Non-parametric estimation of time varying AR(1)–processes with local stationarity and periodicity

Abstract

Extending the ideas of [7], this paper aims at providing a kernel based non-parametric estimation of a new class of time varying AR(1) processes $(X_{t})$, with local stationarity and periodic features (with a known period $T$), inducing the definition $X_{t}=a_{t}(t/nT)X_{t-1}+\xi_{t}$ for $t\in \mathbb{N}$ and with $a_{t+T}\equiv a_{t}$. Central limit theorems are established for kernel estimators $\widehat{a}_{s}(u)$ reaching classical minimax rates and only requiring low order moment conditions of the white noise $(\xi_{t})_{t}$ up to the second order.