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3.7 — Interaction Effects

ECON 480 • Econometrics • Fall 2021

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/metricsF21
metricsF21.classes.ryansafner.com

Outline

Interactions Between a Dummy and Continuous Variable

Interactions Between Two Dummy Variables

Interactions Between Two Continuous Variables

Sliders and Switches

Sliders and Switches

Sliders and Switches

  • Marginal effect of dummy variable: effect on Y of going from 0 to 1

Sliders and Switches

  • Marginal effect of dummy variable: effect on Y of going from 0 to 1

  • Marginal effect of continuous variable: effect on Y of a 1 unit change in X

Interaction Effects

  • Sometimes one X variable might interact with another in determining Y

Interaction Effects

  • Sometimes one X variable might interact with another in determining Y

Example: Consider the gender pay gap again.

Interaction Effects

  • Sometimes one X variable might interact with another in determining Y

Example: Consider the gender pay gap again.

  • Gender affects wages
  • Experience affects wages

Interaction Effects

  • Sometimes one X variable might interact with another in determining Y

Example: Consider the gender pay gap again.

  • Gender affects wages
  • Experience affects wages
  • Does experience affect wages differently by gender?
    • i.e. is there an interaction effect between gender and experience?

Interaction Effects

  • Sometimes one X variable might interact with another in determining Y

Example: Consider the gender pay gap again.

  • Gender affects wages
  • Experience affects wages
  • Does experience affect wages differently by gender?
    • i.e. is there an interaction effect between gender and experience?
  • Note this is NOT the same as just asking: “do men earn more than women with the same amount of experience?”

^wagesi=β0+β1Genderi+β2Experiencei

Three Types of Interactions

  • Depending on the types of variables, there are 3 possible types of interaction effects

  • We will look at each in turn

Three Types of Interactions

  • Depending on the types of variables, there are 3 possible types of interaction effects

  • We will look at each in turn

  1. Interaction between a dummy and a continuous variable: Yi=β0+β1Xi+β2Di+β3(Xi×Di)

Three Types of Interactions

  • Depending on the types of variables, there are 3 possible types of interaction effects

  • We will look at each in turn

  1. Interaction between a dummy and a continuous variable: Yi=β0+β1Xi+β2Di+β3(Xi×Di)
  2. Interaction between a two dummy variables: Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

Three Types of Interactions

  • Depending on the types of variables, there are 3 possible types of interaction effects

  • We will look at each in turn

  1. Interaction between a dummy and a continuous variable: Yi=β0+β1Xi+β2Di+β3(Xi×Di)
  2. Interaction between a two dummy variables: Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)
  3. Interaction between a two continuous variables: Yi=β0+β1X1i+β2X2i+β3(X1i×X2i)

Interactions Between a Dummy and Continuous Variable

Interactions: A Dummy & Continuous Variable

Interactions: A Dummy & Continuous Variable

  • Does the marginal effect of the continuous variable on Y change depending on whether the dummy is “on” or “off”?

Interactions: A Dummy & Continuous Variable I

  • We can model an interaction by introducing a variable that is an interaction term capturing the interaction between two variables:

Yi=β0+β1Xi+β2Di+β3(Xi×Di) where Di={0,1}

Interactions: A Dummy & Continuous Variable I

  • We can model an interaction by introducing a variable that is an interaction term capturing the interaction between two variables:

Yi=β0+β1Xi+β2Di+β3(Xi×Di) where Di={0,1}

  • β3 estimates the interaction effect between Xi and Di on Yi

Interactions: A Dummy & Continuous Variable I

  • We can model an interaction by introducing a variable that is an interaction term capturing the interaction between two variables:

Yi=β0+β1Xi+β2Di+β3(Xi×Di) where Di={0,1}

  • β3 estimates the interaction effect between Xi and Di on Yi

  • What do the different coefficients (β)’s tell us?

