Please explain in detail.

1. (40 pts) Suppose we observe data from the model Yij = XiB tej for i = 1, ..., n and j = 1, ..., m, where 3 is an unknown scalar regression coefficient. We assume X;s are i.i.d, and the random errors cjs are uncorrelated with Xis. For simplicity, we also assume the mean-zero es are i.i.d across i = 1, ...,n (they may be correlated across j unless state otherwise in the subquestions). We are interested in testing the hypothesis Ho : B = 0. (a) (10 pts) Explain how to use permutation test to test the hypothesis. (b) (10 pts) Explain how to use residual bootstrap to test the hypothesis. (c) (10 pts) Suppose eij i.i.d follows N(0, 2) across i = 1, ..., n and j = 1, ..., m, where of is some unknown parameter. Utilize this information and improve the residual bootstrap testing procedure in (b). (d) (10 pts) Suppose Var(eij | Xi) = o'X?, can we still use the residual bootstrap in (b) to test the hypothesis? If not, provide an alternative solution