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Working Paper

Random Walk Forecasts of Stationary Processes Have Low Bias

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We study the use of a zero mean first difference model to forecast the level of a scalar time series that is stationary in levels. Let bias be the average value of a series of forecast errors. Then the bias of forecasts from a misspecified ARMA model for the first difference of the series will tend to be smaller in magnitude than the bias of forecasts from a correctly specified model for the level of the series. Formally, let P be the number of forecasts. Then the bias from the first difference model has expectation zero and a variance that is O(1/P2), while the variance of the bias from the levels model is generally O(1/P). With a driftless random walk as our first difference 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.
Topics
Econometrics, modeling, and forecasting
08.03.2023ISSN 2573-7953 WP 23-18DOI 10.26509/frbc-wp-202318
Keywords ARMA Models, Overdifferenced, Prediction, Macroeconomic Time Series, Simulation
JEL Classification Codes C22, C53, E37, E47

Working Papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to stimulate discussion and critical comment on research in progress. They may not have been subject to the formal editorial review accorded official Federal Reserve Bank of Cleveland publications. The views expressed in this paper are those of the authors and do not represent the views of the Federal Reserve Bank of Cleveland or the Federal Reserve System.

Suggested Citation

Lunsford, Kurt G., and Kenneth West. 2023. “Random Walk Forecasts of Stationary Processes Have Low Bias.” Federal Reserve Bank of Cleveland, Working Paper No. 23-18. https://doi.org/10.26509/frbc-wp-202318