1. Chapter 6: Multiple Regression – Additional Topics

2. Functional Form of Regression: Logs
Logarithmic functional forms
Log-linear: ln(y)=a+bx  Δ ln 𝑦 = Δ𝑦 𝑦 =𝑏∗Δ𝑥 ln(y)=10+.3x  one unit increase in x causes 30% change in y Log-log: ln(y)=a+bln(x)  Δln y =b∗Δ ln 𝑥  Δ𝑦 𝑦 =𝑏∗ Δ𝑥 𝑥 ln(y)=3-.2ln(x)  1% increase in x causes .2% decrease in y.
Slope coefficients on logged variables are invariant to rescaling by a constant.
ln(yi)= 𝑎 0 + 𝑎 1 ln 𝑥 𝑖
Suppose 𝑥 𝑖 ∗ = 𝑥 𝑖 𝑐 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ln 𝑦 = 𝑏 0 + 𝑏 1 ln 𝑥 𝑖 ∗ = 𝑏 0 +𝑏1 ln 𝑥 𝑖 − ln 𝑐
=( 𝑏 0 −𝑏1 ln 𝑐 + 𝑏 1 ln 𝑥 𝑖
Note: intercept changes, but not slope.

3. Functional Form of Regression: Logs
Taking logs often eliminates/mitigates problems with outliers
Taking logs often helps to secure normality and homoskedasticity
Logs must not be used if variables take on zero or negative values

4. Functional Form of Regression: Quadratic
Using quadratic functional forms
Example: Wage equation
Marginal effect of experience
Over what range of experience is marginal effect positive? Negative?
Concave experience profile

5. Functional Form of Regression: Quadratic
Example: Effects of pollution on housing prices
Nitrogen oxide in the air, distance from employment centers, average student/teacher ratio
Does this mean that, at a low number of rooms, more rooms are associated with lower prices?

6. Functional Form of Regression: Quadratic
Calculation of the turnaround point
Turnaround point:
This area can be ignored as it concerns only 1% of the observations.
Increase rooms from 5 to 6:
Increase rooms from 6 to 7:

7. Functional Form of Regression: Interaction Terms
Models with interaction terms
Interaction term
Effect of number of bedrooms, but for a square footage of zero

8. Functional Form of Regression: Interaction Terms
Reparametrization of interaction effects 𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 3 𝑥 1 𝑥 2 +𝑢
Measure ( 𝑥 1 , 𝑥 2 ) as deviations from means 𝜇 1 , 𝜇 2
𝑥 1 ∗ = 𝑥 1 − 𝜇 1 , 𝑥 2 ∗ = 𝑥 2 − 𝜇 2
Re-estimate model with 𝑥 1 ∗ , 𝑥 2 ∗
𝑦= 𝛿 0 + 𝛿 1 𝑥 1 ∗ + 𝛿 2 𝑥 2 ∗ + 𝛿 3 𝑥 1 ∗ 𝑥 2 ∗ +𝑢
Δ𝑦 Δ 𝑥 1 ∗ = 𝛿 1 + 𝛿 3 𝑥 2 ∗
= 𝛿 1 𝑖𝑓 𝑥 2 = 𝜇 2
Advantages of reparametrization
Easy interpretation of coefficients
Standard errors for partial effects at the mean
Interactions may be centered at values other than mean

9. Average Partial Effects
In models with quadratics, interactions, and other nonlinear functional forms, the partial effect depends on the values of one or more explanatory variables
Average partial effect (APE) is a summary measure to describe the relationship between dependent variable and each explanatory variable
After computing the partial effect and plugging in the estimated parameters, average the partial effects for each unit across the sample

10. Goodness of Fit: R2 General remarks on R2 Adjusted vs Unadjusted R2
A high R-squared does not imply that there is a causal interpretation
A low R-squared does not preclude precise estimation of partial effects
Adjusted vs Unadjusted R2
is an estimate for
The adjusted R-squared increases if, and only if, the t-statistic of a newly added regressor is greater than one in absolute value

11. Goodness of Fit: R2
Using adjusted R-squared to choose between nonnested models
Models are nonnested if neither model is a special case of the other
A comparison between the R-squared of both models would be unfair to the first model because the first model contains fewer parameters
In the given example, even after adjusting for the difference in degrees of freedom, the quadratic model is preferred

12. Goodness of Fit: R2
Comparing models with different dependent variables
R-squared or adjusted R-squared must not be used to compare models which differ in their definition of the dependent variable
Other methods must be used (perhaps later)
CEO compensation and firm performance
There is much
less variation
in log(salary)
that needs to
be explained
than in salary

13. Controlling for too many factors
In some cases, certain variables should not be held fixed
In a regression of traffic fatalities on state beer taxes (and other factors) one should not directly control for beer consumption
In a regression of family health expenditures on pesticide usage among farmers one should not control for doctor visits
Different regressions may serve different purposes
In a regression of house prices on house characteristics, one would only include price assessments if the purpose of the regression is to study their validity; otherwise one would not include them

14. Effect of adding variables to regression.

15. Predicting y when log(y) is the dependent variable
Under the additional assumption that is independent of :
Prediction for y

16. Comparing R-squared of a logged and an unlogged specification
These are the R-squareds for the predictions of the unlogged salary variable (squared correlation between salary and predicted salary). Both R-squareds can now be directly compared.