## Impulse Response of Regression Models with ARIMA Errors

The general form of a regression model with ARIMA errors is:

`$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ Η\left(L\right){u}_{t}=Ν\left(L\right){\epsilon }_{t,}\end{array}$`

where

• 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

 ${u}_{t}={Η}^{-1}\left(L\right)Ν\left(L\right){\epsilon }_{t}=\psi \left(L\right){\epsilon }_{t},$ (1)
where ψ(L) = 1 + ψ1L + ψ2L2 + ... is an infinite degree polynomial.

The coefficient ψj is called a dynamic multiplier [1]. 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

 ${\psi }_{j}=\frac{\partial {y}_{t+j}}{\partial {\epsilon }_{t}}.$ (2)
Equation 2 implies that the regression intercept (c) and predictors (Xt) of Equation 1 do not impact the impulse response function. In other words, the impulse response function describes the change in the response that is solely due to the one-time unit shock of the innovation εt.

• If the series {ψj} is absolutely summable, then Equation 1 is a stationary stochastic process [2].

• 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.

## References

[1] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[2] Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.