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Random Walk Forecasts of Stationary Processes Have Low Bias

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  • Kurt Graden Lunsford
  • Kenneth D. West

Abstract

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/P-squared), 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.

Suggested Citation

  • Kurt Graden Lunsford & Kenneth D. West, 2023. "Random Walk Forecasts of Stationary Processes Have Low Bias," Working Papers 23-18, Federal Reserve Bank of Cleveland.
    • Handle: RePEc:fip:fedcwq:96521
    DOI: 10.26509/frbc-wp-202318
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    References listed on IDEAS

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    1. Richardson, Matthew & Stock, James H., 1989. "Drawing inferences from statistics based on multiyear asset returns," Journal of Financial Economics, Elsevier, vol. 25(2), pages 323-348, December.
    2. Croushore Dean, 2010. "An Evaluation of Inflation Forecasts from Surveys Using Real-Time Data," The B.E. Journal of Macroeconomics, De Gruyter, vol. 10(1), pages 1-32, May.
    3. Ulrich K. Müller & Mark W. Watson, 2016. "Measuring Uncertainty about Long-Run Predictions," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 83(4), pages 1711-1740.
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    More about this item

    Keywords

    ARMA Models; Overdifferenced; Prediction; Macroeconomic Time Series; Simulation;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • E37 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Forecasting and Simulation: Models and Applications
    • E47 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Forecasting and Simulation: Models and Applications

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