Consider the linear model yi = ????1xi1 + ????2xi2 + ⋯ + ????40xi,40 + ????i with ????1 = 1 and ????2 = ⋯ = ????40 = 0, where xij = ui + vj with {ui}, {vj}, and

{????i} being iid N(0, 1) random variables.

a. Find the correlation between y and x1, y and xj for j ≠ 1, and xj and xk for j ≠ k, and the multiple correlation between y and the set of explanatory variables.

b. Using this model, randomly generate n = 100 observations on the 41 variables. Use the lasso to select a model, for a variety of ???? smoothing parameter values. Summarize results, and evaluate the effectiveness of this method.

c. Specify alternative values for {????j} for which you would not expect the lasso to be effective. Re-generate y, and summarize results of using the lasso.