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Ph.D. Econometrics I \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2004 \linebreak
\vspace{0.2in} \linebreak
Final
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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.

                      Use the backs of pages if you need more space.

                      Each part of each question is worth 6 points,
                      except for parts 3a-d which are each worth 7 
                      points.

                      

                       
\end{itemize}

\pagebreak
\begin{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.

      \textbf{
      The data extract in this example is different from the homework
      extract.
      }

      In this extract, I have chosen a group of people aged 19-64 in
      1996 and followed them over two years (1996 and 1997).  For
      each person, there are exactly two observations in the dataset,
      one for 1996 and one for 1997.
    
      The following variables appear on the data:

      \begin{tabular}{r r}
      Variable & Meaning \\ \hline\hline
      age      & age of person in years\\
      age2     & age squared\\
      sex      & sex of person, 1=male \& 0=female\\
      yred     & years of education (12=HS diploma)\\
      income   & \textbf{earned} (wages + profits) income in \$\\
      health   & perceived health status, higher is sicker\\
      health\_?? & dummies for perceived health status \\
                & ex=excellent, ... , pr=poor\\
      \end{tabular}

      We will begin with the regression on page 4 of the output,
      and we will begin with the assumption that the classical
      linear regression model is true for these data and this
      model.

  ~\pagebreak
  \begin{enumerate}
  \item Give an estimate and 95\% confidence interval for the effect
        of a year's education on earned income.
        \pagebreak
  \item Please test the hypothesis that people in good and excellent
        health have equal income, other things in the regression
        equal.
        \pagebreak
  \item Please test the hypothesis that age has no effect on income.
        \pagebreak
  \item A critic claims that there is heteroskedasticity in this
        regression.  What will the effect of this be if true, and how 
        would you deal with the critique?
        \pagebreak
  \item Is there any particular reason to believe that any of the 
        RHS variables in this regression might be correlated with 
        the error?  What effect will this have, if it is true?
        \pagebreak
  \end{enumerate}

\item A critic complains that the regression in the previous
      question suffers from the repeated measures problem ---
      that observations from 1996 and 1997 for the same 
      individual are correlated.  He proposes a model like this:

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

      Here, $i$ refers to individual $i$ and $t$ refers to the time
      period.  $i$ runs from $1$ to $N$ and $t$
      from $1$ to $T$, so that there are $NT$ total observations.
      (Here, $t$ would equal 1 for 1996 and 2 for 1997).

      $\epsilon_{it}$ is a well-behaved error, so that 
      $V(\epsilon)=\sigma^2_{\epsilon}I$.  

      The other error term is $\nu_i$.  It is also a random variable,
      and it captures the serial correlation within individuals.
      $\nu_i$ has a mean of zero, a variance of
      $\sigma^2_{\nu}$ and is uncorrelated across individuals.

      To be absolutely clear, the very same $\nu_i$ appears in 
      the two observations for individual $i$.
      \pagebreak

      \begin{enumerate}
      \item What is the covariance between two error terms (error term
            here means $\nu_i+\epsilon_{it}$) from the 
            same individual?
            \pagebreak
      \item What is the covariance between two error terms from 
            different individuals?
            \pagebreak
      \item Does this model create a serial correlation problem?  What
            would be its consequence for the answers to the previous
            question?
            \pagebreak
      \item A helpful fellow researcher suggests that you can eliminate
            your serial correlation problem by only using a single
            year of data --- as in the regressions in the output 
            labeled ``96 only'' or ``97 only''.  That is she suggests
            you throw away half the data.  Will this fix the problem? 
            \pagebreak
      \item Another helpful fellow researcher suggests that you can
            eliminate your serial correlation problem by averaging
            your data.  That is, have one observation per person, 
            with the LHS variable being the average of that person's
            LHS variable for the years 1996 and 1997 and each of the
            RHS variables being the average of the respective RHS variable
            for 1996 and 1997.  Like the regression in the output 
            labeled 96/7 averaged.  Will this fix the problem?
            \pagebreak
      \item Which of these two methods is better and why?
            \pagebreak
      \end{enumerate}

\item Consider the following very simple regression models:

      \begin{equation}
      Y = \beta_1 + \beta_2*\mathrm{Female}+\epsilon
      \end{equation}

      \begin{equation}
      Y = \beta_3 + \beta_4*\mathrm{Male}+\epsilon
      \end{equation}

      For the parts of this question, you can get full credit
      only with proofs of your answers.

      \begin{enumerate}
      \item What will be the relationship between 
            $\hat{\beta}_{2,OLS}$ and
            $\hat{\beta}_{4,OLS}$?
            \pagebreak
      \item What will be the relationship between
            $V(\hat{\beta}_{2,OLS})$ and
            $V(\hat{\beta}_{4,OLS})$?
            \pagebreak
      \item What will be the relationship between the
            $\hat{Y}$ from the two regressions?
            \pagebreak
      \item What will be the relationship between $R^2$ from
            the two regressions and $\hat{\sigma}^2_{OLS}$ from
            the two regressions?
            \pagebreak
      \item What will be the relationship between
            $\hat{V}(\hat{\beta}_{2,OLS})$ and
            $\hat{V}(\hat{\beta}_{4,OLS})$?
      \end{enumerate}

\end{enumerate}

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