Inference in Two-Step Panel Data Models with Weak Instruments and Time-Invariant Regressors: Bootstrap versus Analytic Estimators

The primary advantage to panel data is the ability they afford to control for unobserved heterogeneity. The fixed-effects (FE) estimator is by far the most popular technique for exploiting this advantage, but it eliminates any time-invariant regressors along with the unobserved time-invariant effects. Their partial effects can be easily recovered in a second-step regression of residuals, constructed from the FE estimator and the time means of the data, on the time-invariant variables. Hausman and Taylor (1981) proposed such a two-step estimator, allowing for some of the time-invariant variables to be correlated with the effects. Unfortunately, the Hausman-Taylor procedure appears to have been overlooked in empirical applications where the time-invariant variables are of interest. In this paper, we resurrect the Hausman-Taylor estimator and extend their work by deriving the asymptotic covariance matrix with instruments for the second-step estimators. Because of the complexity of the asymptotic formula, its finite-sample bias, and its inaccuracy in inference with weak instruments, we first adapt the pairs and two wild bootstrapping alternatives (restricted and unrestricted) to standard-error estimation of the second-step estimators. Using Monte Carlo methods, we examine four combinations of weak and strong instruments with weak and strong endogeneity in the second step and find that the wild estimators dominate, producing a minimal error rejection probability (ERP) of computed t-values, defined as the difference between the nominal and actual probability of rejection under the hull hypothesis. Since increasing T has no significant effect, we set T = 5 and find that the ERPs for the wild bootstrap methods are very close to zero at substantially smaller values of N than for the other methods. The dominance of the wild bootstrap methods over the asymptotic and pairs methods is greatest with weak instruments, strong endogeneity, and small N . This dominance diminishes but still obtains even with N = 1600 and T = 5. Weak instruments is a much greater problem for the pairs and asymptotic methods than is strong endogeneity. Power curves for the wild methods and the pairs are highly similar, implying that the wild methods should generally be employed with endogeneity.