Dealing with Weak Instruments: An Application to the Protection for Sale Model

Endogeneity of explanatory variables is now receiving the concern it deserves in the empirical political science literature. Instrumental variables (IVs) estimators, such as two-stage least squares (2SLS), are the primary means for tackling this problem. These estimators solve the endogeneity problem by "instrumenting" the endogenous regressors using exogenous variables (the instruments). In many applications, a problem that the IV approach must overcome is that of weak instruments (WIs), where the instruments only weakly identify the regression coefficients of interest. With WIs, the infinite-sample properties (e.g., consistency) used to justify the use of estimators like 2SLS are on thin ground because these estimators have poor small-sample properties. Specifically, they may suffer from excessive bias and/or Type I error. We highlight the WI problem in the context of empirical testing of "protection for sale" model that predicts the cross-sectional pattern of trade protection as a function of political organization, imports and output. These variables are endogenous. Importantly, the instruments used to solve the endogeneity problem are weak. A method better suited to exact inference with WIs is the limited information maximum likelihood (LIML) estimator. Censoring in the dependent variable in the application requires a nonlinear Tobit LIML estimator.