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Testing for contemporaneous correlation of disturbances in seemingly unrelated regressions with serial dependence
Economics Letters 83 (2004) 69 – 76 www.elsevier.com/locate/econbase
Testing for contemporaneous correlation of disturbances in seemingly unrelated regressions with serial dependence Wen-Jen Tsay * The Institute of Economics, Academia Sinica, 128 Academia Road, Sec. 2, Taipei, Taiwan Received 14 October 2003; accepted 17 October 2003
Abstract This paper develops a multivariate independent (MI) statistic which can conveniently and powerfully test the independence between the disturbances of a seemingly unrelated regressions (SURE) model. The implementation of our test is straightforward and can be carried out by standard statistics packages. As compared to the results in Tables 2 and 3 of Dufour and Khalaf [Journal of Econometrics 106 (2002) 143], our MI test performs reasonably well, even though the sample size is only 25. D 2004 Elsevier B.V. All rights reserved. Keywords: Multivariate independent test; SURE model JEL classification: C30
1. Introduction This paper considers the independence test for the disturbances of the seemingly unrelated regressions (SURE) model proposed by Zellner (1962). It is well known that the generalized least squares (GLS) estimator, which exploits the correlations of the disturbances of the SURE model, may be more efficient than its ordinary least squares (OLS) counterpart. However, the GLS estimator is equivalent to the OLS estimator if the error covariance of the SURE model is diagonal. Therefore, many independence tests for the disturbances of multivariate models have been proposed in the literature, including the Lagrange multiplier (LM) test of Breusch and Pagan (1980), the locally best invariant test of Kariya (1981), and the exact independence test of Harvey and Phillips (1982), among others.1
* Tel.: +886-2-27822791x296; fax: +886-2-27853946. E-mail address: [email protected] (W.-J. Tsay). 1 Please refer to Dufour and Khalaf (2002) for more details. 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2003.10.005
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W.-J. Tsay / Economics Letters 83 (2004) 69–76
Dufour and Khalaf (2002) show that standard asymptotic tests (e.g., Breusch and Pagan’s (1980) LM test and Shiba and Tsurumi’s (1988) likelihood ratio (LR) test) exhibit important size distortions, while the Monte Carlo (MC) exact tests proposed by them achieve complete size control. However, the exact tests considered by Dufour and Khalaf (2002) are more computationally intensive than standard asymptotic tests. To easily and powerfully test that the disturbances across different regressions of a SURE model are independent of each other, we propose a multivariate independent (MI) statistic which can be carried out by standard statistics packages. Moreover, our MI test can be applied to the SURE model with serially dependent disturbances, thus making the coverage of the MI test broader than that of Dufour and Khalaf (2002). Under the null hypothesis that the disturbances are independent of each other, as the sample size T approaches infinity, the MI test possesses a v2 distribution with p( p 1)/2 degrees of freedom as Breusch and Pagan’s (1980) LM test does. Compared to the exact tests of Dufour and Khalaf (2002), our MI test is an asymptotic test. Nevertheless, the small sample performance of our MI test does extremely well. In particular, the size control of our MI test is comparable to that of exact tests even though T is only 25. As T = 100, the power performance of our MI test reaches 100% under all the four alternatives considered in Table 3 of Dufour and Khalaf (2002). This indicates that the MI test is a convenient alternative as opposed to the computationally more intensive exact tests of Dufour and Khalaf (2002) in detecting the independence of the SURE model’s disturbances. The plan of this paper is as follows: Section 2 presents the test statistic and the main results. Section 3 illustrates the small sample performance of the MI test. Section 4 provides a conclusion.
