Lab 5
Interaction terms
Download the CPS data set
Download the data from the website using:
cps <- read.csv("https://rtgodwin.com/data/cps1985.csv")
Estimate a model with an interaction term
model1 <- lm(log(wage) ~ education + gender + education*gender + experience + region
+ occupation + union, data = cps)
summary(model1)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.85222 0.20568 4.143 3.99e-05 ***
education 0.08417 0.01329 6.330 5.27e-10 ***
gendermale 0.53085 0.19880 2.670 0.007815 **
experience 0.01066 0.00166 6.420 3.07e-10 ***
regionsouth -0.10630 0.04159 -2.556 0.010876 *
occupationoffice -0.21713 0.07616 -2.851 0.004531 **
occupationsales -0.34956 0.09161 -3.816 0.000152 ***
occupationservices -0.40104 0.08073 -4.968 9.18e-07 ***
occupationtechnical -0.04780 0.07290 -0.656 0.512373
occupationworker -0.21475 0.07605 -2.824 0.004927 **
unionyes 0.20197 0.05099 3.960 8.52e-05 ***
education:gendermale -0.02449 0.01475 -1.661 0.097386 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4293 on 522 degrees of freedom
Multiple R-squared: 0.3519, Adjusted R-squared: 0.3382
F-statistic: 25.77 on 11 and 522 DF, p-value: < 2.2e-16
The interaction term allows for the effect of education on wage to differ between men and women. For example, the effect of an additional year of education is to increase wagges by 8.4% for women, and to increase wages by (0.08417 - 0.02449 = 0.05968) 6.0% for men.
To test to see if there is a difference in the effect of education between men and women, we need to test the significance of the interaction term (it’s what’s allowing for there to be a difference!). If there is no difference, then the effect of the interaction is 0. The null hypothesis is:
$H_0:$ effect of education on wage is the same for men and women
or equivalently
$H_0: \beta_{education \times gendermale} = 0$
R has already tested this hypothesis for us. The p-value is 0.097386. It is borderline; we fail to reject the null hypothesis at the 5% level, suggesting there are no differences in the effects of education on wage, between men and women.
Instrumental variables estimation
Download the Card (1993) wage data
college <- read.csv("https://rtgodwin.com/data/collegedist.csv")
The variables in the data are:
- wage - the dependent $y$ variable
- education - number of years of education of the worker
- distance - distance from the nearest college. This will be our instrument $z$.
- Other demographic variables used as controls.
IV estimation using ivreg()
To estimate the model:
\[wage = \beta_0 + \beta_1education + \beta_2urban + \beta_3gender + \beta_4ethnicity + \beta_5unemp + \epsilon\]using IV estimation, and where distance is an instrument for education, we can use:
install.packages("ivreg")
library(ivreg)
iv <- ivreg(wage ~ education + urban + gender + ethnicity + unemp |
distance + urban + gender + ethnicity + unemp, data = college)
summary(iv)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.65702 1.83641 -0.358 0.7205
education 0.64710 0.13594 4.760 1.99e-06 ***
urbanyes 0.04614 0.06039 0.764 0.4449
gendermale 0.07075 0.04997 1.416 0.1569
ethnicityhispanic -0.12405 0.08871 -1.398 0.1621
ethnicityother 0.22724 0.09863 2.304 0.0213 *
unemp 0.13916 0.00912 15.259 < 2e-16 ***
IV estimation using 2SLS approach
In the first stage we get the LS predicted values from a regression of the endogenous variable education on the instrument, and all other $x$ variables:
first.stage <- lm(education ~ urban + gender + ethnicity + unemp + distance,
data = college)
education.hat <- first.stage$fitted.values
In the second stage we estimate the original population model, but we replace the variable distance with the predicted values from the first stage:
iv <- lm(wage ~ education.hat + urban + gender + ethnicity + unemp,
data = college)
summary(iv)
The results are the same as from the ivreg() package!