Two-Stage Least Squares as a Mean and Covariance Structure Estimator: Reconciling Econometric and Psychometric Formulations

Instrumental variables regression is increasingly used in psychology and forms the building block of model-implied instrumental variables structural equation modeling (MIIV-SEM), but econometric and psychometric formulations use different statistical representations and inferential terminology. We develop a general theory of multi-stage weighted least squares estimation for mean and covariance structures and show that classical two-stage least squares (2SLS) arises as the second stage of a three-stage procedure. The first stage estimates the predictive equations for the explanatory variables, the second estimates the structural intercept and regression coefficients, and the third estimates the remaining disturbance variance and covariances while propagating uncertainty from the preceding stages. For the structural parameters, the normal-theory and asymptotically distribution-free covariance estimators reduce exactly to the conventional homoscedastic and heteroscedasticity-robust 2SLS covariance estimators. When model-based covariance matrices are used, Browne’s normal-theory goodness-of-fit statistic equals Sargan’s overidentification statistic, and its asymptotically distribution-free counterpart equals Wooldridge’s heteroscedasticity-robust overidentification score statistic. The third-stage disturbance covariances quantify endogeneity and yield Wald tests that, under the null hypothesis, are asymptotically equivalent to the Hausman OLS–2SLS contrast test under normal-theory assumptions and Wooldridge’s robust exogeneity test under asymptotically distribution-free assumptions. By translating IV regression between econometric and psychometric formulations, the article facilitates methodological exchange between the two traditions and places the estimators, standard errors, and tests used in MIIV-SEM within a coherent multi-stage statistical theory.