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578 Chapter 14 Introduction to Time Series Regression and Forecasting
A systematic way to forecast GDP growth, GDPGRt, using the previous quarter’s
value, GDPGRt−1, is to estimate an OLS regression of GDPGRt on GDPGRt−1.
Estimated using data from 1962 to 2012, this regression is
GDPGRt = 1.991 + 0.344 GDPGRt - 1, (14.7)
(0.349) (0.075)
where, as usual, standard errors are given in parentheses under the estimated
coefficients, and GDPGR is the predicted value of GDPGR based on the
estimated regression line. The model in Equation (14.7) is called a first-order
autoregression: an autoregression because it is a regression of the series onto its
own lag, GDPGRt−1, and first-order because only one lag is used as a regressor.
The coefficient in Equation (14.7) is positive, so positive growth of GDP in one
quarter is associated with positive growth in the next quarter.
A first-order autoregression is abbreviated AR(1), where the 1 indicates that
it is first order. The population AR(1) model for the series Yt is
Yt = b0 + b1Yt - 1 + ut, (14.8)
where ut is an error term.
Forecasts and forecast errors. Suppose that you have historical data on Y, and you
want to forecast its future value. If Yt follows the AR(1) model in Equation (14.8)
and if b0 and b1 are known, then the forecast of YT + 1 based on YT is b0 + b1YT.
In practice, b0 and b1 are unknown, so forecasts must be based on estimates
of b0 and b1. We will use the OLS estimators bn0 and bn1, which are constructed
using historical data. In general, Yn T + 10T will denote the forecast of YT + 1 based on
information through period T, using a model estimated with data through period T.
Accordingly, the forecast based on the AR(1) model in Equation (14.8) is
Yn T + 10T = bn0 + bn1YT, (14.9)
where bn0 and bn1 are estimated using historical data through time T.
The forecast error is the mistake made by the forecast; this is the difference
between the value of YT + 1 that actually occurred and its forecasted value based
on YT:
Forecast error = YT + 1 - Yn T + 10T. (14.10)
