We show that the nonstandard limiting distribution of HAR test statistics under fixed-b asymptotics is not pivotal (even after studentization) when the data are nonstationarity. It takes the form of a complicated function of Gaussian processes and depends on the integrated local long-run variance and on on the second moments of the relevant series (e.g., of the regressors and errors for the case of the linear regression model). Hence, existing fixed-b inference methods based on stationarity are not theoretically valid in general. The nuisance parameters entering the fixed-b limiting distribution can be consistently estimated under small-b asymptotics but only with nonparametric rate of convergence. Hence, We show that the error in rejection probability (ERP) is an order of magnitude larger than that under stationarity and is also larger than that of HAR tests based on HAC estimators under conventional asymptotics. These theoretical results reconcile with recent finite-sample evidence in Casini (2021) and Casini, Deng and Perron (2021) who showing that fixed-b HAR tests can perform poorly when the data are nonstationary. They can be conservative under the null hypothesis and have non-monotonic power under the alternative hypothesis irrespective of how large the sample size is.
翻译:我们显示,如果数据不固定,在固定的零星状态下,基于固定的零星状态的现有固定的推断方法并非关键(即使在学生入学后),在固定的零星状态下,进入固定的零星状态分配的干扰参数并非关键(即使在学生入学后),其形式是高山进程的一个复杂功能,取决于当地长期差异的综合变化,取决于相关系列的第二个时刻(例如,递后者和线性回归模型的错误)。因此,基于固定的回归模型的现有固定的推论方法在理论上不具有一般效力。进入固定的零星分布的干扰参数可以在小型的零星状态下持续估算,但只能以非准趋同率的速度估算。因此,我们表明拒绝概率(ERP)的错误是比不固定的幅度大得多,而且也大于基于HAC标准根据传统惯性回归模型的估测者进行的正常测试。这些理论结果与Casini(2021年)和Casini、Deng和Perron(2021年)的最近有限抽样证据并不有效,表明固定的HAR测试在无弹性的假设下,在无弹性的假设下,它们能够在无弹性的假设下如何进行。