\documentclass{article}

\begin{document}
\pagestyle{empty}

\begin{centering}
\large
Intro to Econometric Theory \linebreak
Heinz School, Carnegie Mellon University \linebreak
90-906, Spring 2005-6 \linebreak
\vspace{0.2in} \linebreak
Homework \#2, due Friday, March 10, 2006
\normalsize

\end{centering}
\vspace{0.3in}

\begin{enumerate}
\item Consider a sample of people who were asked to report their
      sex (Everyone answers, and everyone answers either male or
      female).  From their answers, we construct the following 
      matrix:

      \begin{equation}
      X = \left[ \begin{array}{c c c} \mathbf{1} & M & F \end{array} \right]
      \end{equation}

      In the matrix, each row corresponds to one person's response;
      $\mathbf{1}$ is a column of 1s, $M$ is a column with a 1 in
      the $ith$ row if the $ith$ person answered male and a zero 
      otherwise, and $F$ is a column with a 1 in the $ith$ row if
      person $i$ answered female and a zero otherwise.

      Is $X'X$ invertible.  Why or why not? 

\item We discussed in class that variance matrixes are always 
      positive semi-definite but not always positive definite.
      In the case of scalar random variables, a singular 
      variance matrix means a zero variance --- that is, that
      the random variable is not really random at all, but fixed.

      Now, let's think about what a positive semi-definite but
      not positive definite (hereafter PSDNPD) 
      variance matrix means for the random
      vector it corresponds to.  Let x be a random vector with
      PSDNPD variance matrix V.

      \begin{enumerate}
      \item Must there be a fixed non-zero vector $\alpha$, such that
            $\alpha'V\alpha$=0?  Why or why not?
      \item Suppose there is such an $\alpha$, what would the variance
            of $\alpha'x$ be?
      \item What does this tell us about $\alpha'x$?
      \item Does this mean that $\alpha'x=0$?
      \item So, what does it mean for a random vector to have a
            PSDNPD variance matrix?
      \end{enumerate}


\item You are interested in the effect of going to the emergency room on
      a person's life expectancy.  There is a sample of people collected
      on a particular day, some of whom went to an emergency room and
      some of whom did not.  They are followed over time until they die.
      From each person there are two items in the dataset: $T_i$ the
      length of time from the day of the survey until death and 
      $ER_i$ a variable which equals one if they went to the emergency room
      and zero if they did not.  You wish to run a regression, but the
      people who conducted the study will not let you have the data, only
      summary statistics.  What should you ask for and how will you use
      it? \label{q:er}

\item Considering the example in problem \ref{q:er}, consider two 
      linear models:
      \begin{enumerate}
        \item $T_i = \beta_1 + \beta_2ER_i + \epsilon_i$
        \item $ER_i = \beta_3 + \beta_4T_i + \epsilon_i$
      \end{enumerate}
      Let's call the OLS estimates you would get from regressions run
      on these equations $\widehat{\beta}_{1,OLS},\widehat{\beta}_{2,OLS},
      \ldots$.  What relationship will there be between 
      $\widehat{\beta}_{2,OLS},\widehat{\beta}_{4,OLS}$?  What sign ($\pm$)
      do you expect for $\widehat{\beta}_{2,OLS},\widehat{\beta}_{4,OLS}$?
      \label{q:er-models}  

\item We usually interpret regression models as embodying some causal
      story (the RHS causing the LHS); however, OLS does not ``know''
      this: it produces estimates regardless of whether our implicit
      causal story is right.  Discuss this fact in the context of the
      models in question \ref{q:er-models} and your expectations about
      the signs of the relevant coefficients.

\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.

\end{enumerate}


\end{document}
