In this paper we introduce the general setting of a multivariate time series autoregressive model with stochastic time-varying coefficients and time-varying conditional variance of the error process. This allows modeling VAR dynamics for non-stationary times series and estimation of time varying parameter processes by well-known rolling regression estimation techniques. We establish consistency, convergence rates and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular 7 variable data set to analyze evidence of time-variation in empirical objects of interest for the DSGE literature. The results of this paper serve as a starting point for further research on numerous open problems including establishing estimation results of time-varying parameters that are uniform in time t, constructing Bonferroni-type correction to the pointwise standard error bands and developing a valid test of the null hypothesis of no time variation.
Kernel estimation Time-varying VAR Structural change Monetary policy shock