The basic problem that we’re considering is the construction of a lagged regression in which we predict a y-variable at the present time using lags of an x-variable (including lag 0) and lags of the y-variable. In Week 8, we introduced the CCF (cross-correlation function) as an aid to the identification of the model. One difficulty is that the CCF is affected by the time series structure of the x-variable and any “in common” trends the x and y series may have over time.

One strategy for dealing with this difficulty is called “pre-whitening.” The steps are:

1. Determine a time series model for the x-variable and store the residuals from this model.
2. Filter the y-variable series using the x-variable model (using the estimated coefficients from step 1). In this step we find differences between observed y-values and “estimated” y-values based on the x-variable model.
3. Examine the CCF between the residuals from Step 1 and the filtered y-values from Step 2. This CCF can be used to identify the possible terms for a lagged regression.

This strategy stems from the fact that when the input series (say, \(w_{t}\)) is “white noise” the patterns of the CCF between \(w_{t}\) and \(z_{t}\), a linear combination of lags of the \(w_{t}\), are easily identifiable (and easily derived). Step 1 above creates a “white noise” series as the input. Conceptually, Step 2 above arises from a starting point that y-series = linear combination of x-series. If we “transform” the x-series to white noise (residuals from its ARIMA model) then we should apply the transformation to both sides of the equation to preserve an equality of sorts.

It’s not absolutely crucial that we find the model for x exactly. We just want to get close to the “white noise” input situation. For simplicity, some analysts only consider AR models (with differencing possible) for x because it’s much easier to filter the y-variable with AR coefficients than with MA coefficients.

Pre-whitening is just used to help us identify which lags of x may predict y. After identifying possible model from the CCF, we work with the original variables to estimate the lagged regression.

Alternative strategies to pre-whitening include:

• Looking at the CCF for the original variables – this sometimes works
• De-trending the series using either first differences or linear regressions with time as a predictor

In general, we’re trying to identify lags of x and y that might predict y. The following table describes some typical situations with a peak at lag d. Notice that a smooth tapering pattern in the CCF indicates that we should use \(y_{t-1}\) and perhaps even \(y_{t-2}\) along with lags of x.

Table: Patterns in CCF between residuals for x and pre-whitened y

Pattern in CCF	Regression Terms in Model
Non-zero values at lag d and perhaps other lags beyond d, with no particular pattern; correlations near 0 at other lags. Pattern 1 Example Pattern 1 Example Non-zero values at lag d and perhaps other lags beyond d, with no particular pattern; correlations near 0 at other lags.	X: Lag d, \(x_{t-d}\) and perhaps additional lags of x Y: No lags of y
Exponential decay of correlations beginning at lag d; correlations near 0 at lags before lag d. Pattern 2 Example Pattern 2 Example Exponential decay of correlations beginning at lag d; correlations near 0 at lags before lag d.	X: Lag d, \(x_{t-d}\) Y: Lag 1, \(y_{t-1}\)
Non-zero correlations with no particular pattern beginning at lag d, but then exponential decay begins at last used lag of x; correlations near 0 before lag d. Pattern 3 Example Pattern 3 Example Non-zero correlations with no particular pattern beginning at lag d, but then exponential decay begins at last used lag of x; correlations near 0 before lag d.	X: Lag d, \(x_{t-d}\), and perhaps additional lags of x. Y: Lag 1, \(y_{t-1}\)
Sinusoidal pattern beginning at lag d, with decay toward 0; correlations near 0 before lag d. Pattern 4 Example Pattern 4 Example Sinusoidal pattern beginning at lag d, with decay toward 0; correlations near 0 before lag d.	X: Lag d, \(x_{t-d}\) Y: Lags 1 and 2, \(y_{t-1}\), \(y_{t-2}\)
Non-zero correlations with no particular pattern beginning at lag d, but then Sinusoidal pattern beginning at last used lag of x, with decay toward 0; correlations near 0 before lag d. Pattern 5 Example Pattern 5 Example Non-zero correlations with no particular pattern beginning at lag d, but then Sinusoidal pattern beginning at lag d, with decay toward 0; correlations near 0 before lag d.	X: Lag d, \(x_{t-d}\), and perhaps additional lags of x. Y: Lags 1 and 2, \(y_{t-1}\), \(y_{t-2}\)

Transfer Functions

The term “transfer function” applies to models in which we predict y from past lags of both y and x (including possibly lag 0), and we also model x and the errors. For prediction of the future past the end of the series, we might first use the model for x to forecast x. Then we would use the forecasted x when we forecast future y.

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.