Transfer function model

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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 .