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15.2 Dynamic Causal Effects 641

where ut is an error term that includes measurement error in Yt and the effect of
omitted determinants of Yt. The model in Equation (15.3) is called the distributed
lag model relating Xt, and r of its lags, to Yt.

As an illustration of Equation (15.3), consider a modified version of the
tomato/fertilizer experiment: Because fertilizer applied today might remain in the
ground in future years, the horticulturalist wants to determine the effect on tomato
yield over time of applying fertilizer. Accordingly, she designs a 3-year experiment
and randomly divides her plots into four groups: The first is fertilized in only the
first year; the second is fertilized in only the second year; the third is fertilized in
only the third year; and the fourth, the control group, is never fertilized. Tomatoes
are grown annually in each plot, and the third-year harvest is weighed. The three
treatment groups are denoted by the binary variables Xt - 2, Xt - 1, and Xt, where t
represents the third year (the year in which the harvest is weighed), Xt - 2 = 1 if
the plot is in the first group (fertilized two years earlier), Xt - 1 = 1 if the plot was
fertilized 1 year earlier, and Xt = 1 if the plot was fertilized in the final year. In the
context of Equation (15.3) (which applies to a single plot), the effect of being fertil-
ized in the final year is b1, the effect of being fertilized 1 year earlier is b2, and the
effect of being fertilized 2 years earlier is b3. If the effect of fertilizer is greatest in
the year it is applied, then b1 would be larger than b2 and b3.

More generally, the coefficient on the contemporaneous value of Xt, b1, is the
contemporaneous or immediate effect of a unit change in Xt on Yt. The coefficient
on Xt - 1, b2, is the effect on Yt of a unit change in Xt - 1 or, equivalently, the effect
on Yt + 1 of a unit change in Xt; that is, b2 is the effect of a unit change in X on Y
one period later. In general, the coefficient on Xt - h is the effect of a unit change
in X on Y after h periods. The dynamic causal effect is the effect of a change in Xt
on Yt, Yt + 1, Yt + 2, and so forth; that is, it is the sequence of causal effects on cur-
rent and future values of Y. Thus, in the context of the distributed lag model in
Equation (15.3), the dynamic causal effect is the sequence of coefficients b1,
b2, c, br + 1.

Implications for empirical time series analysis. This formulation of dynamic causal
effects in time series data as the expected outcome of an experiment in which dif-
ferent treatment levels are repeatedly applied to the same subject has two implica-
tions for empirical attempts to measure the dynamic causal effect with observational
time series data. The first implication is that the dynamic causal effect should not
change over the sample on which we have data. This in turn is implied by the data
being jointly stationary (Key Concept 14.5). As discussed in Section 14.7, the
hypothesis that a population regression function is stable over time can be tested
using the QLR test for a break, and it is possible to estimate the dynamic causal