    • Again, think logically by examining each group (Di=0 or Di=1)

Interaction Effects as Two Regressions I

Yi=β0+β1Xi+β2Di+β3Xi×Di

Interaction Effects as Two Regressions I

Yi=β0+β1Xi+β2Di+β3Xi×Di

  • When Di=0 (Control group):

^Yi=^β0+^β1Xi+^β2(0)+^β3Xi×(0)^Yi=^β0+^β1Xi

Interaction Effects as Two Regressions I

Yi=β0+β1Xi+β2Di+β3Xi×Di

  • When Di=0 (Control group):

^Yi=^β0+^β1Xi+^β2(0)+^β3Xi×(0)^Yi=^β0+^β1Xi

  • When Di=1 (Treatment group):

^Yi=^β0+^β1Xi+^β2(1)+^β3Xi×(1)^Yi=(^β0+^β2)+(^β1+^β3)Xi

Interaction Effects as Two Regressions I

Yi=β0+β1Xi+β2Di+β3Xi×Di

  • When Di=0 (Control group):

^Yi=^β0+^β1Xi+^β2(0)+^β3Xi×(0)^Yi=^β0+^β1Xi

  • When Di=1 (Treatment group):

^Yi=^β0+^β1Xi+^β2(1)+^β3Xi×(1)^Yi=(^β0+^β2)+(^β1+^β3)Xi

  • So what we really have is two regression lines!

Interaction Effects as Two Regressions II

  • Di=0 group:

Yi=^β0+^β1Xi

  • Di=1 group:

Yi=(^β0+^β2)+(^β1+^β3)Xi

Interpretting Coefficients I

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • To interpret the coefficients, compare cases after changing X by ΔX:

Interpretting Coefficients I

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • To interpret the coefficients, compare cases after changing X by ΔX:

Yi+ΔYi=β0+β1(Xi+ΔXi)β2Di+β3((Xi+ΔXi)Di)

Interpretting Coefficients I

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • To interpret the coefficients, compare cases after changing X by ΔX:

Yi+ΔYi=β0+β1(Xi+ΔXi)β2Di+β3((Xi+ΔXi)Di)

  • Subtracting these two equations, the difference is:

Interpretting Coefficients I

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • To interpret the coefficients, compare cases after changing X by ΔX:

Yi+ΔYi=β0+β1(Xi+ΔXi)β2Di+β3((Xi+ΔXi)Di)

  • Subtracting these two equations, the difference is:

ΔYi=β1ΔXi+β3DiΔXiΔYiΔXi=β1+β3Di

Interpretting Coefficients I

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • To interpret the coefficients, compare cases after changing X by ΔX:

Yi+ΔYi=β0+β1(Xi+ΔXi)β2Di+β3((Xi+ΔXi)Di)

  • Subtracting these two equations, the difference is:

ΔYi=β1ΔXi+β3DiΔXiΔYiΔXi=β1+β3Di

  • The effect of X→Y depends on the value of Di!

  • β3: increment to the effect of X→Y when Di=1 (vs. Di=0)

Interpretting Coefficients II

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • ^β0: E[Yi] for Xi=0 and Di=0

Interpretting Coefficients II

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • ^β0: E[Yi] for Xi=0 and Di=0

  • β1: Marginal effect of Xi→Yi for Di=0

Interpretting Coefficients II

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • ^β0: E[Yi] for Xi=0 and Di=0

  • β1: Marginal effect of Xi→Yi for Di=0

  • β2: Marginal effect on Yi of difference between Di=0 and Di=1

Interpretting Coefficients II

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • ^β0: E[Yi] for Xi=0 and Di=0

  • β1: Marginal effect of Xi→Yi for Di=0

  • β2: Marginal effect on Yi of difference between Di=0 and Di=1

  • β3: The difference of the marginal effect of Xi→Yi between Di=0 and Di=1

Interpretting Coefficients II

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

  • ^β0: E[Yi] for Xi=0 and Di=0

  • β1: Marginal effect of Xi→Yi for Di=0

  • β2: Marginal effect on Yi of difference between Di=0 and Di=1

  • β3: The difference of the marginal effect of Xi→Yi between Di=0 and Di=1

  • This is a bit awkward, easier to think about the two regression lines:

