A Comparison between Successive Estimate of TVAR(1) and TVAR(2) and the Estimate of a TVAR(3) Process

In time series analyses, the auto-regressive (AR) modelling of zero mean data is widely used for system identification, signal decorrelation, detection of outliers and forecasting. An AR process of order p is uniquely defined by p coefficients and the variance in the noise. The roots of the characteristic polynomial can be used as an alternative parametrization of the coefficients, which can be used to construct a continuous covariance function of the AR process or to verify that the AR process is stationary. In a previous study, we introduced an AR process of time variable coefficients (TVAR process) in which the movement of the roots was specified as a polynomial of order one. Until now, this method was analytically derived only for TVAR processes of orders one and two. Thus, higher-level processes had to be assembled by the successive estimation of these process orders. In this contribution, the analytical solution for a TVAR(3) process is derived and compared to the successive estimation using a TVAR(1) and TVAR(2) process. We will apply the proposed approach to a GNSS time series and compare the best-fit TVAR(3) process with the best-fit composition of TVAR(2) and TVAR(1) process.