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\large
Ph.D. Econometrics I \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2004 \linebreak
\vspace{0.2in} \linebreak
Midterm
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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 worth 10 points, except
                      for \ref{enum:biginterp}, which is worth 20.

                       
\end{itemize}

\pagebreak
\begin{enumerate}

  \item Consider the following regression model in which we assume that
        all the classical assumptions are satisfied:

        \begin{equation}
          Y = \beta_1X + \epsilon \label{eqn:OLSmodel}
        \end{equation}

        Consider the following estimator for $\beta_1$:

        \begin{equation*}
        \widetilde{\beta}_1 = \frac{\sum X_i^2Y_i}{\sum X_i^3}
        \end{equation*}

        \begin{enumerate}
          \item Is $\widetilde{\beta}_1$ unbiased?
                \pagebreak
          \item What is the variance of $\widetilde{\beta}_1$?
                \pagebreak
          \item Is the variance of the OLS estimator greater than,
                less than, or the same as the variance of 
                $\widetilde{\beta}_1$?
        \end{enumerate}

        \pagebreak


  \item Consider a factory producing pairs of shoes.  We have data
        on the monthly costs of running the factory and the number
        of shoes produced.  We want to estimate the following model
        by OLS:

        \begin{equation*}
        \mathrm{Cost}_i = \beta_1 + \beta_2\mathrm{Left \: Shoes}
                        + \beta_3\mathrm{Right \: Shoes} + \epsilon
        \end{equation*}

        Is it a good plan to try to estimate this model by OLS?  Why
        or why not?
        \pagebreak

  \item Suppose you are interested in trying to predict how much
        money people will spend on their next new car.  You have
        a dataset documenting how much a large, relevant sample 
        of people spent on their cars and how much they 
        characteristically spend on a bottle of wine.  So, you 
        estimate the following model by OLS:

        \begin{equation*}
        P_{\mathrm{car}} = \beta_1 + \beta_2 P_{\mathrm{wine}}
                         + \epsilon
        \end{equation*}

        Then, for a similar group of people, you are informed of
        each person's $P_{\mathrm{wine}}$, and you use your 
        estimates of $\beta_1$ and $\beta_2$ to predict car 
        purchase price, $P_{\mathrm{car}}$.

        A critic complains that the model above does not conform
        to the assumptions of the classical linear regression
        model.  Is the critic right about this?  

        The critic goes
        on to claim that, since the assumptions of CLRM are 
        violated, you should not use OLS for these purposes.
        Assuming that the critic is right about the assumption
        violation, is she right about the uselessness of OLS here?
        \pagebreak

\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.

      I have prepared an extract of these data for 1996.  The following
      variables appear on the data:

      \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}

  ~\pagebreak
  \begin{enumerate}
  \item \label{enum:biginterp}
        Consider the regression on page 5 of the output.  Please interpret
        each of the estimates there, excluding the constant.  Which ones
        ``make sense'' intuitively?
        \pagebreak
  \item For which of the coefficients can we reject, at the 5\% level,
        the null hypothesis that the coefficient is equal to zero?
        \pagebreak
  \item Suppose an unemployed and uninsured person took a job with health
        insurance.  What is your best estimate of the change in their
        health spending?
        Can we conclude at the 10\% level that this change is different
        from 0?
        \pagebreak
  \item If we were to regress spending on income alone, would the 
        slope coefficient be positive or negative and why?
  \end{enumerate}

\end{enumerate}

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