The estimation of time-varying parameters in multivariate linear time series models

Abstract

We propose a new likelihood-based estimation procedure for time-varying parameters in state space models. The state space model consists of measurement and transition equations and relies on a state vector together with a set of system matrices. In linear Gaussian state space models, the state vector contains the stochastically time-varying components that are linear in the observation vector. The estimation of the state vector can take place via the Kalman filter and related methods. Stochastically time-varying parameters in the system matrices are typically nonlinear in the observation vector. The estimation of such parameters is less straightforward. We propose to specify the nonlinear time-varying parameters by a generalized autoregressive score model. The driving mechanism for this specification is the scaled score of the loglikelihood function. Since the Kalman filter carries out the prediction error decomposition of the likelihood function, it can also provide both the score function and its scaling for the nonlinear time-varying parameters. The computations can be carried out recursively and simultaneously within the Kalman filter. As a result, we obtain an unified filter for all time-varying (linear and nonlinear) parameters in the model. The details of this novel approach are presented in this paper. We further provide test statistics for the null hypothesis that the parameters are not time-varying. We finally present three empirical illustrations of interest to economics and finance.