Robust Procedures for Semiparametric Partly Linear Autoregression

In many situations, a fully nonparametric autoregressive process, $\{y_t\}$, can neglect a possible linear relation between $y_t$ and any lag $y_{t-k}$ and so, it may be sensible to fit a partly linear autoregressive model.

In the simplest partly linear autoregression model, the stationary process $\{y_t: t\geq 3\}$ satisfies \[\begin{equation}y_t=\beta y_{t-1}+ g(y_{t-2})+\epsilon_t, \label{eq:1:507}\end{equation}\] with $\epsilon_t$ i.i.d. independent of $\{y_{t-j}, j\geq 1\}$, $E(\epsilon_t)=0$ finite $E\epsilon_t^2$.

The sensitivity of the least squares estimates to outliers has been extensively described both in the purely parametric and in the nonparametric setting. The sensitivity to outliers of the classical estimates under a partly linear autoregression model ($\ref{eq:1:507}$) is good evidence that robust methods, less sensitive to a single wild spike outlier, would be desirable, since the effect of a single outlier is even worse than in the independent setting.

In this talk, which corresponds to a joint work with Ana Bianco, the problem of obtaining a family of robust estimates for model ($\ref{eq:1:507}$) is addressed introducing a three–step robust procedure whose asymptotic behavior is derived. A robust procedure to choose the smoothing parameter is also discussed. Through a Monte Carlo study, the performance of the proposed estimates is compared with the classical ones. Moreover, a procedure to detect anomalous observations is discussed.