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 xt be the series, and let bias be the sample average of a series of forecast errors. Then, the bias of forecasts of xt generated by a misspecified overdifferenced ARMA model for Δxt will tend to be smaller in magnitude than the bias of forecasts of xt generated by a correctly specified model for xt. Formally, let <i>P</i> be the number of forecasts. The bias from the model for Δxt has a variance that is O(1/P2), while the variance of the bias from the model for xt 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.