2. The MI test and the main results Consider the following SURE model i ¼ 1; : : : ; p; Yi ¼ Xi b þ ei ;
ð1Þ
i
where Yi is a vector of T observations on a dependent variable, Xi is a full-rank T ki matrix of regressors, bi is ki 1 unknown coefficients, and ei is mean zero T 1 random disturbances. The SURE model is first proposed by Zellner (1962) and has been considered by Anderson (1984); Kariya (1981), and Dufour and Khalaf (2002), among others. Model (1) can be rewritten in stacked form Y ¼ X b þ e; where 2 3 Y1 6 7 6 7 6 Y2 7 6 7 7 Y ¼6 6 7; 6]7 6 7 4 5 Yp
2
X1
6 6 6 0 6 X ¼6 6 6 ] 6 4 0
0
:::
X2
:::
]
O
0
:::
0
3
7 7 0 7 7 7; 7 ] 7 7 5 Xp
2
b1
3
6 7 6 7 6 b2 7 6 7 7 b¼6 6 7; 6]7 6 7 4 5 bp
2
e1
3
6 7 6 7 6 e2 7 6 7 7 e¼6 6 7: 6]7 6 7 4 5 ep
W.-J. Tsay / Economics Letters 83 (2004) 69–76
71
We impose the following Assumption 1 throughout this paper. Assumption 1. i) e.t=(e1,t,. . .,ep,t)Vis a p-dimensional disturbance vector for the t-th observation; (ii) each element of e.t has an infinite order moving average (MA) representation which includes stationary autoregressive (AR) and stationary and invertible ARMA processes as the special cases, l i.e., X wi;h ai;th ; ei;t ¼ P h¼0 where l h¼0 Awi;h A < l and wi,0 = 1 for i = 1,. . ., p; (iii) at=(a1,t,. . .,ap,t)V is a p-dimensional inde2 2 ) = ra,i > 0 for pendently and identically distributed (i.i.d.) process with E(at) = 0, and E(ai,t i = 1,. . .,p; (iv) ei,t and Xj,t (the t-th row of Xj) satisfy the conditions in items (ii), (iii), and (iv) of White’s (2001) Theorem 5.17, for all i,j = 1,. . .,p; in particular, we require that {(ei,t , Xj,t)} is a stationary ergodic sequence for all i, j = 1,. . .,p, among other conditions. Assumption 1 reveals that we allow the disturbances of the SURE model to be weakly dependent processes. Accordingly, the coverage of our Assumption 1 is broader than that in (2.5)–(2.7) of Dufour and Khalaf (2002, p. 147). This generalization is especially useful when the observations are time series data. In fact, the conditions in (2.10) of Dufour and Khalaf (2002) are even more restrictive than ours, because Dufour and Khalaf (2002) further assume that e.t is a normal and i.i.d. (n.i.d) process. In the literature the assumption that e.t is an n.i.d process also has been proposed by Breusch and Pagan (1980). Dufour and Khalaf (2002, p. 163) themselves even use a Gaussian distributional assumption in their application to the growth equation. Another feature of Assumption 1 is that it does not assume that e is independent of X as compared to (2.8) of Dufour and Khalaf (2002), where they require X to be strictly exogeneous or fixed. Instead, it assumes that the regressors Xi are also dependent observations, because the disturbances are allowed to be covariance stationary. To attain this generality of the data generating processes (DGP), we require that each pair of the disturbances ei and regressor Xj (i, j = 1,. . .,p) in the SURE model satisfy the conditions imposed in items (ii), (iii), and (iv) of White’s (2001) Theorem 5.17, which has been used to derive the asymptotic properties of the OLS estimator when the DGP are dependent identically distributed observations.2 In the current paper, item 4 ensures that the OLS estimator from Eq. (1) and the statistic defined later are well behaved asymptotically. We shall provide more details later. The objective of this paper is to propose a multivariate independent (MI) statistic which can test the null hypothesis that the elements in e.t are independent of each other. Therefore, under the null hypothesis, Se u E(e.t e.tV) will be a diagonal matrix, i.e., 3 2 2 r1 0 : : : 0 7 6 7 6 2 ::: 6 07 X 6 0 r2 7 7; H0 : ¼6 7 6 6] ] O ]7 e 7 6 5 4 2 : : : 0 0 rp 2
The conditions in items (ii) and (iii) of our Assumption 1 jointly guarantee that ei,t is an adapted mixingale of size 1, as is imposed in item (iii)(a) of White’s (2001) Theorem 5.17.
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W.-J. Tsay / Economics Letters 83 (2004) 69–76
where r2i
¼
Eðe2i;t Þ
¼
l X
! w2i;h
r2a;i > 0:
h¼0
The idea behind the MI test is mainly based on two observations: First, if the elements in e.t are mutually independent, then Zij,t u ei,tej,t will be uncorrelated with Zmn,t u em,ten,t as long as i p m or j p n. Thus, all the p( p 1)/2 combinations of ei,t ej,t (i p j) among these p disturbances e.tV=(e1,t. . ., ep,t) will also be uncorrelated PT with each other. Second, if the elements in e.t are mutually independent, 1=2 then as T ! l, T t¼1 ei;t ej;t Z N ð0; Xij Þ by adapting Theorem 7.7.8 of Anderson (1971, p. 429), where Xij denotes the long run variance of ei,tej,t, and Z denotes convergence in distribution. Given the two preceding observations and that the elements in e.t are mutually independent, we have KuT
