### Load Libraries and Packages
library(foreign)
library(AER)

	# To obtain a consistent estimate of the impact of kids on labor supply, 
	# some authors have suggested using whether a mother had twins on their 
	# first birth as an instrument for the number of children in the household.
	# Twins are in many respects random and by definition, the realization of a 
	# twin increases the number of children in the household. Using data from 
	# the 1980 Public Use Micro Sample 5% Census data files, a sample of women aged 
	# 21-40 with at least one kid is constructed. The 1980 PUMS identifies a 
	# person’s age at the time of the census and their quarter of birth. 
	# Because the census is taken on April 1st, we know a person’s year and quarter 
	# of  birth and we can infer that any two kids in the household with the same 
	# age and quarter of birth are twins. There are roughly 6,000 1st births to 
	# mothers that are twins. There are over 800,000 observations in the original 
	# data set so to make the problem manageable, a random sample is selected of 
	# approximately 6,500 non-twin births for a total of about 12,500 observations. 
	# The data file is called twins1st.dta.

### Load and Confirm Data
df <- read.dta(choose.files())
	# twin1st.dta

head(df)
tail(df)
dim(df)
class(df)
attach(df)

#--------------------------------------- 1. ---------------------------------------# 
# What fraction of women work? 
# What is average weeks worked among women? 
# What is median log of labor earnings for women who worked?

mean(worked)
mean(weeks)
median(lincome)
median(subset(df, worked == 1)$lincome)
median(lincome[worked==1])

# ANSWER:
# 60.4% of women worked last year, average weeks worked is 23 weeks.  
# The median log labor income for women is $1,005 and median log labor income 
# for women who worked is $5,505..

#--------------------------------------- 2. ---------------------------------------#

# Construct an indicator that equals 1 for women that have a second child. 
# Call this variable SECOND. What fraction of women had a second child? 
# Consider a simple bivariate regression where WEEKS of work (Y) is regressed 
# on SECOND (X), Y = ß0 + ß1X + u. What is the coefficient for ß1 in this 
# regression and interpret the coefficient?

second <- kids > 1 
summary(second)

reg.ols <- lm(weeks ~ second)
summary(reg.ols)

# ANSWER: 
# The coefficient on second is -6.8 meaning that, on average, 
# among women with one or more kids, the presence of the second child reduces 
# weeks worked by an average of 6.8 weeks/year, ceteris paribus.

#--------------------------------------- 3. ---------------------------------------#

# Because of the concern that X and u are correlated, use twins on 1st birth (Z) 
# as an instrument for X in an instrumental variables model. What is the 
# first-stage and reduced-form estimates for this model? Interpret these 
# coefficients, that is, what do these coefficients measure? Consider the 
# regression of X on Z. Why is the coefficient on Z not 1; e.g, don't twins 
# increase the number of kids in the house by 1? Run the first stage, does having 
# a twin (Z) increase the kids in the home (X)?

# run the 1st stage model
reg.1st <- lm(second ~ twin1st)
summary(reg.1st)

# run the reduced form, impact of twins (Z) on weeks worked (Y)
reg.red <- lm(weeks ~ twin1st)
summary(reg.red)

# run the 2sls model
iv <- ivreg(weeks ~ second | twin1st)
summary(iv)

### Hausman test `by-hand' for endogeneity of regressors
cf_diff <- coef(iv) - coef(reg.ols)
vc_diff <- vcov(iv) - vcov(reg.ols)
x2_diff <- as.vector(t(cf_diff) %*% solve(vc_diff) %*% cf_diff)
pchisq(x2_diff, df = 2, lower.tail = FALSE)

# Tests: Weak Instruments, Wu-Hausman, Sargan
summary(iv, vcov = sandwich, diagnostics = TRUE)

# On average, the presence of a twin increases the probability of having a second 
# child by 27.5 percentage points, ceteris paribus. Why is this coefficient not 1? 
# At the time of the birth, the presence of the twin increases family size from 1 
# to 2. However, many of the women who had a twin on the 1st birth would have had 
# a second one anyway so that is the reason the twin1st coefficient is less than 1.

# Notice that in the reduced form regression (weeks worked on twin1st) produces 
# a coefficient of -0.99. Women assigned a twin on the first birth are working 
# 1 week fewer per year. Notice that -0.99/0.2746 = -3.605 which is exactly the 
# 2SLS estimate.

# According to the OLS model, the presence of the 2nd kid reduces work by almost 
# 7 weeks per year. In the 2SLS model, however, this number reduces to -3.6. The 
# OLS estimate is too large by a factor of 2 suggesting large omitted variables 
# problems in the OLS model.

#--------------------------------------- 4. ---------------------------------------#

# In this model, we run an OLS model similar to that in part 2 but we add 
# additional covavariates. First, generate dummy variables for mothers that are 
# black and other_race. Notice that the estimated impact of havin a second 
# kid increases in magnitude from -6.8 to -9.26, providing strong evidence that 
# the observed characteristics of the mother are correlated with whether the 
# mother had a second child.

summary(race)
	# 1-3
black <- race==2
other.race <- race==3
summary(black)
summary(other.race)

# OLS with Covariates
reg.ols.2 <- lm(weeks ~ second + agem + agefst + black + other.race + educm + married)
summary(reg.ols.2)

#--------------------------------------- 5. ---------------------------------------#

# Now, use twin1st as an instrument for the second child the model above. 
# Compare these estimates to the results in part 4. Next, compare these results 
# to the simple 2SLS estimates. What has happened to the labor supply 
# impacts of having a second child? 

# IV with Covariates
iv.2 <- ivreg(weeks ~ second + agem + agefst + black + other.race + educm + married 
	| agem + agefst + black + other.race + educm + married + twin1st)
summary(iv.2)