On the least squares estimation of multiple-regime threshold autoregressive models

This paper studies the least squares estimator (LSE) of the multiple-regime threshold autoregressive (TAR) model and establishes its asymptotic theory. It is shown that the LSE is strongly consistent. When the autoregressive function is discontinuous over each threshold, the estimated thresholds are n-consistent and asymptotically independent, each of which converges weakly to the smallest minimizer of a one-dimensional two-sided compound Poisson process. The remaining parameters are n-consistent and asymptotically normal. The theory of Chan (1993) is revisited and a numerical approach is proposed to simulate the limiting distribution of the estimated threshold via simulating a related compound Poisson process. Based on the numerical result, one can construct a confidence interval for the unknown threshold. This issue is not straightforward and has remained as an open problem since the publication of Chan (1993). This paper provides not only a solution to this long-standing open problem, but also provides methodological contributions to threshold models. Simulation studies are conducted to assess the performance of the LSE in finite samples. The results are illustrated with an application to the quarterly U.S. real GNP data over the period 1947–2009.