Second-Order Autoregressive Model

A stationary second-order autoregressive model, abbreviated as AR(2), is defined by

У, = РіУ,-і +р2У,-г + е» t = 0, ±1,±2……………. (5.2.16)

where we assume Assumptions A, C, and

Assumption B’. The roots of z2 — pxz — p2 — 0 lie inside the unit circle. Using the lag operator defined in Section 5.2.1, we can write (5.2.16) as (1 – PlL-p2L2)y, = et. (5.2.17)

(1 – цхЩ – p2L)yt = e„

Подпись: (5.2.18)where Ці and ц2 are the roots of z2 – pxz – p2 = 0. Premultiplying (5.2.18) by (1 —p1L)~l( 1 —fi2L)~l, we obtain

image385

Подпись: Hence, Подпись: j,k* 0

(5.2.19)

Convergence in the mean-square sense of (5.2.19) is ensured by Assumption B’. Note that even if px and p2 are complex, the coefficients on the є,_, are always real.

The values of px and p2 for which the condition |^,|, p2 < 1 is satisfied correspond to the inner region of the largest triangle in Figure 5.1. In the region above the parabola, the roots are real, whereas in the region below it, they are complex.

The autocovariances may be obtained as follows: Multiplying (5.2.16) by y,-i and taking the expectation, we obtain

Подпись: (5.2.20)Уі=РіУо + РгУі-

Squaring each side of (5.2.16) and taking the expectation, we obtain

Подпись: (5.2.21)

Подпись: Рг -1 Figure 5.1 Regions of the coefficients in AR(2)

Уо = (РЇ + Рг)Уо + ЇРхРгУх + о2.

Solving (5.2.20) and (5.2.21) for y0 and yy, we obtain oPi~ 1)

Подпись: Уо!(і+лИ^-О-л)2]

and

Подпись:Подпись:(1+лМ/*-(1-А>*Г

Next, multiplying (5.2.16) by y,-h and taking the expectation, we obtain

Ун – PiVh-1 + PiVh-i, A = 2. (5.2.24)

Note that {%} satisfy the same difference equation as {y,} except for the random part. This is also true for a higher-order autoregressive process. Thus, given the initial conditions (5.2.22) and (5.2.23), the second-order difference equation (5.2.24) can be solved (see Goldberg, 1958) as

image394

(5.2.25)

 

Ун-

 

image395

= Рк~ШУі-рУо)+РУо if Pi =Pi=P – If Цх and fi2 are complex, (5.2.25) may be rewritten as

image396

(5.2.26)

 

where Цх = rew and ц2 = re~w.

Arranging the autocovariances given in (5.2.25) in the form of (5.1.1) yields the autocovariance matrix of AR(2), denoted X2. We shall not write it explic­itly; instead, we shall express it as a function of a transformation analogous to

(5.2.10). If we define a Г-vector ef2) = (аіУі, а.2У + a3y2, e3, e4……… eT)

and a TXT matrix

0

0

• 0

a2

аз

0

~Pi

~Pi

1

0

0

~Pi

-Pi

1 0

1 0

0

<2

1

0

-pi 1

we have

 

(5.2.28)

Now, if we determine a,, a2, and a3 by solving К(а1у,) = <т2, У(а2Уі + аъу2) = ст2, and E[alyl{a2yl + a3y2)] = 0, we have = a21.

Therefore we obtain from (5.2.28)

Подпись: (5.2.29)X2 = a2Rl,(R2)’1-

Higher-order autoregressive processes can be similarly handled.

5.1.3 pth-Order Autogressive Model

A stationary pth-order autoregressive process, abbreviated as AR(p), is de­fined by

Подпись: (5.2.30)yt = 2 PjVt-j + t = 0, ± 1, ±2,… ,

7-І

where we assume Assumptions A and C and

Assumption B". The roots of Х*_0 PjZp~J = 0, />0 = — 1, lie inside the unit circle.

A representation of the TX T autocovariance matrix Xp of AR(p) analo­gous to (5.2.12) or (5.2.29) is possible. The j, /cth element (j, k = 0, 1,. . . , T— 1) of Xj1 can be shown to be the coefficient on &Ck in

image399

where p0 = — 1 (see Whittle, 1983, p. 73).

We shall prove a series of theorems concerning the properties of a general autoregressive process. Each theorem except Theorem 5.2.4 is stated in such a way that its premise is the conclusion of the previous theorem.

Theorem 5.2.1. (yt) defined in (5.2.30) with Assumptions A, B", and C can be written as a moving-average process of the form

Подпись: j- 0у,=І) 2 i^i < °°>

j – о j – 0

where {e,} are i. i.d. with Ее, = 0 and Eej= a2.

image401

Proof. From (5.2.30) we have

where fa, ц2,. . . . , (ip are the roots of p}zp J. Therefore we have

image402(5.2.33)

Equating the coefficients of (5.2.31) and (5.2.33), we obtain

Подпись: (5.2.34)00 ^

Фі=Рі+Рг + ■ • -+Рр

02 = Е PiPj

Подпись: />Bi,B/JB...B/llBl

0»“ . 2 РФн ■ • • /Ч-

Therefore where/Хм = max[|/i,|, p2,. . . . , рр].

Theorem 5.2.2. Let {y,) be any sequence of random variables satisfying (5.2.31). Then

image405(5.2.36)

where yh = Ey, yt+h.

Proof. We have

Yo = ff2(0o + ФІ + • ■ •)

Подпись: and so on.Yi = <т2(ФоФі + Ф1Ф2 + • . .) Уг = о2(ФоФі + ФіФз + ■ • •),


image407

Therefore

 

image408(5.2.37)

from which the theorem follows.

Theorem 5.2.3. Let {y,} be any stationary sequence satisfying (5.2.36). Then the characteristic roots of the T X Г autocovariance matrix X of{y,} are bounded from above.

Proof. Letx = (xq, xx, . . . , xr_,)’ be the characteristic vector of X corre­sponding to a root A. That is,

Xx = Ax. (5.2.38)

Suppose |x,| = max [Uq|, |x,|, . . . , |хг_,[]. The (t + l)st equation of (5.2.38) can be written as

y, Xo + yt-xXy + . . . + yQxt + . . . + yT^_txT-x = Axt. (5.2.39) Therefore, because А Ш 0,

Подпись: (5.2.40)ІУгІ + ІУг-іІ+• • -+ІУ0І+- • . + ІУг-i-ilsA.

Therefore

T

(5.2.41)

from which the theorem follows.

The premise of the next theorem is weaker than the conclusion of the preceding theorem. In terms of the spectral density, the premise of Theorem

5.1.4 is equivalent to its existence, and the conclusion of Theorem 5.2.3 to its continuity.

Theorem 5.2.4. Let {y,} be any sequence of random variables satisfying

image410(5.2.42)

where {€,} are i. i.d with Ее, = 0 and Eef= a2. Then lim yh = 0.

Л-*»

Note that (5.2.31) implies (5.2.42).

Proof. The theorem follows from the Cauchy-Schwartz inequality, namely,

Подпись: (5.2.44)

image412

УІ = оФ<Фк + ФіФи+і + • • У

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