Constant Variance in a Subset of the Sample

The heteroscedastic model we shall consider in this subsection represents the simplest way to restrict the number of estimable parameters to a finite and manageable size. We assume that the error vector u is partitioned into N nonoverlapping subvectors as u = (u’,, Uj,. . . , Uy)’ such that £u, uj = a2lTr We can estimate each a2 consistently from the least squares residuals provided that each T, goes to infinity with Г. Note that this model is a special case of Zellner’s SUR model discussed in Section 6.4, so that the asymptotic results given there also hold here. We shall consider a test of homoscedasticity and the exact moments of FGLS.

If we assume the normality of u, the hypothesis of — <r2 for all і can be tested by the likelihood ratio test in a straightforward manner. Partitioning у and X into N subsets that conform to the partition of u, we can write the constrained and unconstrained log likelihood functions respectively as

CLL—– j log (У# – *А'(Уі ~ *iP) (6.5.7)


ULL—– і f Tt log of – ~ J – jj (у, – Х, Д)'(у, – ЗД. (6.5.8)

2 і-і 2 /Sr{ Of

Therefore —2 times the log likelihood ratio is given by

-2 log LRT = X Tt log (o2/of), (6.5.9)

where o2 and of are the constrained and unconstrained MLE, respectively. The statistic is asymptotically distributed as chi-square with N — 1 degrees of freedom.4

Taylor (1978) has considered a special case in which N= 2 in a model with normal errors and has derived the formulae for the moments of FGLS. By evaluating the covariance matrix of FGLS at various parameter values, Taylor has shown that FGLS is usually far more efficient than LS and is only slightly less efficient than GLS.

We shall sketch briefly the derivation of the moments of FGLS. Let C be a KX К matrix such that

C’XiXjC = ff? I (6.5.10)


C’X’2X2C = oA, (6.5.11)

where A is a diagonal matrix, the elements Als 2^ • • • , 2* of which are the roots of the equation

|<7Г2Х;Х2-2<7Г2Х’1Х1| = 0. (6.5.12)

The existence of such a matrix is guaranteed by Theorem 16 of Appendix 1. With S = C(I + A)~1/2, transform the original equation у = X/f + n to

y = X*y + u, (6.5.13)

where X* = XS and у = The FGLS estimator of y, denoted y, is given by

у = (оГ2ХГХ? + 5їаХГХЇ)-,(ггаХГ’Уі + °ї2*ГУі), (6-5.14)

where a] = (Г, – К)~уі – Х, Д)'(у, – X,#.5 Using аГ2Х?’Х? = (1 + Л)"1 and <rj2f’Xf = Л(І + A)-1, we obtain

у – у = (1 + A)[ff72DX*’u, + <rl2A-‘(I – D)X?’u2], (6.5.15)

where D is a diagonal matrix the ith diagonal element of which is equal to oal{ao + Finally, the moments of у — у can be obtained by mak­

ing use of the independence of D with Xf’u, and XJ’u2 and of known formulae for the moments of D that involve a hypergeometric function.

Kariya (1981) has derived the following inequalities concerning the covar­iance matrices of GLS and FGLS in a two-equation model:

vk S *,* [< + ЗД _V_ 2) + Зд Л-2)] По – (6-5 *6)

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