1=2
T X
Zt Z N ð0; XÞ;
t¼1
where ZtV ¼ ðZ12;t ; Z13;t ;: : :; Z1p;t ; Z23;t ; Z24;t ;: : :; Z2p;t ;: : :; Zðp1Þp;t Þ;
Zij;t ¼ ei;t ej;t :
ð2Þ
Here, X is a p( p 1)/2 p( p 1)/2 matrix such that 2
X12
6 6 60 6 Xu6 6 6] 6 4 0
:::
0
X13 : : :
0
]
O
]
0
:::
Xðp1Þp
0
3 7 7 7 7 7: 7 7 7 5
ð3Þ
In an empirical application, we do not observe ei in Eq. (1) directly. We instead collect the OLS residuals from these p regression models, respectively, i.e., eiV=(ei1,ei2,. . .,eiT), i = 1,. . ., p, or ei ¼ ½IT Xi ðXiVXi Þ1 XiV Yi ;
i ¼ 1;: : :; p;
where IT is a T T identity matrix. With the OLS residuals at hand, we now can define the MI test: ˆ ˆ Xˆ 1 K; MIuKV
ð4Þ
W.-J. Tsay / Economics Letters 83 (2004) 69–76
73
where 1=2 ˆ KuT
T X
Zˆ t ;
t¼1
Zˆ t V ¼ Zˆ 12;t ; Zˆ 13;t ;: : :; Zˆ 1p;t ; Zˆ 23;t ; Zˆ 24;t ;: : :; Zˆ 2p;t ;: : :; Zˆ ðp1Þp;t ; 2
Xˆ 12
:::
0
6 6 60 6 ˆ Xu6 6 6] 6 4
0
Zˆ ij;t ¼ ei;t ej;t ;
3
0
Xˆ 13 : : :
0
]
O
]
0
:::
Xˆ ðp1Þp
7 7 7 7 7; 7 7 7 5
ð5Þ
such that Xˆ ij ¼
T1 X
ð6Þ
Ci;h Cj;h ;
h¼Tþ1
and Ci;h ¼ T 1
X tðhÞ
ei;t ei;tþh ;
Cj;h ¼ T 1
X
ej;t ej;tþh :
ð7Þ
tðhÞ
Here, At(h) is a sum over 1 V t, t + h V T. The estimator Xˆ defined in Eq. (5) is used to estimate X presented in Eq. (3). To derive the asymptoticpproperties of the MI test, we note that the OLS estimator bˆi from Eq. (1) ffiffiffiffi P P T T 1=2 1=2 ˆ possesses that t¼1 ei;t ej;t T t¼1 ei;t ej;t ¼ op ð1Þ, or PT the property PT T ðbi bi Þ ¼ Op ð1Þ and T 1=2 1=2 ˆ holds. When the T t¼1 Zij;t T t¼1 Zij;t ¼ op ð1Þ , given that item (iv) of Assumption 1P T 1=2 ˆ ˆ disturbances are independent of each other and Assumption 1 holds, we have KuT t¼1 Zt Z N ð0; XÞ. Provided that we can consistently estimate X, the asymptotic properties of the MI test when the disturbances are independent of each other can then be obtained. According to Eqs. (6) and (7), we realize that each element in the principle diagonal of Xˆ is based on the covariance matrix estimator proposed by Robinson (1998), where he intended to estimate the long run variance of the product of two independent stationary long memory processes. Because each element in the principle diagonal of Xˆ is constructed with the product of two independent weakly dependent processes, the justification for applying pRobinson’s covariance estimator in the current paper can be established. The details to prove that Xˆ ! X when Assumption 1 holds and that the elements in e.t are independent of each other are omitted. Combining the preceding results, asymptotically we can see that the MI test is v2 distributed and the results are presented in the following Theorem 1.
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W.-J. Tsay / Economics Letters 83 (2004) 69–76
Table 1 Rejection percentages of the MI test when Ae is a diagonal matrix T
k=5
k=6
k=7
k=8
k=9
k = 10
k = 11
k = 12
k = 13
k = 14
k = 15
25 50 100
2.9 3.3 5.5
2.9 3.7 5.2
3.0 3.7 5.2
2.9 3.2 5.2
2.7 4.2 5.7
2.7 3.6 4.1
3.5 4.1 3.9
3.4 2.7 5.1
2.4 3.4 4.9
4.2 4.1 4.4
2.3 3.9 5.7
2 The results are all based on 1000 replications and p = 5. The critical values used are obtained from the vp( p 1)/2 distribution at the 5% level of significance.
Theorem 1. Given that the conditions in Assumption 1 hold and the disturbances across different regressions of a SURE model are independent of each other, then as T ! l, MI Z v2p(p1)/2. Theorem 1 clearly indicates that our MI test is an asymptotic test as opposed to the exact test of Dufour and Khalaf (2002). However, there are several advantages to using our MI test. First, the critical values of the v2p( p 1)/2 distribution can be used for testing, implying that we do not need to calculate the critical values for a specific sample size, although the disturbances are generalized to be weakly dependent processes. Second, the MI test is very flexible, because not only can we test that Ae is a diagonal matrix, but we can also test the hypothesis that Ae is a block-diagonal matrix. Third, the MI test utilizes all the p( p 1)/2 combinations of the p groups of OLS residuals, but we only need to calculate the p groups of autocovariance functions from these p groups of OLS residuals, respectively. Therefore, the burden of calculating Xˆ for the MI test only increases linearly with the magnitude of p, even though there are p( p 1)/2 pairs of long run variances that need to be estimated. Finally, because Xˆ does not use any kernel function, or bandwidth parameter, or AR filter, the trouble with deciding the ‘‘best’’ combination of the kernel function and bandwidth parameter, or the ‘‘optimal’’ AR lag length, does not exist any more.
