QUESTION:

Please explain, in words not equations, the role of pre-whitening and transfer function model identification.

ANSWER:

The procedure for transfer function model identification outlined by Box and Jenkins uses the cross correlations between

two prewhitened series to tentatively identify model form. The first step to this process is to develop an ARIMA model for

each of the user-specified input time series in the equation. Each series must then be made stationary by applying the

appropriate differencing and transformation parameters from its ARIMA model. The stationary time series are, in turn,

"prewhitened". Prewhitening refers to the process of applying a given set of autoregressive and moving average factors

to a stationary series. Each input series is prewhitened by its own ARIMA model AR (autoregressive) and MA (moving

average) factors. The output series is also prewhitened by the input series AR and MA factors. If there is more than one

input series, then the stationary output series is prewhitened once for each different input. Prewhitening is necessary

because it removes the intrarelationship in the individual series. This allows you to more accurately assess the

interrelationship between the input and the output series. The cross correlations between the prewhitened input and

output reveal the extent of this interrelationship. The cross correlations can be converted to estimates of the impulse

response weights or regression weights. The pattern in the impulse response weights indicate can suggest a tentative

transfer function model. By applying these impulse response weights to the input series to predict the output series,

one can generate a preliminary estimate of the noise series. Following the rules for ARIMA model identification, the

patterns in the autocorrelations and partial autocorrelations of the tentative noise process give clues as to the initial

form of the noise model. Given the identified transfer function and noise model, one can proceed to the model

estimation/diagnostic checking phase.

The process of identifying a transfer function model, as outlined by Box and Jenkins, is in itself a multi-step

procedure. The very first step is to develop the optimal ARIMA model for each time series that is to be included in

the transfer function equation. These ARIMA models play a critical role in the analysis of the relationship between

each input series and the output series. AFS suggests that the ARIMA model used in prewhitening be kept as simple

as possible, but not too simple. We suggest that interventions, both mean and variance, should not be included as it is

possible to mask interrelationships by fitting overly complicated intrarelationships. AUTOBOX offers option to use

differencing factors in the "Identification phase". AFS has found that over differencing can lead to poor identification.

Box and Jenkins, faced with "expensive computing costs" took the shortcut and used the ARIMA model differences,

possibly different, in the Transfer Function identification phase. We suggest that the user try it both ways, that is with

and without the differencing factors.