Explain Box-Jenkins intervention analysis.

Box-Jenkins Methodology
PHASE  I : Identification and Preparation

1. Data Preparation 

Differencing the data to achieve stationary series

Transform data to stabalize variance

1. Model Selection/identification

Shortlist suitable models according to the data

Apply ACF and PACF to determine potential models

PHASE  II : Estimation and Testing

1. Estimation

Estimate parameters in potential models

Select best model using suitable criteria

1. Diagnostics

Check ACF/PACF of residuals 

Do portmanteau test of residuals

PHASE  II : Forecasting/Application

Using the derived model to forecast future series

Box-Jenkins intervention analysis (Box and Tiao, 1975) extends an ARIMA model with an extra input term that measures how a known outside event (an "intervention") changed the time series, and whether that change was significant.

Examples of interventions: a new law, a price change, an advertising campaign, a strike, a policy launch, a pandemic lockdown.

Model

\[Y_t = \frac{\omega(B)}{\delta(B)}\,B^{b}\,I_t + N_t\]

• \(N_t\): the usual ARIMA model of the series without the event (the "noise" model).
• \(I_t\): an indicator variable for the event.
• \(\omega(B)/\delta(B)\): transfer function giving the size and shape of the effect; \(b\) is the delay; \(B\) is the backshift operator.

Types of intervention variable

• Step \(S_t\): 0 before time \(T\), 1 from \(T\) onward, for a lasting change (e.g. a new tax).
• Pulse \(P_t\): 1 only at time \(T\), 0 otherwise, for a one-time temporary shock (e.g. a one-day strike).

Common effect shapes

• Abrupt permanent: \(\omega S_t\), a sudden level shift.
• Gradual permanent: \(\frac{\omega}{1-\delta B}S_t\), the level moves slowly to a new value \(\omega/(1-\delta)\).
• Abrupt temporary: \(\frac{\omega}{1-\delta B}P_t\), a jump that decays back to the old level.

Steps

1. Identify an ARIMA model \(N_t\) using data from before the intervention (Box-Jenkins identification with ACF/PACF).
2. Choose the intervention form (step or pulse, abrupt or gradual) from theory and a plot of the data.
3. Estimate the ARIMA and intervention parameters (\(\omega,\ \delta\)) together on the whole series.
4. Check diagnostics: residuals should be white noise; test whether \(\omega\) is significant.
5. Interpret the size and duration of the effect and use the model for forecasting.