A weak instrument [Formula: see text]-test in linear IV models with multiple endogenous variables

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 [Formula: see text] where [Formula: see text] and [Formula: see text] are the parameters in the two reduced-form equations, [Formula: see text] is a vector of constants and [Formula: see text] is the sample size. We investigate the use of a conditional first-stage [Formula: see text]-statistic along the lines of the proposal by Angrist and Pischke (2009) and show that, unless [Formula: see text], the variance in the denominator of their [Formula: see text]-statistic needs to be adjusted in order to get a correct asymptotic distribution when testing the hypothesis [Formula: see text]. We show that a corrected conditional [Formula: see text]-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 [Formula: see text] 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 [Formula: see text]-statistics.