The validity of bootstrap testing in the threshold framework
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门限框架下自助法检验的有效性
- 研究问题
- 在线性自回归与门限自回归之间进行检验时，如何改善小样本中的过度拒绝问题，并证明递归自助法的有效性？
- 方法与思路
- 作者提出上确界拉格朗日乘子统计量，采用递归自助法校准，并建立一致自助大数定律和自助函数型中心极限定理。
- 材料陈述的贡献
- 摘要称证明了自助检验有效性，模拟中尺寸正确且功效未损失，并展示了选择自回归阶数及种群动态应用的表现。
- 阅读关注点
- 应核对原假设下的平稳性和参数范围、门限搜索区间、递归生成方式，以及尺寸、功效与阶数选择结论的适用条件。
- 阅读边界
- 这是 arXiv 预印本，同行评审及接收未核实；摘要未列完整理论条件，不能由非平稳问题的讨论推断其覆盖所有非平稳过程。
原文摘要
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 oversized in the case of small, or even moderate sample sizes, or when the estimated parameters indicate non-stationarity, as often witnessed in the analysis of financial or climate data. To address the issue we propose a supremum Lagrange Multiplier test statistic (sLMb), where the null hypothesis specifies a linear autoregressive (AR) model against the alternative of a TAR model. We consider a recursive bootstrap applied to the sLMb statistic and establish its validity. This result 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. These new results can also be used as a general theoretical framework that can be adapted to other situations, such as regime-switching processes with exogenous threshold variables, or testing for structural breaks. The Monte Carlo evidence shows that the bootstrap test has correct empirical size even for small samples, and also no loss of empirical power when compared to the asymptotic test. Moreover, its performance is not affected if the order of the autoregression is estimated based on information criteria. Finally, we analyse a panel of short time series to assess the effect of warming on population dynamics.