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\noindent{\bf BOSTON COLLEGE} \\
 Department of Economics \\
 EC 771: Econometrics \\
 Spring 2011 \\
 Prof. Baum, Mr. Smith 
 
 \bigskip
\noindent {\sc Problem Set 2: due Monday 21  February at classtime}
 

\medskip\noindent
% DM 2.13
1. Consider the two regressions
 \begin{eqnarray*}
y = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + u \\
y = \alpha_1 z_1 + \alpha_2 z_2 + \alpha_3 z_3 + u
\end{eqnarray*}
where $z_1 = x_1 - 2 x_2$, $z_2 = x_2 + 4 x_3$ and $z_3 = 2 x_1 - 3 x_2 + 5 x_3$. Let $\mathbf{X} = [ x_1~ x_2~ x_3]$ and $\mathbf{Z} = [z_1~ z_2 ~z_3]$. Show that the columns of $\mathbf{Z}$  can be expressed as linear combinations of the columns of $\mathbf{X}$, that is, $\mathbf{Z = X A}$ for some $3 \times 3$ matrix $\mathbf{A}$. Find the elements of this matrix.

Show that the matrix $\mathbf{A}$ is invertible by showing that the columns of $\mathbf{X}$ are linear combinations of the columns of $\mathbf{Z}$. Find the elements of $\mathbf{A^{-1}}$. Show that the two regressions give the same fitted values and residuals.

Precisely how is the OLS\ estimate $b_1$ related to the OLS\ estimate $a_1$?
 
\medskip\noindent
% DM 2.23
2. Use the dataset {\tt http://fmwww.bc.edu/ec-p/data/greene2008/tbrate} in Stata for this exercise. It contains data (1950q1-1996q4) on three series: $r_t$, the three-month Treasury bill rate, $\pi_t$, the rate of inflation and $y_t$, the log of real GDP. For the period 1950q4--1996q4, run the regression 
\[
\Delta r_t = \beta_1 + \beta_2 \pi_{t-1} + \beta_3 \Delta y_{t-1} + \beta_4 \Delta r_{t-1} + \beta_5 \Delta r_{t-2} + u_t
\]
(Hint: use the Stata timeseries operators {\tt D.} and {\tt L.}). Plot the residuals and fitted values against time (hint: {\tt predict} and {\tt tsline}). Regress the residuals on the fitted values and a constant. What do you learn from this second regression? Now regress the fitted values on the residuals and a constant. What do you learn from this third regression?

\medskip\noindent
% DM 2.24
3. For the same sample period, regress $\Delta r_t $ on a constant, $\Delta y_{t-1} $, $\Delta r_{t-1}$ and $\Delta r_{t-2} $. Save the residuals from this regression (call them $\hat{e}_t$). Then regress $\pi_{t-1}$ on the same regressors. Save the residuals from this regression and call them $\hat{v}_t$. Now regress $\hat{e}_t$ on $\hat{v}_t$. How are the estimated coefficients and residuals from this last regression related to anything that you obtained when you ran the regression in exercise 3?

\medskip\noindent
% DM 3.6
4. For any pair of random variables $b_1, b_2$, show, by using the fact that the covariance matrix of $\mathbf{b} = (b_1 ~b_2)$  is positive semidefinite, that
\[ 
(Cov(b_1,b_2))^2 \le Var(b_1) Var(b_2)
\]
 and show that the correlation between $b_1,b_2$ must lie between $-1, +1$. 
 
 \medskip\noindent
% DM 3.20
5. Consider the linear regression model
\[
y_t = \beta_1 + \beta_2 x_{t2} + \beta_3 x_{t3} + u_t
\]
Explain how you could estimate this model subject to the restriction that $b_2 + b_3 = 1$ by running a regression that imposes the restriction. Also explain how you could estimate the unrestricted model in such a way that the value of one of your coefficients would be zero if the restriction held exactly in your data.

 \medskip\noindent
 6. Using Stata routine \texttt{ivreg2} ({\tt ssc install ivreg2}, {\tt ssc install ranktest} if needed; full details on {\tt ivreg2} in Baum--Schaffer--Stillman papers and {\tt help ivreg2}): 

\begin{verbatim}
use http://fmwww.bc.edu/ec-p/data/hayashi/griliches76.dta
xi i.year
\end{verbatim}

a. Estimate the regression of log wage (lw) on experience (expr), years of schooling (s) and iq, considering iq as potentially mismeasured; instrument the equation with age, kww and med (mother's years of education). What is the identification status of this equation?

b. These data are pooled cross-section time-series (but not a panel). Introduce time effects (the year dummies) and reestimate the equation. What effect has this had on the model? What do you conclude?

c. Reestimate the equation of part b using robust standard errors. What effect does this have on the estimated model?

d. Reestimate the equation of part b using generalized method of moments (IV-GMM) with robust standard errors. What effect does this have on the model?  What assumptions have been relaxed vis-\`a-vis the model estimated in part b? How do you interpret the Hansen $J$ statistic for this model? How do you interpret the Kleibergen--Paap  rk statistic?

e. Use the IV-GMM\ {\tt endog} option (the ``C'' or GMM distance test) to test a subset of the orthogonality conditions: the exogeneity/endogeneity of $s$ (years of schooling). What do you conclude?

f. Reestimate the model treating $s$ as endogenous. What does the Hansen J\ test signify in this context?
 
 
 \end{document}  
 
 
  \medskip
 \noindent 
 % DM 2.8
 1. Given the three vectors $x_1, x_2, x_3$:
 \[
\left(
\begin{array}{ccc}
1  & 0   & 1  \\
1  & 4  & 0   \\
1 & 0 & 1 \\
1  & 4  & 0 \\
1 & 0 & 1    
\end{array}
\right)
\]
Show that any vector $z = b_1 x_1 + b_2 x_2$ in the subspace $\delta(x_1,x_2)$ also belongs to the subspace  $\delta(x_1,x_3)$ and  $\delta(x_2,x_3)$, where the subspace $\delta(\cdot)$ of $\mathbb{R}^3$ is the vector space spanned by the arguments. Give explicit formulas for $z$ as a linear combination of $x_1,x_3$ and of $x_2, x_3$.
