ECON 626: Problem Set 2

Published

September 29, 2026

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

Problem 1

Let \(X_{1},...,X_{n}\) be independent and identically distribution and \(P(X_{1}\leq t)=F(t)\) for some function \(F\). Then, write the probability \(P(\max_{1\leq i\leq n}X_{i}\leq t)\) in terms of \(F\).

Problem 2

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.

Problem 3

Suppose \(Y_i = m(X_i' \beta + u_i)\) for \(i=1, ... , n\) with \(X_i \in \R^k\) and \(m:\R \to \R\) is a known function. Throughout, assume that observations are independent across \(i\), \(\Er[u] =0\), and \(u\) is independent of \(X\), and \(\Er[XX']\) is nonsingular. \(Y\) and \(X\) are observed, but \(u\) is not.

  1. If \(m\) is strictly increasing show that \(\beta\) is identified by explicitly writing \(\beta\) as a function of the distribution of \(X\) and \(Y\).

  2. Suppose \(m(z) = 1\{z \geq 0\}\). For simplicitly, let \(k=1\) and \(X_i \in \{-1, 1\}\). Show that \(\beta\) is not identified by finding an observationally equivalent \(\tilde{\beta}\).