Multiple Regression
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Introduction to Multiple Regression
Real estate example
Predict sales price of home from sqft (R2=26%) or distance from downtown (R2=37%)
If combine two predictor variables the new R2 ≠ 26% + 37% because sqft is related to distance from downtown
Instead, we will use multiple regression
Multiple regression
Extension of simple regression that allows us to analyze the relationship between multiple x variables (IV) and a y variable (DV)
We will have to be careful about relationships between x variables
Wont use scatterplots as much in multiple regression
Adapting Basic Concepts
Format of equation very similar to simple regression
Add more x variables sample equation:
y = a + b1x1 + b2x2 + + bkxk
True population equation
y = α + β1×1 + β2×2 + + βkxk + error
Note that the coefficient is different for distance from simple and multiple regression analyses
For simple regression: the coefficient is the change in y (price) due to one unit change in x (distance) [“gross” effect]
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]
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:
Look at p-value to see if x variable (IV) coefficient is significant
Check the sign of the coefficient to see if makes sense given your understanding of the situation
Look at the magnitude of the coefficient to understand the structural relationship with the dependent variable
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
Residual = actual value (y) – predicted value (y hat) (same