What is the Generalized Least Squares model?

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What is the Generalized Least Squares model?

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Answer 1 · 2018-02-23T10:02:48

$The GLS model is used in the presence of heteroscedasticity.$ $"The GLS model is used in the presence of heteroscedasticity."$
$It generalizes the OLS (Ordinary Least Squares) model.$ $"It generalizes the OLS (Ordinary Least Squares) model."$

$If we take the two variable linear regression$ $"If we take the two variable linear regression"$
$Y = β_{1} + β_{2} \cdot X$ $Y = beta_1 + beta_2 * X$
$Then we have the following formulas with OLS :$ $"Then we have the following formulas with OLS :"$

${ˆ β}_{2} = \frac{\sum_{i = 1}^{i = n} x_{i} \cdot y_{i}}{\sum_{i = 1}^{i = n} x_{i}^{2}}$ $hat beta_2 = (sum_{i=1}^{i=n} x_i*y_i) / (sum_{i=1}^{i=n} x_i^2)$

${ˆ β}_{1} = ¯ ¯ ¯ Y - {ˆ β}_{2} \cdot ¯ ¯ ¯ X$ $hat beta_1 = bar Y - hat beta_2 * bar X$

$with x_{i} = X_{i} - ¯ ¯ ¯ X$ $"with "x_i = X_i - bar X$
$and y_{i} = Y_{i} - ¯ ¯ ¯ Y$ $"and "y_i = Y_i - bar Y$
$(¯ ¯ ¯ X and ¯ ¯ ¯ Y being the average values of the observations$ $"("bar X" and "bar Y" being the average values of the observations"$
$(X_{i}, Y_{i})) .$ $(X_i, Y_i)).$

$With GLS we have a weighted sum$ $"With GLS we have a weighted sum"$

${ˆ β}_{2} = \frac{(\sum w_{i}) (\sum w_{i} X_{i} Y_{i}) - (\sum w_{i} X_{i}) (\sum w_{i} Y_{i})}{(\sum w_{i}) (\sum w_{i} X_{i}^{2}) - {(\sum w_{i} X_{i})}_{i}^{2}}$ $hat beta_2 = ((sum w_i)(sum w_i X_i Y_i) - (sum w_i X_i)(sum w_i Y_i))/((sum w_i)(sum w_i X_i^2) - (sum w_i X_i)^2)$

$with w_{i} = \frac{1}{σ_{i}^{2}},$ $"with "w_i = 1/sigma_i^2 ,$
$σ_{i}^{2} being the variances of the deviations u_{i} = Y_{i} - E (Y ∣ X_{i})$ $sigma_i^2" being the variances of the deviations "u_i = Y_i-E(Y|X_i)$

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What is the Generalized Least Squares model?

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