ECON 626: Problem Set 3
\[ \def\R{{\mathbb{R}}} \def\Er{{\mathrm{E}}} \]
Problem 1
In the linear model, \(Y = \alpha + \beta X + \epsilon\), assume \(\mathrm{Var}(X)>0\) and \(\Er[\epsilon] = 0\). Show that without any more assumptions, \(\beta\) is not identified by finding values of \(\beta\) that are observationally equivalent.
Problem 2
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.
If \(m\) is strictly increasing show that \(\beta\) is identified by explicitly writing \(\beta\) as a function of the distribution of \(X\) and \(Y\).
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 \(\tilde{\beta}\).
Problem 3
Song (2021) Chatper 4, exercise 1.3.
Consider the binary choice model in Example 1.3 and assume that \(\tilde{\beta}_0\) and \(\tilde{\beta}_1\) have appropriate estimators \(\hat{\beta}_0\) and \(\hat{\beta}_1\). Provide a sample analogue estimator of the average derivative, replacing \(\tilde{\beta}_0\) and \(\tilde{\beta}_1\) with \(\hat{\beta}_0\) and \(\hat{\beta}_1\)