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\large
Intro to Econometric Theory \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2003-4 \linebreak
\vspace{0.2in} \linebreak
Solution \#5
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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 a model like the one we used on the
midterm:

\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 --- refer to the answers to 
      the questions on homework 4 and be as specific as you can.

      \textbf{
      See the attached output.  On page 6 is the result of White's test
      for heteroskedasticity.  The null hypothesis of homoskedasticity
      is rejected at $p<0.0001$, so we reject homoskedasticity in favor
      of heteroskedasticity.
      }

      \textbf{
      Referring to homework 4, every single hypothesis test and 
      confidence interval in questions 1-4 is wrong.  The paramter
      estimates are
      unbiased but the all the variances used to construct the 
      tests and CI are wrong.  All tests and CIs must be redone.
      }

\item Answer question 1 from homework 4 again, correcting for
      heteroskedasticity.

      \textbf{
      See the test on page 9 of the output.  We would reject the
      null that income and sex do not belong at the 10\% level,
      but we would accept it at the 5\% or 1\% levels.  Whether or
      not sex and income have an effect is thus a matter of 
      researcher judgement --- depending on which confidence level
      you choose.  Notice how different this is from the result on
      page 8 (the test assuming homoskedasticity).  There we would 
      accept the null that income and sex do not belong at conventional
      significance levels.
      }

\item Answer question 3 from homework 4 again, correcting for
      heteroskedasticity.

      \textbf{
      Refer to the output on page 12.  We reject the null that 
      health status may be entered linearly at $p<0.0001$.  In 
      this case, the result is quite similar to the result assuming
      homoskedasticity (appearing on page 11).
      }

\end{enumerate}

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