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14.3 Autoregressions 579

Forecasts versus predicted values. The forecast is not an OLS predicted value,
and the forecast error is not an OLS residual. OLS predicted values are calculated
for the observations in the sample used to estimate the regression. In contrast, the
forecast is made for some date beyond the data set used to estimate the regression,
so the data on the actual value of the forecasted dependent variable are not in the
sample used to estimate the regression. Similarly, the OLS residual is the difference
between the actual value of Y and its predicted value for observations in the sample,
whereas the forecast error is the difference between the future value of Y, which is
not contained in the estimation sample, and the forecast of that future value. Said
differently, forecasts and forecast errors pertain to “out-of-sample” observations,
whereas predicted values and residuals pertain to “in-sample” observations.

Root mean squared forecast error. The root mean squared forecast error
(RMSFE) is a measure of the size of the forecast error—that is, of the magnitude
of a typical mistake made using a forecasting model. The RMSFE is the square
root of the mean squared forecast error:

RMSFE = 3E3(YT + 1 - Yn T + 10T)24. (14.11)

The RMSFE has two sources of error: the error arising because future values of
ut are unknown and the error in estimating the coefficients b0 and b1. If the first
source of error is much larger than the second, as it can be if the sample size is
large, then the RMSFE is approximately 2var(ut), the standard deviation of the
error ut in the population autoregression [Equation (14.8)]. The standard devia-
tion of ut is in turn estimated by the standard error of the regression (SER; see
Section 4.3). Thus, if uncertainty arising from estimating the regression coeffi-
cients is small enough to be ignored, the RMSFE can be estimated by the standard
error of the regression. Estimation of the RMSFE including both sources of fore-
cast error is taken up in Section 14.4.

Application to GDP growth. What is the forecast of the growth rate of GDP in the
first quarter of 2013 (2013:Q1) that a forecaster would have made in 2012:Q4, based
on the estimated AR(1) model in Equation (14.7) (which was estimated using data
through 2012:Q4)? According to Table 14.1, the growth rate of GDP in 2012:Q4
was 0.15% (so GDPGR 2012:Q4 = 0.15). Plugging this value into Equation (14.7),
the forecast of the growth rate of GDP in 2013:Q1 is GDPGR2013:Q1͉2012:Q4 =
1.991 + 0.344 * GDPGR2012:Q4 = 1.991 + 0.344 * 0.15 = 2.0 (rounded to the
nearest tenth).Thus, the AR(1) model forecasts that the growth rate of GDP will
be 2.0% in 2013:Q1.