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340 Chapter 10 Further Topics
10.2 Intervention Analysis
During the period for which a time series is observed, it is sometimes the case that a
change occurs that affects the level of the series. A change in the tax laws may, for
example, have a continuing effect on the daily closing prices of shares on the stock
market. In the same way construction of a dam on a river may have a dramatic effect
on the time series of streamflows below the dam. In the following we shall assume
that the time T at which the change (or “intervention”) occurs is known.
To account for such changes, Box and Tiao (1975) introduced a model for intervention
analysis that has the same form as the transfer function model
∞
Yt τjXt−j + Nt, (10.2.1)
j0
except that the input series {Xt} is not a random series but a deterministic function of
t. It is clear from (10.2.1) that ∞ j0 τjXt−j is then the mean of Yt. The function {Xt}
and the coefficients {τj} are therefore chosen in such a way that the changing level
of the observations of {Yt} is well represented by the sequence ∞ j0 τjXt−j. For a
series {Yt} with EYt 0 for t ≤ T and EYt → 0ast →∞, a suitable input series is
1 if t T,
Xt It(T )
(10.2.2)
0 if t T.
For a series {Yt} with EYt 0 for t ≤ T and EYt → a 0ast →∞, a suitable
input series is
∞
1 if t ≥ T,
Xt Ht(T ) It(k)
(10.2.3)
kT 0 if t