## Impulse Response Function

The general linear model for a time series *y*_{t} is

$${y}_{t}=\mu +{\epsilon}_{t}+{\displaystyle \sum _{i=1}^{\infty}{\psi}_{i}{\epsilon}_{t-i}=\mu +\psi (L){\epsilon}_{t},}$$ | **(5-19)** |

where $$\psi (L)$$ denotes the infinite-degree
lag operator polynomial $$(1+{\psi}_{1}L+{\psi}_{2}{L}^{2}+\dots )$$.

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

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

Provided the series $$\left\{{\psi}_{i}\right\}$$ is
absolutely summable, Equation 5-19 corresponds to a stationary stochastic
process [2]. For a stationary
stochastic process, the impact on the process due to a change in *ε*_{t} is
not permanent, and the effect of the impulse decays to zero. If the
series $$\left\{{\psi}_{i}\right\}$$ is explosive, the process *y*_{t} is
nonstationary. In this case, a one-unit change in *ε*_{t} permanently
affects the process.

The series $$\left\{{\psi}_{i}\right\}$$ describes the
change in future values *y*_{t+i} due
to a one-unit impulse in the innovation *ε*_{t},
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.

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