13.3 Intervention analysis and modeling structural breaks

Intervention Analysis and Structural Breaks

Intervention analysis is a method for measuring how specific external events (like new laws, natural disasters, or product launches) affect a time series. Without it, your model treats these disruptions as unexplained noise, which leads to biased parameter estimates and poor forecasts.

Concept of Intervention Analysis

A time series sometimes gets hit by an outside event that changes its behavior. Intervention analysis gives you a formal way to detect, quantify, and account for that change.

The core goals are:

• Determine significance: Did the intervention actually affect the series, or is the apparent shift just random variation?
• Quantify the effect: How large was the impact, and in which direction?
• Characterize duration: Was the effect temporary (a one-week spike) or permanent (a new baseline level)?
• Improve forecasts: Models that ignore real interventions will systematically over- or underestimate future values.

Interventions are always external to the series itself. A new tax law, an earthquake, or a competitor entering the market are interventions. A gradual trend that was already present in the data is not.

Modeling Structural Breaks

A structural break is an abrupt change in the level, trend, or variance of a time series. You model these breaks by adding dummy variables to your regression or ARIMA specification.

There are two fundamental types of intervention functions:

Pulse function (temporary effect)

$P_t = \begin{cases} 1 & \text{if } t = t_0 \\ 0 & \text{otherwise} \end{cases}$

Use this when the event has a one-time impact that doesn't persist. For example, a factory shutting down for a single week of maintenance would cause a temporary drop in output at $t_0$, with production returning to normal afterward.

Step function (permanent effect)

$S_t = \begin{cases} 1 & \text{if } t \geq t_0 \\ 0 & \text{otherwise} \end{cases}$

Use this when the event shifts the series to a new level permanently. A new emissions regulation that caps factory output going forward is a good example.

Once you've chosen the right function, you incorporate it directly into your model. Here's an ARIMA(1,0,0) example with a pulse intervention:

$y_t = \beta_0 + \beta_1 P_t + \phi_1 y_{t-1} + \varepsilon_t$

• $\beta_0$ is the intercept (baseline level)
• $\beta_1$ captures the size of the intervention effect
• $\phi_1$ is the autoregressive coefficient
• $\varepsilon_t$ is white noise error

If you suspected a permanent break instead, you'd replace $P_t$ with $S_t$. You can also include both types simultaneously if the series experienced multiple interventions.

Assessing the Impact of Interventions

After estimating the model, you need to evaluate whether the intervention coefficients are meaningful.

Steps for assessment:

1. Estimate parameters using your chosen method (e.g., maximum likelihood for ARIMA models).

2. Test significance of each intervention coefficient with a t-test. A p-value below 0.05 suggests the intervention had a real effect on the series.

3. Interpret the coefficient's sign and size. A positive $\beta_1$ means the intervention raised the series level; a negative $\beta_1$ means it lowered it. The magnitude tells you by how much.

   • Example: A new marketing campaign with $\beta_1 = 1000$ means the campaign is associated with an increase of 1,000 units in weekly sales.
   • Example: A competitor's product launch with $\beta_1 = -0.05$ (in a proportion-scaled model) means a 5 percentage point drop in market share.

4. Assess duration. If you used a pulse function and it's significant, the effect was temporary. If a step function is significant, the series has shifted to a new permanent level.

Applying Intervention Analysis in Practice

Putting this together on real data involves a few key decisions:

1. Identify candidate interventions. Use domain knowledge (you know a regulation took effect on a certain date) or visual inspection of the time series plot (a sudden jump or drop that doesn't fit the prior pattern).
2. Choose the right dummy variable. One-time events like a store renovation get a pulse function. Lasting changes like a new tax policy get a step function.
3. Estimate the model and check significance. If $\beta_1$ is significant, the intervention matters. If not, the apparent break may just be noise.
4. Use results for forecasting. Once you've quantified a past intervention (e.g., a regulation reduced factory output by 10%), you can adjust future projections accordingly. For instance, if a similar regulation is expected, you can incorporate that estimated effect into your forecast.

Limitations to keep in mind:

• Exogeneity assumption. The intervention must be independent of past values of the series. If the "intervention" was actually triggered by the series itself (e.g., a policy enacted because prices spiked), the analysis breaks down.
• Confounding events. If two interventions happen at the same time (say, a holiday season and a new product launch overlap), separating their individual effects becomes difficult. You'd need additional information or assumptions to disentangle them.
• Known timing. Intervention analysis assumes you know when the break occurred. If the break point is unknown, you'll need a structural break test (like the Chow test) to detect it first.