Theil proposed two error measures, but at different times and under the same symbol "U,” which has caused some confusion.
better called U1
better called U2
U1 is taken from Theil (1958, pp. 31-42), where he calls U a measure of forecast accuracy. Ai represents the actual observations and Pi the corresponding predictions. He left it open whether A and P should be used as absolute values or as observed and predicted changes. Both possibilities have been taken up in the literature and used by different forecasters, while Theil himself applied U1 to changes.
Theil (1966, chapter 2) proposed U2 as a measure of forecast quality, "where Ai and Pi stand for a pair of predicted and observed changes." Bliemel (1973) analyzed Theil’s measures and concluded that U1 has serious defects and is not informative for assessing forecast accuracy regardless of being applied with absolute values of the changes. For example, when applying U1 to changes, all U1 values will be bounded by 0 (the case of perfect forecasting) and 1 (the supposedly worst case). However, the value of 1 will be obtained when a forecaster applies the simple no-change model (all Pi are zero). All other possible forecasts would lead to a U1 value lower than 1, regardless of whether the forecast method led to better or worse performance than the naive no-change model. U1 should therefore not be used and should be regarded as a historical oddity. In contrast, U2 has no serious defects. It can be interpreted as the RMSE of the proposed forecasting model divided by the RMSE of a no-change model. It has the no-change model (with U2 = 1 for no-change forecasts) as the benchmark. U2 values lower than 1.0 show an improvement over the simple no-change forecast. Some researchers have found Theil’s error decomposition useful. For example, Ahlburg (1984) used it to analyze data on annual housing starts, where a mechanical adjustment provided major improvement in accuracy for the two-quarters-ahead forecast and minor improvements for eight-quarters-ahead. See also Relative Absolute Error.