## Impulse Response Function

The general linear model for a time series yt is

 ${y}_{t}=\mu +{\epsilon }_{t}+\sum _{i=1}^{\infty }{\psi }_{i}{\epsilon }_{t-i}=\mu +\psi \left(L\right){\epsilon }_{t},$ (1)
where $\psi \left(L\right)$ denotes the infinite-degree lag operator polynomial $\left(1+{\psi }_{1}L+{\psi }_{2}{L}^{2}+\dots \right)$.

The coefficients ${\psi }_{i}$ are sometimes called dynamic multipliers [1]. You can interpret the coefficient ${\psi }_{j}$ as the change in yt+j due to a one-unit change in εt,

`$\frac{\partial {y}_{t+j}}{\partial {\epsilon }_{t}}={\psi }_{j}.$`

Provided the series $\left\{{\psi }_{i}\right\}$ is absolutely summable, Equation 1 corresponds to a stationary stochastic process [2]. For a stationary stochastic process, the impact on the process due to a change in εt is not permanent, and the effect of the impulse decays to zero. If the series $\left\{{\psi }_{i}\right\}$ is explosive, the process yt is nonstationary. In this case, a one-unit change in εt permanently affects the process.

The series $\left\{{\psi }_{i}\right\}$ describes the change in future values yt+i due to a one-unit impulse in the innovation εt, with no other changes to future innovations ${\epsilon }_{t+1},{\epsilon }_{t+2},\dots$. As a result, $\left\{{\psi }_{i}\right\}$ is often called the impulse response function.

## 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.