I have modeled daily data using an ARMAX model with seasonal ARMA components. My dependent variable is the amount of web visits to a web site due to daily TV commercial airings. MY variables are stationarized and controlled for seasonality so that my residuals are un-autocorrelated and insignificant white noise. I am easily able to see the affects that my exogenous (advertising) variables have on immediate web traffic, but I would like to incorporate a "growth" factor. That is, the advertising variables should have an immediate impact which continues on for some indefinite time before dying out. The goal is to be able to go back and look at how Web traffic would look if advertising were to cease for x amount of time. 

When I recursively forecast using my models parameters, in the middle of the data set, it follows the observed values for a while but ultimately breaks off and starts to isolate around a mean value well below the observed data. I believe it is because the model is not correctly compounding the continuous affects of the advertising activity; thus, past advertising isn't influencing future web traffic. 

My questions would be, how do I model my data by compounding the effects of advertising into long-term effects? 

How would I essentially use forecast to predict what web visits would be if advertising was stopped at any period within my data set?