INTRODUCTION

This chapter studies some problems concerning two commonly used instrumental variable (IV)-based misspecification tests in linear models. One is Sargan's (1958, 1959) test of overidentifying restrictions, and the other is the Durbin–Hausman–Wu test based on the ordinary least squares – instrumental variables estimator (OLS–IV) contrasts. What causes these problems is the potential rank deficiency of a certain moment matrix. This chapter extensively uses orthogonal groups of matrices and their invariance properties to investigate these issues. As shown in Phillips' (1989a) treatment of “partially identified models,” coordinate rotations by orthogonal matrices are useful in studying a rank deficiency problem. Moreover, the theory of group invariance often simplifies derivations of seemingly complicated distributions as exemplified by a series of papers by Phillips (1984; 1985a,b; 1986a,b; 1989b).

It has been recognized that some IV-based misspecification tests can fail to detect misspecification in certain cases. Newey (1985) provides a unified framework for misspecification tests based on the generalized method of moments (GMM) and points out a potential failure of such tests. That is, if the departure from the moment condition is in certain directions, the GMM misspecification tests have no power. See also Tauchen (1986) and Holly (1982). Because IV estimators are special cases of GMM, this potential failure clearly applies to some misspecification testing procedures based on (linear) IV estimators such as Sargan's test.