20. Let the size n random sample Y Y1;Y2; ... Yn 0 be such that Y xb , where x

20. Let the size n random sample Y ¼ ½  Y1;Y2; ... Yn 0 be such that Y ¼ xb þ «, where x is a n 1 non-zero vector of

“explanatory variable values”, b is an unknown parameter value, and « is an n 1 random vector for which Eð Þ¼ « 0 and covð Þ¼ « s2I.

a. Is tð Þ¼ Y Pn i¼1 xiYi=

Pn i¼1 x2 i the BLUE of b? Explain.

b. Define themean and the variance of the estimator in a).

c. Under what conditions will the estimator in

a) be a consistent estimator of b?

d. Consider an alternative estimatort ð Þ¼ Y x0 ð Þ x þ k 1 x0 Y

for b . (this is an example of a so-called “ridge regression” estimator). Define the mean and variance of this estimator. Is this a linear estimator? Is it unbiased? Is it asymptotically unbiased?

e. Under what conditions will the estimator in

d) be a consistent estimator of b?

f. If you wanted to use the estimator that has the smaller expected squared distance from b , would you prefer one estimator over the other? Explain.