9.2 Intervention Analysis

Suppose that at time \(t\) = T (where T will be known), there has been an intervention to a time series. By intervention, we mean a change to a procedure, or law, or policy, etc. that is intended to change the values of the series \(x_{t}\). We want to estimate how much the intervention has changed the series (if at all). For example, suppose that a region has instituted a new maximum speed limit on its highways and wants to learn how much the new limit has affected accident rates.

Intervention analysis in time series refers to the analysis of how the mean level of a series changes after an intervention, when it is assumed that the same ARIMA structure for the series \(x_{t}\) holds both before and after the intervention.

Overall Intervention Model

Suppose that the ARIMA model for \(x_{t}\) (the observed series) with no intervention is

\(x_t - \mu = \dfrac{\Theta(B)}{\Phi(B)}w_t\)

with the usual assumptions about the error series \(w_{t}\).

\(\Theta(B)\) is the usual MA polynomial and \(\Phi(B)\) is the usual AR polynomial.

Let \(z_{}\)= the amount of change at time \(t\) that is attributable to the intervention. By definition, \(z_{t}\) = 0 before time T (time of the intervention). The value of \(z_{t}\) may or may not be 0 after time T.

Then the overall model, including the intervention effect, may be written as

\(x_t - \mu = z_t + \dfrac{\Theta(B)}{\Phi(B)}w_t\)

There are several possible patterns for how an intervention may affect the values of a series for \(t\) ≥ T (the intervention point). Four possible patterns are as follows:

\(z_{t}\) = the amount of change at time \(t\) that is attributable to the intervention.

Suppose that \(I_{t}\) is an indicator variable such that \(I_{t}\) = 1 when \(t\) ≥ T and \(I_{t}\) = 0 when \(t\) < T.

1. Pattern 1

   Constant permanent change. A constant change of equal to the amount \(\delta_0\) after time T may be written simply as

   \(z_{t}\) = \(\delta_0\)\(I_{t}\).

   Note!\(z_{t}\) = \(\delta_0\) for \(t\) ≥ T, and \(z_{t}\) = 0 and for \(t\) < T. The coefficient \(\delta_0\) will be estimated using the data.
2. Pattern 2

   A temporary (constant change) lasting for d times past the intervention time T, can be described by the intervention effect model

   \(z_{t}\) = \(\delta_0 \left(1 - B^{d} \right) I_{t}\).

   Or, we can redefine the indicator so that \(I_{t}\) = 1 for T ≤ \(t\) ≤ T + d, and \( I_{t} \) = 0 for all other \(t\). Then, we use the model \(z_{t}\) = \(\delta_0\)\(I_{t}\).

   With this intervention effect, \(z_{t}\) = \(\delta_0\) for T ≤ \(t\) ≤ T + d, and \(z_{t}\) = 0 and for all other \(t\). Again, the coefficient \(\delta_0\) will be estimated using the data.

3. Pattern 3

   A gradually increasing effect that eventually levels off can be written as:

   \(z_t = \dfrac{\delta_0}{1-\omega_1B}I_t,\)

   with \(I_{t}\) = 1 for \(t\) ≥ T and \(I_{t}\) = 0 when \(t\) < T. Assume \(\lvert\omega_1\rvert\) < 1.

   This is equivalent to \(z_{t}\) = \(\omega_1\)\(z_{t-1}\) + \(\delta_0\)\(I_{t}\).

   When \(t\) < T, \(z_{t}\) = 0.

   When \(t\) ≥ T, \(I_{t}\) =1 so \(z_{t}\) = \(\omega_1\)\(z_{t-1}\) + \(\delta_0\). Because \(\lvert\omega_1\rvert\) < 1, we can continue to express each past \(z_{t}\) in terms of \(\omega_1\) and \(\delta_0\) as a geometric series to obtain the following:

   \(z_t = \dfrac{\delta_0 (1-\omega_1^{t-T+1})}{1-\omega_1}.\)

   The coefficients \(\omega_1\) and \(\delta_0\) will be estimated using the data.

4. Pattern 4

   An immediate change that eventually returns to 0 can be modeled as follows

   \(z_t = \dfrac{\delta_0}{1-\omega_1B}P_t ,\)

   with \(P_{t}\) = 1 when t = T and \( P_{t} \) = 0 otherwise. Assume \( \lvert\omega_1\rvert\) < 1.

   This is equivalent to \(z_{t}\) = \(\omega_1\)\(z_{t-1}\) + \(\delta_0 P_{t}\).

   When \(t < T\), \(z_{t}\) = 0.

   When \(t = T\), \(z_{t-1}\) = 0 and \(P_{t }\)= 1 so \(z_{t}\) = \(\delta_0\).

   When \(t ≥ T+1\), \(P_{t }\)= 0 so \(z_{t}\) = \(\omega_1\)\(z_{t-1}\).

   The coefficients \(\omega_1\) and \(\delta_0\) will be estimated using the data.

Two parts of the overall model have to be estimated – the basic ARIMA model for the series and the intervention effect. Several approaches have been proposed. One approach has the following steps:

1. Use the data before the intervention point to determine the ARIMA model for the series.
2. Use that ARIMA model to forecast values for the period after the intervention.
3. Calculate the differences between actual values after the intervention and the forecasted values.
4. Examine the differences in step 3 to determine a model for the intervention effect.

What we do after step 4 depends on available software. If the right program is available we can use all of the data to estimate the overall model that combines the ARIMA for the series and the intervention model. Otherwise, we might use only the differences from step 4 above to make estimates of the magnitude and nature of the intervention.