Specification Tests for GARCH Processes with Nuisance Parameters on the Boundary
针对GARCH模型条件方差函数设定检验中真实参数可能位于参数空间边界的问题，提出了基于Kolmogorov-Smirnov和Cramér-von Mises类型的检验统计量，并引入一种新的bootstrap方法，该方法在参数估计向边界收缩时仍保持渐近有效性，模拟和实证表明其有限样本表现良好。
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.