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Asymptotic theory for QMLE for the real-time GARCH\((1,1)\) model. (English) Zbl 1476.62193

Summary: We investigate the asymptotic properties of the Gaussian quasi-maximum-likelihood estimator (QMLE) for the Real-time GARCH(1,1) model of the first author [“Real-time GARCH”, J. Financ. Econom. 15, No. 4, 561–601 (2017; doi:10.1093/jjfinec/nbx008)]. The developed theory relies on the functional dependence measure and recently developed theory for derivative processes in [R. Dahlhaus et al., Bernoulli 25, No. 2, 1013–1044 (2019; Zbl 1427.60057)]. 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 \(\sqrt{T}\) rate. Finally, in our simulations we show that consistency and asymptotic normality holds for typical sample sizes.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 1427.60057
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References:

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