I am trying to build a time series regression forecasting model for an outcome variable, in dollar amount, in terms of other predictors/input variables and autocorrelated errors. This kind of model is also called dynamic regression model. I need to learn how to identify transfer functions for each predictor and would love to hear from you about ways to do just that.
The classic approach, described in Box, Jenkins & Reinsell (4th ed, 2008) involves looking at the cross-correlation function and the various auto-correlation functions, and making a lot of subjective decisions about the orders and lags for the various terms. The approach works ok for a single predictor, but is not really suitable for multiple predictors.
An alternative approach, described in Pankratz (1991), involves fitting lagged regressions with AR errors and determining the appropriate rational lag structure from the fitted coefficients (also a relatively subjective process). Then refitting the entire model with the supposed lag structures and extracting the residuals. The order of the ARMA error process is determined from these residuals (using AIC for example). Then the final model is re-estimated. This approach works well for multiple predictors, and is considerably simpler to apply than the classic approach.
I wish I could say there was this neat automated procedure that did it all for you, but I can't. At least not yet.
Originally the idea of examining pre-whitened cross-correlations was suggested by Box and Jenkins. In 1981, Liu and Hanssens published( L.-M. Liu and D.M. Hanssens (1982). "Identification of Multiple-Input Transfer Function Models." Communications in Statistics A 11: 297-314.) a paper that suggested a common filter approach that would effectively deal with multiple inputs whose pre-whitened series exhibit cross-correlative structure. They even created a 2 input model data set to demonstrate their solution. After we programmed that approach and then compared it to the Box-Jenkins pre-whitening approach iteratively implemented by us we decided to not to use either the Pankratz approach or the Liu-Hanssens approach.We would be glad to share the Liu-Hansens test data with you if you wish me to post it to the list.