In this paper we consider testing for weak instruments in a model with two endogenous variables, but where the instruments in one reduced form equation are weak after having explained the other endogenous variable. This is a setting where the first-stage F-statistics in each of the two first stage equations are high, but the identification of the model parameters is weak. The way we model this is to introduce 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 multivariate first stage F-statistic along the lines of the proposal by Angrist and Pischke (2009) and show that 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 $$H_{0}: \pi_{1} = \delta \pi_{2}$$.

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