A COMPANION TO Theoretical Econometrics

Sampling Theory Estimation and Inference with Unknown Covariance Matrix

Consider again the linear model y = Xp + e with error-covariance matrix V = о2Л. In this section we relax the assumption that Л is known. As we saw in the previous section, there are some circumstances where such an assumption is reasonable. However, there are also many where it is not. For example, in a household expenditure function, we may be willing to assume the variance of expenditure depends on total expenditure and the demographic composition of the household, but not willing to specify the values of parameters that describe the dependence. Thus, we could write, for example,

o2 = 00 + 0Z + 02Z2;, (4.16)

where z1; and z2i are total expenditure and demographic composition, respect­ively, and (0O, 01, 02) are unknown parameters. If (o21, o2,..., oN) are not known, then some kind of reparameterization such as that in (4.16) is necessary to reduce the number of parameters to a manageable number that does not increase with sample size. We will work in terms of the general notation

o2 = o 2hi (a) = o 2h(z' a), (4.17)

where a is an (S x 1) vector of unknown parameters, and h() is a differentiable function of those parameters and an (S x 1) vector z{ which could be identical to or different from x{. To write (4.16) in terms of the general notation in (4.17), we re-express it as

0

In this example, and others which we consider, hi(0) = 1, implying that a = 0 describes a model with homoskedastic errors.

Several alternative specifications of hi(a) have been suggested in the literature. See Judge et al. (1985, p. 422) for a review. One of these is that given in (4.18), namely

hi(a) = 1 + a1z 1; + ... + aSzSi. (4.19)

This model has been considered by, among others, Goldfeld and Quandt (1972) and Amemiya (1977), and, in the context of a random coefficient model, by Hildreth and Houck (1968) and Griffiths (1972). Note that, if (z1;,..., zSl) are non­overlapping dummy variables, then the specification in (4.19) describes a parti­tion of the sample into (S + 1) subsamples, each one with a different error variance. Such a model could be relevant if parts of the sample came from different geo­graphical regions or there exists some other way of naturally creating sample separations. Examples where estimation within this framework has been con­sidered are Griffiths and Judge (1992) and Hooper (1993).

One potential difficulty with the specification in (4.19) is that the requirement hi(a) > 0 can mean that restrictions must be placed on a to ensure that negative variances are not possible. Two possible specifications which avoid this problem are

h(a) = (1 + a 1z у + ... + aSzSi)2 (4.20)

and

h(a) = exp(a 1z n + ... + aSzSi). (4.21)

The specification in (4.20) has received attention from Rutemiller and Bowers (1968) and Jobson and Fuller (1980). The specification in (4.21) was introduced by Harvey (1976) under the heading "multiplicative heteroskedasticity." For applica­tions and extensions, see Griffiths and Anderson (1982), and Hill et al. (1997).

A class of models which has been popular, but which does not fit within the framework of equation (4.17), is that where the location parameter vector в also appears within the variance function. Authors who have considered this class of models under varying degrees of generality include Amemiya (1973), Jobson and Fuller (1980), and Welsh et al. (1994).