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Intro to Econometric Theory \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2004-5 \linebreak
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Homework \#2, due Friday, March 18, 2005
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\item \label{q:reg-covariance}
      You may recall that, in the bivariate regression model,
      \begin{equation}
        Y = \beta_1 + \beta_2X_2 + \epsilon
      \end{equation}
      the OLS estimate of $\beta_2$ is 
      $\hat{\beta}_{2,OLS} = \frac{\widehat{Cov}(Y,X_2)}{\widehat{V(X_2)}}$.
      This means that if the correlation (or covariance) between $Y$
      and $X_2$ is positive, then $\hat{\beta}_2$ is positive.  
      Consider now a slightly more complex model:
      \begin{equation}
        Y = \beta_1X_1 + \beta_2X_2 + \epsilon
      \end{equation}
      Let's suppose that $\overline{Y}=\overline{X_1}=\overline{X_2}=0$.
      Please calculate an expression for $\hat{\beta}_{1,OLS}$ and 
      $\hat{\beta}_{2,OLS}$ in terms of the variances and covariances
      of $Y,X_1,X_2$.  Using your formula, discuss whether or not it
      can happen that $\widehat{Cov}(Y,X_1)>0$ while 
      $\hat{\beta}_{1,OLS}<0$.  If it cannot, why not?  If it can,
      then how?  Your goal is both to construct a correct argument
      and to explain in \emph{intuitive} English why your result
      holds.

\item Do problem 9 on page 115 of Greene

\item Do problem 10 on page 115 of Greene


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