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:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“... 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)