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1. Suppose we want to fit the no-intercept model y Br + e using weighted least squares. Assume the the observations are uncorrelated but have unequal variances. Find the weighted least-squares estimator for B, and its variance. 2. (MSSC) Suppose we fit the model y = X 8. + when the true model is actually y XB.+XB2 + . For both models, assume E() = 0 and Var(e) = oI. Find the expected value and variance of the ordinary least-squares estimate b. Under what conditions is the estimate unbiased? 3. (MSSC) Let W and y be the design matrix and response vector after unit length scaling of the design matrix X with no column of ones and y. That is wy and VEM (tis-*,) -9) i = 1,2,...n, j = 1,2,...,k. Show that W'W is a correlation matrix [rukxk, where ry is the simple correlation coefficient between regressor , and xy, and W'y is (ry, Tayy..., Tky)', where ray is the simple correlation coefficient between regressor x, and the response y. 4. (MSSC) Ridge regression is one of the penalized regression methods. Show that the ridge regression estimator br = (X'X +81) - 'X'y is the minimizer of the (y - XS) (y - X 8) +88'8 op = 9