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

Regression extensions

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

Curvilinear Relationships Techniques for capturing nonlinear relationships Dummy variables Transformation Polynomials

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

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

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

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

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

Time Series

Time Series

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 = 5837+556x1+ 1,200x2+ 202x1x2 R2 = .65

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?

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 = 4.4279 + .81X1+.10X2 +.01X1X2 + e Se1 = .10 se2 = .05 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

Does Family Size Moderate Preference for more children? Y = desired total number of children X1= Current number of children X2= Annual Income Y = 4.4279 + .81X1+.10X2 +.01X1X2 + e Se1 = .10 se2 = .05 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

Interactions Stata example

Interaction between Gender and Income?

Interaction between Gender and Income?

Interaction between Gender and Income?

Interaction between Gender and Income?