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Weighted least squares (WLS), also known as weighted linear regression, [1] [2] is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations (heteroscedasticity) is incorporated into the regression.
The result of fitting a set of data points with a quadratic function Conic fitting a set of points using least-squares approximation. In regression analysis, least squares is a parameter estimation method based on minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each ...
EViews: EViews version 8 offers three different methods for robust least squares: M-estimation (Huber, 1973), S-estimation (Rousseeuw and Yohai, 1984), and MM-estimation (Yohai 1987). [ 14 ] Julia : the CovarianceMatrices package offers several methods for heteroskedastic robust variance covariance matrices.
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.
A very important application of the variance function is its use in parameter estimation and inference when the response variable is of the required exponential family form as well as in some cases when it is not (which we will discuss in quasi-likelihood). Weighted least squares (WLS) is a special case of generalized least squares. Each term ...
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. Then the distribution of y conditionally on X is
Ordinary least squares can be interpreted as maximum likelihood estimation with the prior that the errors are independent and normally distributed with zero mean and common variance. In GLS, the prior is generalized to the case where errors may not be independent and may have differing variances .
In ordinary least squares, the definition simplifies to: =, =, where the numerator is the residual sum of squares (RSS). When the fit is just an ordinary mean, then χ ν 2 {\displaystyle \chi _{\nu }^{2}} equals the sample variance , the squared sample standard deviation .