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16.2 Multiperiod Forecasts 693

The two-quarter-ahead forecast of the growth rate of GDP in 2013:Q2 based
on data through 2012:Q4 is computed by substituting the values of GDPGR2012:Q4,
GDPGR2012:Q3, TSpread2012:Q4, and TSpread2012:Q3 into Equation (16.13); this yields

GDPGR2013:Q2∙2012:Q4 = 0.57 + 0.34GDPGR2012:Q4 + 0.03GDPGR2012:Q3

+ 0.62TSpread2012:Q4 - 0.01TSpread2012:Q3 = 1.68.
(16.14)

The three-quarter-ahead direct forecast ofGDPGRT+3 is computed by lagging all
the regressors in Equation (16.13) by one additional quarter, estimating that
regression, and then computing the forecast. The h-quarter-ahead direct forecast
of GDPGRT+h is computed by using GPDGRt as the dependent variable and the
regressors GPDGRt−h and TSpreadt−h, plus additional lags of GPDGRt−h and
TSpreadt−h, as desired.

Standard errors in direct multiperiod regressions. Because the dependent vari-
able in a multiperiod regression occurs two or more periods into the future, the
error term in a multiperiod regression is serially correlated. To see this, consider
the two-period-ahead forecast of the growth rate of GDP and suppose that a
surprise jump in oil prices occurs in the next quarter. Today’s two-period-ahead
forecast of the growth rate of GDP will be too low because it does not incorporate
this unexpected event. Because the oil price rise was also unknown in the previous
quarter, the two-period-ahead forecast made last quarter will also be too low.
Thus the surprise oil price jump next quarter means that both last quarter’s and
this quarter’s two-period-ahead forecasts are too low. Because of such intervening
events, the error term in a multiperiod regression is serially correlated.

As discussed in Section 15.4, if the error term is serially correlated, the usual
OLS standard errors are incorrect or, more precisely, they are not a reliable basis
for inference. Therefore, heteroskedasticity- and autocorrelation-consistent
(HAC) standard errors must be used with direct multiperiod regressions. The
standard errors reported in Equation (16.13) for direct multiperiod regressions
therefore are Newey–West HAC standard errors, where the truncation parameter
m is set according to Equation (15.17); for these data (for which T = 128), Equa-
tion (15.17) yields m = 4. For longer forecast horizons, the amount of overlap—
and thus the degree of serial correlation in the error—increases: In general, the
first h - 1 autocorrelation coefficients of the errors in an h-period-ahead regres-
sion are nonzero. Thus larger values of m than indicated by Equation (15.17) are
appropriate for multiperiod regressions with long forecast horizons.

Direct multiperiod forecasts are summarized in Key Concept 16.3.