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.*
A_{i} represents the actual observations and P_{i} 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 A_{i} and P_{i} 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 P_{i} 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.

- Theil,
H. (1958),
*Economic Forecasts and Policy.*Amsterdam: North Holland. - Thiel,
H. (1966),
*Applied Economic Forecasting*. Chicago: Rand McNally. - Bliemel,
F.W. (1973), “Theil’s forecast accuracy coefficient: A clarification,”
*Journal of Marketing Research,*10, 444-446. - Ahlburg, D. (1984), “Forecast evaluation and improvement
using Theil’s decomposition,”
*Journal of Forecasting*, 3, 345-351.