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\large
Intro to Econometric Theory \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2000 \linebreak
\vspace{0.2in} \linebreak
Homework \#4, due Tuesday, May 2, 2000
\normalsize

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\begin{enumerate}
\item Suppose that a dataset you are using has the ``repeated measures''
      problem.  The observations in your dataset come from measuring
      some quantity several times for each observable unit.

      You have a dataset consisting of 100 families each of whom you
      observe in 1990, 1991, 1992, 1993, 1994, 1995.  So, there are
      600 observations.  Your LHS variable is, say, family income and
      you also have some RHS variables.  It seems reasonable to believe
      that the six obs for each family will be correlated with one
      another but that each family's observations will be uncorrelated
      with the other families' obs.  To capture this idea, you use the
      following ``repeated measures'' or ``random effects'' model:

      \begin{eqnarray}
        Y_{it}        &amp;=&amp; X^{it}\beta + \epsilon_{it} \\
        \epsilon_{it} &amp;=&amp; \xi_i + \nu_{it}
      \end{eqnarray}

      Assume that $\nu$ is a well behaved error with mean 0 and 
      variance $\sigma^2_\nu$, uncorrelated with anything.  Similarly,
      assume that $\xi$ is a well behaved error with mean 0 and
      variance $\sigma^@_\xi$, uncorrelated with anything.

      \begin{enumerate}
        \item What are $\sigma^2$ and $\Omega$ in 
              $V(\epsilon)=\sigma^2\Omega$?
        \item What would be a reasonable estimator for $\Omega$?
        \item \label{q:strategies}
              Consider the following strategies for dealing with
              this problem:
              \begin{enumerate}
                \item ``First difference'' the data.  Instead of 
                      regressing $Y$ on $X$, create a new dataset
                      with $\Delta Y_{it}=Y_{it}-Y_{it-1}$ and
                      $\Delta X_{it}=X_{it}-X_{it-1}$.  Then regress
                      $\Delta Y$ on $\Delta X$.
                \item ``Differences from means'' method.   Instead of 
                      regressing $Y$ on $X$, create a new dataset 
                      with $Y^0_{it} = Y_{it} - \overline{Y}_i$ and 
                      $X^0_{it} = X_{it} - \overline{X}_i$.  
                      $\overline{Y}_i$ contains the mean level of $Y$
                      for family $i$.  Then regress $Y^0$ on $X^0$
              \end{enumerate}
              How many observations  are usable in each of these two 
              methods?
        \item Considering the two strategies in \ref{q:strategies}, what
              is $\Omega$ in each of the two transformed regressions?
        \item In a model like this, $N$ can go to infinity either by
              the number of obs per individual going to infinity or by
              the number of individuals going to infinity.  How does 
              $\Omega$ change as $N$ goes to infinity in each of these
              two ways for the two new models above?
        \item What are the strengths and weaknesses of ``full'' GLS and
              each of the two methods above to deal with the econometric
              problem of repeated measures?
      \end{enumerate}


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

      \begin{eqnarray}
         lnC &amp;=&amp; \beta_1 + \beta_2lnD + \beta_3lnV + \beta_4time + \epsilon
               \label{eqn:logs}\\
      \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.

      You suspect that the errors in this model follow an AR(1) process
      within hospitals but are uncorrelated across hospitals.

      \begin{enumerate}
        \item What does $\Omega$ look like in this case?
        \item Propose a test for serial correlation here.
        \item Propose a method to estimate the AR(1) parameter.  Perform
              your test.
        \item Use a Cochrane-Orcutt like method to estimate the model
              accounting for serial correlation.
        \item Estimate the scale economies 
              (\% costs rise if all inputs
              rise by 1\%) --- with a confidence interval ---
              using the uncorrected and corrected models.
       \end{enumerate}

\end{enumerate}

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