## Autoregressive Models

5.1.2 First-Order Autoregressive Model

Consider a sequence of random variables {y,}, t = 0, ± 1, ±2,. . . , which follows

Уі = РУі- i + e(, (5-2.1)

where we assume

Assumption А. {є,}, t = 0, ± 1, ±2,…………. are i. i.d. with Ее, = 0 and

Ee}= a2 and independent of y,-i, yf_2,….

Assumption B. p < 1.

Assumption C. Ey, — 0 and Ey, yt+h = yh for all t. (That is, {}>,} are weakly stationary.)

Model (5.2.1) with Assumptions A, B, and C is called a stationaryfirst-order autoregressive model, abbreviated as AR(1).

From (5.2.1) we have

У, = Psy,-s + 2 fa-i – (5-2.2)

j-о

But 1іт,_ю E(psy,_s)2 = 0 because of Assumptions В and C. Therefore we have

Уг-^pbt-,, (5.2.3)

which means that the partial summation of the right-hand side converges to y, in the mean square. The model (5.2...

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