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Nonparametric estimation of volatility models with serially dependent innovations. (English) Zbl 1107.62109

Summary: We are interested in modelling the time series process \(y_t=\sigma (x_t) \varepsilon_t\), where \(\varepsilon_t=\varphi_0\varepsilon_{t-1}+ v_t\). This model is of interest as it provides a plausible linkage between risk and expected return of financial assets. Further, the model can serve as a vehicle for testing the martingale difference sequence hypothesis, which is typically uncritically adopted in financial time series models. When \(x_t\) has a fixed design, we provide a novel nonparametric estimator of the variance function based on the difference approach and establish its limiting properties. When \(x_t\) is strictly stationary on a strongly, mixing base (hereby allowing for ARCH effects) the nonparametric variance function estimator by J. Fan and Q. Yao [Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, No. 3, 645–660 (1998; Zbl 0918.62065)] can be applied and seems very promising. We propose a semiparametric estimator of \(\varphi_0\) that is \(\sqrt T\)-consistent, adaptive, and asymptotic normally distributed under very general conditions on \(x_t\).

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0918.62065
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References:

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