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The general form of a multiple regression model can be specified as: [ \mathbf{Y} = \mathbf{X}\beta + \mathbf{u} ] Where: * (\mathbf{Y}) is an (n \times 1) vector of the dependent variable observations, * (\mathbf{X}) is an (n \times k) matrix of the independent variables (with (k) being the number of explanatory variables, including the constant term), * (\beta) is a (k \times 1) vector of the regression coefficients, * (\mathbf{u}) is an (n \times 1) vector of the error terms. To estimate the parameters (\beta), we use Ordinary Least Squares (OLS) estimation, where the estimator is: [ \hat{\beta}_{OLS} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{Y} ] However, OLS estimators assume that the error terms (\mathbf{u}) have certain properties: they must have zero mean, constant variance (homoscedasticity), and no autocorrelation. When these assumptions are violated—specifically when there is autocorrelation (the error terms are correlated over time) or heteroscedasticity (the variance of the error terms is not constant)—the OLS estimators can still be unbiased, but they will no longer be efficient, meaning they will not have the minimum variance among the class of linear estimators. In such cases, Generalized Least Squares (GLS) can be used to correct for both autocorrelation and heteroscedasticity. GLS is a more general estimation technique that adjusts for these violations by transforming the model in such a way ________ _________ ____ ___ _____ ___.
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