The unconditional distributions of the OLS,TSLS and LIML estimators in a simple structural equations model

The exact distributions of the standard estimators of the structural coe?cients in a linear structural equations model conditional on the exogenous variables have been shown to have some unexpected and quirky features. Since the argument for conditioning on exogenous (ancillary) variables has been weakened over the past 20 years by the discovery of an “ancillarity paradox,” it is natural to wonder whether such finite sample properties are in fact due to conditioning on the exogenous variables. This article studies the exact distributions of the ordinary least squares (OLS), two-stage least squares (TSLS), and limited information maximum likelihood (LIML) estimators of the structural coe?cients in a linear structural equation without conditioning on the exogenous variables.