Robust Estimation in Threshold Autoregression

In this paper we study robust properties of M-estimates of the parameters of self-excited threshold autoregression model. The loss function that determines M-estimates was supposed to be convex and twice differentiable. The threshold of the autoregressive model was considered to be known and unique. We proved the asymptotic normality of M-estimates of autoregressive equation parameters. Moreover, we found the asymptotic relative efficiency of M-estimates, the least square and least absolute deviation estimates with respect to each other. First, we calculated the asymptotic relative efficiency values for the normal distribution, as well as the values of the double exponential distribution (Laplace distribution) and contaminated normal distribution (Tukey distribution). Then, we described the dependence of the asymptotic relative efficiency of these estimates on Tukey distribution parameters (the proportion and level of contamination). Next, for all three estimates in the space of Tukey distribution parameters, we built lines of equal efficiency, which made it possible to single out the preference areas for each pair of estimates considered. Findings of the research show that M-estimates are more efficient than the least squares and least absolute deviation estimates if the distribution of the innovation process slightly deviates from the normal distribution. Finally, we give recommendations on the use of these estimates in practical applications.

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