The threshold GARCH model: estimation and density forecasting for financial returns

Abstract

<jats:title>Abstract</jats:title><jats:p>We consider multiple threshold value-at-risk (VaRt) estimation and density forecasting for financial data following a threshold GARCH model. We develop an α-quantile quasi-maximum likelihood estimation (QMLE) method for VaRt by showing that the associated density function is an α-quantile density and belongs to the tick-exponential family. This establishes that our estimator is consistent for the parameters of VaRt. We propose a density forecasting method for quantile models based on VaRt at a single nonextreme level, which overcomes some limitations of existing forecasting methods with quantile models. We find that for heavy-tailed financial data our α-quantile QMLE method for VaRt outperforms the Gaussian QMLE method for volatility. We also find that density forecasts based on VaRt outperform those based on the volatility of financial data. Empirical work on market returns shows that our approach also outperforms some benchmark models for density forecasting of financial returns.</jats:p>