The validity of bootstrap testing for threshold autoregression

We consider bootstrap-based testing for threshold effects in non-linear threshold autoregressive (TAR) models. It is well-known that classic tests based on asymptotic theory tend to be biased in case of small, or even moderate sample sizes, especially when the estimated parameters indicate non-stationarity, or in presence of heteroskedasticity, as often witnessed in the analysis of financial or climate data. To address the issue we propose a supremum Lagrange Multiplier test statistic (sLM), where the null hypothesis specifies a linear autoregressive (AR) model against the alternative of a TAR model. We consider both the classical recursive residual i.i.d. bootstrap (sLMi) and a wild bootstrap (sLMw), applied to the sLM statistic, and establish their validity under the null hypothesis. The framework is new, and requires the proof of non-standard results for bootstrap analysis in time series models; this includes a uniform bootstrap law of large numbers and a bootstrap functional central limit theorem. The Monte Carlo evidence shows that the bootstrap tests have correct empirical size even for small samples; the wild bootstrap version (sLMw) is also robust against the presence of heteroskedasticity. Moreover, there is no loss of empirical power when compared to the asymptotic test and the size of the tests is not affected if the order of the tested model is selected through AIC. Finally, we use our results to analyse the time series of the Greenland ice sheet mass balance. We find a significant threshold effect and an appropriate specification that manages to reproduce the main non-linear features of the series, such as the asymmetric seasonal cycle, the main periodicities, and the multimodality of the probability density function.