Consider the model \(y=\alpha+\beta x+\varepsilon\), where \(\alpha, \beta\), and \(x\) are scalars and \(\varepsilon \sim\) \(\mathcal{N}\left[0, \sigma^{2}\right]\). Generate a sample of size \(N=20\) with \(\alpha=2, \beta=1\), and \(\sigma^{2}=1\) and suppose that \(x \sim \mathcal{N}[2,2]\). We wish to test \(H_{0}: \beta=1\) against \(H_{a}: \beta eq 1\) at level 0.05 using the \(t\)-statistic \(t=(\widehat{\beta}-1) / \operatorname{se}[\widehat{\beta}]\). Do as much of the following as your software permits. Use \(B=499\) bootstrap replications.

(a) Estimate the model by OLS, giving slope estimate \(\widehat{\beta}\).

(b) Use a paired bootstrap to compute the standard error and compare this to the original sample estimate. Use the bootstrap standard error to test \(H_{0}\).

(c) Use a paired bootstrap with asymptotic refinement to test \(H_{0}\).

(d) Use a residual bootstrap to compute the standard error and compare this to the original sample estimate. Use the bootstrap standard error to test \(H_{0}\).

(e) Use a residual bootstrap with asymptotic refinement to test \(H_{0}\).