On the conditional likelihood ratio test for several parameters in IV regression

<p>For the problem of testing the hypothesis that all <i>m</i> coefficients of the RHS endogenous variables in an IV regression are zero, the likelihood ratio (LR) test can, if the reduced form covariance matrix is known, be rendered similar by a conditioning argument. To exploit this fact requires knowledge of the relevant conditional <i>cdf</i> of the LR statistic, but the statistic is a function of the smallest characteristic root of an (<i>m</i> + 1)&#8722;square matrix, and is therefore analytically difficult to deal with when <i>m</i> > 1. We show in this paper that an iterative conditioning argument used by Hillier (2006) and Andrews, Moreira, and Stock (2007) to evaluate the cdf in the case <i>m</i> = 1 can be generalized to the case of arbitrary <i>m</i>. This means that we can completely bypass the difficulty of dealing with the smallest characteristic root. Analytic results are obtained for the case <i>m</i> = 2, and a simple and efficient simulation approach to evaluating the <i>cdf</i> is suggested for larger values of <i>m</i>.</p>