A weak instrument $F$-test in linear IV models with multiple endogenous variables

Abstract

We consider testing for weak instruments in a model with multiple endogenous variables. Unlike Stock and Yogo (2005), who considered a weak instruments problem where the rank of the matrix of reduced form parameters is near zero, here we consider a weak instruments problem of a near rank reduction of one in the matrix of reduced form parameters. For example, in a two-variable model, we consider weak instrument asymptotics of the form ${\pi }_1=\delta {\pi }_2+c/\sqrt{n}$ where ${\pi }_1$ and ${\pi }_2$ are the parameters in the two reduced-form equations, $c$ is a vector of constants and $n$ is the sample size. We investigate the use of a conditional first-stage $F$-statistic along the lines of the proposal by Angrist and Pischke (2009) and show that, unless $\delta =0$, the variance in the denominator of their $F$-statistic needs to be adjusted in order to get a correct asymptotic distribution when testing the hypothesis $H_0:{\pi }_1=\delta {\pi }_2$. We show that a corrected conditional $F$-statistic is equivalent to the Cragg and Donald (1993) minimum eigenvalue rank test statistic, and is informative about the maximum total relative bias of the 2SLS estimator and the Wald tests size distortions. When $\delta =0$ in the two-variable model, or when there are more than two endogenous variables, further information over and above the Cragg–Donald statistic can be obtained about the nature of the weak instrument problem by computing the conditional first-stage $F$-statistics.