Impulse Response of Regression Models with ARIMA Errors

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

$\begin{matrix} y_{t} = c + X_{t} β + u_{t} \\ Η (L) u_{t} = Ν (L) ε_{t,} \end{matrix}$

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} = Η^{- 1} (L) Ν (L) ε_{t} = ψ (L) ε_{t},

(1)

where ψ(L) = 1 + ψ₁L + ψ₂L² + ... is an infinite degree polynomial.

The coefficient ψ_j is called a dynamic multiplier [1]. You can interpret ψ_j as the change in the future response (y_t+j) due to a one-time unit change in the current innovation (ε_t) and no changes in future innovations (ε_t+1,ε_t+2,...). That is, the impulse response function is

ψ_{j} = \frac{\partial y_{t + j}}{\partial ε_{t}} .

(2)

Equation 2 implies that the regression intercept (c) and predictors (X_t) 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 ε_t.

If the series {ψ_j} 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 ε_t 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 ε_t 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.