Intervention Analysis

Definition

Intervention analysis (Box-Tiao 1975) estimates the effect of a known event — a policy change, a new law, a natural disaster — on a time series, correctly accounting for the serial dependence and trend in the series. The observed series is modeled as a transfer function applied to an indicator of the intervention, added to an ARIMA noise process: $Y_t = \nu(B)\,I_t + N_t$. It is the interrupted-time-series method, and the known-date counterpart of change-point analysis.

Key Ideas

• Indicator variable: step vs pulse. A step $S_t$ (0 before the event, 1 after) models a permanent change; a pulse $P_t$ (1 at the event date, 0 otherwise) models a temporary one. The choice encodes the hypothesized shape of the effect.
• Transfer function shapes the dynamics. The indicator enters through a rational lag polynomial $\nu(B)=\omega(B)/\delta(B)$, so the response can be abrupt or gradual and permanent or decaying: e.g. $\dfrac{\omega}{1-\delta B}S_t$ rises geometrically to a limiting level shift $\omega/(1-\delta)$; a pulse through the same filter gives a spike that decays back.
• ARIMA noise for the counterfactual. $N_t$ is a Box-Jenkins ARIMA process representing what the series would have done absent the intervention — the model of dependence and trend that a naive before/after comparison ignores.
• Maximum-likelihood estimation and testing. The intervention and noise parameters are estimated jointly by ML; because serial correlation invalidates the ordinary two-sample $t$-test, this is what delivers valid significance tests and standard errors for the level-change parameter.
• Known date, by assumption. The intervention time is given (not estimated) — the crucial contrast with change-point/structural-break methods, which infer whether and when a break occurred.

Why It Matters

• Policy and program evaluation. Intervention analysis is the classical tool for interrupted time series: measuring the effect of regulations, price changes, interventions, or shocks on an outcome series (public health, economics, environment).
• Correct uncertainty under dependence. By modeling the noise explicitly, it avoids the overstated significance that plagues naive before/after tests on autocorrelated data.
• Foundational template. It generalizes to transfer-function/dynamic-regression modeling and is the ancestor of modern interrupted-time-series and much of quasi-experimental time-series policy evaluation.

Open Questions

• Model dependence. Conclusions hinge on the identified ARIMA noise model and the assumed transfer-function form; a mis-specified response shape can smear or mask a genuine effect.
• Unknown or anticipated timing. If the effective date is uncertain or the effect is anticipated, the known-date assumption breaks and change-point methods or lead/lag specifications are needed.
• Multiple and confounded interventions. Overlapping interventions and concurrent confounders are hard to disentangle from a single series.

Related

• Change-Point Analysis
• Structural Break Testing
• Autoregressive Model
• Box-Tiao (1975) Intervention Analysis with Applications to Economic and Environmental Problems
• George E. P. Box
• George C. Tiao