Model Estimation 277
Another generalization involves the case when regressors are determin-
istic but we relax the assumption that all errors e have the same variance.
As explained in Chapter 4, this is the homoscedasticity property. In prac-
tice, many financial time series are not homoscedastic, but have different
variance. Time series with different variances are said to heteroscedastic.
In Chapter 11 we explained time series whose heteroscedasticity can be
described through an autoregressive process. Here we consider the case
where the variances of each residual are perfectly known.
Consider the regression equation given by equation (13.4) and consider
the covariance matrix of residuals:
WE==cov(εε)'()ε (13.6)
W is an nn× matrix. Under the standard assumptions of regression given
in Chapter 3, W is proportional to the identity matrix C where σ^2 is the
constant variance of the residuals. Let’s relax this assumption and assume
that W is a diagonal matrix but with different diagonal terms:
=σ=
WV
V
vv00
00
(^00) n
2
1
2
2
(13.7)
In the above case, the usual OLS formula does not work. It has been
demonstrated, however, that a modified version of OLS called weighted
least squares (WLS) works. WLS seeks the minimum of the sum of squared
weighted residuals. Instead of minimizing the sum of squared residu-
als given by equation (13.2), it minimizes the sum of weighted squared
residuals:
WS weii
i
n
=
=∑
2
1(13.8)
Given the regression equation yx=+ββ 011 ++ βεkkx+ , if the variance of
each residual term is known exactly and is given by the matrix in equation
(13.7), then WLS seeks to minimize
WS e
i
i
in
=
=∑
1
22
1 ν