\documentclass{article}

\usepackage{LectureNotes}

\begin{document}
\pagestyle{empty}

\begin{centering}
\large
Intro to Econometric Theory \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2004-5 \linebreak
\vspace{0.2in} \linebreak
Homework \#3 (counts as 2 homeworks), due Thursday, April 28\\
\normalsize

\end{centering}
\vspace{0.3in}

\begin{enumerate}

\item Consider the generalized regression model.  Suppose we try to run
      GLS using a matrix $\Omega_1$ which we believe is equal to 
      (or a good estimator of) $\Omega$.  Suppose further that we are 
      wrong, and that $\Omega_1 \ne \Omega$.  Let's call the estimator
      we calculate using $\Omega_1$, $\hat{\beta}_{WGLS}$ (W for wrong).
      \begin{enumerate}
        \item Is $\hat{\beta}_{WGLS}$ unbiased?  
        \item What is its variance?  
        \item Can you think of a reasonable estimator for this variance?  
        \item Is this variance bigger or smaller than the GLS variance?
        \item What should we conclude about the dangers of trying to 
              correct for a non-scalar covariance matrix, but screwing it
              up?
       \end{enumerate}

\item 
      With respect to the hospital cost dataset from the web site, please
      estimate the following two models:

      \begin{eqnarray}
         lnC &=& \beta_1 + \beta_2lnD + \beta_3lnV + \beta_4time + \epsilon
               \label{eqn:logs}\\
         C   &=& \alpha_1 + \alpha_2D + \alpha_3V + \alpha_4time + \epsilon
               \label{eqn:levels}
      \end{eqnarray}

      In the above, $D$ is inpatient days, $V$ is outpatient visits, and 
      $time$ is the number of quarters since the first quarter of 1991.

      \begin{enumerate}
        \item Discuss how the estimates that these two equations provide
              for marginal costs differ.
        \item Test for non-linearity in equation~\ref{eqn:levels}.  Discuss
              the meaning of your findings.
        \item Think carefully about the variable $C$.  Think carefully 
              about equation~\ref{eqn:levels} (not the estimated 
              version, but the theoretical version).  Are there 
              circumstances in which equation~\ref{eqn:levels} can give 
              obviously wrong predictions for$C$?  Does 
              equation~\ref{eqn:logs} suffer from a similar problem?
        \item Calculate predicted values for $C$ from both models.  
              Calculate the mean squared error for both models: 
              $\sum(C-\hat{C})$.  Which model is better by this criterion?  
        \item Would it make sense to compare the $R^2$ from the two 
              equations in order to decide which is best?  Why or why not?
        \item If we care about getting the right answer, which equation should
              we use and why? (You may do more analysis if you like to answer
              this question)
       \end{enumerate}

\item 
The Medical Expenditure Panel Survey is an annual survey
      which collects information about medical expenditures, income,
      employment, demographics, health information, \&c for a 
      representative sample of Americans.

      I have prepared an extract of these data for 1996, and it is available
      on the course website.  The following are the columns in the data, in
      order:\vspace{2\baselineskip}

      \begin{tabular}{r r}
      Variable & Meaning \\ \hline\hline
      age      & age of person in years\\
      sex      & sex of person, 1=male \& 0=female\\
      income   & income in 1996 \$\\
      employed & 1=employed, 0=not employed\\
      insured  & 1=had health insurance, 0=not\\
      health   & perceived health status, higher is sicker\\
      spending & spending on health care, 1996 \$
      \end{tabular}\vspace{2\baselineskip}

To begin with, let's consider this model:

\begin{eqnarray}
\mathrm{spending}_i &=& \beta_1 + \beta_2 \mathrm{income} 
                     +  \beta_3 \mathrm{age} + \beta_4 \mathrm{sex}
                    \nonumber\\
                    &+& \beta_4 \mathrm{employed} + \beta_5 \mathrm{insured}
                     +  \beta_6 \mathrm{health}
                    \label{eqn:model}
\end{eqnarray}


\begin{enumerate}
\item Test this model for heteroskedasticity.  Discuss your findings and
      and what they mean for inference.
\item How much more do people in very good health status spend than
      do people in excellent health status (estimate and CI).
      Make sure your answer is correct in the presence of heteroskedasticity.
\item Test whether insurance affects spending for people of different
      health statuses differently and discuss.
      Make sure your anser is correct in the presence of heteroskedasticity.
\end{enumerate}

\item Do problems 5 and 6 on pp 248-9 of Greene.

\end{enumerate}

\end{document}
