Comparison results for Garch processes

TL;DR

This paper investigates stochastic comparison methods for Garch-like processes, identifying how various stochastic orders are propagated through the process and their implications for different parameters and innovation distributions.

Contribution

It introduces new stochastic comparison results for Garch processes, especially regarding the convex order and parameter effects, extending existing theoretical frameworks.

Findings

Convex order is propagated in symmetric innovations.

Multivariate comparison results are established.

Ordering with respect to Garch(1,1) parameters is discussed.

Abstract

We consider the problem of stochastic comparison of general Garch-like processes, for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the Garch process itself, and discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the Garch process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular order. Finally we discuss ordering with respect to the parameters in the Garch (1,1) case. Key words: Garch, Convex Order, Peakedness, Kurtosis, Supermodularity.