• asymptotic normality;
  • consistency;
  • estimated likelihood estimator;
  • nuisance parameter;
  • quasi-maximum likelihood estimator;
  • threshold GARCH model

Abstract.  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.