Restricted normal mixture QMLE for non-stationary TGARCH(1, 1) models

The threshold GARCH (TGARCH) models have been very useful for analyzing asymmetric volatilities arising from financial time series. Most research on TGARCH has been directed to the stationary case. This paper studies the estimation of non-stationary first order TGARCH models. Restricted normal mixture quasi-maximum likelihood estimation (NM-QMLE) for non-stationary TGARCH models is proposed in the sense that we estimate the other parameters with any fixed location parameter. We show that the proposed estimators (except location parameter) are consistent and asymptotically normal under mild regular conditions. The impact of relative leptokursis and skewness of the innovations’ distribution and quasi-likelihood distributions on the asymptotic efficiency has been discussed. Numerical results lend further support to our theoretical results. Finally, an illustrated real example is presented.