Time-Varying-Parameter Models with Endogenous Regressors: A Two-Step MLE Approach and an Augmented Kalman Filter

Abstract

In this paper, we provide a unified framework for LIML (limited information maximum likelihood) IV (instrumental variables) estimation to deal with endogeneity problems in the time-varying parameter models. For this purpose, we derive a Heckman-type (1976) two-step maximum likelihood estimation (MLE) procedure. The proposed two-step procedure, based on the conventional Kalman filter, provides consistent estimates of the hyper-parameters, as well as correct inferences on the time-varying coefficients. However, the use of the conventional Kalman filter in the second step would result in an invalid conditional covariance matrix for the time-varying coefficients. The correction for the conditional covariance matrix can be made by employing an augmented Kalman filter proposed in this paper. The basic model and the two-step procedure is also extended to handle the issue of heteroscedasticity in the disturbance terms. This is done by considering a time-varying parameter model for Campbell and Mankiw's (1989) consumption function.