\documentclass{article} \usepackage{amsmath} \begin{document} \section*{1. Consider the following model:} \begin{align*} y_t &= x_t'\beta + \varepsilon_t = \beta_0 + \beta_1x_{1t} + \varepsilon_t, \quad t = 1, \ldots, T \\ Y &= X\beta + \varepsilon \\ E(\varepsilon_t) &= 0, \quad E(\varepsilon_t|x_t) = 0, \quad E(\varepsilon_t^2) = \sigma^2 < \infty, \quad t = 1, \ldots, T \\ E(\varepsilon_t^2|x) &= \sigma^2 \end{align*} where \( x_t = [1, x_{1t}]' \), \( X = \begin{bmatrix} 1 & x_{11} \\ \vdots & \vdots \\ 1 & x_{1T} \end{bmatrix} \) is a \( T \times 2 \) matrix of regressors, \( \varepsilon = \begin{bmatrix} \varepsilon_1 \\ \vdots \\ \varepsilon_T \end{bmatrix} \) is a \( T \times 1 \) matrix of iid errors, \( Y = \begin{bmatrix} y_1 \\ \vdots \\ y_T \end{bmatrix} \), \( \beta = \begin{bmatrix} \beta_0 \\ \beta_1 \end{bmatrix} \) is a \( 2 \times 1 \) matrix of parameters and \( E(x_{1t}) = 0 \). \subsection*{a) The researcher would like to test whether there is a break in \(\beta_0\) at an unknown point in time. Write the \(QLR_{RT}\) Wald test statistic for testing the null hypothesis that \(\beta_0\) is constant against the alternative of a one-time break in \(\beta_0\) at an unknown point in time \(\tau\). Explicitly describe the estimate of the variance you are going to use.} Hint: let \( X_\tau = \begin{bmatrix} 1 & x_{11} \\ \vdots & \vdots \\ 1 & x_{1\tau} \end{bmatrix} \), \( Y_\tau = \begin{bmatrix} y_1 \\ \vdots \\ y_\tau \end{bmatrix} \), \( X_{T-\tau} = \begin{bmatrix} 1 & x_{\tau+1} \\ \vdots & \vdots \\ 1 & x_{1T} \end{bmatrix} \), \( Y_{T-\tau} = \begin{bmatrix} y_{\tau+1} \\ \vdots \\ y_T \end{bmatrix} \), and let \( Wald(\tau) \) denote the Wald test for testing the null hypothesis at a known point in time \(\tau\). The problem requires you to write explicitly the expression of \( Wald(\tau) \). If you get lost, write the test \( QLR_{RT} \) as a function of \( Wald(\tau) \). \end{document} Solve this