3. Monte Carlo experiment As shown previously, the MI test can be applied to the SURE model with serially dependent disturbances. For the purpose of comparison, in this section, we follow the design of Dufour and Khalaf (2002) where the disturbances are i.i.d. processes so as to demonstrate the small sample performance of the MI test. We also thank an anonymous referee for pointing out that our Monte Carlo design does not allow for serial dependence in each disturbance, because the experimental design of Dufour and Khalaf (2002) is used as the benchmark of our studies. Specifically, our Monte Carlo experiment is based on 1000 replications with different sample sizes. Two hundred additional values are generated in order to obtain random starting values. In each experiment, the design matrices Xi, i = 1,. . .,p, contain a constant term and an equal number of regressors (ki = k, i = 1,. . ., p). The value of k considered is k = 5,. . ., 15. The variables Xi are generated using a multivariate normal distribution and are kept constant over all replications. The disturbances are also generated from multivariate normal distributions.3 Table 1 illustrates the size performance of the MI test when the critical values used are obtained from 2 the vp( p 1)/2 distribution at the 5% level of significance. The experimental design of our Table 1 is 3
Please check Dufour and Khalaf (2002, p. 159) for the details of their experimental design.
W.-J. Tsay / Economics Letters 83 (2004) 69–76
75
Table 2 Rejection percentages of the MI test when Ae is not a diagonal matrix K=p=5
A0(H0)
A1
A2
A3
A4
T = 25 T = 50 T = 100
2.9 3.3 5.5
99.8 100 100
29.5 86.1 100
81.9 100 100
54.1 98.6 100
2 The results are all based on 1000 replications. The critical values used are obtained from the vp( p 1)/2 distribution at the 5% level of significance.
identical to that in Table 2 of Dufour and Khalaf (2002, p.161), where Dufour and Khalaf (2002) apply the technique of the MC tests of Dwass (1957) and Barnard (1963) to implement exact tests. As compared to the results in Table 2 of Dufour and Khalaf (2002), we find that there is some size distortion of our MI test. However, this size distortion is very mild (about 2% as T = 25), and the difference between our MI test and the MC tests decreases as the sample size increases. When T = 100, the size control of the MI test is equivalent to that of the MC exact tests. Table 2 demonstrates the power performance of the MI test under the four alternatives, A1, A2, A3, and A4, considered in Table 3 of Dufour and Khalaf (2002).4 As T = 25, the power performance of the MI test is close to that of the MC exact tests of Dufour and Khalaf (2002) under three alternatives, A1, A3, and A4. The power performance of the MI test is not that well when the sample size is only 25 and Ae = A2. However, as the sample size increases to be 100, the rejection rates of the MI test under these four alternatives all reach 100%.5 Combining the results in Tables 1 and 2, we note that the small sample performance of our MI test is quite satisfactory, even though the sample size is moderately small.
4. Conclusion This paper develops a multivariate independent (MI) test which can conveniently and powerfully test the independence between the disturbances of the SURE model. The coverage of the MI test is broader than the existing independence tests for the disturbances of the SURE model, because the disturbances are now extended to be weakly dependent processes. Moreover, the implementation of our MI test is straightforward, because it does not use any kernel function, or bandwidth parameter, or AR filter. The Monte Carlo experiment conducted in this paper also shows that the small sample performance of the MI test is comparable to that of exact tests considered in Tables 2 and 3 of Dufour and Khalaf (2002, p. 161). This indicates that the MI test can be viewed as a convenient alternative as opposed to the computationally more intensive exact tests in detecting the independence of the SURE model’s disturbances. Nevertheless, as pointed out by an anonymous referee, for a less regular regressor Xi, the performance of our MI test might deteriorate. Furthermore, under Assumption 1, according to the results of Dufour (2003), Faust (1996, 1999), and Po¨tscher (2002), the size of our asymptotic test will converge to one, which is not the size under the asymptotic distribution. Although to find the sufficient conditions for solving the asymptotic size problem of the MI test will be left for our future studies, the value of using the exact tests of Dufour and Khalaf (2002) is still important in empirical applications. 4 5
The exact specifications of A1, A2, A3, and A4 are displayed in Table 1 of Dufour and Khalaf (2002). Dufour and Khalaf (2002) only consider the power performance of their exact tests as T = 25.
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W.-J. Tsay / Economics Letters 83 (2004) 69–76
Acknowledgements We thank an anonymous referee for his or her valuable comments and suggestions.
