Specification Tests for GARCH Processes with Nuisance Parameters on the Boundary

This article develops tests for the correct specification of the conditional variance function in GARCH models when the true parameter may lie on the boundary of the parameter space. The test statistics considered are of Kolmogorov-Smirnov and Cramér-von Mises type, and are based on empirical processes marked by centered squared residuals. The limiting distributions of the test statistics depend on unknown nuisance parameters in a nontrivial way, making the tests difficult to implement. We therefore introduce a novel bootstrap procedure which is shown to be asymptotically valid under general conditions, irrespective of the presence of nuisance parameters on the boundary. The proposed bootstrap approach is based on shrinking of the parameter estimates used to generate the bootstrap sample toward the boundary of the parameter space at a proper rate. It is simple to implement and fast in applications, as the associated test statistics have simple closed form expressions. Although the bootstrap test is designed for a data generating process with fixed parameters (i.e., independent of the sample size <i>n</i>), we also discuss how to obtain valid inference for sequences of DGPs with parameters approaching the boundary at the n−1/2 rate. A simulation study demonstrates that the new tests: (i) have excellent finite sample behavior in terms of empirical rejection probabilities under the null as well as under the alternative; (ii) provide a useful complement to existing procedures based on Ljung-Box type approaches. Two data examples illustrate the implementation of the proposed tests in applications.

本文针对真实参数可能位于参数空间边界的广义自回归条件异方差（GARCH）模型，构建了条件方差函数正确设定的检验方法。所考虑的检验统计量为柯尔莫哥洛夫-斯米尔诺夫（Kolmogorov-Smirnov）与克莱默-冯·米塞斯（Cramér-von Mises）型统计量，其构建基于以中心化平方残差为标记的经验过程。检验统计量的极限分布以非平凡方式依赖于未知冗余参数，使得此类检验的实际实施存在一定难度。为此本文提出一种全新的自助法（bootstrap），该方法被证明在一般条件下具备渐近有效性，无需考虑冗余参数是否位于参数空间边界。所提出的自助法以恰当速率将用于生成自助样本的参数估计量向参数空间边界收缩，由于对应的检验统计量具备简洁的闭合形式表达式，因此该方法实施简便、应用高效。尽管本文的自助检验针对固定参数（即不随样本量$n$变化）的数据生成过程（DGP, Data Generating Process）设计，但本文同时探讨了如何对参数以$n^{-1/2}$速率趋近边界的数据生成过程序列获得有效的统计推断。蒙特卡洛模拟研究表明，所提新检验具备两项优良特性：其一，在原假设与备择假设下，其经验拒绝概率均展现出优异的有限样本表现；其二，可作为现有基于Ljung-Box型方法的有益补充。文末通过两个数据实例演示了所提检验在实际应用中的具体实施流程。