Generate a sample of size 20 according from the following dgp. The two regressors are generated by \(x_{1} \sim \chi^{2}(4)-4\) and \(x_{2} \sim 3.5+\mathcal{U}[1,2]\); the error is from a mixture of normals with \(u \sim \mathcal{N}[0,25]\) with probability 0.3 and \(u \sim \mathcal{N}[0,5]\) with probability 0.7 ; and the dependent variable is \(y=1.3 x_{1}+0.7 x_{2}+0.5 u\).

(a) Estimate by OLS the model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+u\).

(b) Suppose we are interested in estimating the quantity \(\gamma=\beta_{1}+\beta_{2}^{2}\) from the data. Use the least-squares estimates to estimate this quantity. Use the delta method to obtain approximate standard error for this function.

(c) Then estimate the standard error of \(\widehat{\gamma}\) using a paired bootstrap. Compare this to se \([\widehat{\gamma}]\) from part

(b) and explain the difference. For the bootstrap use \(B=25\) and \(B=200\).

(d) Now test \(H_{0}: \gamma=1.0\) at level 0.05 using a paired bootstrap with \(B=999\). Perform bootstrap tests without and with asymptotic refinement.