A Note on "Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach"
Frederik Krabbe
TL;DR
The note extends stationary ergodic solution results for stochastic difference equations $Y_t = \Phi_t(Y_{t-1})$ to settings where the state space $Y$ is only a complete subspace of a separable Banach space. Under Lyapunov-type contraction conditions, specifically $E[ \log^{+} \|\Phi_0(y) - y\| ] < \infty$, $E[ \log^{+} \Lambda(\Phi_0) ] < \infty$, and $-\infty \le E[ \log \Lambda(\Phi_0^{(r)}) ] < 0$ for some $r$, there exists a unique stationary ergodic solution given by $Y_t = \lim_{n\to\infty} \Phi_t^{(n)}(y)$. The paper also proves stability with respect to initial conditions and perturbations: $\|\bar{Y}_t - Y_t\| \to 0$ a.s., and if a perturbed driver $\hat{\Phi}_t$ satisfies $\|\hat{\Phi}_t(y) - \Phi_t(y)\| \to 0$ and $\Lambda(\hat{\Phi}_t - \Phi_t) \to 0$ a.s., then $\|\hat{Y}_t - Y_t\| \to 0$ a.s. These results rely on, and extend, the classical Bougerol (1993) and Straumann & Mikosch (2006) framework by accommodating state spaces that are complete subspaces rather than full Banach spaces, via a perturbation/approximation argument and standard lemmata on products and Lyapunov-type contractions. The findings broaden applicability to models with non-Banach state spaces, enabling robust quasi-maximum-likelihood-type analyses in conditional heteroskedastic time series.
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 $Y_t \overset{a.s.}{=} Φ_t (Y_{t-1}), t \in \mathbb{Z}$ where $(Φ_t)_{t \in \mathbb{Z}}$ 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 $\hat{Y}_t \overset{a.s.}{=} \hatΦ_t (\hat{Y}_{t-1}), t \in \mathbb{N}$ with $\hat{Y}_0$ given where $(\hatΦ_t)_{t \in \mathbb{N}}$ is only a sequence of random Lipschitz continuous functions from $(Y,|| \cdot ||)$ to $(Y,|| \cdot ||)$ satisfies $γ^t || \hat{Y}_t - Y_t || \overset{a.s.}{\rightarrow} 0$ as $t \rightarrow \infty$ for some $γ> 1$. 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).