Homework4.tex

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\begin{document}
\begin{center}
  {\Large \textsc{Problem Set 4}}

  MGMT 737
\end{center}


\begin{enumerate}
\item \textbf{Quantile Regression}. This analysis will use the dataset
  from Problem Set 1, \texttt{lalonde\_nsw.csv} (which I will refer to
  as NSW), as well as the dataset from Problem Set 2,
  \texttt{lalonde\_psid.csv} (which I will call PSID).
  \begin{enumerate}
  \item We will begin by defining an estimation approach for doing
    quantile regression that doesn't require linear programming. This
    approach comes from Gary Chamberlain (in Chamberlain (1994), and
    discussed in Angrist et al. (2006)). 

    Let $X$ be a (discrete) right hand side variable with $J$ discrete
    values.  For each $j$ value of $X = x_{j}$, calculate
    $\hat{\pi}_{\tau}(x) = Q_{\tau}(Y|X_{j})$, which is the $\tau$
    percentile of the outcome variable, conditional on the value of
    $X$, and $\hat{p}_{j}$, which is the empirical probability of
    $X = x_{j}$.  Do so using the PSID dataset for
    $X = \textrm{education}$, for $\tau = (0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)$
  \item Given these inputs, the quantile regression slope
    estimates is just
    $$ \hat{\beta}_{\tau} = \arg\min_{b} \sum_{j} (\hat{\pi}_{\tau}(x_{j}) - x_{j}b)^{2}\hat{p}_{j}.$$
    This is a simple (weighted) linear regression (or minimum distance
    problem), with the diagonal weight matrix
    $\hat{W} = diag(\hat{p}_{1}, \ldots, \hat{p}_{J})$. Estimate
    $\hat{\beta}_{\tau}$ for the education example above.
  \item Our variance estimator is the sum of two terms (coming from
    uncertainty in the QCF, and the estimation of the slope
    conditional on those terms), $V$ and $D$:
    \begin{align*}
  V &= \left(\boldsymbol{x}'\hat{W}\boldsymbol{x}\right)^{-1}\boldsymbol{x}'\hat{W}\Sigma\hat{W}\boldsymbol{x}\left(\boldsymbol{x}'\hat{W}\boldsymbol{x}\right)^{-1},\\
  \Sigma &= diag(\sigma^{2}_{\tau,1}/p_{1}, \ldots, \sigma^{2}_{\tau,J}/p_{J})\\
  D &= \left(\boldsymbol{x}'\hat{W}\boldsymbol{x}\right)^{-1}\boldsymbol{x}'\hat{W}\Delta\hat{W}\boldsymbol{x}\left(\boldsymbol{x}'\hat{W}\boldsymbol{x}\right)^{-1},\\
  \Delta &= diag((\pi_{\tau,1} - x_{1}\beta_{\tau})^{2}/p_{1}, \ldots, (\pi_{\tau,J} - x_{J}\beta_{\tau})^{2}/p_{J}).
\end{align*}
Everything here should be straight forward to estimate, except for
$\sigma^{2}_{\tau,j}$. To do this, define the following order
statistics:
\begin{align*}
  b_{j} &= \max \left\{1, \mathrm{round}\left(\tau N_{j} - z_{1-\alpha/2}\sqrt{\tau(1-\tau)N_{j}}\right)\right\}\\
  t_{j} &= \min \left\{N_{j}, \mathrm{round}\left(\tau N_{j} + z_{1-\alpha/2}\sqrt{\tau(1-\tau)N_{j}}\right)\right\},
\end{align*}
where $\mathrm{round}(a)$ rounds to the closest integer, and
$z_{1-\alpha/2} = 1.96$, typically, and $N_{j}$ is the number of
observations in the $j$th bin of $X$. Then,
\begin{equation}
  \hat{\sigma}^{2}_{\tau,j} = N_{j}\left(\frac{y_{j(t_{j})}-y_{j(b_{j})}}{2 z_{1-\alpha}}\right)^{2}.
\end{equation}
Report the standard error on your estimates, which is calculated as $\sqrt{(V+D)/N}$
\item Finally, using the NSW dataset, calculate the
  $\tau = (0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)$ treatment effects, and their
  standard errors.
\end{enumerate}
\item \textbf{Unconditional Quantile Effects} This analysis will walk
  through an alternative way of approaching quantile effects, using
  results from Firpo, Fortin and Lemieux (2009). Instead of focusing
  on conditional quantile functions, we will consider the
  \emph{uncondtional} (or marginal) distribution of the outcome, $Y$,
  and how that changes when you shift an exogeneous covariate.

