Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z\)
\(b_2 =\) the effect of \(Z\) on \(Y\) after controlling for the effects of \(X\)
\(\color{#000000}{\widehat{y}} = \color{#000000}{a} + \color{#000000}{b_1}\color{#000000}{x} + \color{#000000}{b_2}\color{#000000}{z}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X = 0\) and \(Z = 0\)
\(z\) is the value of \(Z\) for the individual
Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z\)
\(b_2 =\) the effect of \(Z\) on \(Y\) after controlling for the effects of \(X\)
\(\color{#a68100}{\widehat{y}} = \color{#000000}{a} + \color{#000000}{b_1}\color{#000000}{x} + \color{#000000}{b_2}\color{#000000}{z}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X = 0\) and \(Z = 0\)
\(z\) is the value of \(Z\) for the individual
Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z\)
\(b_2 =\) the effect of \(Z\) on \(Y\) after controlling for the effects of \(X\)
\(\color{#a68100}{\widehat{y}} = \color{#754cbc}{a} + \color{#000000}{b_1}\color{#000000}{x} + \color{#000000}{b_2}\color{#000000}{z}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X = 0\) and \(Z = 0\)
\(z\) is the value of \(Z\) for the individual
Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z\)
\(b_2 =\) the effect of \(Z\) on \(Y\) after controlling for the effects of \(X\)
\(\color{#a68100}{\widehat{y}} = \color{#754cbc}{a} + \color{#e93cac}{b_1}\color{#000000}{x} + \color{#000000}{b_2}\color{#000000}{z}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X = 0\) and \(Z = 0\)
\(z\) is the value of \(Z\) for the individual
Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z\)
\(b_2 =\) the effect of \(Z\) on \(Y\) after controlling for the effects of \(X\)
\(\color{#a68100}{\widehat{y}} = \color{#754cbc}{a} + \color{#e93cac}{b_1}\color{#1b8883}{x} + \color{#000000}{b_2}\color{#000000}{z}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X = 0\) and \(Z = 0\)
\(z\) is the value of \(Z\) for the individual
Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z\)
\(b_2 =\) the effect of \(Z\) on \(Y\) after controlling for the effects of \(X\)
\(\color{#a68100}{\widehat{y}} = \color{#754cbc}{a} + \color{#e93cac}{b_1}\color{#1b8883}{x} + \color{#690c48}{b_2}\color{#000000}{z}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X = 0\) and \(Z = 0\)
\(z\) is the value of \(Z\) for the individual
Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z\)
\(b_2 =\) the effect of \(Z\) on \(Y\) after controlling for the effects of \(X\)
\(\color{#a68100}{\widehat{y}} = \color{#754cbc}{a} + \color{#e93cac}{b_1}\color{#1b8883}{x} + \color{#690c48}{b_2}\color{#6e8e13}{z}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X = 0\) and \(Z = 0\)
\(z\) is the value of \(Z\) for the individual
When we control for a variable, we are holding its effect constant (i.e., removing its influence)
Note that we are ADDING variables to the model in order to REMOVE their influence from the effects of \(X\).
When they are not in the model, we are ignoring their influence
Multivariate regression
\(x\) is the value of \(X\) for the individual
\(b_1 =\) the effect of \(X\) on \(Y\) after controlling for the effects of \(Z_1\) and \(Z_k\)
\(b_2 =\) the effect of \(Z_1\) on \(Y\) after controlling for the effects of \(X\) and \(Z_k\)
\(b_k =\) the effect of \(Z_k\) on \(Y\) after controlling for the effects of \(X\) and \(Z_1\)
\(\color{#a68100}{\widehat{y}} = \color{#754cbc}{a} + \color{#e93cac}{b_1}\color{#1b8883}{x} + \color{#690c48}{b_2}\color{#6e8e13}{z} + \text{. . . } + \color{#bf5700}{b_k}\color{#808080}{z_k}\)
\(\widehat{y}\) is the predicted value of \(Y\)
\(a = Y-intercept\) (i.e., constant);
predicted value of \(Y\) when \(X\), \(Z_1\), and \(Z_k\) all equal \(0\)
\(z_1\) is the value of \(Z_1\) for the individual
\(z_k\) is the value of \(Z_k\) for the individual
Purpose 2: developing better predictions
Can use the multivariate regression model to predict values of the dependent variable for any combination of values of the predictors.
\[
\widehat{Income} = 799.22 + 623.35(educ) + 0.31(\text{par_wealth})
\]
Example: What is the predicted level of income for a person with 14 years of education and whose parents have total wealth of $250,000?
\[
\begin{aligned}
\widehat{Income} &= 799.22 + 623.35(\color{#e93cac}{14}) + 0.31(\color{#e93cac}{250000}) \\
&= \$87,025.12
\end{aligned}
\]
Example: What is the predicted level of income for a person with 16 years of education and whose parents have total wealth of $50,000?
\[
\begin{aligned}
\widehat{Income} &= 799.22 + 623.35(\color{#1b8883}{16}) + 0.31(\color{#1b8883}{50000}) \\
&= \$26,271.82
\end{aligned}
\]