Identification through Heteroscedasticity: What If We Have the Wrong Form of Heteroscedasticity?

Abstract Klein and Vella (2010) and Lewbel (2012) respectively propose estimators that utilize the heteroscedasticity of the error terms to identify the coefficient of the endogenous regressor in a standard linear model, even when there are no exogenous excluded instruments. The assumptions on the form of heteroscedasticity are different for these two estimators, and whether they are robust to misspecification is an important issue because it is not straightforward how to justify which form of heteroscedasticity is true. This paper presents some simulation results for the finite-sample performance of the two estimators under various forms of heteroscedasticity. The results reveal that both estimators can be substantially biased when the form of heteroscedasticity is of the wrong type, meaning that they lack robustness to misspecification of the form of heteroscedasticity. Moreover, the J statistics of the over-identification test for the Lewbel (2012) estimator has low power under the wrong form of heteroscedasticity in the cases considered. The results suggest that it is not enough for researchers to justify only the existence of heteroscedasticity when using the proposed estimators.