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HWx_yourNetID.pdf. For example,
HW01_rqzhu.pdf. Please note that this must be a
.pdf file. .html format
cannot be accepted. Make all of your R
code chunks visible for grading..Rmd file
as a template, be sure to remove this instruction
section.R is \(\geq
4.0.0\). This will ensure your random seed generation is the same
as everyone else. Please note that updating the R version
may require you to re-install all of your packages.For this problem, we use the iv_health.csv data provided
. The dataset aims to study the effects of having health insurance on
medical expenses. For this problem, we are only interested in a
particular set of variables. Load the data and restrict the dataset to
the following variables. In addition, the ssiratio variable
should have values between 0 and 1, which is not true for some entries
in the dataset. Correct this by changing those values
that are greater than 1 to 1.
| Variable | Description |
|---|---|
| logmedexpense | Medical Expenses (log form) |
| healthinsu | Having health insurance (=1) or not (=0) |
| age | Age |
| logincome | Annual Income (log form) |
| illnesses | Severity of Illnesses |
| ssiratio | Ratio of Social Security Income |
This dataset was originally from Medical Expenditure Panel Survey and was used as an example here. However, for our purpose, we will assume that we do not have unobserved confounders. After you process the data, perform the following tasks:
Doubly Robust Estimation: Implement the doubly robust estimator
on the given dataset. Treat logmedexpense as the outcome
\(Y\), healthinsu as the
treatment \(A\), and other variables as
the covariates \(X\). Estimate the ATE
of having health insurance on medical expenses.
Does the result in part a) align with your intuition or understanding of having health insurance? Provide a briefly explain of your thoughts.
Use the idea of bootstrap to estimate the distribution of your doubly robust estimator. For this question, you should use 1000 bootstrap samples. After obtaining the distribution, report the mean and standard deviation of your estimator, provide a histogram of your bootstrap samples, and then calculate a two-sided 95% confidence interval for your estimator. Based on this result, would you reject the following hypothesis of ATE?
\[ \text{H}_0: \tau = 0 \quad \text{vs.} \quad \text{H}_1: \tau \neq 0 \]
In our lecture, we provided a statement that
\[ n \text{Var}[\hat{\tau}_\text{iv}^w] = \frac{\text{Var}[\epsilon]\text{Var}[w(Z)]}{[\text{Cov}(A, w(Z)]^2} \]
where \(A\) is the treatment variable, \(Z\) is a multivariate vector of instrument, and \(w(Z)\) is some function of \(Z\). And we also stated that when \(w(Z)\) takes the optimal form
\[ w(z) = \mathbb{E}[A | Z = z] \]
The variance of the IV estimator is simplifed into
\[ n \text{Var}[\hat{\tau}_\text{iv}^w] = \frac{\text{Var}[\epsilon]}{\text{Var}[\mathbb{E}[A | Z]]} \]
Prove this by showing that
\[ \text{Var}[\mathbb{E}[A | Z]] = \text{Cov}[A, E(A|Z)] \]
Hint: start by the definition of covariance and use the law of total expectation.
In this problem, we will using the card dataset from wooldridge package.
library(AER)
library(wooldridge)
data("card")The dataset contains 3030 observations collected in 1966/1976, aiming to estimate the impact of education on wage. For this exercise, we only use the following variables, you can find a description of all variables here.
| Variable | Description |
|---|---|
| lwage | Annual wage (log form) |
| educ | Years of education |
| nearc4 | Living close to college (=1) or far from college (=0) |
| smsa | Living in metropolitan area (=1) or not (=0) |
| exper | Years of experience |
| expersq | Years of experience (squared term) |
| black | Black (=1), not black (=0) |
| south | Living in the south (=1) or not (=0) |
Among the variables provided above, we are interested in finding
the effect of education on wage. Thus, lwage should be
selected as the response variable \(Y\), and educ is the treatment
\(A\). When estimating the relationship
between wage and education, scholars have encountered issues in dealing
with the so-called “ability bias”: individuals who have higher ability
are more likely both to stay longer in school and get a higher paid job.
Ability, however, is difficult to measure and cannot be included in the
model. Thus, we consider “ability” as an unobserved confounder.
nearc4 would be a good choice as the instrumental variable
in this case. Do you think this is a good choice as instrumental
variable? Does it satisfy the three requirements of instrumental
variables? What could be the potential issues?
Fit a model using linear regression (without using instrumental
techniques) by treating lwage as the response variable,
and all other variables as predictors. What is the
estimated effect of educ in this case after controlling
other variables?
Treat nearc4 as the instrumental variable, and
implement the two-stage least square (2SLS) to fit the model. Implement
the 2SLS method using both your own code, and AER package, does
the result match? What is the estimated effect of educ? How
does it compare with linear regression, what does this suggest?