Extended Glivenko–Cantelli theorem and L₁ strong consistency of innovation density estimator for time-varying semiparametric ARCH model

ABSTRACT

This paper extends the classical Glivenko–Cantelli theorem for the empirical cumulative distribution function based on the innovations in the ARCH model with a slowly time-varying trend. In this semiparametric time-varying model, $L_1$ strong consistency for the innovation density estimator via kernel smoothing method is established, given that the trend and ARCH parameter estimators meet some mild conditions. Besides, the strong consistency for the Gaussian quasi maximum likelihood estimator (QMLE) in the time-varying ARCH parameter is established as well. Moreover, in terms of the existence of the trend in the data, two major nonparametric trend estimators, B-spline and kernel estimators, are shown to be appropriate for the strong consistency results.

KEYWORDS:

• ARCH(p)
• B-spline
• Glivenko–Cantelli theorem
• Kernel density estimator
• time-varying trend

AMS SUBJECT CLASSIFICATIONS:

• 62G05
• 62G07
• 62M10

Acknowledgments

Constructive and insightful comments from two reviewers have led to substantial improvement in the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).