Asymptotic theory for QMLE for the real‐time GARCH(1,1) model

Ekaterina Smetanina and Wei Biao Wu

Journal of Time Series Analysis, 2021, vol. 42, issue 5-6, 752-776

Abstract: We investigate the asymptotic properties of the Gaussian quasi‐maximum‐likelihood estimator (QMLE) for the Real‐time GARCH(1,1) model of Smetanina (2017, Journal of Financial Econometrics, 15(4), 561–601). The developed theory relies on the functional dependence measure and recently developed theory for derivative processes in Dahlhaus etal. (2019, Bernoulli, 25(2), 1013–1044). We prove stationarity and ergodicity of the underlying processes and consistency for the QMLE estimator under mild conditions. Furthermore, under normality of the error term, we also establish asymptotic normality for QMLE, which then becomes MLE, at the usual T rate. Finally, in our simulations we show that consistency and asymptotic normality holds for typical sample sizes.

Date: 2021
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