References Anderson, T.W., 1971. The Statistical Analysis of Time Series Wiley, New York. Anderson, T.W., 1984. An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley, New York. Barnard, G.A., 1963. Comment on ‘The spectral analysis of point processes’ by M.S. Bartlett. Journal of the Royal Statistical Society. Series B 25, 294. Breusch, T.S., Pagan, A.R., 1980. The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies 47, 239 – 254. Dufour, J.-M., 2003. Identification, weak instruments and statistical inference in econometrics. Canadian Journal of Economics 36, 767 – 808. Dufour, J.-M., Khalaf, L., 2002. Exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions. Journal of Econometrics 106, 143 – 170. Dwass, M., 1957. Modified randomization tests for nonparametric hypotheses. Annals of Mathematical Statistics 28, 181 – 187. Faust, J., 1996. Near observational equivalence problems and theoretical size problems with unit root tests. Econometric Theory 12, 724 – 732. Faust, J., 1999. Theoretical confidence level problems with confidence intervals for the spectrum of a time series. Econometrica 67, 629 – 637. Harvey, A.C., Phillips, G.D.A., 1982. Testing for contemporaneous correlation of disturbances in systems of regression equations. Bulletin of Economic Research 34, 79 – 81. Kariya, T., 1981. Tests for the independence between two seemingly unrelated regression equations. Annals of Statistics 9, 381 – 390. Po¨tscher, B., 2002. Lower risk bounds and properties of confidence sets for ill-posed estimation problems with applications to spectral density and persistence estimation, unit roots and estimation of long memory parameters. Econometrica 70, 1035 – 1065. Robinson, P.M., 1998. Inference-without-smoothing in the presence of nonparametric autocorrelation. Econometrica 66, 1163 – 1182. Shiba, T., Tsurumi, H., 1988. Bayesian and non-Bayesian tests of independence in seemingly unrelated regressions. International Economic Review 29, 377 – 389. White, H., 2001. Asymptotic Theory for Econometricians, revised ed. Academic Press, London. Zellner, A., 1962. An efficient method for estimating seemingly unrelated regressions and tests for aggregate bias. Journal of the American Statistical Association 57, 348 – 368.
Testing for contemporaneous correlation of disturbances in seemingly unrelated regressions with serial dependence Wen-Jen Tsay * The Institute of Economics, Academia Sinica, 128 Academia Road, Sec. 2, Taipei, Taiwan Received 14 October 2003; accepted 17 October 2003
Abstract This paper develops a multivariate independent (MI) statistic which can conveniently and powerfully test the independence between the disturbances of a seemingly unrelated regressions (SURE) model. The implementation of our test is straightforward and can be carried out by standard statistics packages. As compared to the results in Tables 2 and 3 of Dufour and Khalaf [Journal of Econometrics 106 (2002) 143], our MI test performs reasonably well, even though the sample size is only 25. D 2004 Elsevier B.V. All rights reserved. Keywords: Multivariate independent test; SURE model JEL classification: C30
1. Introduction This paper considers the independence test for the disturbances of the seemingly unrelated regressions (SURE) model proposed by Zellner (1962). It is well known that the generalized least squares (GLS) estimator, which exploits the correlations of the disturbances of the SURE model, may be more efficient than its ordinary least squares (OLS) counterpart. However, the GLS estimator is equivalent to the OLS estimator if the error covariance of the SURE model is diagonal. Therefore, many independence tests for the disturbances of multivariate models have been proposed in the literature, including the Lagrange multiplier (LM) test of Breusch and Pagan (1980), the locally best invariant test of Kariya (1981), and the exact independence test of Harvey and Phillips (1982), among others.1
* Tel.: +886-2-27822791x296; fax: +886-2-27853946. E-mail address: [email protected] (W.-J. Tsay). 1 Please refer to Dufour and Khalaf (2002) for more details. 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2003.10.005
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W.-J. Tsay / Economics Letters 83 (2004) 69–76
Dufour and Khalaf (2002) show that standard asymptotic tests (e.g., Breusch and Pagan’s (1980) LM test and Shiba and Tsurumi’s (1988) likelihood ratio (LR) test) exhibit important size distortions, while the Monte Carlo (MC) exact tests proposed by them achieve complete size control. However, the exact tests considered by Dufour and Khalaf (2002) are more computationally intensive than standard asymptotic tests. To easily and powerfully test that the disturbances across different regressions of a SURE model are independent of each other, we propose a multivariate independent (MI) statistic which can be carried out by standard statistics packages. Moreover, our MI test can be applied to the SURE model with serially dependent disturbances, thus making the coverage of the MI test broader than that of Dufour and Khalaf (2002). Under the null hypothesis that the disturbances are independent of each other, as the sample size T approaches infinity, the MI test possesses a v2 distribution with p( p 1)/2 degrees of freedom as Breusch and Pagan’s (1980) LM test does. Compared to the exact tests of Dufour and Khalaf (2002), our MI test is an asymptotic test. Nevertheless, the small sample performance of our MI test does extremely well. In particular, the size control of our MI test is comparable to that of exact tests even though T is only 25. As T = 100, the power performance of our MI test reaches 100% under all the four alternatives considered in Table 3 of Dufour and Khalaf (2002). This indicates that the MI test is a convenient alternative as opposed to the computationally more intensive exact tests of Dufour and Khalaf (2002) in detecting the independence of the SURE model’s disturbances. The plan of this paper is as follows: Section 2 presents the test statistic and the main results. Section 3 illustrates the small sample performance of the MI test. Section 4 provides a conclusion.
