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

Подпись: y, = (l — pL)~1e, =image370(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.

image371 image372 Подпись: (5.2.9)

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

Подпись: -ґуа 0 0 • 0 -P 1 0 0 -p 1 . • 0 -p 1 0 0 • • 0 -p 1 Подпись: (5.2.10)
image376

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

Подпись: (5.2.12)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

Подпись: ffil —Подпись: (5.2.13)T2T

1 – P2′

Inverting both sides of (5.2.12) yields

Подпись: 1 -p 0 0 • 0 -p 1 +p2 -p 0 • 0 1 0 -p 1 +p2 • • • t72 * • • • • * • -p 1 +p2 -p * • 0 -p 1 (5.2.14)

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

Подпись:Подпись:/Л")=ТТЛ E

1 P A—»

=у^ [i+J; (pe^f+J; (/*-*? ]

_ g2 Г. peia і

1 — p2 l 1 — 1 — pe~iaJ

Подпись:1 — 2p COS G) + /)

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