ORTHOGONALITY TESTS IN LINEAR MODELS SEUNG CHAN AHN ARIZONA STATE UNIVERSITY ABSTRACT

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1 ORTHOGONALITY TESTS IN LINEAR MODELS SEUNG CHAN AHN ARIZONA STATE UNIVERSITY ABSTRACT This paper considers several tests of orthogonality conditions in linear models where stochastic errors may be heteroskedastic or autocorrelated. It is shown that these tests can be performed with Wald statistics obtained from simple auxiliary regressions. JEL Classification Number: C20 Word Count: 1460 Address for correspondence: Seung Chan Ahn, Department of Economics, Arizona State University, Tempe, AZ , USA, Phone: , Fax: , AASCA.ASUACAD.BITNET August, 1994 Revised March, 1995 Revised August, 1995 The author gratefully acknowledges the financial support of the College of Business, Arizona State University.

2 I. INTRODUCTION This paper considers tests of orthogonality conditions for linear models. The tests of interest are the GMM (generalized method of moments) tests of Hansen (1982), Newey (1985), and Eichenbaum, Hansen and Singleton (1988, hereafter, EHS). These tests are convenient for models in which stochastic error terms are independently and identically distributed (i.i.d.), because each of the test statistics can be computed as the number of observations times the uncentered R 2 from an auxiliary regression. [See Newey (1985) and Pesaran and Smith (1990).] This paper considers cases in which the errors are not i.i.d., and shows that the GMM tests can be easily performed with Wald statistics obtained from auxiliary regressions. This paper is organized as follows. Section II considers the Hansen test, and derives its Wald-type representation. Section III considers the tests of Newey (1985) and EHS (1988), establishes the numerical equivalence of the two statistics, and shows how they can be obtained by a simple auxiliary regression. The relation of these two tests to the Hausman test (1978, 1984) is also examined. Some concluding remarks are contained in Section IV. II. THE HANSEN TEST AND ITS WALD-TYPE ALTERNATIVE Consider the linear regression model: (1) where y is the T 1vector of a dependent variable, X is the T pmatrix of regressors, and ε is the vector of errors. Here we allow the errors to be heteroskedastic and/or autocorrelated. Some variables in X are assumed to be correlated with ε, so that consistent estimation of β requires instrumental-variables methods. Let Z be at q(q p) matrix of instrumental

3 2 variables which satisfy the orthogonality hypothesis: (2) Under H o, the optimal GMM estimator is β =(X ZV -1 Z X) -1 X ZV -1 Z y, where V = cov(z ε). Following White (1982), we will call any estimator of this form a two-stage instrumentalvariables (2SIV) estimator. In practice, a consistent estimator of V should be used for the computation of β. For the appropriate estimates of V, see Newey and West (1987b). Once β is computed, H o can be tested by the Hansen (1982) statistic, which is defined by J T =(y- Xβ ) ZV -1 Z (y-xβ ). Under H o, this statistic has a χ 2 distribution with (q - p) degrees of freedom. For an alternative way to compute J T, consider an auxiliary regression model: (3) where Z A isanyt (q-p)submatrix of Z whose columns are not in X. Since model (2) is exactly identified under H o, the 2SLS estimator of ξ A equals ξˆ A =[βˆa,δˆa ] =(Z X A ) -1 Z y. Define W A T = δˆ A [cov(δˆ A)] -1 δˆ A, which is the Wald statistic for testing the restriction δ A =0. Then, we obtain the following: Proposition 1. W A T=J T, if the same estimator of V is used for both statistics. It is important to note that the numerical equivalence of W A T and J T may not hold if different estimators of V are used. However, the difference between these statistics is asymptotically negligible. In practice, W A T is easier to compute than J T. Most available software packages can compute 2SLS estimates and their covariance matrix by the method of Newey and West (1987b). Using such software, researchers can perform the Hansen test with

4 3 the Wald statistic for the significance of Z A in model (3). Testing for the orthogonality of the whole set X is further simplified. For this case, Z coincides with X A, so that the Hansen statistic can be computed as a Wald statistic based on the OLS estimator of model (3). III. TESTING A SUBSET OF ORTHOGONALITY CONDITIONS Researchers may wish to test the orthogonality of a subset of instrumental variables, when prior information about the orthogonality of other instruments is available. For such cases, assume that the alternative hypothesis of H o is given by: (4) where Z=[Z 1,Z 2 ], and Z 1 and Z 2 include q 1 and q 2 (=q-q 1 ) instruments, respectively. According to Z 1 and Z 2, partition V into [V ij ], where i, j =1, 2. We assume q 1 p, so that β can be estimated by 2SIV using Z 1 only. We denote this 2SIV estimator by βˆ =[X Z 1 (V 11 ) - 1 Z 1 X] -1 X Z 1 (V 11 ) -1 Z 1 y. Newey (1985) considers an optimal GMM statistic for testing H o against HA, s which is a Wald statistic based on r=z 2 (y-xβˆ)-v 21 (V 11 ) -1 Z 1 (y-xβˆ); that is, M T =r R -1 r, where F = Z 2 X-V 21 (V 11 ) -1 Z 1 X and R (= cov(r)) = V 22 -V 21 (V 11 ) -1 V 12 + F[X Z 1 (V 11 ) -1 Z 1 X] -1 F. Under H o, this statistic has a χ 2 distribution with q 2 degrees of freedom. The advantage of M T over J T is that its power is focused on the orthogonality of Z 2. Following EHS (1988), we can also consider a statistic of the likelihood-ratio type, which is given by D T =J T -J 1 T, where J 1 T =(y-xβˆ) Z 1 (V 11 ) -1 Z 1 (y-xβˆ). Proposition 1 suggests an alternative way to compute D T. Similarly to J T,J 1 T can be obtained by a Wald statistic from

