How to write an AR(2) stationary process in the Wold representation

I managed to write an AR(1) process in the Wold representation with help from the geometric series.

I am having trouble with a stationary AR(2). How could I do?

Answer

Let Xt be a zero-mean covariance-stationary time series such that
Xt=φ1Xt1+φ2Xt2+εt
where εt is white noise.

Using L to mean the lag (backshift) operator, the above can be expressed as
(1φ1Lφ2L2)Xt=εt.

Since Xt is a covariance-stationary AR(2) process, the roots of its characteristic polynomial (1φ1zφ2z2)=0 must lie outside the unit circle. Thus, Equation (1) can be written as
(1λ1L)(1λ2L)Xt=εt
where |λ1|<1 and |λ2|<1. The last two inequalities are true of a covariance-stationary AR(2) process since then the roots of the characteristic polynomial, z1=1/λ1 and z2=1/λ2, will lie outside the unit circle. Therefore,
Xt=1(1λ1L)1(1λ2L)εt.

Expand the two fractions on the right-hand side in the equation above using the geometric series. and you'll have the Wold decomposition of an AR(2).

Attribution
Source : Link , Question Author : Monolite , Answer Author : JoOkuma

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