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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, \textbf{Solution}
\normalsize

\end{centering}
\vspace{0.3in}


\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? \vspace{3\baselineskip}

                \textbf{
                Yes, it is. Observe:
                }

                \begin{eqnarray}
                \widetilde{\beta}_1 
                   &=& \frac{\sum X_i^2Y_i}{\sum X_i^3}
                   \nonumber\\
                   &=& \frac{\sum X_i^2(\beta_1X_i + \epsilon_i)}{\sum X_i^3}
                   \nonumber\\
                   &=& \frac{\sum X_i^2(\beta_1X_i)}{\sum X_i^3} 
                        + \frac{\sum X_i^2 \epsilon_i}{\sum X_i^3}
                   \nonumber\\
                   &=& \beta_1 \frac{\sum X_i^2(X_i)}{\sum X_i^3} 
                        + \frac{\sum X_i^2 \epsilon_i}{\sum X_i^3}
                   \nonumber\\
                   &=& \beta_1 
                        + \frac{\sum X_i^2 \epsilon_i}{\sum X_i^3}
                   \nonumber\\
                   \nonumber\\
                E \left( \widetilde{\beta}_1 \right) 
                   &=& \beta_1 
                        + \frac{\sum X_i^2 E(\epsilon_i)}{\sum X_i^3}
                   \nonumber\\
                   &=& \beta_1 
                   \nonumber
                \end{eqnarray}

                \pagebreak
          \item What is the variance of $\widetilde{\beta}_1$? 
                \vspace{3\baselineskip}

                \begin{eqnarray}
                V \left( \widetilde{\beta}_1 \right) 
                   &=& V \left( \beta_1 
                        + \frac{\sum X_i^2 \epsilon_i}{\sum X_i^3} \right)
                   \nonumber\\
                   &=& V \left(
                         \frac{\sum X_i^2 \epsilon_i}{\sum X_i^3} 
                         \right)
                   \nonumber\\
                   &=&  \frac{V \left( \sum X_i^2 \epsilon_i \right)}
                        {\left( \sum X_i^3 \right)^2}
                   \nonumber\\
                   &=&  \frac{\sum X_i^4 V(\epsilon_i) 
                              + \underset{\sum \sum X_i^2 X_j^2 
                                           Cov \left(\epsilon_i, \epsilon_j 
                                               \right)
                                         }{i \ne j}
                             }
                        {\left( \sum X_i^3 \right)^2}
                   \nonumber\\
                   &=&  \frac{\sum X_i^4 \sigma^2} 
                        {\left( \sum X_i^3 \right)^2}
                   \nonumber\\
                   &=&  \sigma^2 \frac{\sum X_i^4} 
                        {\left( \sum X_i^3 \right)^2}
                   \nonumber
                \end{eqnarray}

                \pagebreak
          \item Is the variance of the OLS estimator greater than,
                less than, or the same as the variance of 
                $\widetilde{\beta}_1$?
                \vspace{3\baselineskip}

                \textbf{
                Well, $\widetilde{\beta}_1$ is a linear unbiased 
                estimator of $\beta_1$.  Thus, by the Gauss-Markov
                Theorem, its variance must be no less than the variance
                of $\widehat{\beta}_{1,OLS}$.  And, since the two
                generally have different variances, generally OLS will
                have a lower variance.
                }

        \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?
        \vspace{3\baselineskip}

        \textbf{
        It is a bad idea.  Since shoes are produced in pairs, 
        Right Shoes = Left Shoes.  This is a linear dependence in
        $X$.  This violates one of the assumptions of the CLRM.
        Furthermore, it means that OLS cannot be estimated, since
        $X'X$ is not invertible.
        }

        \textbf{
        Notice a critical distinction.  It may be that left shoes
        are more expensive to produce than are right shoes, so
        that the model above may make a kind of sense.  However,
        since shoes are always produced in pairs, there is no way
        we can estimate the difference in the shoe costs this way.
        If we could find a factory which did not always produce 
        shoes in pairs, we could find out about the difference in
        costs for left and right shoes.
        }

