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Concept# Linear least squares

Summary

Linear least squares (LLS) is the least squares approximation of linear functions to data.
It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals.
Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.
The three main linear least squares formulations are:
Ordinary least squares (OLS) is the most common estimator. OLS estimates are commonly used to analyze both experimental and observational data. The OLS method minimizes the sum of squared residuals, and leads to a closed-form expression for the estimated value of the unknown parameter vector β: where is a vector whose ith element is the ith observation of the dependent variable, and is a matrix whose ij element is the ith observation of the jth independent variable. The estimator is unbiased and consistent if the errors have finite variance and are uncorrelated with the regressors: where is the transpose of row i of the matrix It is also efficient under the assumption that the errors have finite variance and are homoscedastic, meaning that E[εi2xi] does not depend on i. The condition that the errors are uncorrelated with the regressors will generally be satisfied in an experiment, but in the case of observational data, it is difficult to exclude the possibility of an omitted covariate z that is related to both the observed covariates and the response variable. The existence of such a covariate will generally lead to a correlation between the regressors and the response variable, and hence to an inconsistent estimator of β. The condition of homoscedasticity can fail with either experimental or observational data. If the goal is either inference or predictive modeling, the performance of OLS estimates can be poor if multicollinearity is present, unless the sample size is large.
Weighted least squares (WLS) are used when heteroscedasticity is present in the error terms of the model.

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