3SLS

Three-stage least squares (3SLS) is an estimator used for systems of simultaneous equations in which some explanatory variables are endogenous and the error terms are correlated across equations. It extends two-stage least squares (2SLS) by exploiting cross-equation disturbances to gain efficiency, making it a preferred method when multiple related outcomes are modeled together.

In a typical system, each equation has the form y_g = X_g β_g + u_g for g = 1,…,G, where y_g are endogenous dependent variables, X_g contains exogenous variables and possibly endogenous regressors, and u_g are the equation errors that may be contemporaneously correlated across equations. A set of exogenous instruments Z, which are uncorrelated with the u_g but correlated with the endogenous regressors, is available to identify the system.

The estimation procedure proceeds in three stages. Stage 1 estimates reduced-form equations for the endogenous variables as functions of the exogenous variables and instruments, yielding fitted values Ŷ for the endogenous regressors. Stage 2 uses a seemingly unrelated regressions (SUR) framework to estimate the structural equations using the fitted endogenous variables, combining information across equations and obtaining an estimate of the cross-equation error covariance. Stage 3 then performs a feasible generalized least squares (GLS) estimation of the original structural equations using the estimated cross-equation covariance matrix, yielding 3SLS estimates. In some presentations, the first stage provides 2SLS-type instruments, and Stage 3 refines the estimates by GLS with the estimated error structure. Iterate if desired.

Assumptions include correct model specification, validity and relevance of instruments, and limited forms of heteroskedasticity (with robust variants available). 3SLS reduces to 2SLS when errors are uncorrelated across equations and to SUR when there are no endogenous regressors. Applications span economics and other social sciences for systems with endogeneity and cross-equation error correlation.