Box-Tiao (1975) Intervention Analysis with Applications to Economic and Environmental Problems

Summary

Box and Tiao study the effect of a known intervention on a time series when the observations are serially dependent. They model both the dynamic response to the intervention and the noise with difference-equation (ARIMA) models: the series is an ARIMA noise process plus a transfer function applied to an indicator variable marking the intervention. Because the ordinary $t$-test is invalid under serial correlation, level changes are estimated by maximum likelihood within the combined model. The method is illustrated on Los Angeles photochemical-smog (ozone) data and on changes in the consumer price index.

Key Claims

• The question. "Given a known intervention, is there evidence that a change of the expected kind actually occurred, and if so what can be said of its nature and magnitude?" — the interrupted-time-series problem.
• Model. $Y_t = \nu(B)\,I_t + N_t$: an indicator $I_t$ (a step for a permanent change, a pulse for a temporary one) is passed through a rational transfer function $\nu(B)=\omega(B)/\delta(B)$ to produce a flexible dynamic response, added to an ARIMA noise process $N_t$.
• Dynamic responses. The transfer function lets the effect be abrupt or gradual, permanent or decaying — e.g. $\omega/(1-\delta B)$ builds up geometrically to a limiting level change $\omega/(1-\delta)$.
• Correct inference under dependence. Serial correlation makes the naive before/after $t$-test invalid; embedding the intervention in an ARIMA model and estimating by maximum likelihood gives valid tests and standard errors for the level-change parameters.
• Applications. L.A. ozone (the 1960 Golden State Freeway opening plus a regulation reducing reactive hydrocarbons) and the CPI.

Concepts Introduced or Extended

• Intervention Analysis

Entities Mentioned

• George E. P. Box
• George C. Tiao

Quotes

"This article discusses the effect of interventions on a given response variable in the presence of dependent noise structure. Difference equation models are employed to represent the possible dynamic characteristics of both the interventions and the noise."

My Take

The paper that made "did the policy work?" a well-posed time-series question. Its lasting contribution is methodological discipline: it recognises that a before/after comparison is confounded by serial dependence and trend, and that the fix is to model the counterfactual noise (an ARIMA process) and the intervention's dynamic response (a transfer function) jointly, so the level-change parameter is estimated cleanly. That template is the direct ancestor of modern interrupted-time-series and much of policy evaluation, and it is the known-date sibling of change-point analysis (where the break time is itself unknown and estimated) and of structural-break testing. The main caveats are the ones Box-Jenkins modelling always carries — the conclusion is only as good as the identified ARIMA noise model and the assumed shape of the transfer function, and a mis-specified response can smear or hide a real effect.