Example to illustrate interaction effects in a Mincer wage equation: (employees)

Read data and create dummy variable for married:

# Mincer wage equation
rm(list = ls())

w <- read.csv2("https://www.hsto.info/econometrics2/dl/mincer.csv", stringsAsFactors = TRUE)

w$married <- rep(NA, nrow(w))
w$married[w$status_ == "Married living together"] <- 1 
w$married[w$status_ == "Single"] <- 0 

eq <- lm(log(wage) ~ educ + I(educ * female) + exper + married + female + I(married * female), data = w)
b <- coef(eq)

Dependent variable:
log(wage)
Constant 2.109***
(0.033)
educ 0.041***
(0.002)
I(educ * female) -0.011***
(0.003)
exper 0.008***
(0.001)
married 0.086***
(0.018)
female 0.039
(0.042)
I(married * female) -0.120***
(0.024)
Observations 4,147
Adjusted R2 0.168
Note: p<0.1; p<0.05; p<0.01

 

How would you describe the association between wages, gender and marital status?

 

The estimated equation is: \[y = b_1 + b_2 \text{educ} + b_3 (\text{educ * female}) + b_4 \text{exper} + b_5 \text{married} + b_6 \text{female} + b_7 (\text{married * female}) \]

For single men: (married = 0, female = 0) \[y = b_1 + b_2 \text{educ} + b_4 \text{exper} \]

For married men: (married = 1, female = 0) \[\begin{align*} y &= b_1 + b_2 \text{educ} + b_4 \text{exper} + b_5 \text{married} \\[1.5ex] &= b_1 + b_5 + b_2 \text{educ} + b_4 \text{exper} \end{align*}\]

For single women: (married = 0, female = 1) \[\begin{align*} y &= b_1 + b_2 \text{educ} + b_3 \text{educ * female} + b_4 \text{exper} + b_6 \text{female} \\[1.5ex] &= b_1 + b_6 + (b_2 + b_3)\text{educ} + b_4 \text{exper} \end{align*}\]

For married women: (married = 1, female = 1) \[\begin{align*} y &= b_1 + b_2 \text{educ} + b_3 \text{educ * female} + b_4 \text{exper} + b_5 \text{married} + b_6 \text{female} + b_7 \text{married * female} \\[1.5ex] &= b_1 + b_5 + b_6 + b_7 + (b_2 + b_3)\text{educ} + b_4 \text{exper} \end{align*}\]

Difference between single and married men:
\(= b_5 =\) 0.086, i.e. married men earn c.p. on average \((\exp(b_5) - 1)*100\% =\) 9% more than single man.

Difference between single and married women:
\(= b_5 + b_7 =\) -0.034, i.e. married women earn on average c.p. \((\exp(b_5 + b7) - 1)*100\% =\) -3.35% LESS than single woman!

Difference between single men and single women:
\(= b_3 \text{educ} + b_6=\) -0.011educ + 0.039.
Attn.: However, note that \(b_6\) (coefficient of female) is not statistically significantly different from zero at conventional significance levels.
However, the difference in the slope of educ is statistically highly significantly different from zero, the expected wage of women increases approx. by -1.14% less per year of education than that of men (= 4.12%), i.e. average women wages increase c.p. by approx. 2.97% per year of education.

Difference between married men and married women:
\(= b_3 \text{educ} + b_6 + b_7=\) -0.011educ + -0.081, i.e., married women earn c.p. on average -7.79% less than married men with equal education! (Again, note that \(b_6\) is not statistically significantly different from zero, and since \(b_6\) is surprisingly positive the difference might even be larger!)

 

We have seen, that women profit on average less from years of education than men. Is this related to marital status?

What do you conclude from the following estimate?

eq1 <- lm(log(wage) ~ educ * female * married + exper, data = w)
stargazer(eq1, intercept.bottom = FALSE, type = "html", omit.stat=c("LL","ser","f", "rsq"))
Dependent variable:
log(wage)
Constant 2.178***
(0.045)
educ 0.036***
(0.003)
female -0.150**
(0.065)
married -0.027
(0.055)
exper 0.008***
(0.001)
educ:female 0.003
(0.005)
educ:married 0.009**
(0.004)
female:married 0.174**
(0.082)
educ:female:married -0.023***
(0.006)
Observations 4,147
Adjusted R2 0.170
Note: p<0.1; p<0.05; p<0.01