ECON 626: Problem Set 2

Published

September 21, 2023

\[ \def\R{{\mathbb{R}}} \def\Er{{\mathrm{E}}} \]

Problem 1

Let \(f: \R^2 \to \R\). Assume \(f(\cdot, y): \R \to \R\) is measurable for all \(y : |y - y_0| < \epsilon\) for some \(\epsilon>0\). Also assume that \(f(x, \dot): \R \to \R\) is differentiable at \(y_0\) for all \(x \in A \subset \R\) and \(\left\vert \frac{\partial f}{\partial y}(x, y_0) \right \vert \leq M(x)\) for all \(x \in A\) and \(M(x)\) is integrable. Show that \[ \frac{d}{dy} \int_A f(\cdot, y) d\mu \vert_{y=y_0} = \int_A \frac{\partial f}{\partial y} f(\cdot, y_0) d\mu. \]

Problem 2

  1. Show that if \(E[Y |X] = 0\), then \(Y\) and \(g(X)\) are uncorrelated for any (Borel) measurable function \(g\). Do you have the same conclusion if we weaken the condition \(E[Y |X] = 0\) to \(E[Y |X] = a\) for some fixed constant \(a \in \R\)?

  2. Suppose that \(E[Y^2 |X] < a^2\) for some constant \(a > 0\). Then, show that for any \(b > 0\), \[ P\left(|Y − EY | > b\right) \leq \frac{a^2}{b^2}. \]

  3. Show that if \(E[Y | \exp(X)] = E[Y]\), then the correlation between \(E[Y | cos(X)]\) and \(cos(X)\) is zero.

Problem 3

Let \(X: \Omega \to \R\), \(W: \Omega \to \R\), \(Z: \Omega \to \R\), and \(D: \Omega \to \{0, 1\}\) be random variables. Let \(Y = D X + (1-D) W\). Suppose that \(Y\), \(D\), and \(Z\) are observed, but \(X\) and \(W\) are not.

  1. Suppose \(D\) is independent of \(X\), \(W\). Then show that \(\Er[X - W]\) is identified.

  2. Suppose \(Z\) is independent of \(X\) and \(W\), and \(\exists E_1, E_0 \in \sigma(Z)\) such that \(P(D=1 | E_1) = 1\) and \(P(D=0|E_0) = 1\). Then show that \(\Er[X-W]\) is identified.