A regression model of the heteroscedastic error variance

317

Economics Letters 10 (1982) 317-319 North-Holland Publishing Company

A REGRESSION VARIANCE

MODEL OF THE HETEROSCEDASTIC

ERROR

Bong Joon YOON Southent Illtnois University, Carbondale, IL 62901, USA Received

4 January

1982

We introduce a regression model of the heteroscedastic error variance. A repetitive the least squares method is shown to provide the best linear unbiased estimator parameter vector of the model.

use of of the

It is well-known that the least squares estimator of the coefficient vector is inefficient, when the error term of a regression model is heteroscedastic. In this case, an efficient estimator of the coefficient vector requires a proper estimator of the error variance. Normally, the nature of this heteroscedastic error variance is specified a priori in order to ensure a sufficiently large number of degrees of freedom. To illustrate, consider the following regression model: Y, =

(1)

x:P + u,

where /? is a (k x 1) vector of coefficient parameters. The disturbances U, are assumed to be normally and independently distributed (NID) with mean zero and variance of heteroscedastic nature

012 =f(z:v).

(2)

It is also assumed that x, and z, are exogeneous. Suppose our primary concern is to estimate the parameter vector y rather than /?. A simple procedure to estimate y, suggested by Glejser (1969), is to regress the absolute values of the least squares residuals on 01651765/82/0000-0000/$02.75

0 1982 North-Holland

B.J. Yom / Regression model of the heteroscedastic

318

error variance

z,. The properties of the Glejser estimator are not clear. The maximum likelihood estimator, as in Rutemiller and Bowers (1968) and Breusch and Pagan (1979), provides an efficient estimator. But this requires a specifically designed computer program. In this note, we present a model in which a repetitive use of the widely available least squares method can provide a best linear unbiased estimator of y. For a large data set, a grouping may be found so that the error terms are homoscedastic within each group but heteroscedastic across the groups. This type of restrictive model may occur in demographic studies dealing with the census data, although the exact grouping would depend on the nature of specific problems. This restrictive model can be written as Y,, =

4,P + u'J ’

i=

1,,..,

N,

j=l,...,

n,,

(3)

where u,, are NID with mean zero and variance a*‘I = a,*

over all j.

Apply the ordinary least squares sample data over j = 1,. . . , n,. Then distribution with (n, - k) degrees and e, is the least squares residual

(OLS) to (3) for the ith group of the w = (n, - k)S,*/a,* has a Chi-square of freedom where S,’ = e:e,/(n, - k) vector. Hence,

ln( n, - k)S,* = In u,* + In fl. Note that In y

44~) = (Wk)bx

has expected

(4) value +(a,)

and variance

$‘(a,),

where

r(a,)),

+‘(a,) = (d2/‘d$)Oog r(a,)), and (Y,= f(n, - k). [See Johnson be rewritten as

= In a,2/J+‘(+(n,

- k))

+ c,,

and Kotz (1970, p. 196).] Thus, (4) can

(5)

B.J. Yom / Regression model

of the heteroscedastic error varrnnce

319

where e, = In W, - +(f( n, - WJ$Wt,

- k)) .

Then e, are identically and independently distributed with zero mean and unit variance for i = 1, 2,. . . , M. Now, assume the following specific form of heteroscedasticity:

u,2= exp(

z:y),

(6)

where z, is an (r x 1) vector representing the characteristics of the i th group of the sample. Combining (5) and (6) we obtain the following regression model of the heteroscedastic variance:

where the left-hand

side variable

Notice that the regression standard linear model. Thus, tion method to (7) provides parameter vector of the error

d, is defined

to be

model (7) satisfies the assumptions of the an application of the least squares estimathe best linear unbiased estimator of y, the variance.

References Breusch, T.S. and A.R. Pagan, 1979, A simple test for heteroscedasticity and random coefficient variation, Econometrica 47, 1287-1294. Glejser, H., 1969, A new test for heteroscedasticity, Journal of the American Statistical Association 64, 316-323. Johnson, N.L. and S. Kotz, 1970, Continuous univariate distributions - 1 (Houghton Mifflin, Boston, MA). Rutemiller, H.C. and D.A. Bowers, 1968, Estimation in a heteroscedastic regression model, Journal of the American Statistical Association 63, 552-557.