        \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?
        \vspace{3\baselineskip}

        \textbf{
        The critic is right about the assumption violation.  Both
        the price of wine and the price of the car are caused by
        underlying characteristics of the person (say income and/or
        taste for luxury).  However, the critic is wrong about the
        use of OLS.  All we are trying to do here is \emph{predict}
        car price based on wine price.  Since OLS is essentially
        always the best linear predictor, it is perfectly appropriate
        to use it here.
        }

        \textbf{
        This is an important distinction to get right.  If all we care
        about is predicting $Y$ using $X$ and we have no good reason 
        to believe that a linear predictor is a bad thing, we should
        use OLS.  If, on the other hand, we are trying to discover
        the size of a causal relationship between $Y$ and $X$ (that
        is, we care about the magnitude and meaning of $\beta$) we
        have a persuasive argument in favor of using OLS only when 
        the CLRM is true.
        }

        \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?
        \vspace{3\baselineskip}

        \textbf{
        They all make intuitive sense.  A one dollar increase in income 
        (holding const the other variables) is associated with a 0.5 cent
        increase in health spending.  A one year increase in age (...)
        is associated with a 23 dollar increase in health spending.  Male
        sex is associated with 48 dollars less in health spendingl.
        Employment is associated with 1502 dollars less in health spending
        (sensible since very sick people are usually not employed).  
        Insurance is associated with 1588 dollars more in health spending
        (sensible since you consume more of something that is less
        expensive to you and insurance lowers the price of health spending).
        Finally, people who perceive themselves as sicker spend more.
        }

        \pagebreak
  \item For which of the coefficients can we reject, at the 5\% level,
        the null hypothesis that the coefficient is equal to zero?
        \vspace{3\baselineskip}

        \textbf{
        Look for t value $\ge$ 1.96 or P value $<$ 0.05.  This is true
        for age, employment, insurance, and health status.
        }

        \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?
        \vspace{3\baselineskip}

        \textbf{
        My best estimate (best by G-M Thm) would be 1588-1502=86.  To
        make a 90\% CI, I need the variance of 
        $\widehat{\beta}_{emp,OLS} + \widehat{\beta}_{ins,OLS}$ and 
        this variance is equal to
        \begin{math}
        V \left( \widehat{\beta}_{emp,OLS} \right)
        + V \left( \widehat{\beta}_{ins,OLS} \right)
        + 2 Cov \left( \widehat{\beta}_{emp,OLS}, \widehat{\beta}_{ins,OLS} 
                \right)
        \end{math}.
        }

        \textbf{
        Looking at the output on page 5, we see that 
        $V \left( \widehat{\beta}_{emp,OLS} \right) = 44934$, 
        $V \left( \widehat{\beta}_{ins,OLS} \right) = 48773$, and
        \begin{math}
        Cov \left( \widehat{\beta}_{emp,OLS}, \widehat{\beta}_{ins,OLS}
            \right) = -556 
        \end{math}.
        }

        \textbf{
        Thus, the variance of the sum is $44934+48773-1112=92595$,
        and the standard error of the sum is 304.  So, our 90\%
        confidence interval is $86 \pm 1.65(304)$ or 
        $86 \pm 502$.  Since zero is inside this confidence interval,
        we cannot conclude at the 10\% significance level that the
        effect of going from unemployed and uninsured to employed
        and insured on health spending is different from zero.
        }

        \pagebreak
  \item If we were to regress spending on income alone, would the 
        slope coefficient be positive or negative and why?
        \vspace{3\baselineskip}

        \textbf{
        Negative.  Look at page 3 at the correlation matrix.  The 
        correlation between income and spending is -0.02.  Since the
        bivariate regression coefficient for the slope
        has the same sign as the 
        simple correlation, we know that the bivariate regression
        coefficient will be negative.
        }

  \end{enumerate}

\end{enumerate}

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