Simultaneous Equations and Weak Instruments under Conditionally Heteroscedastic Disturbances

In this paper we extend the setting analysed in Hahn and Hausman (2002a) by allowing for conditionally heteroscedastic disturbances. We start by considering the type of conditional variance-covariance matrices proposed by Engle and Kroner (1995) and we show that, when we impose a GARCH specification in the structural model, some conditions are needed to have a GARCH process of the same order in the reduced form equations. Later, we propose a modified-2SLS and a modified-3SLS procedures where the conditional heteroscedasticity is taken into account, that are more asymptotically efficient than the traditional 2SLS and 3SLS estimators. We recommend to use these modified-2SLS and 3SLS procedures in practice instead of alternative estimators like LIML/FIML, where the non-existence of moments leads to extreme values (in case we are interested in the structural form). We show theoretically and with simulation that in some occasions 2SLS, 3SLS and our proposed 2SLS and 3SLS procedures can have very severe biases, and we present the bias correction mechanisms to apply in practice