Definition
Assume that we are regressing the linear regression model
where X is the design matrix and β is a k × 1 column vector of parameters to be estimated.
The ordinary least squares (OLS) estimator is
If the sample errors have equal variance σ2 and are uncorrelated, then the least-squares estimate of β is BLUE (best linear unbiased estimator), and its variance is easily estimated with
where are regression residuals.
When the assumptions of are violated, the OLS estimator loses its desirable properties. Indeed,
where .
While the OLS point estimator remains unbiased, it is not "best" in the sense of having minimum mean square error, and the OLS variance estimator does not provide a consistent estimate of the variance of the OLS estimates.
Read more about this topic: Heteroscedasticity-consistent Standard Errors
Famous quotes containing the word definition:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)