We consider a model $Y_t=\sigma_t\eta_t$ in which $(\sigma_t)$ is not independent of the noise process $(\eta_t)$, but $\sigma_t$ is independent of $\eta_t$ for each $t$. We assume that $(\sigma_t)$ is stationary and we propose an adaptive estimator of the density of $\ln(\sigma^2_t)$ 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 $(\sigma_t)$ and $(\eta_t)$ are independent, in all cases where these minimax rates are known. The results apply to various linear and non linear ARCH processes.