 1 0

0 0.5 0 . 0 1

4

In (4), we set u11 5u12 5u23 51 and, to obtain just identified cointegrating vectors, u13 5u21 50. To 2 examine the size of the modified x test we set u22 50. DGP (4) has two endogenous and – in the long-run sense of Dolado (1992) – three weakly exogenous variables. From (4) we generate 10 000 samples of size T5100 and perform a usual x 2 test for the null hypothesis u22 50 after imposing zero restrictions on u13 and u21 as just identifying restriction. Fig. 1 presents the center of a normal probability plot of the 10 000 u22 estimates. A comparison of

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Fig. 1. Normal probability plot of estimated cointegrating coefficients.

the empirical distribution, represented by the fat curve, and the normal distribution, represented by the straight line, shows that normality is clearly violated. In accordance with Phillips’ (Phillips, 1994) analysis we see that the empirical distribution is fat-tailed and appears, indeed, Cauchy-like. Using critical values of the usual x 2 distribution for the Wald test we find an excessive number of (wrong) rejections of the null. For example, at the 5% level we would reject the null in 24.8% of the 10 000 cases. When using critical values of the Cauchy-based C x 2 distribution we reject 4.8% of the cases.1 The simulation results indicate that the use of Cauchy-based rather than normal-based critical x 2 values leads to a much more appropriately sized test.

4. Empirical example To briefly illustrate how the heavy-tailedness of the estimator of cointegrating coefficients can affect conclusions in applied work, we consider a four-dimensional VEC consisting of quarterly German unemployment, inflation, real wages, and import prices. The sample covers the period 1967:I–1994:IV.2 We specify a VAR(1) model and perform the Wald test of Toda and Phillips (1994) in a reparameterized VEC with one cointegrating vector. Testing the null hypothesis of Grangernoncausality from unemployment to import prices we obtain a value of 15.42 for the Wald statistic. This corresponds to a p-value of 8.61310 25 when using the normal-based x 2 distribution. Thus, at all practically relevant significance levels we would reject what seems to be a rather reasonable null hypothesis, namely that German unemployment does not Granger-cause import prices. Taking the Cauchy-like distribution of the estimator into account and employing response surface (3) to 1

We obtain similar results when considering different sample sizes and alternative specifications in (4). ¨ Wirtschaftsforschung, Berlin, The data are from the quarterly national accounts as published by the Deutsches Institut f ur 1995.

2

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approximate the Cauchy-based C x 2 distribution we obtain a p-value of 0.108. Thus, we would not reject the null at usual significance levels.

5. Concluding remarks In the empirical example in Section 4 we specified a VAR(1) model. Note that in a first-order cointegrated VAR we have only long-run effects and no short-run dynamics so that the Toda and Phillips (1994) test statistic involves only long-run coefficients. For higher-order systems one should test long-run and short-run restrictions separately and use the x 2 distribution for testing the former and the usual x 2 distribution for the latter. More detailed analyses of size and power properties of the modified test proposed here are under investigation.

Acknowledgements Research support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

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