ECON 626: Problem Set 7

Published

December 6, 2022

\[ \def\Er{{\mathrm{E}}} \def\En{{\mathbb{En}}} \def\cov{{\mathrm{Cov}}} \def\var{{\mathrm{Var}}} \def\R{{\mathbb{R}}} \newcommand\norm[1]{\left\lVert#1\right\rVert} \def\rank{{\mathrm{rank}}} \newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}} \def\inprob{{\,{\buildrel p \over \rightarrow}\,}} \def\indist{\,{\buildrel d \over \rightarrow}\,} \DeclareMathOperator*{\plim}{plim} \]

Problem 1

Consider the following linear regression model such that \[ Y_i = β_0 + X_i β_1 + u_i , \] where \(X_i\) and \(Y_i\) are observed random variables. Let us assume that \(\Er [u_i ] = 0\) but \(\cov(X_i , u_i ) \neq 0\). Suppose that there exists a variable \(Z_i\) such that \(\cov(X_i , Z_i ) > 0\) and \(\cov(Z_i , u_i ) > 0\).

Find the asymptotic bias of the 2SLS estimator of \(\hat{\beta}_1\). (Recall that the asymptotic bias of an estimator is its probability limit minus the true parameter.) Can you determine unambiguously whether the 2SLS estimator tends to underestimate or overestimate the parameter \(β_1\) ? If so, give explanations how.

Problem 2

In the linear model, \[ Y_i = \beta_0 + X_i \beta_1 + u_i \] assume that \(\Er[u_i] = 0\) and \(X_i \in \R^1\). Suppose that \(\Er[X_i u_i] \neq 0\), but, somewhat strangely, you assume \(\Er[u_i^2|X_i] = \sigma^2\).

  1. Show that a set of two elements that contains \(\beta_1\) is identified. Denote this set by \(B_1\). Hint: use the moment condition \(\Er[u_i^2 (X_i-\Er[X_i])]\).

  2. Describe an estimator for \(B_1\) and show that it is consistent. State any additional assumptions needed.

  3. Find the asymptotic distribution of your estimator for \(B_1\). State any additional assumptions needed.