A0669
Title: Robust estimation of conditional variance of time series using density power divergences
Authors: T.N. Sriram - University of Georgia (United States) [presenting]
Abstract: Suppose $Z_{t}$ is the square of a time series $Y_{t}$ whose conditional mean is zero. We do not specify a model for $Y_{t}$, but assume that there exists a $p \times 1$ parameter vector ${\bf \Phi}$ such that the conditional distribution of $Z_t | {\mathbb Z}_{t-1}$ is the same as that of $Z_t | {\mathbb\Phi}^{T} {\mathbb Z}_{t-1}$, where ${\mathbb Z}_{t-1} = (Z_{t-1}, \ldots, Z_{t-p})^{T}$ for some lag $p \ge 1$. Consequently, the conditional variance of $Y_{t}$ is some function of ${\mathbb \Phi}^T {\mathbb Z}_{t-1}$. To estimate ${\mathbb \Phi}$, we propose a robust estimation methodology based on Density Power Divergences (DPD) indexed by a tuning parameter $\alpha \in [0,1]$, which yields a continuum of estimators, $\{{\widehat {\mathbb\Phi}}_{\alpha}; \alpha \in [0,1]\} $, where $\alpha$ controls the trade-off between robustness and efficiency of the DPD estimators. For each $\alpha$, ${\widehat {\mathbb\Phi}}_{\alpha}$ is shown to be strongly consistent. We develop data-dependent criteria for the selection of optimal $\alpha$ and lag $p$ in practice. We illustrate the usefulness of our DPD methodology via simulation studies for ARCH-type models, where the errors are drawn from a gross-error contamination model and the conditional variance is a linear and/or nonlinear function of ${\mathbb\Phi}^T {\mathbb Z}_{t-1}$.
