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The connection of maximum likelihood estimation to OLS arises when this distribution is modeled as a multivariate normal. Specifically, assume that the errors ε have multivariate normal distribution with mean 0 and variance matrix σ 2 I .
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one [clarification needed] effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values ...
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) [1] states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. [2]
The model is estimated by OLS or another consistent (but inefficient) estimator, and the residuals are used to build a consistent estimator of the errors covariance matrix (to do so, one often needs to examine the model adding additional constraints; for example, if the errors follow a time series process, a statistician generally needs some ...
blup vs blue [ edit ] In contrast to the case of best linear unbiased estimation , the "quantity to be estimated", Y ~ k {\displaystyle {\widetilde {Y}}_{k}} , not only has a contribution from a random element but one of the observed quantities, specifically Y k {\displaystyle Y_{k}} which contributes to Y ^ k {\displaystyle {\widehat {Y}}_{k ...
Alternative estimators have been proposed in MacKinnon & White (1985) that correct for unequal variances of regression residuals due to different leverage. [11] Unlike the asymptotic White's estimator, their estimators are unbiased when the data are homoscedastic.
If the experimental errors, , are uncorrelated, have a mean of zero and a constant variance, , the Gauss–Markov theorem states that the least-squares estimator, ^, has the minimum variance of all estimators that are linear combinations of the observations. In this sense it is the best, or optimal, estimator of the parameters.
When f(x, β 0, β 1, ,,,, β p) is a linear function of the parameters and the x-values are known, least square estimators will be best linear unbiased estimator (BLUE). Again, if we assume that the least square estimates are independently and identically normally distributed, then a linear estimator will be minimum-variance unbiased estimator ...