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 is the basis of model‐implied instrumental variables structural equation modelling (MIIV‐SEM), yet econometric and psychometric formulations use different statistical representations and 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) is the second stage of a three‐stage procedure. The first stage estimates 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, propagating uncertainty from earlier stages. For structural parameters, the normal‐theory and asymptotically distribution‐free covariance estimators reduce exactly to conventional homoscedastic and heteroscedasticity‐robust 2SLS estimators. With model‐based covariance matrices, Browne's normal‐theory goodness‐of‐fit statistic equals Sargan's overidentification statistic, and its ADF counterpart equals Wooldridge's heteroscedasticity‐robust score statistic. Third‐stage disturbance covariances quantify endogeneity and yield Wald tests asymptotically equivalent, under the null, to the Hausman OLS–2SLS contrast and Wooldridge's robust exogeneity test, respectively. The results place MIIV‐SEM estimators, standard errors, and tests within a coherent multi‐stage statistical theory.