A data-driven P-spline smoother and the P-Spline-GARCH-models

TL;DR

This paper introduces a data-driven P-spline smoothing method for time series, extending Spline-GARCH models with asymptotic properties, practical applications, and improved error process characteristics.

Contribution

It develops a novel data-driven smoothing parameter selection algorithm and proposes a semiparametric P-Spline-GARCH model with proven asymptotic normality and strong mixing properties.

Findings

The P-Spline-GARCH model effectively captures time series volatility.

The smoothing parameter selection algorithm improves model fitting.

Applications to risk measures demonstrate practical utility.

Abstract

Penalized spline smoothing of time series and its asymptotic properties are studied. A data-driven algorithm for selecting the smoothing parameter is developed. The proposal is applied to define a semiparametric extension of the well-known Spline-GARCH, called a P-Spline-GARCH, based on the log-data transformation of the squared returns. It is shown that now the errors process is exponentially strong mixing with finite moments of all orders. Asymptotic normality of the P-spline smoother in this context is proved. Practical relevance of the proposal is illustrated by data examples and simulation. The proposal is further applied to value at risk and expected shortfall.