Nonlinear Regression Functions
1.What are the implications of a population regression function that is
linear?-
: The effect on Y of a one unit change in X does not depend on the value
of X
2.What are the implications of a population regression function that is non-
linear?: If the effect on Y of a unit change in X does depend on the
value of X.
3.What are two ways to model nonlinearities?: 1. Allow the average effect
on Y of a unit change in X1 to depend on the value of X1 (polynomials,
logarithms).
2. Allow the average effect on Y of a unit change in X1 to depend on the
value of another independent variable X2 (dummy variables, interaction
terms).
4.If we have a dummy variable, say Gender on Wage, how can we test if
wages are significantly different for different genders?: Test if the
coefficient on gender is equal to 0
5.Wage = B0 + B1Gender + B2Yrseduc + u: The regression function for
males is B0 + B2Yrseduc + u
The regression function for females is (B0 + B1) + B2 Yrseduc + u
Graphically, this means they have the same slope but different
intercepts
6.Interaction Term: Gender * Yrseduc
The new regression is B0 + B1Gender + B2Yrseduc + B3Gender *
Yrseduc + u The new regression for males is B0 + B2Yrseduc + u
The new regression for females is (B0 + B1) + (B2 + B3) Yrseduc +
u Graphically, they may have both different intercepts and different
slopes
7.How do you test for equal intercepts with dummies and interactions?:
Test if B1 is equal to 0
8.How do you test for equal slopes with dummies and interactions: Test if
B3 is equal to 0
9.How do you test for equal slopes and intercepts with dummies and
interac- tions?: Test if B1 = 0 and B3 = 0, using an F-test
10.Dummy Variables v Categorical Variables: We should use dummies as
op- posed to categorical variables because if we use a single variable,
we have to have some reason to believe that the west earns 2X as
much as the midwest, etc. We force the vertical differences to the line
to be the same
11.Interactions between two continuous variables: Ex: effect on expected
1/
9
, Nonlinear Regression Functions
earn- ings of an increase in years of education depends on years of
experience. Add an interaction term Yrseduc * Exper, B3 is the effect of
a unit increase in education and experience, above and beyond the sum
of the individual effects of a unit increase in each of them alone.
12.What does the linearity of OLS refer to?: Linearity of the parameters,
not the variables
13.Quadratic Regression: B0 + B1X1 + B2X1^2 + ui
2/
9
1.What are the implications of a population regression function that is
linear?-
: The effect on Y of a one unit change in X does not depend on the value
of X
2.What are the implications of a population regression function that is non-
linear?: If the effect on Y of a unit change in X does depend on the
value of X.
3.What are two ways to model nonlinearities?: 1. Allow the average effect
on Y of a unit change in X1 to depend on the value of X1 (polynomials,
logarithms).
2. Allow the average effect on Y of a unit change in X1 to depend on the
value of another independent variable X2 (dummy variables, interaction
terms).
4.If we have a dummy variable, say Gender on Wage, how can we test if
wages are significantly different for different genders?: Test if the
coefficient on gender is equal to 0
5.Wage = B0 + B1Gender + B2Yrseduc + u: The regression function for
males is B0 + B2Yrseduc + u
The regression function for females is (B0 + B1) + B2 Yrseduc + u
Graphically, this means they have the same slope but different
intercepts
6.Interaction Term: Gender * Yrseduc
The new regression is B0 + B1Gender + B2Yrseduc + B3Gender *
Yrseduc + u The new regression for males is B0 + B2Yrseduc + u
The new regression for females is (B0 + B1) + (B2 + B3) Yrseduc +
u Graphically, they may have both different intercepts and different
slopes
7.How do you test for equal intercepts with dummies and interactions?:
Test if B1 is equal to 0
8.How do you test for equal slopes with dummies and interactions: Test if
B3 is equal to 0
9.How do you test for equal slopes and intercepts with dummies and
interac- tions?: Test if B1 = 0 and B3 = 0, using an F-test
10.Dummy Variables v Categorical Variables: We should use dummies as
op- posed to categorical variables because if we use a single variable,
we have to have some reason to believe that the west earns 2X as
much as the midwest, etc. We force the vertical differences to the line
to be the same
11.Interactions between two continuous variables: Ex: effect on expected
1/
9
, Nonlinear Regression Functions
earn- ings of an increase in years of education depends on years of
experience. Add an interaction term Yrseduc * Exper, B3 is the effect of
a unit increase in education and experience, above and beyond the sum
of the individual effects of a unit increase in each of them alone.
12.What does the linearity of OLS refer to?: Linearity of the parameters,
not the variables
13.Quadratic Regression: B0 + B1X1 + B2X1^2 + ui
2/
9
