Forecast Error Symmetry in ARIMA Models

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Abstract
Forecasts and residuals from estimated autoregressive integrated moving average (ARIMA) models are investigated with respect to the symmetry of their distributions. For models estimated with and without intercept terms and, more generally, for regression models with correlated errors, it is shown that both the set of residuals and the set of forecast errors at leads 1, 2, …, l have joint distributions that are symmetric about 0 under mild symmetry conditions on the true process generating the observations. We consider most common estimation methods including maximum likelihood, conditional least squares, and unconditional least squares. We also show that the forecast error t statistics are symmetrically distributed about 0.

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