Mesure de la fonction de réponse impulsionnelle dans les modèles autorégressifs

Abstract EN

In this paper, the explicit calculation required to obtain impulse response function (IRF) and the accumulated impulse response (AIR) of a variable t X generated by an autoregressive process were carried out.

First the AR(m) model ( ) t t 1 − ϕ B m X = ε where ε t a white noise was considered.

The real root 1/ϕ of the autoregressive polynomial has a multiplicity order m.

Second, a general situation of an AR model (1 −ϕ1B)(1−ϕ 2B)....(1−ϕ mB)Xt = ε t was considered where 1/ϕi ,i = 1,...,m are distinct real numbers and each root has multiplicity order which is equal to one.

The results of this research reveal that the accumulated impulse response (AIR) depends on the position of the autoregressive root with respect to the unit circle and the multiplicity order m of this root: for given m, the AIR(ϕ ) increases with ϕ .

For given ϕ , the sense of variation of the AIR(m ) depends of the sign of ϕ .

Finally an AR(2) process was simulated and the impulse response functions (IRF) and the accumulated impulse responses (AIR) were calculated.

The empiric average of the (AIR) proved nearly to be equal to the value proposed by the theoretical calculation.