Interpretting Coefficients III

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

Interpretting Coefficients III

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

For Di=0 Group: ^Yi=^β0+^β1Xi

  • Intercept: ^β0
  • Slope: ^β1

Interpretting Coefficients III

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

For Di=0 Group: ^Yi=^β0+^β1Xi

  • Intercept: ^β0
  • Slope: ^β1

For Di=1 Group: ^Yi=(^β0+^β2)+(^β1+^β3)Xi

  • Intercept: ^β0+^β2
  • Slope: ^β1+^β3

Interpretting Coefficients III

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

For Di=0 Group: ^Yi=^β0+^β1Xi

  • Intercept: ^β0
  • Slope: ^β1

For Di=1 Group: ^Yi=(^β0+^β2)+(^β1+^β3)Xi

  • Intercept: ^β0+^β2
  • Slope: ^β1+^β3

  • ^β2: difference in intercept between groups

  • ^β3: difference in slope between groups

Interpretting Coefficients III

Yi=β0+β1Xi+β2Di+β3(Xi×Di)

For Di=0 Group: ^Yi=^β0+^β1Xi

  • Intercept: ^β0
  • Slope: ^β1

For Di=1 Group: ^Yi=(^β0+^β2)+(^β1+^β3)Xi

  • Intercept: ^β0+^β2
  • Slope: ^β1+^β3

  • ^β2: difference in intercept between groups

  • ^β3: difference in slope between groups

  • How can we determine if the two lines have the same slope and/or intercept?
    • Same intercept? t-test H0: β2=0
    • Same slope? t-test H0: β3=0

Example I

Example: ^wagei=^β0+^β1experi+^β2femalei+^β3(experi×femalei)

Example I

Example: ^wagei=^β0+^β1experi+^β2femalei+^β3(experi×femalei)

  • For males (female=0): ^wagei=^β0+^β1exper

Example I

Example: ^wagei=^β0+^β1experi+^β2femalei+^β3(experi×femalei)

  • For males (female=0): ^wagei=^β0+^β1exper

  • For females (female=1): ^wagei=(^β0+^β2)⏟intercept+(^β1+^β3)⏟slopeexper

Example II

interaction_plot <- ggplot(data = wages)+
  aes(x = exper,
      y = wage,
      color = as.factor(Gender))+ # make factor
  geom_point(alpha = 0.5)+
  scale_y_continuous(labels=scales::dollar)+
  labs(x = "Experience (Years)",
       y = "Wage")+
  scale_color_manual(values = c("Female" = "#e64173",
                                "Male" = "#0047AB")
                     )+ # setting custom colors
  guides(color=F)+ # hide legend
  theme_slides
interaction_plot
  • Need to make sure color aesthetic uses a factor variable
    • Can just use as.factor() in ggplot code

Example II

interaction_plot+
  geom_smooth(method="lm")

Example II

interaction_plot+
  geom_smooth(method="lm")+
  facet_wrap(~Gender)

Example Regression in R I

  • Syntax for adding an interaction term is easy in R: var1 * var2
    • Or could just do var1 * var2 (multiply)
# both are identical in R
interaction_reg <- lm(wage ~ exper * female, data = wages)
interaction_reg <- lm(wage ~ exper + female + exper * female, data = wages)
ABCDEFGHIJ0123456789
term
<chr>
estimate
<dbl>
std.error
<dbl>
statistic
<dbl>
p.value
<dbl>
(Intercept)6.158275490.3416740818.0238307.998534e-57
exper0.053604760.015437163.4724505.585255e-04
female-1.546546770.48186030-3.2095341.411253e-03
exper:female-0.055069890.02217496-2.4834271.332533e-02
4 rows

