# 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: ${\ displaystyle Y_ {t}}$${\ displaystyle Z_ {t}}$${\ displaystyle X_ {m} t}$${\ displaystyle X_ {m} t}$${\ displaystyle Y_ {t}}$

${\ displaystyle Y_ {t} = \ nu (L) X_ {t} + {\ frac {\ Theta (L)} {\ Phi (L)}} Z_ {t}}$

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 . ${\ displaystyle X_ {t}}$${\ displaystyle Y_ {t}}$${\ displaystyle Y_ {t} = \ nu (L)}$${\ displaystyle Y_ {t}}$${\ displaystyle X_ {t}}$

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:

${\ displaystyle \ rho (l) = {\ frac {\ sigma _ {X}} {\ sigma _ {Y}}} [\ nu _ {0} \ rho _ {X} (l) + \ nu _ { 1} \ rho _ {X} (l-1) + \ nu _ {2} \ rho _ {X} (l-2) + ...]}$.

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

${\ displaystyle \ nu = {\ frac {\ sigma _ {Y}} {\ sigma _ {X}}} \ rho _ {XY} (l)}$.

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: ${\ displaystyle \ nu _ {l}}$${\ displaystyle \ rho _ {XY} (l)}$${\ displaystyle X_ {t}}$

${\ displaystyle \ Phi _ {Y} (L) X_ {t} = \ Theta _ {X} (L) \ alpha _ {t}}$. After forming, the pre-whitened input series results:

${\ displaystyle \ alpha _ {t} = {\ frac {\ Phi _ {X} (L)} {\ Theta _ {X} (L)}} X_ {t}}$

Now the same transformation has to be applied to the output variable : ${\ displaystyle Y_ {t}}$

${\ displaystyle \ beta _ {t} = {\ frac {\ Phi _ {X} (L)} {\ Theta _ {X} (L)}} Y_ {t}}$.

The original transfer function model can now be described as:

${\ displaystyle \ beta _ {t} = \ nu (L) \ alpha _ {t} + \ varepsilon _ {t}}$grasp. there is white noise. and are usually not white noise. For the cross-correlation coefficient of the transformed time series one obtains: ${\ displaystyle \ alpha _ {t}}$${\ displaystyle \ beta _ {t}}$${\ displaystyle \ varepsilon _ {t}}$

${\ displaystyle \ nu = {\ frac {\ sigma _ {\ beta}} {\ sigma _ {\ alpha}}} \ rho _ {\ alpha \ beta} (l)}$.

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