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

$$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ {\rm H}(L){u}_{t}={\rm N}(L){\epsilon}_{t,}\end{array}$$

where

*t*= 1,...,*T*.*H*(*L*) is the compound autoregressive polynomial.*N*(*L*) is the compound moving average polynomial.

Solve for *u _{t}* in the ARIMA error model to obtain

$${u}_{t}={{\rm H}}^{-1}(L){\rm N}(L){\epsilon}_{t}=\psi (L){\epsilon}_{t},$$ | (1) |

The coefficient *ψ _{j}* is called a

$${\psi}_{j}=\frac{\partial {y}_{t+j}}{\partial {\epsilon}_{t}}.$$ | (2) |

If the series {

*ψ*} is absolutely summable, then Equation 1 is a stationary stochastic process [2]._{j}If the ARIMA error model is stationary, then the impact on the response due to a change in

*ε*is not permanent. That is, the effect of the impulse decays to 0._{t}If the ARIMA error model is nonstationary, then the impact on the response due to a change in

*ε*persists._{t}

[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.