In the koyck literature and even the ADL literature in general,  the defintion of the long multiplier is usually presented as the total effect of a unit increase in the predictor at at  $t = 0$. So, for example, in the one predictor Koyck ADL, the multiplier is  Beta/(1-rho) where rho is the coefficient on the lagged dependent variable. What I never understood ( and still pretty much don't ) is the following:

One can think of it this way but what about, if in the real data, this "description" never happens. By this I mean that, in the data, there's never just one impulse like that and then everything but the other ones after that are zero.

If my basic understanding is correct( and maybe I'm missing something here)then the multiplier is the sum ( or integral ) of the effect of an impulse response. So, it should be the place where the y_t variable levels off from where it was originally  ( so an S-curve assuming it started at zero ). 

But, from a time domain or frequency domain standpoint, if such a case never happens in the data, then how is the model identified ? In other words, what is the interpretation say, if there was an impulse response and then another impulse response right after that one. Then, nothing happens after that say. Thanks for any references or insights.

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