Impulse Response of Regression Models with ARIMA Errors
The general form of a regression model with ARIMA errors is:
t = 1,...,T.
H(L) is the compound autoregressive polynomial.
N(L) is the compound moving average polynomial.
Solve for ut in the ARIMA error model to obtain
The coefficient ψj is called a dynamic multiplier . You can interpret ψj as the change in the future response (yt+j) due to a one-time unit change in the current innovation (εt) and no changes in future innovations (εt+1,εt+2,...). That is, the impulse response function is
If the ARIMA error model is stationary, then the impact on the response due to a change in εt is not permanent. That is, the effect of the impulse decays to 0.
If the ARIMA error model is nonstationary, then the impact on the response due to a change in εt persists.
 Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
 Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.