Example Regression in R III

library(huxtable)
huxreg(interaction_reg,
       coefs = c("Constant" = "(Intercept)",
                 "Experience" = "exper",
                 "Female" = "female",
                 "Experience * Female" = "exper:female"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 2)
(1)
Constant6.16 ***
(0.34)   
Experience0.05 ***
(0.02)   
Female-1.55 ** 
(0.48)   
Experience * Female-0.06 *  
(0.02)   
N526       
R-Squared0.14    
SER3.44    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0:

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0: Men with 0 years of experience earn 6.16

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0: Men with 0 years of experience earn 6.16

  • ^β1:

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0: Men with 0 years of experience earn 6.16

  • ^β1: For every additional year of experience, men earn $0.05

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0: Men with 0 years of experience earn 6.16

  • ^β1: For every additional year of experience, men earn $0.05

  • ^β2:

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0: Men with 0 years of experience earn 6.16

  • ^β1: For every additional year of experience, men earn $0.05

  • ^β2: Women with 0 years of experience earn $1.55 less than men

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0: Men with 0 years of experience earn 6.16

  • ^β1: For every additional year of experience, men earn $0.05

  • ^β2: Women with 0 years of experience earn $1.55 less than men

  • ^β3:

Example Regression in R: Interpretting Coefficients

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

  • ^β0: Men with 0 years of experience earn 6.16

  • ^β1: For every additional year of experience, men earn $0.05

  • ^β2: Women with 0 years of experience earn $1.55 less than men

  • ^β3: Women earn $0.06 less than men for every additional year of experience

Interpretting Coefficients as 2 Regressions I

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Interpretting Coefficients as 2 Regressions I

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Regression for men (female=0) ^wagei=6.16+0.05Experiencei

Interpretting Coefficients as 2 Regressions I

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Regression for men (female=0) ^wagei=6.16+0.05Experiencei

  • Men with 0 years of experience earn $6.16 on average

Interpretting Coefficients as 2 Regressions I

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Regression for men (female=0) ^wagei=6.16+0.05Experiencei

  • Men with 0 years of experience earn $6.16 on average

  • For every additional year of experience, men earn $0.05 more on average

Interpretting Coefficients as 2 Regressions II

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Regression for women (female=1) ^wagei=6.16+0.05Experiencei−1.55(1)−0.06Experiencei×(1)=(6.16−1.55)+(0.05−0.06)Experiencei=4.61−0.01Experiencei

Interpretting Coefficients as 2 Regressions II

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Regression for women (female=1) ^wagei=6.16+0.05Experiencei−1.55(1)−0.06Experiencei×(1)=(6.16−1.55)+(0.05−0.06)Experiencei=4.61−0.01Experiencei

  • Women with 0 years of experience earn $4.61 on average

Interpretting Coefficients as 2 Regressions II

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

Regression for women (female=1) ^wagei=6.16+0.05Experiencei−1.55(1)−0.06Experiencei×(1)=(6.16−1.55)+(0.05−0.06)Experiencei=4.61−0.01Experiencei

  • Women with 0 years of experience earn $4.61 on average

  • For every additional year of experience, women earn $0.01 less on average

Example Regression in R: Hypothesis Testing

  • Are slopes & intercepts of the 2 regressions statistically significantly different?

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

## # A tibble: 4 × 5
##   term         estimate std.error statistic  p.value
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    6.16      0.342      18.0  8.00e-57
## 2 exper          0.0536    0.0154      3.47 5.59e- 4
## 3 female        -1.55      0.482      -3.21 1.41e- 3
## 4 exper:female  -0.0551    0.0222     -2.48 1.33e- 2

Example Regression in R: Hypothesis Testing

  • Are slopes & intercepts of the 2 regressions statistically significantly different?

  • Are intercepts different? H0:β2=0

    • Difference between men vs. women for no experience?
    • Is ^β2 significant?
    • Yes (reject) H0: p-value = 0.00

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

## # A tibble: 4 × 5
##   term         estimate std.error statistic  p.value
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    6.16      0.342      18.0  8.00e-57
## 2 exper          0.0536    0.0154      3.47 5.59e- 4
## 3 female        -1.55      0.482      -3.21 1.41e- 3
## 4 exper:female  -0.0551    0.0222     -2.48 1.33e- 2

Example Regression in R: Hypothesis Testing

  • Are slopes & intercepts of the 2 regressions statistically significantly different?

