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

Attn: this is a simplified version, your model in the online exercise is different!

 

Variables:
wage: Gross hourly wage of employees
educ: years of education (potential)
exper: years of professional experience
female: 1 if female and 0 otherwise
married:  1 if married and 0 if single, divorced or widowed

Read data:

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

w <- read.csv("https://econometrics.hsto.info/data/kap02_mincer23.csv")
eq <- lm(log(wage) ~ educ + I(educ * female) + exper + married + female + I(married * female), data = w)
# or equivalent and shorter
# eq1 <- lm(log(wage) ~ educ * female + exper + married * female, data = w)
b <- coef(eq)

Dependent variable:
log(wage)
Constant 2.364***
(0.031)
educ 0.033***
(0.002)
I(educ * female) -0.004
(0.003)
exper 0.008***
(0.001)
married 0.113***
(0.017)
female -0.030
(0.039)
I(married * female) -0.119***
(0.023)
Observations 4,532
Adjusted R2 0.151
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:
\(= b_5 =\) 0.113, i.e. married men earn c.p. on average \((\exp(b_5) - 1)*100\% =\) 11.91% more than single man.

Difference between single and married women:
\(= b_5 + b_7 =\) -0.007, i.e. married women earn on average c.p. \((\exp(b_5 + b_7) - 1)*100\% =\) -0.65%, i.e., in this example married woman earn marginally LESS than single woman!

Difference between single men and single women:
\(= b_3 \text{educ} + b_6=\) -0.004educ + -0.03.
Attn.: Note that \(b_6\) (coefficient of female) is not statistically significant 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 -0.43% less per year of education than that of men (= 3.29%), i.e. average women wages increase c.p. by approx. 2.87% per year of education.

Difference between married men and married women:
\(= b_3 \text{educ} + b_6 + b_7=\) -0.004educ + -0.15, i.e., married women earn c.p. on average (exp(-0.03 + -0.12) - 1)*100 = -13.89% 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!)

Notice that the coefficient of the female-dummy is no longer statistically different from zero once the interaction effect between female and married is considered! What could be possible reasons for this?

 

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

What are your conclusions from the following estimation?

eq1 <- lm(log(wage) ~ educ * female * married + exper, data = w)
se_OLS = sqrt(diag(vcov(eq1)))
se_HC3 = sqrt(diag(vcovHC(eq1, type = "HC3")))
stargazer(eq1, eq1, 
          se=list(se_OLS, se_HC3),
          column.labels=c("se(OLS)", "se(HC3)"),
          model.numbers=FALSE,
          intercept.bottom = FALSE, type = "html", 
          omit.stat=c("LL","ser","f", "rsq")
          )
Dependent variable:
log(wage)
se(OLS) se(HC3)
Constant 2.387*** 2.387***
(0.040) (0.045)
educ 0.031*** 0.031***
(0.003) (0.003)
female -0.023 -0.023
(0.053) (0.062)
married 0.068 0.068
(0.050) (0.053)
exper 0.008*** 0.008***
(0.001) (0.001)
educ:female -0.005 -0.005
(0.004) (0.004)
educ:married 0.003 0.003
(0.004) (0.004)
female:married -0.134* -0.134
(0.073) (0.086)
educ:female:married 0.001 0.001
(0.005) (0.007)
Observations 4,532 4,532
Adjusted R2 0.151 0.151
Note: p<0.1; p<0.05; p<0.01