Information quantity evaluation of nonlinear time series processes and applications

Among several models proposed in the time series literature, the Self-Exciting Threshold Autoregressive (SETAR) model is non-linear and considers threshold values to model time series affected by regimes. Beyond the linear models, the computation of information and dependence metrics in non-linear time series is of great interest to compare processes and test non-linearity. This paper considers the stationary marginal density of a SETAR(2;1,1) process to compute explicit differential entropy and Kullback–Leibler and Jeffrey's divergences. In addition, an asymptotic homogeneity test for statistical significance of the disparity between two SETAR(2;1,1) processes was built. Numerical illustrations and applications to fish condition factor and Chilean economic perception time series are presented to illustrate the proposed methodology. Besides, a numerical algorithm based on Riemann–Stieltjes integral definition is implemented to calculate information measures based on stationary cumulative density functions of other types of nonlinear stationary stochastic processes, such as the first-order double autoregressive bilinear threshold moving average and threshold autoregressive moving average processes.