Multiple Linear Regression (MLR) Assumptions and Equations

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Omitted variable bias

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Occurs when a relevant variable is left out of a model, potentially leading to biased estimates.

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Effect of smaller variance

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A smaller variance can sometimes overcompensate for omitted variable bias in a misspecified model.

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Trade-off between bias and variance

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Trade-off between bias and variance; bias will not vanish even in large samples.

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Error variance

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Error variance is a measure of the variability of the error term in a regression model.

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Estimated sampling variation of the estimated βj

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se(β̂j) = √Var(β̂j) = √σ̂² / [SSTj(1 - R²j)]

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SSTj is

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Sum of squares total for the jth variable in the regression model.

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OLS Estimators

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OLS estimators are the best linear unbiased estimators under the Gauss-Markov Theorem.

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Heteroskedasticity Impact

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If heteroskedasticity is present, OLS is not the best estimator.

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Importance of MLR Assumptions

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MLR assumptions ensure the efficiency and unbiasedness of OLS estimators.

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Multicollinearity

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Multicollinearity occurs when independent variables in a regression model are highly correlated, making it difficult to isolate the individual effect of each variable.