E wi e wix e wixi
and its variance is given by varOs*) = ------ (11.3.9)
CHAPTER ELEVEN: HETEROSCEDASTICITY 397
Difference between OLS and GLS
Recall from Chapter 3 that in OLS we minimize
but in GLS we minimize the expression (11.3.7), which can also be written as
where wi = 1 /of [verify that (11.3.11) and (11.3.7) are identical].
Thus, in GLS we minimize a weighted sum of residual squares with wi = 1 /of acting as the weights, but in OLS we minimize an unweighted or (what amounts to the same thing) equally weighted RSS. As (11.3.7) shows, in GLS the weight assigned to each observation is inversely proportional to its oi, that is, observations coming from a population with larger oi will get relatively smaller weight and those from a population with smaller oi will get proportionately larger weight in minimizing the RSS (11.3.11). To see the difference between OLS and GLS clearly, consider the hypothetical scat-tergram given in Figure 11.7.
In the (unweighted) OLS, each U2 associated with points A, B, and C will receive the same weight in minimizing the RSS. Obviously, in this case the U2 associated with point C will dominate the RSS. But in GLS the extreme observation C will get relatively smaller weight than the other two observations. As noted earlier, this is the right strategy, for in estimating the
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