Threshold autoregressive models for interval-valued time series data

Modeling and forecasting symbolic data, especially interval-valued time series (ITS) data, has received considerable attention in statistics and related fields. The core of available methods on ITS analysis is based on various applications of conventional linear modeling. However, few works have considered possible nonlinearities in ITS data. In this paper, we propose a new class of threshold autoregressive interval (TARI) models for ITS data. By matching the interval model with interval observations, we develop a minimum-distance estimation method for TARI models, and establish the asymptotic theory for the proposed estimators. We show that the threshold parameter estimator is T-consistent and follows an asymptotic compound Poisson process as the sample size $T\to \infty$. And the estimators for other TARI model parameters are root-T consistent and asymptotically normal. Simulation studies show that the proposed TARI model provides more accurate out-of-sample forecasts than the existing center–radius self-exciting threshold (CR-SETAR) model for ITS data in the literature. Empirical applications to the S&P 500 Price Index document significant asymmetric reactions of the stock markets in Japan, U.K. and France to shocks from the U.S. stock market and that incorporating this asymmetric effect yield better out-of-sample forecasts than a variety of popular models available in the literature.