Permutation-Enhanced Two-Stage Least Squares Estimation for Robust Endogeneity Analysis: A Systematic Methodological Review

Abstract

Endogeneity is one of the basic problems in econometric modelling as conventional regression estimates may be inconsistent and the conclusions from these estimates may be erroneous. Two-Stage Least Squares (2SLS) is an existing Instrumental Variable (IV) solution, but it has a proven finite sample unreliability when the data exhibits multicollinearity, weak identification, non-normal distribution, and strong inference assumptions. Permutation methods are distribution-light methods for statistical inference that have proven to be robust in regression, structural-change, unit-root, multivariate, classification, and experimental applications. This review summarizes the methodological and empirical advances of the 2SLS estimation and permutation inference methods and concentrates on the possibility of combining them for a powerful endogeneity analysis. Conventional and modified 2SLS procedures, panel instrumental-variable models, studentized permutation tests, regression permutation procedures, structural-break methods, high-dimensional permutation inference, and recent permutation-enhanced 2SLS estimation: the literature reviewed covered these procedures. Results indicate that 2SLS is still applicable when it comes to endogenous regressors, and that permutation procedures provide enhanced inferential flexibility in non-normal and/or heterogeneous settings. Selected empirical studies in the permutation applications used 10,000-100,000 resamples, whereas high-dimensional procedures proved to be effective for Type I error control and increased simulation power. Conventional 2SLS is found to be superior for some parameters and permutation variants are found to perform better in the estimation of the endogenous coefficients based on recent permutation-enhanced 2SLS evidence. The review finds that permutation methods are not necessarily seen as a panacea to the classic instrumental-variable approaches. Important future directions of research include standardized simulation designs, weak-instrument analysis, sufficient-statistic conditioning, and finite-sample benchmarking.