From 4fcc218d6c33555773ec9a5e4197a30f2d5571ff Mon Sep 17 00:00:00 2001 From: Jacob Fleming Date: Tue, 25 Nov 2025 14:06:01 -0500 Subject: [PATCH] Typo in how multiple predictors affects R^2 B1^2 was pulled out, but it should be B2^2 --- public/latex_notes/unit4/unit4.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/public/latex_notes/unit4/unit4.tex b/public/latex_notes/unit4/unit4.tex index 14a9fc9..f28f02e 100644 --- a/public/latex_notes/unit4/unit4.tex +++ b/public/latex_notes/unit4/unit4.tex @@ -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 \[