2. The MI test and the main results Consider the following SURE model i ¼ 1; : : : ; p; Yi ¼ Xi b þ ei ;
ð1Þ
i
where Yi is a vector of T observations on a dependent variable, Xi is a full-rank T ki matrix of regressors, bi is ki 1 unknown coefficients, and ei is mean zero T 1 random disturbances. The SURE model is first proposed by Zellner (1962) and has been considered by Anderson (1984); Kariya (1981), and Dufour and Khalaf (2002), among others. Model (1) can be rewritten in stacked form Y ¼ X b þ e; where 2 3 Y1 6 7 6 7 6 Y2 7 6 7 7 Y ¼6 6 7; 6]7 6 7 4 5 Yp
2
X1
6 6 6 0 6 X ¼6 6 6 ] 6 4 0
0
:::
X2
:::
]
O
0
:::
0
3
7 7 0 7 7 7; 7 ] 7 7 5 Xp
2
b1
3
6 7 6 7 6 b2 7 6 7 7 b¼6 6 7; 6]7 6 7 4 5 bp
2
e1
3
6 7 6 7 6 e2 7 6 7 7 e¼6 6 7: 6]7 6 7 4 5 ep
W.-J. Tsay / Economics Letters 83 (2004) 69–76
71
We impose the following Assumption 1 throughout this paper. Assumption 1. i) e.t=(e1,t,. . .,ep,t)Vis a p-dimensional disturbance vector for the t-th observation; (ii) each element of e.t has an infinite order moving average (MA) representation which includes stationary autoregressive (AR) and stationary and invertible ARMA processes as the special cases, l i.e., X wi;h ai;th ; ei;t ¼ P h¼0 where l h¼0 Awi;h A < l and wi,0 = 1 for i = 1,. . ., p; (iii) at=(a1,t,. . .,ap,t)V is a p-dimensional inde2 2 ) = ra,i > 0 for pendently and identically distributed (i.i.d.) process with E(at) = 0, and E(ai,t i = 1,. . .,p; (iv) ei,t and Xj,t (the t-th row of Xj) satisfy the conditions in items (ii), (iii), and (iv) of White’s (2001) Theorem 5.17, for all i,j = 1,. . .,p; in particular, we require that {(ei,t , Xj,t)} is a stationary ergodic sequence for all i, j = 1,. . .,p, among other conditions. Assumption 1 reveals that we allow the disturbances of the SURE model to be weakly dependent processes. Accordingly, the coverage of our Assumption 1 is broader than that in (2.5)–(2.7) of Dufour and Khalaf (2002, p. 147). This generalization is especially useful when the observations are time series data. In fact, the conditions in (2.10) of Dufour and Khalaf (2002) are even more restrictive than ours, because Dufour and Khalaf (2002) further assume that e.t is a normal and i.i.d. (n.i.d) process. In the literature the assumption that e.t is an n.i.d process also has been proposed by Breusch and Pagan (1980). Dufour and Khalaf (2002, p. 163) themselves even use a Gaussian distributional assumption in their application to the growth equation. Another feature of Assumption 1 is that it does not assume that e is independent of X as compared to (2.8) of Dufour and Khalaf (2002), where they require X to be strictly exogeneous or fixed. Instead, it assumes that the regressors Xi are also dependent observations, because the disturbances are allowed to be covariance stationary. To attain this generality of the data generating processes (DGP), we require that each pair of the disturbances ei and regressor Xj (i, j = 1,. . .,p) in the SURE model satisfy the conditions imposed in items (ii), (iii), and (iv) of White’s (2001) Theorem 5.17, which has been used to derive the asymptotic properties of the OLS estimator when the DGP are dependent identically distributed observations.2 In the current paper, item 4 ensures that the OLS estimator from Eq. (1) and the statistic defined later are well behaved asymptotically. We shall provide more details later. The objective of this paper is to propose a multivariate independent (MI) statistic which can test the null hypothesis that the elements in e.t are independent of each other. Therefore, under the null hypothesis, Se u E(e.t e.tV) will be a diagonal matrix, i.e., 3 2 2 r1 0 : : : 0 7 6 7 6 2 ::: 6 07 X 6 0 r2 7 7; H0 : ¼6 7 6 6] ] O ]7 e 7 6 5 4 2 : : : 0 0 rp 2
The conditions in items (ii) and (iii) of our Assumption 1 jointly guarantee that ei,t is an adapted mixingale of size 1, as is imposed in item (iii)(a) of White’s (2001) Theorem 5.17.
