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Typo in how multiple predictors affects R^2 #35

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2 changes: 1 addition & 1 deletion public/latex_notes/unit4/unit4.tex
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Expand Up @@ -284,7 +284,7 @@ \subsection{Relationship between single and two predictor regression coefficient
Instead, $\epsilon'$ absorbs the variation in $Y$ that is correlated with $X_2$ but unaccounted for by $X_1$.
Because $X_1$ and $X_2$ are generally correlated, part of the systematic variation explained by $X_2$ is treated as noise in the single-predictor model. Rather we have
\begin{equation}
{\rm var}(\epsilon') = {\rm var}(Y|X_1) = {\rm var}(\beta_1 X_1 + \beta_2 X_2 + \epsilon|X_1) = \beta_1^2{\rm var}(X_2|X_1) + \sigma_{\epsilon}^2 > \sigma_{\epsilon^2}
{\rm var}(\epsilon') = {\rm var}(Y|X_1) = {\rm var}(\beta_1 X_1 + \beta_2 X_2 + \epsilon|X_1) = \beta_2^2{\rm var}(X_2|X_1) + \sigma_{\epsilon}^2 > \sigma_{\epsilon^2}
\end{equation}
hence
\[
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