  • Are intercepts different? H0:β2=0

    • Difference between men vs. women for no experience?
    • Is ^β2 significant?
    • Yes (reject) H0: p-value = 0.00

  • Are slopes different? H0:β3=0

    • Difference between men vs. women for marginal effect of experience?
    • Is ^β3 significant?
    • Yes (reject) H0: p-value = 0.01

^wagei=6.16+0.05Experiencei−1.55Femalei−0.06(Experiencei×Femalei)

## # A tibble: 4 × 5
##   term         estimate std.error statistic  p.value
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    6.16      0.342      18.0  8.00e-57
## 2 exper          0.0536    0.0154      3.47 5.59e- 4
## 3 female        -1.55      0.482      -3.21 1.41e- 3
## 4 exper:female  -0.0551    0.0222     -2.48 1.33e- 2

Interactions Between Two Dummy Variables

Interactions Between Two Dummy Variables

Interactions Between Two Dummy Variables

  • Does the marginal effect on Y of one dummy going from “off” to “on” change depending on whether the other dummy is “off” or “on”?

Interactions Between Two Dummy Variables

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

  • D1i and D2i are dummy variables

Interactions Between Two Dummy Variables

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

  • D1i and D2i are dummy variables

  • ^β1: effect on Y of going from D1i=0 to D1i=1 when D2i=0

Interactions Between Two Dummy Variables

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

  • D1i and D2i are dummy variables

  • ^β1: effect on Y of going from D1i=0 to D1i=1 when D2i=0

  • ^β2: effect on Y of going from D2i=0 to D2i=1 when D1i=0

Interactions Between Two Dummy Variables

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

  • D1i and D2i are dummy variables

  • ^β1: effect on Y of going from D1i=0 to D1i=1 when D2i=0

  • ^β2: effect on Y of going from D2i=0 to D2i=1 when D1i=0

  • ^β3: effect on Y of going from D1i=0 to D1i=1 when D2i=1

    • increment to the effect of D1i going from 0 to 1 when D2i=1 (vs. 0)

Interactions Between Two Dummy Variables

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

  • D1i and D2i are dummy variables

  • ^β1: effect on Y of going from D1i=0 to D1i=1 when D2i=0

  • ^β2: effect on Y of going from D2i=0 to D2i=1 when D1i=0

  • ^β3: effect on Y of going from D1i=0 to D1i=1 when D2i=1

    • increment to the effect of D1i going from 0 to 1 when D2i=1 (vs. 0)

  • As always, best to think logically about possibilities (when each dummy =0 or =1)

2 Dummy Interaction: Interpretting Coefficients

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

2 Dummy Interaction: Interpretting Coefficients

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

  • To interpret coefficients, compare cases:
    • Hold D2i constant (set to some value D2i=d2)
    • Plug in 0s or 1s for D1i

E(Yi|D1i=0,D2i=d2)=β0+β2d2E(Yi|D1i=1,D2i=d2)=β0+β1(1)+β2d2+β3(1)d2

2 Dummy Interaction: Interpretting Coefficients

Yi=β0+β1D1i+β2D2i+β3(D1i×D2i)

  • To interpret coefficients, compare cases:
    • Hold D2i constant (set to some value D2i=d2)
    • Plug in 0s or 1s for D1i

E(Yi|D1i=0,D2i=d2)=β0+β2d2E(Yi|D1i=1,D2i=d2)=β0+β1(1)+β2d2+β3(1)d2

  • Subtracting the two, the difference is:

β1+β3d2

  • The marginal effect of D1i→Yi depends on the value of D2i
    • ^β3 is the increment to the effect of D1 on Y when D2 goes from 0 to 1

Interactions Between 2 Dummy Variables: Example

Example: Does the gender pay gap change if a person is married vs. single?

