Robust tests against smooth transition autoregressive models.

*(English)*Zbl 1019.62085Summary: We consider testing for linearity against a well-known class of regime switching models known as the smooth transition autoregressive (STAR) models. Apart from the model selection issues, one reason for interest in testing for linearity in time-series models is that nonlinear models such as the STAR are considerably more difficult to use. This testing problem is non-standard because a nuisance parameter becomes unidentified under the null hypothesis. We further explore the class of tests proposed by R. Luukkonen, P. Saikkonen and T. Teräsvirta [Biometrika 75, No. 3, 491-499 (1988; Zbl 0657.62109)]. They proposed LM tests for linearity against STAR models. A potential difficulty here is that the linear approximation introduces high leverage points, and hence outliers are likely to be quite influential.

To overcome this difficulty, we use the same approximating linear model of Luukkonen et al., but we apply Wald and \(F\)-tests based on \(l_1\)- and bounded influence estimates. The efficiency gains of this procedure cannot be easily deduced from the existing theoretical results because the test is based on a misspecified model under \(H_1\). Therefore, we carried out a simulation study, in which we observed that the robust tests have desirable properties compared to the test of Luukkonen et al. for a range of error distributions in the STAR model, in particular the robust tests have power advantages over the LM test.

To overcome this difficulty, we use the same approximating linear model of Luukkonen et al., but we apply Wald and \(F\)-tests based on \(l_1\)- and bounded influence estimates. The efficiency gains of this procedure cannot be easily deduced from the existing theoretical results because the test is based on a misspecified model under \(H_1\). Therefore, we carried out a simulation study, in which we observed that the robust tests have desirable properties compared to the test of Luukkonen et al. for a range of error distributions in the STAR model, in particular the robust tests have power advantages over the LM test.

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62F35 | Robustness and adaptive procedures (parametric inference) |

62F03 | Parametric hypothesis testing |