Stata Implementation of Alternative Residual Inclusion Estimators for Models with Endogenous Regressors

Empirical analyses often require implementation of nonlinear models whose regressors include one or more endogenous variables – regressors that are correlated with the unobserved random component of the model. Failure to account for such correlation in estimation leads to bias and produces results that are not causally interpretable. Terza et al. (2008) discuss a relatively simple framework designed to explicitly account for such endogeneity – the residual inclusion (RI) framework. They also give the analytic details of a corresponding two stage estimator that yields consistent parameter estimates in wide variety of nonlinear regression contexts – two-stage residual inclusion (2SRI). The 2SRI estimates can be obtained using packaged Stata commands but the corresponding asymptotically correct standard errors (ACSE) require some analytic derivation and Mata coding [for details see Terza (2016)]. In the proposed presentation, we will discuss two alternative estimation approaches for RI models with a view toward broadening the menu of Stata implementation options for users who may not prefer to program in Mata or may be inclined to avoid analytic derivations: generalized method of moments (GMM) [StataCorp (2015)]; and quasi limited information maximum likelihood (QLIML) [Wooldridge (2014)]. GMM can be applied using the packaged Stata gmm command and, although it requires that the user supply analytic formulae for the relevant moment conditions, it frees the user from the Mata coding required by 2SRI for calculation of the ACSE. QLIML is implemented via the Mata optimize command so it does require some knowledge of Mata coding. On the other hand, it does not place any analytic demands on the user for calculation of the parameter estimates or their ACSE. We will detail all three of these approaches and, in the context of an empirical example, we will give template code for their Stata implementation (including calculation of the ACSE). We note that, although the methods are essentially asymptotically equivalent the methods yield different results in the context of our example. We offer analytic explanations for these differences. We also apply the methods to Monte Carlo simulated samples to further elucidate their implementation. Such simulations also serve to validate their large sample properties and reveal aspects of finite sample performance. Because the methods are essentially asymptotically equivalent, we conclude that one’s choice of approach should depend solely on the user’s coding preferences and his proclivity for analytic derivation. It is hoped that this presentation will broaden Stata users’ access to this important class of models (the RI framework) for the specification and estimation of econometric models involving endogenous regressors. Terza, J., Basu, A. and Rathouz, P. (2008): “Two-Stage Residual Inclusion Estimation: Addressing Endogeneity in Health Econometric Modeling,” Journal of Health Economics, 27, 531-543. Terza, J.V. (2016): “Simpler Standard Errors for Two-Stage Optimization Estimators,” Stata Journal, 16, 368-385. Wooldridge, J.M. (2014): Quasi-Maximum Likelihood Estimation and Testing for Nonlinear Models with Endogenous Explanatory Variables,” Journal of Econometrics, 182, 226-234.