^wagei=^β0+^β1femalei+^β2marriedi+^β3(femalei×marriedi)

Interactions Between 2 Dummy Variables: Example

Example: Does the gender pay gap change if a person is married vs. single?

^wagei=^β0+^β1femalei+^β2marriedi+^β3(femalei×marriedi)

  • Logically, there are 4 possible combinations of femalei={0,1} and marriedi={0,1}

Interactions Between 2 Dummy Variables: Example

Example: Does the gender pay gap change if a person is married vs. single?

^wagei=^β0+^β1femalei+^β2marriedi+^β3(femalei×marriedi)

  • Logically, there are 4 possible combinations of femalei={0,1} and marriedi={0,1}

1) Unmarried men (femalei=0,marriedi=0) ^wagei=^β0

Interactions Between 2 Dummy Variables: Example

Example: Does the gender pay gap change if a person is married vs. single?

^wagei=^β0+^β1femalei+^β2marriedi+^β3(femalei×marriedi)

  • Logically, there are 4 possible combinations of femalei={0,1} and marriedi={0,1}

1) Unmarried men (femalei=0,marriedi=0) ^wagei=^β0

2) Married men (femalei=0,marriedi=1) ^wagei=^β0+^β2

Interactions Between 2 Dummy Variables: Example

Example: Does the gender pay gap change if a person is married vs. single?

^wagei=^β0+^β1femalei+^β2marriedi+^β3(femalei×marriedi)

  • Logically, there are 4 possible combinations of femalei={0,1} and marriedi={0,1}

1) Unmarried men (femalei=0,marriedi=0) ^wagei=^β0

2) Married men (femalei=0,marriedi=1) ^wagei=^β0+^β2

3) Unmarried women (femalei=1,marriedi=0) ^wagei=^β0+^β1

Interactions Between 2 Dummy Variables: Example

Example: Does the gender pay gap change if a person is married vs. single?

^wagei=^β0+^β1femalei+^β2marriedi+^β3(femalei×marriedi)

  • Logically, there are 4 possible combinations of femalei={0,1} and marriedi={0,1}

1) Unmarried men (femalei=0,marriedi=0) ^wagei=^β0

2) Married men (femalei=0,marriedi=1) ^wagei=^β0+^β2

3) Unmarried women (femalei=1,marriedi=0) ^wagei=^β0+^β1

4) Married women (femalei=1,marriedi=1) ^wagei=^β0+^β1+^β2+^β3

Looking at the Data

# get average wage for unmarried men
wages %>%
  filter(female == 0,
         married == 0) %>%
  summarize(mean = mean(wage))
##       mean
## 1 5.168023
# get average wage for married men
wages %>%
  filter(female == 0,
         married == 1) %>%
  summarize(mean = mean(wage))
##       mean
## 1 7.983032
# get average wage for unmarried women
wages %>%
  filter(female == 1,
         married == 0) %>%
  summarize(mean = mean(wage))
##       mean
## 1 4.611583
# get average wage for married women
wages %>%
  filter(female == 1,
         married == 1) %>%
  summarize(mean = mean(wage))
##       mean
## 1 4.565909

Two Dummies Interaction: Group Means

^wagei=^β0+^β1femalei+^β2marriedi+^β3(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57

Interactions Between Two Dummy Variables: In R I

reg_dummies <- lm(wage ~ female + married + female:married, data = wages)
reg_dummies %>% tidy()
## # A tibble: 4 × 5
##   term           estimate std.error statistic  p.value
##   <chr>             <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)       5.17      0.361     14.3  2.26e-39
## 2 female           -0.556     0.474     -1.18 2.41e- 1
## 3 married           2.82      0.436      6.45 2.53e-10
## 4 female:married   -2.86      0.608     -4.71 3.20e- 6

Interactions Between Two Dummy Variables: In R II

library(huxtable)
huxreg(reg_dummies,
       coefs = c("Constant" = "(Intercept)",
                 "Female" = "female",
                 "Married" = "married",
                 "Female * Married" = "female:married"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 2)
(1)
Constant5.17 ***
(0.36)   
Female-0.56    
(0.47)   
Married2.82 ***
(0.44)   
Female * Married-2.86 ***
(0.61)   
N526       
R-Squared0.18    
SER3.35    
*** p < 0.001; ** p < 0.01; * p < 0.05.

