Weakly universally consistent static forecasting of stationary and ergodic time series via local averaging and least squares estimates

• We estimate the conditional expectation of a stationary and ergodic time series.
• Four estimates are constructed as a convex combination of so-called experts.
• The experts are based on local averaging and least squares.
• The weights of the experts in the convex combination depend on the past performance.
• For all estimates the weak universal consistency is proven.

Given a stationary and ergodic time series the problem of estimating the conditional expectation of the dependent variable at time zero given the infinite past is considered. It is shown that the mean squared error of a combination of suitably defined local averaging or least squares estimates converges to zero for all distributions whenever the dependent variable is square integrable.