Heteroscedasticity

In the classical regression model it is assumed that the variance of the error term is constant (homoscedastic). Here we relax this assumption and specify more generally that

0^3.1.12) Vut = пі t = 1, 2, . . . , T.

This assumption of nonconstant variances is called heteroscedasticity. The other assumptions remain the same. If the variances are known, this model is a special case of the model discussed in Section 13.1.1. In the present case, X is a diagonal matrix whose tth diagonal element is equal to erf. The GLS estimator in this case is given a special name, the weighted least squares estimator.

If the variances are unknown, we must specify them as depending on a finite number of parameters. There are two main methods of parameteri­zation.

In the first method, the variances are assumed to remain at a constant value, say, af, in the period t = 1, 2, . . . , Тг and then change to a new constant value of erf in the period t = T + 1, T + 2, . . . , T. If Tx is known, this is the same as (12.4.26). There we suggested how to estimate o-j and о. Using these estimates, we can define the FGLS estimator by the formula (13.1.11). If Tj is unknown, as well as oy and cr2 can be still estimated, but the computation and the statistical inference become much more complex. See Goldfeld and Quandt (1976) for further discus­sion of this case. It is not difficult to generalize to the case where the variances assume more than two values.

In the second method, it is specified that

(13.1.13) a? = g( Z(‘a),

where g(-) is a known function, zt is a vector of known constants, not necessarily related to x(, and a is a vector of unknown parameters. Gold­feld and Quandt (1972) considered the case where g(-) is a linear function

о

and proposed estimating a consistently by regressing щ on zt, where ut) are the least squares residuals defined in (12.2.12). If g( -) is nonlinear, щ must be treated as the dependent variable of a nonlinear regression model—see Section 13.4 below.

o

Even if we do not specify cr, as a function of a finite number of parameters, we can consistently estimate the variance-covariance matrix of the LS estimator given by (13.1.9). Let {ut} be the least squares residuals, and define the diagonal matrix D whose tth diagonal element is equal to щ. Then the heteroscedasticity-consistent estimator of (13.1.9) is defined by

(13.1.14) Up = (X’X) ~ xX’DX (X’X)-1.

Under general conditions 7Vj3 can be shown to converge to 7Vp. See Eicker (1963) and White (1980).