Impulse Response Function

The general linear model for a time series yt is

yt=μ+εt+i=1ψiεti=μ+ψ(L)εt,(5-19)

where ψ(L) denotes the infinite-degree lag operator polynomial (1+ψ1L+ψ2L2+).

The coefficients ψi are sometimes called dynamic multipliers [1]. You can interpret the coefficient ψj as the change in yt+j due to a one-unit change in εt,

yt+jεt=ψj.

Provided the series {ψi} 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 εt is not permanent, and the effect of the impulse decays to zero. If the series {ψi} is explosive, the process yt is nonstationary. In this case, a one-unit change in εt permanently affects the process.

The series {ψi} 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 εt+1,εt+2,. As a result, {ψi} 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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