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:
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“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)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)