A Note on "Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach"

Abstract

Bougerol (1993) and Straumann and Mikosch (2006) gave conditions under which there exists a unique stationary and ergodic solution to the stochastic difference equation $Y_t\stackrel{a.s.}{=}{\Phi }_t(Y_{t-1}),t\in \mathbb{Z}$ where $({\Phi }_t{)}_{t\in \mathbb{Z}}$ is a sequence of stationary and ergodic random Lipschitz continuous functions from $(Y,||\cdot ||)$ to $(Y,||\cdot ||)$ where $(Y,||\cdot ||)$ is a complete subspace of a real or complex separable Banach space. In the case where $(Y,||\cdot ||)$ is a real or complex separable Banach space, Straumann and Mikosch (2006) also gave conditions under which any solution to the stochastic difference equation ${\hat{Y}}_t\stackrel{a.s.}{=}{\hat{\Phi }}_t({\hat{Y}}_{t-1}),t\in \mathbb{N}$ with ${\hat{Y}}_0$ given where $({\hat{\Phi }}_t{)}_{t\in \mathbb{N}}$ is only a sequence of random Lipschitz continuous functions from $(Y,||\cdot ||)$ to $(Y,||\cdot ||)$ satisfies ${\gamma }^t||{\hat{Y}}_t-Y_t||\stackrel{a.s.}{\to }0$ as $t\to \infty$ for some $\gamma >1$. In this note, we give slightly different conditions under which this continues to hold in the case where $(Y,||\cdot ||)$ is only a complete subspace of a real or complex separable Banach space by using close to identical arguments as Straumann and Mikosch (2006).