Non-invertibility in Some Heteroscedastic Models

TL;DR

This paper investigates the conditions under which certain heteroscedastic models, like GARCH, are invertible, revealing that some models such as EGARCH and VGARCH can be non-invertible, affecting volatility estimation consistency.

Contribution

It provides necessary and sufficient conditions for invertibility in heteroscedastic models and identifies classes that can be non-invertible, impacting volatility inference.

Findings

GARCH($p$, $q$) models are always invertible.

EGARCH and VGARCH models can be non-invertible.

Non-invertibility can lead to inconsistent volatility estimates.

Abstract

In order to calculate the unobserved volatility in conditional heteroscedastic time series models, the natural recursive approximation is very often used. Following \cite{StraumannMikosch2006}, we will call the model \emph{invertible} if this approximation (based on true parameter vector) converges to the real volatility. Our main results are necessary and sufficient conditions for invertibility. We will show that the stationary GARCH($p$, $q$) model is always invertible, but certain types of models, such as EGARCH of \cite{Nelson1991} and VGARCH of \cite{EngleNg1993} may indeed be non-invertible. Moreover, we will demonstrate it's possible for the pair (true volatility, approximation) to have a non-degenerate stationary distribution. In such cases, the volatility estimate given by the recursive approximation with the true parameter vector is inconsistent.