ECON 626: Problem Set 5
\[ \def\indep{{\perp\!\!\!\perp}} \def\R{{\mathbb{R}}} \def\Er{{\mathrm{E}}} \newcommand\norm[1]{\left\lVert#1\right\rVert} \]
Problem 1
Suppose \(X_n = O_p(a_n)\) and \(Y_n - c = O_p(a_n)\) for \(c \neq 0\) and \(a_n \to 0\). Show that \(\frac{X_n}{Y_n} = O_p(a_n)\).
Hint: \(\frac{X_n}{Y_n} = \frac{X_n}{c} + \frac{X_n}{Y_n} (c - Y_n) \frac{1}{c}\), use Lemma 3.3 and Exericse 3.1 from Song (2021).
Problem 2
Suppose that each individual person \(i\) has potential outcomes \(Y_i(1)\) and \(Y_i(0)\) depending on the treated state or the untreated state. The average treatment effect is defined to be \[ \tau = \Er[Y_i(1) - Y_i(0)]. \] We assume that the econometrician observes \((Y_i,D_i)\), where \(Y_i = D_i Y_i(1) + (1 - D_i)Y_i(0)\), \(D_i\) is a binary variable representing the treatment status, and that \((Y_(1),Y_i(0))\) is independent of \(D_i\). Finally, assume that \(0 < \Er[D_i] <1\), \(Y_i \in [0,1]\) for all \(i =1,...,n\), and that \((Y_i(1),Y_i(0),D_i)\) are i.i.d. across \(i\)’s.
Show that \[ \hat{\tau} = \frac{\sum_{i=1}^n Y_i D_i}{\sum_{i=1}^n D_i} - \frac{\sum_{i=1}^n Y_i (1-D_i)}{\sum_{i=1}^n (1-D_i)} \] is a consistent estimator for \(\tau\).
Show that \(\hat{\tau} - \tau = O_p(n^{-1/2})\)
Show that \(\sqrt{n}(\hat{\tau}-\tau) \to^d N(0,\sigma_\tau^2)\) and calculate \(\sigma_\tau^2\).