Two-stage least squares model averaging for instrumental variable models with exogenous variables

Instrumental variable (IV) methods are widely used to address unmeasured confoundings in structural equation models. In this paper, we focus on the settings where a possibly large number of instruments and a weak correlation between the instruments and the endogenous variable exist. Specifically, we propose a novel two-stage least squares (2SLS) model averaging approach to estimate the coefficient of an endogenous variable. Differing from existing literature, our model averaging estimation allows multiple exogenous variables to be included in both stages simultaneously. Theoretically, we study the consistency and asymptotic distributions of the estimated weights and the proposed model averaging estimator. Importantly, we discover that the proposed model averaging estimator produces an asymptotic bias when the endogenous variable and exogenous variables are correlated. Then, we construct a debiased estimator and establish its consistency and asymptotic normality to make statistical inference. Furthermore, we present an equivalent interpretation of the debiased estimator from another construction. Finally, numerical simulations and a real data analysis are conducted to illustrate our proposal.