Econometrics

The Essence of Econometrics

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Ordinary Least Squares (OLS)

\[\mathbf{b} = \begin{bmatrix} b_0 \\ b_1 \\ \vdots \\ b_{p-1} \end{bmatrix} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}\]
StepExplanation
\(\mathbf{y} = \mathbf{X}\mathbf{b}\)The linear regression model
\(\mathbf{X}'\mathbf{y} = \mathbf{X}'\mathbf{X}\mathbf{b}\)Pre-multiply both sides by \(\mathbf{X}’\)
\((\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y} = (\mathbf{X}'\mathbf{X})^{-1}(\mathbf{X}'\mathbf{X})\mathbf{b}\)Pre-multiply both sides by the inverse \((\mathbf{X}’\mathbf{X})^{-1}\)
\((\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y} = \mathbf{I}\mathbf{b}\)A matrix multiplied by its inverse is the identity matrix
\(\mathbf{b} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}\)The OLS estimator

Classical OLS Assumptions

AssumptionExpressionIf violated
1. Linearity\(Y_t = \alpha + \beta X_t + \varepsilon_t\)Misspecified model
2. Zero mean error\(E(\varepsilon \mid X) = 0\)Biased estimates
3. Exogenous regressors\(\operatorname{Cov}(X_s, \varepsilon_t) = 0\)Endogeneity bias
4. No serial correlation\(\operatorname{Cov}(\varepsilon_s, \varepsilon_t) = 0,\ s \neq t\)Serial correlation
5. Homoskedasticity\(\operatorname{Var}(\varepsilon_t) = \sigma^2\)Heteroskedasticity
6. No perfect multicollinearityNo regressor is an exact linear combination of the othersMulticollinearity
7. Normal errors\(\varepsilon_t \sim N(0, \sigma^2)\)Invalid small-sample inference