ECON 626: Problem Set 8

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

Problem 1

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.

Problem 2

¹

Suppose that we observe \(Y_1,...,Y_n\) and \(X_1,...,X_n\) such that \[ \begin{eqnarray*} Y_i = f(X_i;\beta) + u_i, \end{eqnarray*} \] where, \(\beta = (\beta_0,...,\beta_{k-1})' \in \mathbf{R}^k\), \[ \begin{eqnarray*} f(x;b) = b_0 + b_1 x + b_2 x^2+....+ b_{k-1}x^{k-1}, \quad b = (b_0,b_1,...,b_{k-1})', \end{eqnarray*} \] and \(X_i, u_i\) are i.i.d. with \(\Er[|X_i|^{2(k-1)}]<\infty\) and \(\Er[|X_i^{k-1} u_i|^2] < \infty\). Assume that $is nonsingular, where \(\tilde X_i = [1,X_i,X_i^2,...,X_i^{k-1}]'\). Our focus on the predicted value of \[ \begin{eqnarray*} \theta_0 = f(2.5;\beta). \end{eqnarray*}\medskip \]

Asymptotic Distribution

Let \(\hat \beta\) be the least squares estimator of \(\beta\). Find out the asymptotic distribution of \(\hat \theta = f(2.5;\hat \beta)\).

Confidence Interval

Construct a confidence interval \(C\) for the predicted value \(\theta_0\) using \(\hat \theta\) such that \[ \begin{eqnarray*} P\{\theta_0 \in C\} = 0.95. \end{eqnarray*} \]

Testing Linearity

We would like to test whether \(f(x;b)\) is linear in \(x\) or not. Let \(\gamma = (\beta_2,\beta_3,...,\beta_{k-1})'\), and consider the following hypothesis: \[ \begin{eqnarray*} H_0&:& \gamma = 0, \text{ against}\\ H_1&:& \gamma \ne 0. \end{eqnarray*} \] Provide a test with size 5%. Give a brief explanation of why the test works (i.e., why it has size 5% and has nontrivial power under the alternative hypothesis).

Footnotes

1. Adapted from the final for this course in 2019.