Inference for Box-Cox Transformed Threshold GARCH Models with Nuisance Parameters

Generalized autoregressive conditional heteroscedastic (GARCH) models have been widely used for analyzing financial time series with time-varying volatilities. To overcome the defect of the Gaussian quasi-maximum likelihood estimator (QMLE) when the innovations follow either heavy-tailed or skewed distributions, Berkes & Horváth (Ann. Statist., 32, 633, 2004) and Lee & Lee (Scand. J. Statist. 36, 157, 2009) considered likelihood methods that use two-sided exponential, Cauchy and normal mixture distributions. In this paper, we extend their methods for Box-Cox transformed threshold GARCH model by allowing distributions used in the construction of likelihood functions to include parameters and employing the estimated quasi-likelihood estimators (QELE) to handle those parameters. We also demonstrate that the proposed QMLE and QELE are consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.