Prediction and nonparametric estimation for time series with heavy tails

Motivated by prediction problems for time series with heavy‐tailed marginal distributions, we consider methods based on `local least absolute deviations' for estimating a regression median from dependent data. Unlike more conventional `local median' methods, which are in effect based on locally fitting a polynomial of degree 0, techniques founded on local least absolute deviations have quadratic bias right up to the boundary of the design interval. Also in contrast to local least‐squares methods based on linear fits, the order of magnitude of variance does not depend on tail‐weight of the error distribution. To make these points clear, we develop theory describing local applications to time series of both least‐squares and least‐absolute‐deviations methods, showing for example that, in the case of heavy‐tailed data, the conventional local‐linear least‐squares estimator suffers from an additional bias term as well as increased variance.