Joint Mean-Vector and Var-Matrix estimation for Locally Stationary VAR(1) processes

During the last two decades, locally stationary processes have been widely studied in the time series literature. In this paper we consider the locally-stationary vector-auto-regression model of order one, or LS-VAR(1), and estimate its parameters by weighted least squares. The LS-VAR(1) we consider allows for a smoothly time-varying non-diagonal VAR matrix, as well as for a smoothly time-varying non-zero mean. The weighting scheme is based on kernel smoothers. The time-varying mean and the time-varying VAR matrix are estimated jointly, and the definition of the local-linear weighting matrix is provided in closed-from. The quality of the estimated curves is illustrated through simulation results.