Random Walk Forecasts of Stationary Processes Have Low Bias

We study the use of a misspecified overdifferenced model to forecast the level of a stationary scalar time series. Let x(t) be the series, and let bias be the sample average of a series of forecast errors. Then, the bias of forecasts of x(t) generated by a misspecified overdifferenced ARMA model for Δx(t) will tend to be smaller in magnitude than the bias of forecasts of x(t) generated by a correctly specified model for x(t). Formally, let P be the number of forecasts. The bias from the model for Δx(t) has a variance that is O(1/P^2), while the variance of the bias from the model for x(t) generally is O(1/P). With a driftless random walk as our baseline overdifferenced model, we confirm this theoretical result with simulations and empirical work: random walk bias is generally one-tenth to one-half that of an appropriately specified model fit to levels.