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Ph.D. Econometrics I \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2005 \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.

                      The parts of question 1 are worth 10 points each.
                      The parts of question 2 are worth 15 points each.
                      Question 3 is worth 15 points.


                       
\end{itemize}

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\begin{enumerate}

\item Consider the following very simple wage regression:

      \begin{equation}
      W_i = \beta_1 + \beta_2 HS_i + \epsilon_i
      \label{eqn:levels}
      \end{equation}

      In these equations, $HS$ is a dummy variable taking the value
      1 if a person has graduated from high school and a value of
      0 otherwise, $W$ is wage.

      As we discussed in class, it seems reasonable to believe that
      $\hat{\beta}_{2,OLS} = \overline{W}_{HS=1}-\overline{W}_{HS=0}$,
      where $\overline{W}_{HS=1}$ means average wage among high
      school grads (ie among observations where $HS=1$).  

      In the following parts, we will try to prove this fact.  If you
      fail to prove any of the individual parts, please continue on
      anyway, assuming that you have proved the previous parts.

      \begin{enumerate}
      \item Use the first-order conditions for the OLS minimization
            problem in the context of the model of 
            equation~\ref{eqn:levels} to prove that
            \begin{math}
            \hat{\beta}_{2,OLS}=\overline{W}_{HS=1}-
             \hat{\beta}_{1,OLS}
            \end{math}
            \pagebreak
      \item Prove that 
            \begin{math}
            \hat{\beta}_{1,OLS} = \overline{W}-
                \hat{\beta}_{2,OLS}\overline{HS}
            \end{math}
            \pagebreak
      \item Prove that
            \begin{math}
            \hat{\beta}_{2,OLS} = \frac{\overline{W}_{HS=1}-\overline{W}}
                                       {1-\overline{HS}}
            \end{math}
            \pagebreak
      \item Prove that 
            \begin{math}
            \hat{\beta}_{2,OLS} = \overline{W}_{HS=1}-\overline{W}_{HS=0}
            \end{math}
            \pagebreak
      \end{enumerate}

\item Consider the following estimator for $\beta$ in the classical
      linear regression model:

      \begin{math}
      \tilde{\beta} = \left( \left( X+Z  \right)'X \right)^{-1}
                      \left( X+Z \right)'Y
      \end{math}

      Assume $Z$ is an $N \times K$ fixed matrix, and that
      $\left( X+Z \right)'X$ is positive definite.

      \begin{enumerate}
      \item Is $\tilde{\beta}$ biased?
            \pagebreak
      \item What is the variance of $\tilde{\beta}$?
            \pagebreak
      \item Is this variance greater than less than or equal
            to the variance of $\hat{\beta}_{OLS}$?
            \pagebreak
      \end{enumerate} 

\item Consider the following regression model, which we will estimate
      by OLS:

      \begin{equation}
      Y = X\beta + \epsilon
      \end{equation}

      Now, suppose we multiply $X_3$ by 100 (say by measuring it in
      cents rather than in dollars).  What effect will this have
      on the OLS estimates.  Please answer and prove your answer.

\end{enumerate}

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