Generalized multivariate threshold autoregressive models with linearly partitioned threshold space

Abstract

We consider a k-dimensional multiple-regime vector threshold autoregressive model, in which the regime-switching mechanism is governed by a bivariate threshold variable. Specifically, the regimes are induced by a partition of the threshold space by an unknown number of threshold lines. Within each regime, the process follows a specific vector autoregressive (VAR) model. We formulate the model selection and parameter estimation into a minimization problem based on the Minimum Description Length (MDL) principle and estimate the number of threshold lines, parametric forms of threshold lines and VAR model parameters in each regime simultaneously. Theoretically, we show that the MDL estimators of threshold lines are n-consistent and characterize their limiting distribution. This requires novel proving techniques of introducing a new functional space G for the local MDL difference functions and establishing weak convergence results therein. Finally, we conduct some empirical studies with simulated datasets and perform real data analyses on U.S. interest rates and U.S. GNP Data.