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The Threshold GARCH Model: Estimation and Density Forecasting for Financial Returns*

Yuzhi Cai Orcid Logo, Julian Stander

Journal of Financial Econometrics, Volume: 18, Issue: 2, Pages: 395 - 424

Swansea University Author: Yuzhi Cai Orcid Logo

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DOI (Published version): 10.1093/jjfinec/nbz014

Abstract

We consider multiple threshold value-at-risk (VaR\(_t\)) estimation and density forecasting for financial data following a threshold GARCH model. We develop an \(\alpha\)-quantile quasi-maximum likelihood estimation (QMLE) method for VaR\(_t\) by showing that the associated density function is an \(...

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Published in: Journal of Financial Econometrics
ISSN: 1479-8409 1479-8417
Published: Oxford University Press (OUP) 2019
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URI: https://cronfa.swan.ac.uk/Record/cronfa49769
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Abstract: We consider multiple threshold value-at-risk (VaR\(_t\)) estimation and density forecasting for financial data following a threshold GARCH model. We develop an \(\alpha\)-quantile quasi-maximum likelihood estimation (QMLE) method for VaR\(_t\) by showing that the associated density function is an \(\alpha\)-quantile density and belongs to the tick-exponential family. This establishes that our estimator is consistent for the parameters of VaR\(_t\). We propose a density forecasting method for quantile models based on VaR\(_t\) at a single non-extreme level, which overcomes some limitations of existing forecasting methods with quantile models. We find that for heavy-tailed financial data our \(\alpha\)-quantile QMLE method for VaR\(_t\) outperms the Gaussian QMLE method for volatility. We also find that density forecasts based on VaR\(_t\) 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.
Keywords: \(\alpha\)-quantile density, density forecasting, QMLE, threshold, value-at-risk (VaR)
College: Faculty of Humanities and Social Sciences
Issue: 2
Start Page: 395
End Page: 424

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