Other Models

In the preceding subsections we have discussed models in which cross-sec­tion-specific components are independent across individuals and time-spe­cific components are independent over time periods. We shall cite a few references for each of the other types of random coefficients models. They are classified into three types on the basis of the type of regression coefficients: (1) Regression coefficients are nonstochastic, and they either continuously or discretely change over cross-section units or time periods. (2) Regression coefficients follow a stochastic, dynamic process over time. (3) Regression coefficients are themselves dependent variables of separate regression models. Note that type 1 is strictly not a RCM, but we have mentioned it here because of its similarity to RCM. Types 1 and 2 together constitute the varying param­eter regression models.

References for type 1 are Quandt (1958), Hinkley (1971), and Brown, Durbin, and Evans (1975). References for type 2 are Rosenberg (1973), Cooley and Prescott (1976), and Harvey (1978). A reference for type 3 is Amemiya (1978b).

Exercises

1. (Section 6.1.2)

Consider a classic regression model
у = ax + /?z + u,

where a and fi are scalar unknown parameters; x and z are Г-component vectors of known constants such that x’l = z’l = 0, where 1 is a Г-compo­nent vector of ones; and u is a Г-component vector of unobservable i. i.d. random variables with zero mean and unit variance. Suppose we are given an estimator Д such that ЕД=/?, V0 = T~ and£u/i = T l/2pl, where pis

a known constant with 0 S p < 1. Write down the expression for the best estimator of a you can think of. Justify your choice.

2. (Section 6.1.2)

In the model of Exercise 17 of Chapter 3, assume that the exact distribu­tion of P is known to be N{p, T~l).

a. Obtain the mean squared error of a.

b. Find an estimator of a whose mean squared error is smaller than that of a.

3. (Section 6.1.3)

Prove that statement D => statement C in Theorem 6.1.1.

4. (Section 6.1.3)

If К = 1 in Model 6, the efficiency of LS relative to GLS can be defined by

Подпись:(x’x)2

(x’X-‘xXx’Sx)’

Watson (1955) showed Eff Ш Ak, XJ{kt + ks)2, where k, and Xs are the larg­est and smallest characteristic roots of 2, respectively. Evaluate this lower bound for the case where 2 is given by (5.2.9), using the approximation of the characteristic roots by the spectral density (cf. Section 5.1.3).

5. (Section 6.1.3)

In Model 6 assume К = 1 and X = 1, a vector of ones. Also assume 2 is equal to 2, given in (5.2.9). Calculate the limit of the efficiency of LS as Г—»oo. (Efficiency is defined in Exercise 4.)

6. (Section 6.1.3)

Prove (6.1.6) directly, without using the fact that GLS is BLUE.

7. (Section 6.1.5)

Consider a regression model

у = X0 + u,

where Ей = 0 and £uu’ = P = Z(Z’Z)-1Z’. We assume that X and Z are TX К and TX G matrices of constants, respectively, such that rank(X) = K, rank(Z) = G <T, and PX = X. Find a linear unbiased esti­mator of fi the variance-covariance matrix of which is smaller than or equal to that of any other linear unbiased estimator. Is such an estimator unique?

8. (Section 6.2)

Suppose у — N(Xfi, X), where there is no restriction on X except that it is positive definite. Can you obtain the MLE of X by setting the derivative of the log likelihood function with respect to X equal to 0?

9. (Section 6.3.2)

Show that A, and A2 given in (6.3.5) and (6.3.6) converge to 0 in probabil­ity under the assumption of the text.

10. (Section 6.3.2)

Combining y, = x’,fi + u, and u, = pu,_, + e„ we can write У, = РУ,-і +

Durbin (1960) proposed estimating/? by the least squares coefficient on y,-i in the regression of y, on x„ and In vector notation

– _ yli[I-Z(Z’Z)->Z’]y 90 yiJI-ZfZ’Zr^’Jy,.’

where y = (h, y2.- • • .УгУ. • • .Уг-іУ* Z =

(X, X_j), X = (x1,x2>. . . ,xTy, X_1 = (x0,x1). . . , Xj-_i)/. Show that pD has the same asymptotic distribution as XjLiKf_iMf/XfLiM?_i if limr_„ T~l Z’Z is a finite nonsingular matrix.

11. (Section 6.3.3)

In Model 6 suppose К = 1 and {ut) follow AR(1), u, = put-x + e,. If we thought {u,} were i. i.d., we would estimate the variance of LS jiL by V = <r2/x’x, where a2 = Г-1[у’у — (х’х)_1(у’х)2]. But the true variance is V = x’Xx/(x’x)2, where X is as given in (5.2.9). What can you say about the sign of V— VI

12. (Section 6.3.3)

Consider у, = Дх, + u„ ut = pu,-x + e,, where x, is nonstochastic, p < 1, (e,) are i. i.d. with Ее, = 0, Ve, = a2, {u,) are stationary, and Д, p, and cr2 are unknown parameters. Given a sample (y,, x,), t = 1, 2,. . . ,T, and given xT+l, what do you think is the best predictor of yT+ x?

