Advanced Econometrics Takeshi Amemiya
Unrestricted Heteroscedasticity
When heteroscedasticity is unrestricted, the heteroscedasticity is not parameterized. So we shall treat each of the Г variances {a}} as an unknown parame
ter. Clearly, we cannot consistently estimate these variances because we have but one observation per variance. Nevertheless, it is possible to estimate the regression coefficients 0 consistently and more efficiently than by LS and to test for heteroscedasticity.
The idea of obtaining estimates of the regression coefficients that are more efficient than LS in an unrestricted heteroscedastic regression model has been developed independently by White (1982) and by Chamberlainj(1982), following the idea of Eicker (1963), who suggested an estimator of V0(6.1.5) that does not require the consistent estimation of 1 The following discussion follows Amemiya (1983c).
Let W be a TX(T— K) matrix of constants such that [X, W] is nonsingular and W'X = 0. Then, premultiplying (6.1.1) by [X, W]', GLS estimation of 0 can be interpreted as GLS applied simultaneously to
X'y = X'X0 + X'u (6.5.1)
and
W'y = W'u.
Thus, applying Theorem 13 of Appendix 1 to
[X'XX X'XW]-'
we obtain
0o = 0- (X, X)-,X, XW(W,2W)-,W'y. (6.5.3)
Of course, it is also possible to derive (6.5.3) directly from (6.1.3) without regard to the interpretation given above. An advantage of (6.5.3) over (6.1.3) is that the former does not depend on X-1. Note that one cannot estimate 7’-lX'X-1X consistently unless X can be consistently estimated. To transform (6.5.3) into a feasible estimator, one is tempted to replace X by a diagonal matrix D whose fth element is (y, — x',0f. Then it is easy to prove that under general assumptions plim r^X'DW = plim r-,X'XW and plim r_1W'DW = plim r_,W'XW element by element. However, one difficulty remains: Because the size of these matrices increases with T, the resulting feasible estimator is not asymptotically equivalent to GLS.
We can solve this problem partially by replacing (6.5.2) with
W'y = W',u, (6.5.4)
where W, consists of N columns of W, N being a fixed number. When GLS is
applied to (6.5.1) and (6.5.4), it is called the partially generalized least squares (PGLS) and the estimator is given by
Д, = Д - (X, X)-1X/2W,(W/12W,)-,W',y. (6.5.5)
PGLS is more efficient than LS because
Vfl-V^ (X/X)-,X/2W1(W12W,)"1W12X(X, X)-1, (6.5.6)
which is clearly nonnegative definite, but it is less efficient than GLS. An asymptotically equivalent feasible version of ft. is obtained by replacing the 2 in (6.5.5) by the D defined in the preceding paragraph.
White (1980a) proposed testing the hypothesis a? = a2 for all t by comparing (X'X)-IX'DX(X'X)-1 with ct2(X'X)_1, where D is as defined earlier and a2 is the least squares estimator of a2 defined in (1.2.5). Equivalently, White considered elements of X'DX — a2X'X. If we stack the elements of the upper triangular part of this matrix, we obtain a vector of {K2 + K)/2 dimension defined by S'(n2 — <r2l), where fi2 is a Г-vector the /th element of which is fi?, 1 is a Г-vector of ones, and S is а ГХ (К2 + K)/2 matrix, the columns of which are (xnxn, хахп,. . . , xiTxjT)' for 1 ё i, j S K, and і ^ j. It is easy to show that r_1/2S'(fi2 — <r2l) —► iV(0, A), where A = lim(r-,S, AS + T~2YM • S'll'S - r-2S'Al • VS - T~2S'l • l'AS) and Л = Ди2 - a4) • (и2 — <тЧ)'. The test statistic proposed by White is
(u2 - ff2l),S(rA)-1S'(a2 - o2),
where A is obtained from A by eliminating lim and replacing Л with D{(u2 — a2)2}. This statistic is asymptotically distributed as chi-square with (K2 + K)/2 degrees of freedom under the null hypothesis. It can be simply computed as TR2 from the regression of fi? on the products and cross products of xf.