  We define the \emph{uncondtional} (marginal) distribution of $Y$:
  $$ F_{Y}(y) = \int F_{Y|X}(y | X = x) \cdot dF_{X}(x),$$

  and consider small changes to $F_{X}$, holding fixed
  $F_{Y|X}$. Intuitively, we consider the effect of changing $F_{X}$
  to $G_{X}$ using a simple additive shift, and consider the effect
  that has on the marginal distiribution of $F$. This will give us the
  \emph{uncondtional quantile partial effect} (UQPE), analogous to the
  unconditional average partial effect (UAPE), $E(dE(Y|X)/dx)$. The
  estimation approach for this uses influence functions. I will walk
  you through the estimation approach, using our example above and the
  approach they call RIF-OLS.
  \begin{enumerate}
  \item Our goal is to estimate $UQPE(\tau)$. We need to estimate
    three pieces. Let $Y_{i}$ be income (\texttt{re78}) and $X_{i}$ be
    years of education using the PSID dataset.
  \item First, we estimate the $\tau$ quantile of
    $Y_{i}$ (unconditionally), which we denote as $\hat{q}_{\tau}$
  \item Second, we need to estimate a constant
    $c_{1,\tau} = 1/f_{Y}(q_\tau)$. To do so, we need an estimate of
    the density of $\hat{q}_{\tau}$. Calculate this directly using the
    following estimator:
    $$\hat{f}_{Y}(\hat{q}_{\tau}) = \frac{1}{Nb} \sum_{i = 1}^{N}K(\frac{Y_{i} - \hat{q}_{\tau}}{b}$$
    Assume $b = 2534.263$ (this bandiwdth depending on the outcome),
    and let $K$ be the Gaussian kernel,
    $$K(u) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{u^{2}}{2}\right)$$.
  \item Finally, we need to estimate $E(dPr(Y > q_{\tau} | X) /
    dX)$. Firpo, Fortin and Lemiux (2009) show that you can do this as
    follows. Calculate
    $\hat{RIF}(Y_{i}, \hat{q}_{\tau}) = \hat{c}_{1,\tau} \cdot 1(Y_{i}
    > \hat{q}_{\tau}) + \hat{c}_{2,\tau}$, where
    $\hat{c}_{2,\tau} = \hat{q}_{\tau} -
    \hat{c}_{1,\tau}(1-\tau)$. Regress
    $\hat{RIF}(Y_{i}, \hat{q}_{\tau})$ against $X_{i}$, and extract
    the coefficient on $X_{i}$. This is the estimate for the $UQPE$ at
    the $\tau$ quantile.
  \item Implement the above estimation procedure for
      $\tau = (0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9)$. Compare these results to your
    results in Part I.
  \end{enumerate}
\item \textbf{Debiased Lasso Regression}. We'll now consider estimation of
  Double/De-biased Lasso. Using the treated units in the NSW dataset
  for the treatment group, and PSID data for the control group, let
  $Y$ denote \texttt{re78}, $D$ denote the treatment indicator, and
  $X$ denote a matrix of indicator variables for all values of
  \texttt{education}, \texttt{hispanic}, \texttt{black}, an indicator
  variable if \texttt{re74} is zero, an indicator if \texttt{re75} is
  zero, and a linear, quadratic, and cubic term for \texttt{age},
  \texttt{re74} and \texttt{re75}. This should be $K = 30$ terms. 
  \begin{enumerate}
  \item Using \texttt{rlasso} (available in R in the package ``hdm''
    and in Stata in Lassopack), implement the DML codebook to identify
    the subset of $X$ to control for in the regression of $Y$ on $D$
    and controls. Report the coefficient on D.
  \item Now reimplement this using NSW dataset. What variables get
    selected in $X$?
  \end{enumerate}
\item \textbf{Puffer Lasso} Finally, we consider a new dataset,
  \texttt{health\_ins}. This dataset has commuting zone level measures
  of health insurance (\texttt{has\_insurance}), and additional
  explanatory variables (the remaining variables in the dataset,
  except for \texttt{czone}). We will now consider how our estimates
  change when we use Puffer vs. not in Lasso. Denote
  $Y = \text{has\_insurance}$ and let the matrix $X$ be the remaining
  variables in the dataset (except for \texttt{czone}).
  \begin{enumerate}
  \item Estimate the OLS regression of Y on X and report the coefficient on \texttt{cs\_frac\_hisp}
  \item Now estimate Lasso (using rLasso as before), and report the coefficient on \texttt{cs\_frac\_hisp}, and the number of non-zero coefficients
  \item Implement the Puffer transformation of $Y$ and $X$
    using the singular value decomposition of $X$ (you should use a
    built-in package to get these values) to construct a
    pre-multiplying matrix $F$. Report the coefficient on
    \texttt{cs\_frac\_hisp} and the number of non-zero coefficients of
    the regressino of $FY$ on $FX$.
  \item Take the non-zero coefficients from the Puffer lasso
    regression, and rerun OLS using those selected coefficients (using
    the non-puffered data). Report the coefficient on
    \texttt{cs\_frac\_hisp}, and the standard error, and compare to the
    original OLS regression and standard error.
  \end{enumerate}
\end{enumerate}

\end{document}