A Transfer function approach to modeling
a time series is a multivariate way of modeling the various lag
structures found in the data. It is similar to a distributed lag
model in traditional econometrics. The transfer function model,
however, makes use of the ratio between what are called Numerator
and Denominator polynomials. The Numerator polynomials are the
different lags of the exogoneous variables. The Denominator polynomials
are the coefficients associated with lags of the forecast itself
(i.e. the traditional dependent variable). Something called a
Delay Factor or "dead time" is associated with the exogenous variables.
This is the amount of time elapsed before the numerator polynomial
begins to impact the forecasted series. The error term is modeled
simultaneously as an ARMA process. Stationarity requirements hold
the same for each exogenous series as in the univariate endogenous
case. Difference operations must be used if the autocorrelation
function of each series is to exhibit stationarity. These difference
operations need not be exactly the same across all variables.
There may seem to be a close relationship between the Transfer
Function models and multiple regression (OLS). At a glance, one
might think that the regression models are a special case of the
transfer function. However, they are quite different.