Effect of regressor forecast error on the variance of regression forecasts

It is well understood that the standard formulation for the variance of a regression-model forecast produces interval estimates that are too narrow, principally because it ignores regressor forecast error. While the theoretical problem has been addressed, there has not been an adequate explanation of the effect of regressor forecast error, and the empirical literature has supplied a disparate variety of bits and pieces of evidence. Most business-forecasting software programs continue to supply only the standard formulation. This paper extends existing analysis to derive and evaluate large-sample approximations for the forecast error variance in a single-equation regression model. We show how these approximations substantially clarify the expected effects of regressor forecast error. We then present a case study, which (a) demonstrates how rolling out-of-sample evaluations can be applied to obtain empirical estimates of the forecast error variance, (b) shows that these estimates are consistent with our large-sample approximations and (c) illustrates, for ‘typical’ data, how seriously the standard formulation can understate the forecast error variance.