Multiple Regression

Introduction to Multiple Regression

* Real estate example

o Predict sales price of home from sqft (R2=26%) or distance from downtown (R2=37%)

o If combine two predictor variables the new R2 ≠ 26% + 37% because sqft is related to distance from downtown

o Instead, we will use multiple regression

* Multiple regression

o Extension of simple regression that allows us to analyze the relationship between multiple x variables (IV) and a y variable (DV)

o We will have to be careful about relationships between x variables

o Won't use scatterplots as much in multiple regression

Adapting Basic Concepts

* Format of equation very similar to simple regression

o Add more x variables sample equation: y = a + b1x1 + b2x2 + ...+ bkxk

o True population equation y = α + β1x1 + β2x2 + ... + βkxk + error

* Note that the coefficient is different for distance from simple and multiple regression analyses

o For simple regression: the coefficient is the change in y (price) due to one unit change in x (distance) ["gross" effect]

o For multiple regression: the coefficient is the change in y (price) due to one unit change in x (distance) controlling for other x variables (i.e., the houses have the same sqft) ["net" effect]

o Note in this example, distance has a negative effect on price (because this increases commuting time) but houses farther away tend to be bigger (the correlation between the x variables) and bigger houses tend to cost more--so, multiple regression has to balance these two predictor variables

* 4 steps to interpreting coefficients:

o Look at p-value to see if x variable (IV) coefficient is significant

o Check the sign of the coefficient to see if makes sense given your understanding of the situation

o Look at the magnitude of the coefficient to understand the structural relationship with the dependent variable

o Note which other x variables are included in the regression so you can interpret the coefficient as an appropriate gross or net effect

* gives you instructions on how to conduct multiple regression in Excel

* Residual analysis

o Residual = actual value (y) - predicted value (y hat) (same

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