ECON 626: Problem Set 6
\[ \def\Er{{\mathrm{E}}} \def\En{{\mathbb{En}}} \def\cov{{\mathrm{Cov}}} \def\var{{\mathrm{Var}}} \def\R{{\mathbb{R}}} \newcommand\norm[1]{\left\lVert#1\right\rVert} \def\rank{{\mathrm{rank}}} \newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}} \def\inprob{{\,{\buildrel p \over \rightarrow}\,}} \def\indist{\,{\buildrel d \over \rightarrow}\,} \DeclareMathOperator*{\plim}{plim}\]
Problem 2
In the linear model, \[ y = \underbrace{X}_{n \times k} \beta + \epsilon, \] partition \(X\) as \[ X= \begin{pmatrix} \underbrace{X_1}_{n \times k_1} & \underbrace{ X_2}_{n \times (k - k_1)} \end{pmatrix} \]
and \(\beta = \begin{pmatrix} \beta_1\\ \beta_2 \end{pmatrix}\).
Let \(\hat{\beta} = \begin{pmatrix} \hat{\beta}_1\\ \hat{\beta}_2 \end{pmatrix}\) be the OLS estimator.
Show that \[ \hat{\beta}_1 = \textrm{arg}\min_{b_1} \norm{M_{X_2} y - M_{X_2} X_1 b_1 }^2 \] where \(M_{X_2} = I - X_2 (X_2' X_2)^{-1} X_2'\).
Problem 3
Consider estimating the linear model,1 \[ y_i = \beta D_i + x_i' \alpha + \epsilon_i \] where \(\beta \in \R\), \(D_i \in \{0,1\}\) and \(x_i \in \R^k\).
Show that the OLS estimate for \(\beta\) can be written as \[ \hat{\beta} = \frac{1}{n} \sum_{i=1}^n y_i \hat{\omega}(x_i,D_i) \] where \(\frac{1}{n} \sum_{i=1}^n D_i \hat{\omega}(x_i,D_i) = 1\), and \(\frac{1}{n} \sum_{i=1}^n (1-D_i) \hat{\omega}(x_i,D_i) = -1\). [Hint: use the result of problem 2.]
Let \[ \pi \in \mathrm{arg}\min_{\pi} \Er[(D_i - x_i'\pi)^2]. \] Show that \[ \hat{\omega}(x,D) \inprob \frac{D - x'\pi}{\Er[(D - x'\pi)^2]} \] If needed, state additional assumptions about dependence or moments.
Suppose that the linear model being estimated might not be the data generating process. Instead, assume the data comes from a potential outcomes framework. There are potential outcomes \((y_i(1),y_i(0))\). You observe \(y_i = y_i(1)D_i + y_i(0)(1-D_i)\). Show that \[ \hat{\beta} \inprob \Er[y_i(0) \omega(x_i,D_i)] + \Er\left[\left(y_i(1) - y_i(0)\right) D_i \omega(x_i,D_i) \right]. \]
If you assume that \((y_i(1),y_i(0))\) are independent of \(D_i\) conditional on \(x_i\), does \(\hat{\beta}\) have any nice interpretation? [Hint: what can be the range of \(\omega(x,1)\), especially if the range of \(x\) is large?]