13. (Section 6.3.3)

Let y, = pt+ u„ where {u,} follow AR(1), u, = + €,. Define the

following two predictors of ут+і – 9т+ і = (T + 1)Д and ут+і = (T+ 1)Д, where Д and P are the LS and GLS estimators of P based on yx, y2,. . . , yT, respectively. In defining GLS, p is assumed known. Assum­ing p> 0, compare the two mean squared prediction errors.

14. (Section 6.3.5)

Suppose Ey = Xfl and Fy = X( Д), meaning that the covariance matrix of у is a function of Д. The true distribution of у is unspecified. Show that minimizing (у — ХД)’Х(Д)~‘(У — ХД) with respect to /? yields an incon­sistent estimator, whereas minimizing (у — Х)?)’Х0?)-1(У ~ХД) + log|X(>3)| yields a consistent estimator. Note that the latter estimator would be MLE if у were normal.

15. (Section 6.3.6)

Show limr_o. (EdL — 2)2 = 0, where dL is defined in (6.3.24). Use that to show plimr_„ dL = 2.

16. (Section 6.3.6)

In the table of significance points given by Durbin and Watson (1951), du — dh gets smaller as the sample size increases. Explain this phe­nomenon.

17. (Section 6.3.7)

Verify (6.3.30).

18. (Section 6.3.7)

Prove (6.3.33).

19. (Section 6.3.7)

In the model (6.3.31), show plimp = ap(a + p)/( 1 + exp) (cf. Malinvaud, 1961).

20. (Section 6.3.7)

Consider model (6.3.27). Define the Г-vector x_, = (ль, xx,. . . , хт-іУ and the ГХ 2 matrix S = (x, x_,). Show that the instrumental variables estimator у = (S’Z)_1S’y is consistent and obtain its asymptotic distribution. Show that the estimator p of p obtained by using the residuals у — Zy is consistent and obtain its asymptotic distri­bution.

21. (Section 6.4)

In the SUR model show that FGLS and GLS have the same asymptotic distribution as T goes to infinity under appropriate assumptions on u and X. Specify the assumptions.

22. (Section 6.5.1)

Consider a regression model

yt = x’fi+u„ ut = put-l + et, t= 1,2,. . . ,T.

Assume

(A) x, is a vector of known constants such that jxpr,’ is a nonsingu­lar matrix.

(B) щ = 0.

(C) {€,} are independent with Ее, = 0 and Ve, = a*.

(D) {of} are known and bounded both from above and away from 0.

(E) fi and p are unknown parameters.

Explain how you would estimate fi. Justify your choice of the estimator.

23. (Section 6.5.3)

Show that the density of a multinomial model with more than two re­sponses or the density of a normal model where the variance is propor­tional to the square of the mean cannot be written in the form of (6.5.17).

24. (Section 6.5.4)

Show that ‘ffioti — a), wherea, is defined in(6.5.21),has the same limit distribution as VT(Z, Z)_1Z’v1.

25. (Section 6.5.4)

Show (6.5.25).

26. (Section 6.5.4)

Consider a heteroscedastic nonlinear regression model

У, =I(fio) + Щ,

where {u,} are independent and distributed as N(0, z,’ao), Д, is a scalar unknown parameter, and cto is a vector of unknown parameters. Assume

(A) If fi is the NLLS estimator of fi0, 4T{fi — fiQ) —»N{ 0, c), where 0 < c< ».

(B) {zt) are vectors of known constants such that z, < h for some

finite vector h (meaning that every element of z, is smaller than the corresponding element of h), zis positive and bounded away from 0, and T~1 , z?!t is nonsingular for every T and converges to a finite

nonsingular matrix.

(C) df/dfi < M for some finite M for every t and every fi. Generalize the Goldfeld-Quandt estimator of Oq to the nonlinear case and derive its asymptotic distribution.

27. (Section 6.6.2)

Consider an error components model

yit = x’J + pi + eu, i= 1,2,. . . ,N and

t=l,2,. . . ,T,

where xit is a А-vector of constants and p, and eit are scalar random variables with zero means. Define p = (px, p2, ■ • • ,PnY> ei~ (en, ea,. . . ,eiT)’, and e = (e;,e^,. . . , e’N)’. Then we assume Epp’ = ap. N, Eee’ = Ijy © 2, and Epe’ = 0, where 2 is a Г-dimensional diagonal matrix the rth diagonal element of which is a] > O. Assume and 2 are known. Obtain the formulae for the following two estimators of#

1. The generalized least squares estimator fiG;

2. The fixed-effects estimator fi (it is defined here as the generalized least squares estimator of fi treating p, as if they were unknown parameters rather than random variables).

Подпись:
fi: КXI, unknown constants v: scalar, unobservable, Ev = 0, Vv = a2 u: ГХ 1, unobservable, £u = 0, Auu’ =1.

Assume v and u are independent and a2 is a known constant. Also assume X’z Ф 0 and rank[X, z] = K+ 1. Rank the following three estimators in terms of the mean squared error matrix:

LS0§l): (X’X)-‘X’y

GLSO? G): (X’2 *X) ‘X’2 *y, where 2 = E(vz + u)(vz + u)’

a 77/

QLS(0Q): (X’JX)_1X’Jy, where J = I-^

z z

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