# Transfer function model

In time series analysis, a **transfer function model** is understood to be a univariate time series model in which the target variable can be dynamically dependent on other observable variables in addition to itself and an unobservable shock variable . In contrast to vector processes, there is only an influence of the on and not the other way around. Such a model can also be viewed as a univariate dynamic model with input variables. The following representation can be selected formally:

It is the input variable. In contrast to the intervention model , this input variable can have more characteristics than the indicator function (only 0 and 1). can be referred to as an output variable. is called the transfer function. This function is comparable in its effect on the time series with the impulse-response function of the intervention model. The transfer function model is stable if the impulse-response weights can be summed up absolutely. Thus, a limited input would also produce a limited output. The model is called causal if there is no leading function of . *X* with respect to *Y *exogenously , and there is no feedback relationship from *Y* to *X* .

The instrument of the cross-correlation function is used to identify the model . In contrast to the autocorrelation function, this function is not symmetrical about *l* . The relationship between the cross-correlation and transfer functions is quite complicated:

.

The simultaneous system of equations to be solved here is quite complicated. It would be easier if the following connection could be established:

.

This would be proportional to the cross-correlation coefficient . This can be achieved by transforming the input so that it becomes white noise . In the case of the transformation referred to as "pre-white" , it is assumed that the input series is understood as an ARMA process:

. After forming, the pre-whitened input series results:

Now the same transformation has to be applied to the output variable :

.

The original transfer function model can now be described as:

grasp. there is white noise. and are usually not white noise. For the cross-correlation coefficient of the transformed time series one obtains:

.

With this result, the estimate can be made as in the ARMA model .