1. Curvilinear Relationships
Age
Age Squared
Combined effect 25
625
15500 45
2025
17100 65
4225
9100 80
6400
-3200

2. Regression extensions

3. Curvilinear Relationships
Regression assumes a linear relationship between x and y
This is not always true
Consider age and income

4. Curvilinear Relationships
Techniques for capturing nonlinear relationships
Dummy variables
Transformation
Polynomials

5. Curvilinear Relationships
Relationship between age and income
The two age terms are interpreted together

6. Logarithmic transformations
Ln(ŷ) = (a) + b ln(x1)
Interpretation
Percentage change in y associated with a percentage change in x

7. Logarithmic transformations
Ln(ŷ) = (a) + b (x1)
Interpretation
Percentage change in y associated with a unit change in x

8. Logarithmic transformations
ŷ = (a) + b ln(x1)
Interpretation
Unit change in y associated with a percentage change in x

9. Time Series Yt =a + bxt + et Subscript denotes time
Time intervals are equal
Example
US population as a function of time

10. Time Series

11. Time Series

12. Interactions
The relationship of x to y might vary for different levels of x.
There is an interaction between x1 and x2
Create xy variable by multiplying x1 *x2
Interaction between gender, education in years and income. X1=education, X2 = gender male = 1
Multiply gender*education
Ŷ = x1+ 1,200x2+ 202x1x R2 = .65

13. Interactions Ŷ = 5837+556x1+ 1,200x2+ 202x1x2 R2 = .65
Sb1 = 4.44 Sb2 = 2.7 Sb12=7.5
Interpretation:
Men earn greater returns from years of schooling
Discrimination?
Women earn less in school?

14. Does Family Size Moderate Preference for more children?
100 religious catholics surveyed on desired total number of children
Y = desired total number of children
X1= Current number of children
X2= Annual Income
Y = X1+.10X2 +.01X1X2 + e
Se1 = .10 se2 = se12 = .001
The interaction term is significant
In the presence of an interaction the coefficients (slopes) components for the variable that form the interaction are “average” effects

15. Does Family Size Moderate Preference for more children?
Y = desired total number of children
X1= Current number of children
X2= Annual Income
Y = X1+.10X2 +.01X1X2 + e
Se1 = .10 se2 = se12 = .001
The interaction term is significant
In the presence of an interaction the coefficients (slopes) components for the variable that form the interaction are “average” effects
The effect of the number of income depends on the number of children
For every one dollar increase in income the impact of the current number of children on the desired number of children increase by .01

16. Interactions
Stata example

17. Interaction between Gender and Income?

18. Interaction between Gender and Income?

19. Interaction between Gender and Income?

20. Interaction between Gender and Income?