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • Wage for unmarried men: ^β0=5.17

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • Wage for unmarried men: ^β0=5.17
  • Wage for married men: ^β0+^β2=5.17+2.82=7.98

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • Wage for unmarried men: ^β0=5.17
  • Wage for married men: ^β0+^β2=5.17+2.82=7.98
  • Wage for unmarried women: ^β0+^β1=5.17−0.56=4.61

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • Wage for unmarried men: ^β0=5.17
  • Wage for married men: ^β0+^β2=5.17+2.82=7.98
  • Wage for unmarried women: ^β0+^β1=5.17−0.56=4.61
  • Wage for married women: ^β0+^β1+^β2+^β3=5.17−0.56+2.82−2.86=4.57

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • ^β0: Wage for unmarried men

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • ^β0: Wage for unmarried men
  • ^β1: Difference in wages between men and women who are unmarried

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • ^β0: Wage for unmarried men
  • ^β1: Difference in wages between men and women who are unmarried
  • ^β2: Difference in wages between married and unmarried men

2 Dummies Interaction: Interpretting Coefficients

^wagei=5.17−0.56femalei+2.82marriedi−2.86(femalei×marriedi)

Men Women
Unmarried $5.17 $4.61
Married $7.98 $4.57
  • ^β0: Wage for unmarried men
  • ^β1: Difference in wages between men and women who are unmarried
  • ^β2: Difference in wages between married and unmarried men
  • ^β3: Difference in:
    • effect of Marriage on wages between men and women
    • effect of Gender on wages between unmarried and married individuals

Interactions Between Two Continuous Variables

Interactions Between Two Continuous Variables

Interactions Between Two Continuous Variables

  • Does the marginal effect of X1 on Y depend on what X2 is set to?

Interactions Between Two Continuous Variables

Yi=β0+β1X1i+β2X2i+β3(X1i×X2i)

Interactions Between Two Continuous Variables

Yi=β0+β1X1i+β2X2i+β3(X1i×X2i)

  • To interpret coefficients, compare changes after changing ΔX1i (holding X2 constant):

Yi+ΔYi=β0+β1(X1+ΔX1i)β2X2i+β3((X1i+ΔX1i)×X2i)

Interactions Between Two Continuous Variables

Yi=β0+β1X1i+β2X2i+β3(X1i×X2i)

  • To interpret coefficients, compare changes after changing ΔX1i (holding X2 constant):

Yi+ΔYi=β0+β1(X1+ΔX1i)β2X2i+β3((X1i+ΔX1i)×X2i)

  • Take the difference to get:

Interactions Between Two Continuous Variables

Yi=β0+β1X1i+β2X2i+β3(X1i×X2i)

  • To interpret coefficients, compare changes after changing ΔX1i (holding X2 constant):

Yi+ΔYi=β0+β1(X1+ΔX1i)β2X2i+β3((X1i+ΔX1i)×X2i)

  • Take the difference to get:

ΔYi=β1ΔX1i+β3X2iΔX1iΔYiΔX1i=β1+β3X2i

Interactions Between Two Continuous Variables

Yi=β0+β1X1i+β2X2i+β3(X1i×X2i)

  • To interpret coefficients, compare changes after changing ΔX1i (holding X2 constant):

Yi+ΔYi=β0+β1(X1+ΔX1i)β2X2i+β3((X1i+ΔX1i)×X2i)

  • Take the difference to get:

ΔYi=β1ΔX1i+β3X2iΔX1iΔYiΔX1i=β1+β3X2i

  • The effect of X1→Yi depends on X2
    • β3: increment to the effect of X1→Yi for every 1 unit change in X2

Continuous Variables Interaction: Example

Example: Do education and experience interact in their determination of wages?

