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1、I. Intervention ModelsThere is often a qualitative change in events which can have an effect on quantitative measures. For example, price may increase sharply at a point in time, affecting demand. An example of such a situation is discussed in the Vandaele textbook for telephone directory assistance
2、 in Cincinnati. Legislation may be introduced, such as per se legislation making blood alcohol content for drivers, per se, illegal, despite the lack of an accident or lack of specifically observed behavior, such as driving dangerously. Such legislation was introduced in Great Britain before it was
3、introduced in California. Is per se legislation effective in reducing casualties? This issue is examined in the assigned reading authored by Phillips, Ray and Votey. Another example is whether fatal airplane accidents affect the price of the stock of the carriers. The capital asset pricing model is
4、used to assess this question in an article by Borenstein and Zimmerman in the American Economic Review, Dec. 1988, and referenced in the text, Microeconomics for Business Decisions (1992) by Eric Solberg.If the date when the event that is suspected of having an impact on the data in question is know
5、n, then the behavior of the data, for example, a time series, can be examined. If the impact of this event is large, then visual inspection will often reveal the effect. If the impact of the event is small, it may take more sophisticated methods to see if the impact was statistically significant. If
6、 the series being studied is noisy or highly variable, distinguishing the impact of the event from other variations may require great care.A useful approach is to model the behavior of the time series before the event, model the event, model the time series after the event, and splice these together
7、. How to model the event raises the question of the nature of the impact of the event. Is it immediate, but transitory, affecting only the moment? Is it immediate and permanent with a lasting change? Is the effect anticipated, with the impact building or fading in? Is the impact immediate but not pe
8、rmanent, perhaps fading with time? How would we model these alternative events?II. Modeling the EventA. A Once and For All Change in Levels: The Step FunctionY(t) =Z4 S(t)-B. A One Period Transitory Effect in Levels, or the Effect in Differences of a Once and For All Change in Levels: The Pulse Func
9、tion Y(t) = Z4 P(t) = Z4 (1-Z) S(t)-C. A Change in Levels with a Partial Fallback: The Quasi-DifferenceY(t) = (0.8-0.4Z) Z4 S(t) = 0.8(1-0.5Z) Z4 S(t)-D. Zero Long Run Impact, the Perfect Offset: First Difference of the Pulse FunctionY(t) = (0.8-0.8Z) Z4 P(t) = 0.8 Z4 (1-Z) P(t)-E. The Fade-In: A Di
10、stributed Lag of the Step FunctionY(t) = 0.8/(1-0.5Z) Z4 S(t)-F. The Fade-Out or Decay: A Distributed Lag of the PulseY(t) = 0.8/(1-0.5Z) Z4 P(t)-G. The Ramp or Trend: A Distributed Sum of the Step FunctionY(t) = 0.8/(1-Z) Z4 S(t)-III. An Intervention Model of Telephone Directory AssistancePursuing
11、the telephone assistance model, we see that it is important to divide the sample into at least two parts, before the intervention and after the intervention. Failure to do so may mean the event itself will so increase the unexplained variance as to disguise the nature of the time series, making iden
12、tification and modelling difficult. Separating the data and analyzing the time series before intervention, we difference assistance, A(t), to remove trend. Identifying the difference in assistance, we find a seasonal component and so seasonally difference the data to obtain:SDDA(t) = (1-Z12)(1-Z)A(t
13、) .Identifying this series we find an MA(12) is appropriate, leading to the following estimates:(1-Z12)(1-Z)A(t) = e(t) - 0.843 e(t-12) =1-0.843Z12e(t).The intervention is modeled in levels as a step function at the 127th observation, Z127S(t).The next step is to combine these two models, expressed
14、in levels. From above, we have that assistance is:A(t) = 1-0.843Z12e(t)/(1-Z12)(1-Z),and adding the delayed step function,A(t) = 1-0.843Z12e(t)/(1-Z12)(1-Z) + Z127S(t).Rather than estimate this model in levels to test its adequacy, we estimate the equivalent model for the assistance series, seasonally differenced and first differenced:(1-Z1