72
W.-J. Tsay / Economics Letters 83 (2004) 69–76
where r2i
¼
Eðe2i;t Þ
¼
l X
! w2i;h
r2a;i > 0:
h¼0
The idea behind the MI test is mainly based on two observations: First, if the elements in e.t are mutually independent, then Zij,t u ei,tej,t will be uncorrelated with Zmn,t u em,ten,t as long as i p m or j p n. Thus, all the p( p 1)/2 combinations of ei,t ej,t (i p j) among these p disturbances e.tV=(e1,t. . ., ep,t) will also be uncorrelated PT with each other. Second, if the elements in e.t are mutually independent, 1=2 then as T ! l, T t¼1 ei;t ej;t Z N ð0; Xij Þ by adapting Theorem 7.7.8 of Anderson (1971, p. 429), where Xij denotes the long run variance of ei,tej,t, and Z denotes convergence in distribution. Given the two preceding observations and that the elements in e.t are mutually independent, we have KuT
1=2
T X
Zt Z N ð0; XÞ;
t¼1
where ZtV ¼ ðZ12;t ; Z13;t ;: : :; Z1p;t ; Z23;t ; Z24;t ;: : :; Z2p;t ;: : :; Zðp1Þp;t Þ;
Zij;t ¼ ei;t ej;t :
ð2Þ
Here, X is a p( p 1)/2 p( p 1)/2 matrix such that 2
X12
6 6 60 6 Xu6 6 6] 6 4 0
:::
0
X13 : : :
0
]
O
]
0
:::
Xðp1Þp
0
3 7 7 7 7 7: 7 7 7 5
ð3Þ
In an empirical application, we do not observe ei in Eq. (1) directly. We instead collect the OLS residuals from these p regression models, respectively, i.e., eiV=(ei1,ei2,. . .,eiT), i = 1,. . ., p, or ei ¼ ½IT Xi ðXiVXi Þ1 XiV Yi ;
i ¼ 1;: : :; p;
where IT is a T T identity matrix. With the OLS residuals at hand, we now can define the MI test: ˆ ˆ Xˆ 1 K; MIuKV
ð4Þ
W.-J. Tsay / Economics Letters 83 (2004) 69–76
73
where 1=2 ˆ KuT
T X
Zˆ t ;
t¼1
Zˆ t V ¼ Zˆ 12;t ; Zˆ 13;t ;: : :; Zˆ 1p;t ; Zˆ 23;t ; Zˆ 24;t ;: : :; Zˆ 2p;t ;: : :; Zˆ ðp1Þp;t ; 2
Xˆ 12
:::
0
6 6 60 6 ˆ Xu6 6 6] 6 4
0
Zˆ ij;t ¼ ei;t ej;t ;
3
0
Xˆ 13 : : :
0
]
O
]
0
:::
Xˆ ðp1Þp
7 7 7 7 7; 7 7 7 5
ð5Þ
such that Xˆ ij ¼
T1 X
ð6Þ
Ci;h Cj;h ;
h¼Tþ1
and Ci;h ¼ T 1
X tðhÞ
ei;t ei;tþh ;
Cj;h ¼ T 1
X
ej;t ej;tþh :
ð7Þ
tðhÞ
Here, At(h) is a sum over 1 V t, t + h V T. The estimator Xˆ defined in Eq. (5) is used to estimate X presented in Eq. (3). To derive the asymptoticpproperties of the MI test, we note that the OLS estimator bˆi from Eq. (1) ffiffiffiffi P P T T 1=2 1=2 ˆ possesses that t¼1 ei;t ej;t T t¼1 ei;t ej;t ¼ op ð1Þ, or PT the property PT T ðbi bi Þ ¼ Op ð1Þ and T 1=2 1=2 ˆ holds. When the T t¼1 Zij;t T t¼1 Zij;t ¼ op ð1Þ , given that item (iv) of Assumption 1P T 1=2 ˆ ˆ disturbances are independent of each other and Assumption 1 holds, we have KuT t¼1 Zt Z N ð0; XÞ. Provided that we can consistently estimate X, the asymptotic properties of the MI test when the disturbances are independent of each other can then be obtained. According to Eqs. (6) and (7), we realize that each element in the principle diagonal of Xˆ is based on the covariance matrix estimator proposed by Robinson (1998), where he intended to estimate the long run variance of the product of two independent stationary long memory processes. Because each element in the principle diagonal of Xˆ is constructed with the product of two independent weakly dependent processes, the justification for applying pRobinson’s covariance estimator in the current paper can be established. The details to prove that Xˆ ! X when Assumption 1 holds and that the elements in e.t are independent of each other are omitted. Combining the preceding results, asymptotically we can see that the MI test is v2 distributed and the results are presented in the following Theorem 1.
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Table 1 Rejection percentages of the MI test when Ae is a diagonal matrix T
k=5
k=6
k=7
k=8
k=9
k = 10
k = 11
k = 12
k = 13
k = 14
k = 15
25 50 100
2.9 3.3 5.5
2.9 3.7 5.2
3.0 3.7 5.2
2.9 3.2 5.2
2.7 4.2 5.7
2.7 3.6 4.1
3.5 4.1 3.9
3.4 2.7 5.1
2.4 3.4 4.9
4.2 4.1 4.4
2.3 3.9 5.7
2 The results are all based on 1000 replications and p = 5. The critical values used are obtained from the vp( p 1)/2 distribution at the 5% level of significance.
