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Title: Bootstrap specication tests for GARCH processes with nuisance parameters on the boundary Authors:  Indeewara Perera - University of Sheffield (United Kingdom) [presenting]
Giuseppe Cavaliere - University of Bologna (Italy)
Anders Rahbek - University of Copenhagen (Denmark)
Abstract: Tests are developed 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 Cramer-von Mises type, and are based on a certain empirical process marked by centered squared residuals. The limiting distributions of the test statistics depend on unknown nuisance parameters in a non-trivial 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 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 $n$), we also discuss how to obtain valid inference for sequences of DGPs with parameters approaching the boundary. A simulation study demonstrates that the new tests have excellent finite sample properties. Two data examples illustrate the implementation of the proposed tests in applications.