^wagei=^β0+^β1educi+^β2experi+^β3(educi×experi)

  • Estimated effect of education on wages depends on the amount of experience (and vice versa)!

ΔwageΔeduc=^β1+β3experi

ΔwageΔexper=^β2+β3educi

  • This is a type of nonlinearity (we will examine nonlinearities next lesson)

Continuous Variables Interaction: In R I

reg_cont <- lm(wage ~ educ + exper + educ:exper, data = wages)
reg_cont %>% tidy()
## # A tibble: 4 × 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept) -2.86      1.18       -2.42  1.58e- 2
## 2 educ         0.602     0.0899      6.69  5.64e-11
## 3 exper        0.0458    0.0426      1.07  2.83e- 1
## 4 educ:exper   0.00206   0.00349     0.591 5.55e- 1

Continuous Variables Interaction: In R II

library(huxtable)
huxreg(reg_cont,
       coefs = c("Constant" = "(Intercept)",
                 "Education" = "educ",
                 "Experience" = "exper",
                 "Education * Experience" = "educ:exper"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 3)
(1)
Constant-2.860 *  
(1.181)   
Education0.602 ***
(0.090)   
Experience0.046    
(0.043)   
Education * Experience0.002    
(0.003)   
N526        
R-Squared0.226    
SER3.259    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Continuous Variables Interaction: Marginal Effects

^wagesi=−2.860+0.602educi+0.047experi+0.002(educi×experi)

Continuous Variables Interaction: Marginal Effects

^wagesi=−2.860+0.602educi+0.047experi+0.002(educi×experi)

Marginal Effect of Education on Wages by Years of Experience:

Experience ΔwageΔeduc=^β1+^β3exper
5 years 0.602+0.002(5)=0.612
10 years 0.602+0.002(10)=0.622
15 years 0.602+0.002(15)=0.632

Continuous Variables Interaction: Marginal Effects

^wagesi=−2.860+0.602educi+0.047experi+0.002(educi×experi)

Marginal Effect of Education on Wages by Years of Experience:

Experience ΔwageΔeduc=^β1+^β3exper
5 years 0.602+0.002(5)=0.612
10 years 0.602+0.002(10)=0.622
15 years 0.602+0.002(15)=0.632
  • Marginal effect of education → wages increases with more experience

Continuous Variables Interaction: Marginal Effects

^wagesi=−2.860+0.602educi+0.047experi+0.002(educi×experi)

Continuous Variables Interaction: Marginal Effects

^wagesi=−2.860+0.602educi+0.047experi+0.002(educi×experi)

Marginal Effect of Experience on Wages by Years of Education:

Education ΔwageΔexper=^β2+^β3educ
5 years 0.047+0.002(5)=0.057
10 years 0.047+0.002(10)=0.067
15 years 0.047+0.002(15)=0.077

Continuous Variables Interaction: Marginal Effects

^wagesi=−2.860+0.602educi+0.047experi+0.002(educi×experi)

Marginal Effect of Experience on Wages by Years of Education:

Education ΔwageΔexper=^β2+^β3educ
5 years 0.047+0.002(5)=0.057
10 years 0.047+0.002(10)=0.067
15 years 0.047+0.002(15)=0.077
  • Marginal effect of experience → wages increases with more education

Continuous Variables Interaction: Marginal Effects

^wagesi=−2.860+0.602educi+0.047experi+0.002(educi×experi)

Marginal Effect of Experience on Wages by Years of Education:

Education ΔwageΔexper=^β2+^β3educ
5 years 0.047+0.002(5)=0.057
10 years 0.047+0.002(10)=0.067
15 years 0.047+0.002(15)=0.077

  • Marginal effect of experience → wages increases with more education

  • If you want to estimate the marginal effects more precisely, and graph them, see the appendix in today’s class page

Outline

Interactions Between a Dummy and Continuous Variable

Interactions Between Two Dummy Variables

Interactions Between Two Continuous Variables

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