Variance as a Linear Function of Regressors

In this subsection we shall consider the model <rj = z,’a, where z, is a 6-vector of known constants and a is a 6-vector of unknown parameters unrelated to the regression coefficients fi. Such a parametric heteroscedastic regression model has been frequently discussed in the literature. This is not surprising, for this specification is as simple as a linear regression model and is more general than it appears because the elements of z, may be nonlinear transfor­mations of independent variables. The elements of г, can also be related to x, without affecting the results obtained in the following discussion. We shall consider the estimation of a and the test of the hypothesis aj = a2 under this model.

Hildreth and Houck (1968) and Goldfeld and Quandt (1972) were among the first to study the estimation of a. We shall call these estimators HH and GQ for short. We shall follow the discussion of Amemiya (1977b), who pro­posed an estimator of a asymptotically more efficient under normality than the HH and GQ estimators.

Hildreth and Houck presented their model as a random coefficients model (cf. Section 6.7) defined by

У, = *№ + &, (6-5.18)

where {£,} are АГ-dimensional i. i.d. vectors with E£, — 0 and = D{aJ. The last matrix denotes aKXK diagonal matrix, the/th element of which is at. Thus the Hildreth and Houck model is shown to be a special case of the model where the variance is a linear function of regressors by putting z, = (•*u, *i> • • * , *«)’•

We shall compare the HH, GQ, and Amemiya estimators under the as­sumption of the normality of u. All three estimators are derived from a regression model in which u] serves as the dependent variable. Noting й, = и, — x^X’XJ-‘X’u, we can write

Hj = x’ta + vu — 2v2t + v3t, (6.5.19)

where vu = u2- a2, v2t = «^(X’X^X’u, and v3t = [x^X’XJ^X’u]2. We can write (6.5.19) in vector notation as

u2 = Za + v, – 2v2 + v3. (6.5.20)

We assume that X fulfills the assumptions of Theorem 3.5.4 and that limr_„ r_1Z’Z is a nonsingular finite matrix. (Amemiya, 1977b, has presented more specific assumptions.)

Equation (6.5.20) is not strictly a regression equation because Ev2 Ф 0 and E3 Ф 0. However, they can be ignored asymptotically because they are 0( T~l/2) and 0( T~ *), respectively. Therefore the asymptotic properties of LS, GLS, and FGLS derived from (6.5.20) can be analyzed as if v, were the only error term. (Of course, this statement must be rigorously proved, as has been done by Amemiya, 1977b.)

The GQ estimator, denoted a,, is LS applied to (6.5.20). Thus

d, = (Z’ZT’Z’V. (6.5.21)

It can be shown that Vr(di — a) has the same limit distribution as

VT(Z’Z) ‘Z’y, . Consequently, a, is consistent and asymptotically normal with the asymptotic covariance matrix given by

V, = 2(Z’Z)-,Z’D2Z(Z’Z)-1, (6.5.22)

where D s £uu’ = D(z,’a}.

The HH estimator d2 is LS applied to a regression equation that defines Ей}:

fi2 = MZa + w, (6.5.23)

where M is obtained by squaring every element of M — l — X(X’X)^1X’ and Ew — 0. Thus

d2 = (Z’MMZ)-1Z’Mu2. (6.5.24)

It can be shown that

= 2Q, (6.5.25)

where Q = MDM * MDM. The symbol * denotes the Hadamard product defined in Theorem 23 of Appendix 1. (Thus we could have written M as M * M.) Therefore the exact covariance matrix of d2 is given by

V2 = (6.5.26)

To compare the GQ and the HH estimators we shall consider a modifica­tion of the HH estimator, called MHH, which is defined as

dj = (Z’MMZ)-‘Z’MMfl2. (6.5.27)

Because v2 and v3 in (6.5.20) are of 0(Т~Ш) and 0(T~l), respectively, this estimator is consistent and asymptotically normal with the asymptotic covar­iance matrix given by

V? = 2(Z’MMZ)-1Z’MMD2MMZ(Z’MMZ)‘I. (6.5.28)

Amemiya (1978a) showed that

Q ё MIWVI, (6.5.29)

• •

or that Q — MIrM is nonnegative definite, by showing A’A * A’A ё (A * A)'(A * A) for any square matrix A (take A = D1/2M). The latter in­equality follows from Theorem 23 (i) of Appendix 1. Therefore V2 ё VJ, thereby implying that MHH is asymptotically more efficient than HH.

Now, from (6.5.27) we note that MHH is a “wrong” GLS applied to

(6.5.20) , assuming ZTvjv’, were (MM)’1 when, in fact, it is 2D2. Therefore we cannot make a definite comparison between GQ and MHH, nor between GQ and HH. However, this consideration does suggest an estimator that is asymptotically more efficient than either GQ or MHH, namely, FGLS ap­plied to Eq. (6.5.20),

d3 = (Z’D72Z)~‘Z’D72u2, (6.5.30)

where Dj = D(zt’di}. Amemiya (1977b) showed that the estimator is con­sistent and asymptotically normal with the asymptotic covariance matrix given by

V3 = 2(Z’D“2Z)-1. (6.5.31)

Amemiya (1977) has also shown that the estimator has the same asymptotic distribution as MLE.7

It should be noted that all four estimators defined earlier are consistent even if u is not normal, provided that the fourth moment of u, is finite. All the formulae for the asymptotic covariance matrices in this subsection have been obtained under the normality assumption, and, therefore, the ranking of estimators is not preserved in the absence of normality.

So far we have concentrated on the estimation of a. Given any one of the estimators of a defined in this subsection, say d3, we can estimate)? by FGLS

k – (X’6j‘ХПХ’б^У. (6-5-32)

We can iterate back and forth between (6.5.30) and (6.5.32). That is, in the next round of the iteration, we can redefine & as у —XfiF and obtain a new estimate of a3, which in turn can be used to redefine D3, and so on. Buse (1982) showed that this iteration is equivalent under normality to the method of scoring. Goldfeld and Quandt (1972) presented a Monte Carlo study of the performance of LS, FGLS, and MLE in the estimation of)? in the model where У, — fti + fat + Щ an<3 Vu, = a, + a2x, + a3xf.

We shall conclude this subsection with a brief account of tests of homosce – dasticity in the model where the variance is a linear function of regressors. Assuming that the first column of Z is a vector of ones, the null hypothesis of homoscedasticity can be set up as H0: a = (a2, 0, 0, . . . ,0)’ in this model. Breusch and Pagan (1979) derived Rao’s score test (cf. Section 4.5.1) of H0 as

Rao = ^lI(fi2-^2l),Z(Z, Z)-1Z,(fi2-ff2l), (6.5.33)

2Cr

where 1 is the Г-vector of ones. It is asymptotically distributed asxb-1 • Because the asymptotic distribution may not be accurate in small sample, Breusch and

Pagan suggested estimating P(Rao > c) by simulation. They pointed out that since Rao’s score test depends only on fi, the simulation is relatively simple. For a simple way to make (6.5.33) robust to nonnormality, see Koenker (1981b).

Goldfeld and Quandt (1965) proposed the following nonparametric test of homoscedasticity, which they called the peak test: First, order the residuals й, in the order of z’,a for a given estimator a. This defines a sequence {£<,•)} where j ^ к if and only if zja § z*a. (Instead of г’, a, we can use any variable that we suspect influences cr,2 most significantly.) Second, define that a peak occurs at j > 1 if and only if I Qj > йк for all к < j. Third, if the number of peaks exceeds the critical value, reject homoscedasticity. Goldfeld and Quandt (1965,1972) presented tables of critical values.

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