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\begin{centering}
\large
Intro to Econometric Theory \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2003-4 \linebreak
\vspace{0.2in} \linebreak
Solution \#4
\normalsize

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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 What do you think of the claim that income and sex do not belong
      in this model?

      \textbf{
      To test this, we must test the null hypothesis that income and
      sex do not belong in the model, that is that both their 
      coefficients are zero.  We will conduct an F-test comparing
      the equation above to the restricted model with coefficients
      on sex and income constrained to be zero.
      }

      \textbf{
      The unrestricted model appears in the SAS output on page 3.  The
      restricted model appears in the SAS output on page 5.
      }

      \begin{eqnarray}
      \mathrm{F-stat} 
             &=& \frac{(5.6944 \times 10^{11} - 5.6932 \times 10^{11})/2}
                      {(5.6932 \times 10^{11})/9504}
                 \nonumber\\
             &=& 1.00 \nonumber
       \end{eqnarray}

       \textbf{
       Since an F-stat of 1 does not lead to rejection at a conventional
       significance level, we accept the null hypothesis and conclude
       that we do not have compelling evidence in these data to reject
       the theory that sex and income do not affect health care spending,
       once the other variables in the regression are controlled for.
       }

       \textbf{
       Notice, on page 4 of the output, I use SAS's test command to 
       perform a Wald test of the same null hypothesis, leading to the
       same conclusion (as we discussed in class).
       }

\item Consider the health status variable.  Respondents were asked to
      rate their health status; their choices were excellent, 
      very good, good, fair, or poor.  These were assigned the numerical
      values 1-5.  Does it make sense to enter health status as a 
      single continuous variable as in equation~\ref{eqn:model}?

      Enter health status into the model as a set of dummies, and then
      test whether they belong.

      \textbf{
      No, it does not make a lot of sense to enter this variable 
      linearly.  The underlying variable is ordinal (the order 1-5
      has meaning) but not cardinal (the distance between the responses
      does not have meaning).  Entering the variable linearly amounts
      to the assumption that the difference between spending between a 
      poor \& fair person is the same as the difference in spending 
      between a fair \& good person is the same as the difference
      in spending between a good \& a very good person is the same
      as the difference in spending between a very good \& excellent
      person, all controlling for the other variables in the model.
      There is no \textit{a priori} reason to believe that this is
      true.
      }

      \textbf{
      Whether the health status variables do not belong can be tested
      by comparing (with an F-test) the regressions on output pages
      6 and 9 or by examining the Wald test on page 7.  The Wald-like
      test produces an F-stat of 33, which clearly rejects the null
      hypothesis that the health status variables do not belong.  So,
      we conclude that we are very confident that they do belong.
      }


\item How much more do people in very good health status spend than
      do people in excellent health status (estimate and CI).

      \textbf{
      This question can be answered by looking at the coefficient
      on hsvryg in the regression on page 6.  Since excellent health
      is the omitted contrast, the coefficient on hsvryg in the 
      model estimated on this page is the difference in spending 
      between people in very good and excellent health status, 
      controlling for the other variables in the equation.
      }

      \textbf{ Estimate and 95\% CI: }

      \begin{eqnarray}
      \beta_{\mathrm{hsvryg}} &=& 286 \pm (1.96)202 \nonumber\\
                              &=& 286 \pm 396 \nonumber
      \end{eqnarray}

\item Test whether it was correct to enter health status linearly.

      \textbf{
      As discussed above, entering health status linearly means
      assuming that the differences in spending between each adjacent
      health category (poor vs fair, fair vs good, etc) are equal.
      The null hypothesis that this is true can be written as a 
      set of linear restrictions:  
      }

      \begin{eqnarray}
      \beta_{\mathrm{vryg}} &=& \beta_{\mathrm{good}}-\beta_{\mathrm{vryg}}
                                 \nonumber\\
      \beta_{\mathrm{good}}-\beta_{\mathrm{vryg}} 
                            &=& \beta_{\mathrm{fair}}-\beta_{\mathrm{good}}
                                 \nonumber\\
      \beta_{\mathrm{good}}-\beta_{\mathrm{vryg}} 
                            &=&  \beta_{\mathrm{poor}}-\beta_{\mathrm{fair}}
                                 \nonumber
      \end{eqnarray}

      \textbf{
      We can write these another way (substituting in the above):
      }

      \begin{eqnarray}
      2\beta_{\mathrm{vryg}} &=& \beta_{\mathrm{good}} \nonumber\\
      3\beta_{\mathrm{vryg}} &=& \beta_{\mathrm{fair}} \nonumber\\
      4\beta_{\mathrm{vryg}} &=& \beta_{\mathrm{poor}} \nonumber
      \end{eqnarray}

      \textbf{
      There are a few ways to test this.  For example, we can run the
      regression reported on output page 6 and examine the Wald-like
      F-test on page 8.  Or, we can compare the regression on page
      6 with the constrained regression on page 10.  Or, we can
      compare the regression on page 6 with the constrained regression
      on page 3 --- Notice that the sum of squared errors and many param
      estimates are identical for the regressions on pages 3 and 10:
      you should think about why this is so.  I'll do the F-test comparing
      pages 6 and 10.
      }

      \begin{eqnarray}
      \mathrm{F-stat} 
             &=& \frac{(5.6932 \times 10^{11} - 5.6724 \times 10^{11})/3}
                      {(5.6724 \times 10^11)/9497}
                 \nonumber\\
             &=& 11.61 \nonumber
       \end{eqnarray}

      \textbf{
      This leads to rejection at any conventional significance level.  We
      conclude that we are very confident that it is wrong to enter
      health status linearly here.
      }

\item Test whether insurance affects spending for people of different
      health statuses differently and discuss.

      \textbf{
      To get insurance to affect spending differently for people of
      different health statuses, we need to interact health status
      with insurance.  This is done in the model estimated on output
      page 11.  It looks as if insurance increases the health spending
      of sick people more than it does the health spending of well
      people.  To test this, we can compare the regressions on pages
      11 and 6 with an F-test or we can look at the Wald-like F-test
      on page 12.  Looking at page 12, we can reject the null of no
      interactions at better than the 1\% level, so we conclude that
      these interactions belong in the regression.
      }

\end{enumerate}

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