Adaptive density estimation for general ARCH models

Abstract

We consider a model $Y\_t=σ\_tη\_t$ in which $(σ\_t)$ is not independent of the noise process $(η\_t)$, but $σ\_t$ is independent of $η\_t$ for each $t$. We assume that $(σ\_t)$ is stationary and we propose an adaptive estimator of the density of $\ln(σ^2\_t)$ based on the observations $Y\_t$. Under various dependence structures, the rates of this nonparametric estimator coincide with the minimax rates obtained in the i.i.d. case when $(σ\_t)$ and $(η\_t)$ are independent, in all cases where these minimax rates are known. The results apply to various linear and non linear ARCH processes.