Consider the following model: y_t = x_t' \beta + \epsilon_t = \beta_0 + \beta_1 x_{1t} + \epsilon_t, t = 1, ..., T \\ Y = X \beta + \epsilon, \\ E(\epsilon_t) = 0, E(\epsilon_t|x_{1t}) = 0, E(\epsilon_t^2) = \sigma^2 < \infty, t = 1, ..., T \\ E(\epsilon_t^2 | x) = \sigma^2 where x_t = [1, x_{1t}]', X = \begin{pmatrix} 1 & x_{11} \\ ... & .... \\ 1 & x_{1T} \end{pmatrix} is a T \times 2 matrix of regressors, \epsilon = \begin{pmatrix} \epsilon_1 \\ ... \\ \epsilon_T \end{pmatrix} is a T \times 1 matrix of iid errors, Y = \begin{pmatrix} y_1 \\ ... \\ y_T \end{pmatrix}, \beta = [\beta_0, \beta_1]' is a 2 \times 1 matrix of parameters and E(x_{1t}) = 0. The researcher would like to test whether there is a break in \beta_0 at an unknown point in time. Write the QLR Wald test statistic QLR_T 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{pmatrix} 1 & x_{11} \\ ... & ... \\ 1 & x_{1\tau}\end{pmatrix}, Y_\tau = \begin{pmatrix} y_1 \\ ... \\ y_\tau \end{pmatrix}, X_{T-\tau} = \begin{pmatrix} 1 & x_{1\tau + 1} \\ ... & ... \\ 1 & x_{1T}\end{pmatrix}, Y_{T-\tau} = \begin{pmatrix} y_{\tau + 1} \\ ... \\ y_T \end{pmatrix} and let Wald_T(\tau) denotes 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_T(\tau). If you get lost, write the test QLR_T as a function of Wald_T(\tau)