Theorem 1. Given that the conditions in Assumption 1 hold and the disturbances across different regressions of a SURE model are independent of each other, then as T ! l, MI Z v2p(p1)/2. Theorem 1 clearly indicates that our MI test is an asymptotic test as opposed to the exact test of Dufour and Khalaf (2002). However, there are several advantages to using our MI test. First, the critical values of the v2p( p 1)/2 distribution can be used for testing, implying that we do not need to calculate the critical values for a specific sample size, although the disturbances are generalized to be weakly dependent processes. Second, the MI test is very flexible, because not only can we test that Ae is a diagonal matrix, but we can also test the hypothesis that Ae is a block-diagonal matrix. Third, the MI test utilizes all the p( p 1)/2 combinations of the p groups of OLS residuals, but we only need to calculate the p groups of autocovariance functions from these p groups of OLS residuals, respectively. Therefore, the burden of calculating Xˆ for the MI test only increases linearly with the magnitude of p, even though there are p( p 1)/2 pairs of long run variances that need to be estimated. Finally, because Xˆ does not use any kernel function, or bandwidth parameter, or AR filter, the trouble with deciding the ‘‘best’’ combination of the kernel function and bandwidth parameter, or the ‘‘optimal’’ AR lag length, does not exist any more.
3. Monte Carlo experiment As shown previously, the MI test can be applied to the SURE model with serially dependent disturbances. For the purpose of comparison, in this section, we follow the design of Dufour and Khalaf (2002) where the disturbances are i.i.d. processes so as to demonstrate the small sample performance of the MI test. We also thank an anonymous referee for pointing out that our Monte Carlo design does not allow for serial dependence in each disturbance, because the experimental design of Dufour and Khalaf (2002) is used as the benchmark of our studies. Specifically, our Monte Carlo experiment is based on 1000 replications with different sample sizes. Two hundred additional values are generated in order to obtain random starting values. In each experiment, the design matrices Xi, i = 1,. . .,p, contain a constant term and an equal number of regressors (ki = k, i = 1,. . ., p). The value of k considered is k = 5,. . ., 15. The variables Xi are generated using a multivariate normal distribution and are kept constant over all replications. The disturbances are also generated from multivariate normal distributions.3 Table 1 illustrates the size performance of the MI test when the critical values used are obtained from 2 the vp( p 1)/2 distribution at the 5% level of significance. The experimental design of our Table 1 is 3
Please check Dufour and Khalaf (2002, p. 159) for the details of their experimental design.
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Table 2 Rejection percentages of the MI test when Ae is not a diagonal matrix K=p=5
A0(H0)
A1
A2
A3
A4
T = 25 T = 50 T = 100
2.9 3.3 5.5
99.8 100 100
29.5 86.1 100
81.9 100 100
54.1 98.6 100
2 The results are all based on 1000 replications. The critical values used are obtained from the vp( p 1)/2 distribution at the 5% level of significance.
identical to that in Table 2 of Dufour and Khalaf (2002, p.161), where Dufour and Khalaf (2002) apply the technique of the MC tests of Dwass (1957) and Barnard (1963) to implement exact tests. As compared to the results in Table 2 of Dufour and Khalaf (2002), we find that there is some size distortion of our MI test. However, this size distortion is very mild (about 2% as T = 25), and the difference between our MI test and the MC tests decreases as the sample size increases. When T = 100, the size control of the MI test is equivalent to that of the MC exact tests. Table 2 demonstrates the power performance of the MI test under the four alternatives, A1, A2, A3, and A4, considered in Table 3 of Dufour and Khalaf (2002).4 As T = 25, the power performance of the MI test is close to that of the MC exact tests of Dufour and Khalaf (2002) under three alternatives, A1, A3, and A4. The power performance of the MI test is not that well when the sample size is only 25 and Ae = A2. However, as the sample size increases to be 100, the rejection rates of the MI test under these four alternatives all reach 100%.5 Combining the results in Tables 1 and 2, we note that the small sample performance of our MI test is quite satisfactory, even though the sample size is moderately small.
4. Conclusion This paper develops a multivariate independent (MI) test which can conveniently and powerfully test the independence between the disturbances of the SURE model. The coverage of the MI test is broader than the existing independence tests for the disturbances of the SURE model, because the disturbances are now extended to be weakly dependent processes. Moreover, the implementation of our MI test is straightforward, because it does not use any kernel function, or bandwidth parameter, or AR filter. The Monte Carlo experiment conducted in this paper also shows that the small sample performance of the MI test is comparable to that of exact tests considered in Tables 2 and 3 of Dufour and Khalaf (2002, p. 161). This indicates that the MI test can be viewed as a convenient alternative as opposed to the computationally more intensive exact tests in detecting the independence of the SURE model’s disturbances. Nevertheless, as pointed out by an anonymous referee, for a less regular regressor Xi, the performance of our MI test might deteriorate. Furthermore, under Assumption 1, according to the results of Dufour (2003), Faust (1996, 1999), and Po¨tscher (2002), the size of our asymptotic test will converge to one, which is not the size under the asymptotic distribution. Although to find the sufficient conditions for solving the asymptotic size problem of the MI test will be left for our future studies, the value of using the exact tests of Dufour and Khalaf (2002) is still important in empirical applications. 4 5
The exact specifications of A1, A2, A3, and A4 are displayed in Table 1 of Dufour and Khalaf (2002). Dufour and Khalaf (2002) only consider the power performance of their exact tests as T = 25.
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Acknowledgements We thank an anonymous referee for his or her valuable comments and suggestions.
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