Matematika i Matematičeskoe Modelirovanie (Nov 2017)
Threshold Estimation in Threshold Autoregression
Abstract
One of the most important models to describe nonlinear time series is that of the threshold auto-regression. In solving a problem of the threshold autoregressive model identification, there is a need to estimate the threshold value. The most common method for estimating the threshold is a least-squares estimate technique. The least absolute deviation method of estimation is less widespread. Both methods are the special cases of the M-method.The paper objective is to compare the accuracy of the above three threshold estimation methods in the threshold auto-regression model through computer simulation. For simplicity, we study a stationary threshold model at two modes and one threshold.The estimates of the autoregressive threshold were pairwise compared among themselves by calculating their relative effectiveness being equal to the inverse ratio of the estimate variances. The exact and asymptotic variances of the threshold estimates under study are still unknown. Therefore, the variance of estimates was determined through computer simulation.The paper focuses on studying the impact of the probability distribution type of the updating process of a threshold equation on the estimate accuracy. Considers the normal distribution as the most common in practice and also the typical deviations from the normal distribution: contaminated normal distribution, two-sided exponential distribution, logistic distribution, and Student distribution.The simulation results have shown that the least-squares estimate exceeds the other estimates only with the normal distribution of the updating sequence. Even with a light contamination of the normal distribution, the least-squares estimate is worse than the M-estimate, and with increasing contamination, it gets worse than the least absolute deviation estimate as well.For the logistic distribution and Student distribution with a large number of degrees of freedom, which in practice are easily confused with the normal distribution, the M-estimate is more effective than the least-squares estimate. If the updating sequence has a Student distribution with a small number of degrees of freedom, for example, with four, then the least-squares estimate is inferior not only to the M-estimate, but also to the least absolute deviation estimate. For the Laplace distribution, as in most other statistical models of time series, the least absolute deviation estimate is the best.
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