Implementing Tests with Correct Size in the Simultaneous Equations Model

Abstract

In this paper, we propose a fix to the size distortions of tests for structural parameters in the simultaneous equations model by computing critical value functions based on the conditional distribution of test statistics. The conditional tests can then be used to construct informative confidence regions for the structural parameter with correct coverage probability. Commands to implement these tests in Stata are also introduced.

Tests with correct size when instruments can be arbitrarily weak

“…This unifies the theory of similar tests of Anderson and Rubin (1949), Dufour and Jasiak (2001), Kleibergen (2002), and Moreira (2002Moreira ( , 2003. The class of similar tests is large and includes all unbiased tests.…”

“…Reject H β if the score statistic is larger than the (1 − α)-quantile of a chi-square-one distribution: Moreira (2002) shows that T β is independent of S β and, consequently, the null distribution of LM β is chi-square-one. Interestingly, LM β is just a particular case of a score-type statistic proposed by Breusch and Pagan (1980) in a general framework; see Appendix A.…”

“…Three examples of similar tests are the Anderson and Rubin (1949) test, the score test independently proposed by Kleibergen (2002) and Moreira (2002) for the weak-instrument setting, and the conditional likelihood ratio test of Moreira (2003). It is convenient to write these tests in terms of the standardized statistics…”

Bootstrap validity for the score test when instruments may be weak

“…On the other hand, when instruments are strong, rejection rates for the Wald test are much closer to the nominal size, and bootstrapping the Wald test offers improvements over 8 There is a slight difference between Moreira's (2002Moreira's ( , 2003 design and ours. In analogy with our results, our design takes Z as being random whereas Moreira's (2002Moreira's ( , 2003 design takes Z as being fixed. 9 The first-stage F-statistic corresponds to the concentration parameter λ λ/k in the notation of Staiger and Stock (1997).…”

“…In particular, under the weak-instrument asymptotics of Staiger and Stock (1997), the limiting distribution of the Wald statistic is not standard normal. An alternative statistic is the score (LM) used by Kleibergen (2002) and Moreira (2002): In practice, the covariance matrix Ω is typically unknown, so we replace it with the consistent estimator Ω = Y M Z Y /n:…”