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\large
Ph.D. Econometrics I \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2006 \linebreak
\vspace{0.2in} \linebreak
Final
\normalsize

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\begin{itemize}
  \item[Instructions] You may use any books, notes, calculators, and
                      other aids you like.  You may not converse, nor
                      may you cooperate.

                      Please complete all questions.

                      Please show all relevant work.

                      Please interpret your results in plain English.

                      \textbf{You have until 8:20 to complete the test.}


                       
\end{itemize}

\pagebreak
\begin{enumerate}

\item  Consider the hospital cost dataset we used in homework 4.
       Remember that this is a dataset of California hospitals
       with data measured quarterly from 1991 to 1995.  The 
       data elements are:

       \begin{tabular}{l l}
       C & is hospital costs for the quarter\\
       D & is number of inpatient days for the quarter\\
       V & is number of outpatient visits for the quarter\\
       time & is number of quarters since Q1, 1991
       \end{tabular}

       Let's think about the two regressions:

       \begin{eqnarray}
       \ln{C_{it}} &=& \beta_1 + \beta_2 \ln{D_{it}} 
                       + \beta_3 \ln{V_{it}} + \beta_4 t
                       + \epsilon_{it} \label{eqn:linearC}\\
       && \nonumber\\
       \ln{C_{it}} &=& \beta_1 + \beta_2 \ln{D_{it}} 
                       + \beta_3 \ln{V_{it}} + \beta_4 t
                       + \beta_5 (\ln{D_{it}})^2 + \epsilon_{it}
                       \label{eqn:squaredC}\\
       \end{eqnarray}

       \begin{enumerate}
       \item Considering equation~\ref{eqn:linearC} and assuming that
             the assumptions of the CLRM hold, what is your best 
             estimate and 95\% confidence interval for the effect
             of a 1\% increase in inpatient days on hospital costs?
       \item How does your answer change if you suspect heteroskedasticity?
       \item Still on equation~\ref{eqn:linearC} and worrying about
             heteroskedasticity, what 
             is the \uline{annual} growth rate of hospital costs in CA
             (holding constant days and visits)?  Estimate and 
             95\% confidence interval.
       \item Still considering equation~\ref{eqn:linearC} and worrying
             about heteroskedasticity, we are interested in assessing
             whether hospitals are a constant returns to scale business.
             A business is constant returns to scale if a simultaneous 
             1\% increase in all outputs leads to exactly a 1\% increase
             in costs.  Test the theory that California hospitals are
             a constant returns business.
       \item Now, considering equation~\ref{eqn:squaredC} and continuing
             to worry about heteroskedasticity, at what level of 
             days (not log days) are California hospitals exactly 
             constant returns to scale (that is  %
             $\frac{\partial \ln{C}}{\partial \ln{D}}+%
              \frac{\partial \ln{C}}{\partial \ln{V}}=1$)?  Please set
             up completely but do not calculate the standard error
             of your estimate (You don't have to multiply, invert, etc
             matrixes, just set up the relevant equation(s)).
       \end{enumerate}
\item When people talk about the repeated measures problem, they 
      often talk in terms of how many observations you ``really''
      have in your data, as opposed to how many observations you
      seem to have.  The intuition is that if you have,say, two obs
      on each person, but those obs are highly correlated with
      each other, that there is not really 2 obs worth of information
      in them.

      Consider a panel dataset with 2 observations for each person.
      Suppose further that each person's X variables are the same
      for each of their 2 observations, so that we can write:

      \begin{equation}
      Y_{it} = X^i\beta + \epsilon_{it}
      \end{equation}

      Suppose further that each individual's two observations
      $\epsilon_{i1}$ and $\epsilon_{i2}$ have correlation equal
      to $1>\rho>0$.  Otherwise, suppose the usual CLRM assumptions
      hold.

      Now, $X$ is a $2N \times K$ matrix.  However, since each 
      individual is in the matrix twice with the exact same values,
      we can write the $X$ matrix as:

      \begin{equation}
      X = \left[ \begin{array}{c} \tilde{X} \\ 
                                  \tilde{X} \end{array} \right]
      \end{equation}

      Here the data are sorted by time and then individual, so that
      $\tilde{X}$ is an $N \times K$ matrix with one obs for each
      individual.

      \begin{enumerate}
      \item Prove that the usual (and correct) OLS variance 
            for a regression using only one observation per 
            individual is:
            $\widehat{\sigma^2}_{OLS}(\tilde{X}'\tilde{X})^{-1}$
      \item Prove that the usual OLS variance matrix for the
            regression using two observations per individual is: 
            $\frac{\widehat{\sigma^2}_{OLS}}{2}(\tilde{X}'\tilde{X})^{-1}$
      \item Calculate the correct variance matrix for this regression
            and relate it to the discussion of the ``effective''
            number of observations above.  How does the number of effective
            observations change with $\rho$?
      \end{enumerate}

\item Suppose that you are interested in the effect of a person 
      turning 65 on income.  Suppose further that, the true model
      is:

      \begin{equation}
      I_{it} = \beta_1 + \beta_2 \text{age}_{it}
             + \beta_3 (\text{age}_{it})^2 
             + \beta_4 \text{senior}_{it} + \epsilon_{it}
             \label{eqn:65true}
      \end{equation}

      but that you estimate:

      \begin{equation}
      I_{it} = \beta_1 + \beta_2 \text{age}_{it}
             + \beta_4 \text{senior}_{it} + \epsilon_{it}
             \label{eqn:65false}
      \end{equation}

      Here, I am assuming you have a panel of people followed
      over time and that senior is a dummy variable =1 for 
      $\text{age}\ge65$.
      
      To make things easy, assume that, in truth, $\beta_4=0$,
      $\beta_2>0$, and $\beta_3<0$.

      Please discuss how $\beta_4$ is likely to be biased.  First,
      draw a picture and discuss in words how you think the bias
      will work.  Then, do the necessary algebra.  

      If it helps you
      to make some assumptions about the dataset, that's OK.  For
      example, you might want to consider only a single person
      in the panel or you might want to make particular assumptions
      about which ages for the person are in the data (say they
      are observed from age 30-70 or something).

      Ultimately, however, a correct answer is one that proves your
      claims with algebra.  Just pictures and words will not get
      you full credit.

\end{enumerate}

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