# 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 (5.2.1) with Assumptions A, B, and C is equivalent to the model (5.2.3) with Assumptions A and B. The latter is called the moving-average representation of the former.

A quick mechanical way to obtain the moving-average representation

(5.2.3) of (5.2.1) and vice-versa is to define the lag operator L such that Ly, — L2y, = y,_2 Then (5.2.1) can be written as

(1 – pL)y, = e„ (5.2.4)

where 1 is the identity operator such that ly, — y,. Therefore

(5.2.5)

which is (5.2.3).

An AR(1) process can be generated as follows: Define as a random variable independent of €,, e2,. . . , with Ey0 — 0 and Ey% = a2/(1 — p2). Then define y, by (5.2.2) after putting s = t.

The autocovariances (yA) can be expressed as functions of p and a2 as follows: Multiplying (5.2.1) with y,_A and taking the expectation yields

yh = pyh-, h =1,2,…. (5.2.6)

From (5.2.1), E(y, — py,_, )2 = Ее2, so that we have

{І+р^Уо-гру^а2. (5.2.7)

Solving (5.2.6) and (5.2.7), we obtain a2of1

Vh = jZTf’ Л-0,1,2,———– (5.2.8)

Note that Assumption C implies y_A = yh.

Arranging the autocovariances in the form of a matrix as in (5.1.1), we obtain the autocovariance matrix of AR(1),

Now let us examine an alternative derivation of 2, that is useful for deriving the determinant and the inverse of 2, and is easily generalizable to higher – order processes. Define Г-vectors у = (Уі, y2> • • • . Утї and б(*0 = [(1 – рі*УІ2Уі, e2, €3,. . . , ет and а ГХ Tmatrix

Then we have

(5.2.11)

But, because £€*)€*) = cr2I, we obtain

2j = <t2R71(R’i)-,>

which can be shown to be identical with (5.2.9). Taking the determinant of both sides of (5.2.12) yields

T2T

1 – P2′

Inverting both sides of (5.2.12) yields

(5.2.14)

By inserting (5.2.8) into (5.1.2), we can derive the spectral density of AR( 1):

/Л")=ТТЛ E

1 P A—»

=у^ [i+J; (pe^f+J; (/*-*? ]

_ g2 Г. peia і

1 — p2 l 1 — 1 — pe~iaJ

1 — 2p COS G) + /)