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14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 583

Table 14.2 presents autoregressive models of returns are all zero in the AR(2) or AR(4) model.
the excess return on a broad-based index of stock In fact, the adjusted R2 of one of the models is neg-
prices, called the CRSP value-weighted index, using ative and the other two are only slightly positive,
monthly data from 1960:M1 to 2002:M12, where suggesting that none of these models is useful for
“M1” denotes the first month of the year (January), forecasting.
“M2” denotes the second month, and so forth. The
monthly excess return is what you earn, in percentage These negative results are consistent with the
terms, by purchasing a stock at the end of the previ- theory of efficient capital markets, which holds
ous month and selling it at the end of this month, that excess returns should be unpredictable
minus what you would have earned had you pur- because stock prices already embody all currently
chased a safe asset (a U.S. Treasury bill). The return available information. The reasoning is simple: If
on the stock includes the capital gain (or loss) from market participants think that a stock will have a
the change in price plus any dividends you receive positive excess return next month, then they will
during the month. The data are described further in buy that stock now, but doing so will drive up the
Appendix 14.1. price of the stock to exactly the point at which
there is no expected excess return. As a result, we
Sadly, the results in Table 14.2 are negative. The should not be able to forecast future excess returns
coefficient on lagged returns in the AR(1) model by using past publicly available information, and
is not statistically significant, and we cannot reject we cannot do it, at least using the regressions in
the null hypothesis that the coefficients on lagged Table 14.2.

14.4 Time Series Regression with Additional
Predictors and the Autoregressive
Distributed Lag Model

Economic theory often suggests other variables that could help forecast a variable
of interest. These other variables, or predictors, can be added to an autoregression
to produce a time series regression model with multiple predictors. When other
variables and their lags are added to an autoregression, the result is an autoregressive
distributed lag model.

Forecasting GDP Growth Using the Term Spread

Interest rates on long-term and short-term bonds move together, but not one for
one. Figure 14.3a plots interest rates on 10-year U.S. Treasury bonds and 3-Month
Treasury bills from 1960 to 2012. Both interest rates show the same long-run
tendencies: both were low in the 1960s, both rose through the 1970s and peaked