5 4 an artificial 2SLS regression. Thus, D T can be computed as the difference between two Wald statistics. Another interesting and novel property of D T, which we formally state below, is that it is in fact numerically equivalent to the optimal GMM statistic M T. Alternatively, both of M T and D T can be obtained from a 2SIV estimation of the model: (5) where Q 1 =I T -Z 1 (Z 1 Z 1 ) -1 Z 1. Pesaran and Smith (1990) consider 2SLS of the same model with i.i.d. errors, and find that the Wald statistic of the restriction δ B = 0 is equivalent to M T. Their result can be extended to the cases with non-i.i.d. errors. Denoting the 2SIV estimator of ξ B by ˆξ B =[ˆβ B,δˆB ] =[X B ZV -1 Z X B ] -1 X B ZV -1 Z y, define the Wald statistic based on δˆ B by W B T = δˆ B [cov(δˆ B)] -1 δˆ B. Then, we obtain the following result: Proposition 2. M T =D T =W B T, if the same estimator of V is used for the statistics. Finally, we consider a Hausman (1978, 1984) statistic for testing H o against HA, s which is given by H T =(ˆβ-β ) [cov(ˆβ - β )] - (ˆβ - β ), where ( ) - is a g-inverse. Under H o,h T has a χ 2 distribution with degrees of freedom equal to Rank[cov( ˆβ - β )]. Newey (1985) has shown that H T =M T if Rank[cov(ˆβ - β )]=q 2. The following proposition reveals a general link between H T and M T (= D T =W B T): Proposition 3. Let r H =F R -1 r and R H (= cov(r H ))=F R -1 F. Define a Wald statistic based on r H by M H T =r H (R H ) - r H. Then, H T =M H T, when the same estimator of V is used. Comparing the forms of M H T and M T, we can see that M H T (and H T ) is the statistic for testing the hypothesis, H H o: E[F R -1 E(Z 2 ε Z)] = 0. Indeed, it can be shown that β is consistent under H H o. This implies that H T is for testing the consistency of β, not H o (against HA) s itself. Ruud (1984) finds a similar result in a maximum-likelihood framework.

6 5 IV. CONCLUSION This paper has considered several popular GMM tests of orthogonality conditions in linear models. A remarkable feature of the GMM tests is that they are equivalent to Wald tests of exclusive restrictions imposed in artificial models. The test strategies considered in this paper are convenient for models with heteroskedastic and/or autocorrelated errors. APPENDIX: Proofs of Propositions We briefly sketch the proofs of Propositions 1-3. More detailed proofs can be found in the earlier version of this paper. Proposition 1. Define J A T(ξ A )=(y-x A ξ A ) ZV -1 Z (y-x A ξ A ). ξˆ A is the unrestricted GMM estimator minimizing J A T(ξ A ) [because Z (y-x A ξˆ A) = 0], while ξ A =( β,0 ) is the restricted GMM estimator that minimizes J A T(ξ A ) subject to the restriction δ A = 0. Then, Proposition 4 of Newey and West (1987a) implies that W A T =J A T(ξ A)-J A T(ξˆA)=J T. Proposition 2. Applying the separability result of Ahn and Schmidt (1995), we can show that δˆ B =L -1 r, where L=Z 2 Q 1 Z 2. Some tedious but straightforward algebra also shows that cov(δˆ B) =C B (X B ZV -1 Z X B ) -1 C B =L -1 RL -1, where C B =[0 q p,i q q2 ]. Substituting these results into W B T yields M T =W B T. The equality of D T and W B T can be shown similarly to the proof of Proposition 1. Proposition 3. Letting A=X Z 1 (V 11 ) -1 Z 1 X, we can show that βˆ - β = -A -1 F R -1 r and cov(βˆ - β) = cov(βˆ) - cov( β) =[X Z 1 (V 11 ) -1 Z 1 X] -1 -[X ZV -1 Z X] -1 =A -1 F R -1 FA -1. Substituting these results into H T gives us the desired equality.

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