The Almon Lag

Almon (1965) proposed a distributed-lag model

Подпись: N

yl==’ZPjX,+i-j+v»

1-І

in which 0X. . , 0N lie on the curve of a gth-order polynomial; that is,

/«, j=l,2,…,N. (5.6.6)

,Y and S=(S0,Sl,. . . , Sq)’, we

(5.6.7)

The estimation of 6 can be done by the least squares method. Let X be a TXN matrix, the /,jth element of which is Then

6 = (J’X’XJ^’J’X’y and 0 = JS.3 Note that 0 is a special case of the con­strained least squares estimator (1.4.11) where R = J and c = 0.

By choosing N and q judiciously, a researcher can hope to attain both a reasonably flexible distribution of lags and parsimony in the number of pa­rameters to estimate. Amemiya and Morimune (1974) showed that a small order of polynomials {q — 2 or 3) works surprisingly well for many economic time series.

Some researchers prefer to constrain the value of the polynomial to be 0 at j = N+ 1. This amounts to imposing another equation

S0 + 3AN+ 1) + S2(N+ l)2 + . . . + SjiN+ 1)« – 0 (5.6.8)

in addition to (5.6.6). Solving (5.6.8) for S0 and inserting it into the right-hand side of (5.6.7) yields the vector equation

0 = J*S* (5.6.9)

where S* = (S1, S2,. . . , SqY and J* should be appropriately defined.

Exercises

1. (Section 5.2.1)

Prove that model (5.2.1) with Assumptions A, B, and C is equivalent to model (5.2.3) with Assumptions A and B.

2. (Section 5.2.1)

Show that the process defined in the paragraph following Eq. (5.2.5) is AR(1).

3. (Section 5.3)

Find the exact inverse of the variance-covariance matrix of MA( 1) using

(5.3.12) and compare it with the variance-covariance matrix of AR(1).

4. (Section 5.3)

In the MA(1) process defined by (5.3.6), defineyf = ef — p~lef-,, where (ef) are i. i.d. with Eef = 0, Vef = p2a2. Show that the autocovariances of y, and yf are the same.

5. (Section 5.4)

lfX, Y, and Z are jointly normal and if X is independent of either ForZ, then EXYZ = EXEYZ (Anderson, 1958, p. 22). Show by a counterexam­ple that the equality does not in general hold without the normality assumption.

6. (Section 5.4)

Show that ‘ІТ (pA — p) and ІТ (pM — p) have the same limit distribution.

7. (Section 5.4)

In the AR(1) process defined by (5.2.1), define the first differences yf = У,~ У,-1 and derive plimr_. 2£.3 yf-, yfl 2£.3 yf-2!.

8. (Section 5.4)

In the AR(2) process defined by (5.2.16), derive plimr_oo E£.2 Уі-іУі/ *Г-2 УЇ-і-

9. (Section 5.5)

Derive (5.5.2) from the general formula of (5.5.5).

10. (Section 5.5)

In the MA(1) process defined by (5.3.6), obtain the optimal predictor of yt+ngiveny„y,-i,- . . .

11. (Section 5.6)

Show that (5.6.7) can be written in the equivalent form Q’fi = 0, where Q is an NX(N— q— 1) matrix such that [Q, J] is nonsingular and Q’J = 0. Find such a Q when N=4 andq = 2.

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