## Autogressive Models with Moving-Average Residuals

A stationary autoregressive, moving-average model is defined by

2) РіУг-і= 2) Pjet-j> A> = /?o = – l. t = 0, ± 1, ±2,. . . ,

j – о j – о

(5.3.1)

where we assume Assumptions А, В", C, and

Assumption D. The roots of 2J_0 fyz 4~J = 0 lie inside the unit circle.

Such a model will be called ARMA(p, q) for short.

We can write (5.3.1) as

p(L)yt = p(L)€t, (5.3.2)

where p(L) = Sf_0 PjJJ and fi(L) = 2J_0 PjLj. Because of Assumptions B" and C, we can express y, as an infinite moving average

yt = p~l(L)P(L)et = ф(Це„ (5.3.3)

where ф(Ь) = ‘2jL0 ф]и. Similarly, because of Assumption D, we can express y, as an infinite autoregressive process

y{L)yt = p~l{L)p{L)yt = et, (5.3.4)

where ydL) = 2jl0 y/jV.

The spectral density of ARMA(p, q) is given by

where |z|2 = zz for a comp...

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