ECO 4451 London School Prediction with Regressors and Big Data Questions

14.8 Let X and Y be two random variables. Denote the mean of Y given X = x byµ,(x) and the variance of Yby u2(x).

a. Show that the best (minimum MSPE) prediction of Y given X = xisµ,(x) and the resulting MSPE is u 2(x). (Hint: Review Appendix 2.2.)

b. Suppose Xis chosen at random. Use the result in (a) to show that thebest prediction of Y is µ,(X) and the resulting MSPE is E[Y – µ,(X)]2 =E[u2(X)].

14.9 You have a sample of size n = 1 with data Y I = 2 and xI = 1. You areinterested in the value of f3 in the regression Y = X f3 + u. (Note there is nointercept.)

a. Plot the sum of squared residuals (Yi – bxI ) 2 as function of b .

b. Show that the least squares estimate of f3 is ffi0 Ls = 2.

c. Using “-Ridge = 1, plot the ridge penalty term “-Ridgeb 2 as a function of b.

d. Using “-Ridge = 1, plot the ridge penalized sum of squared residuals(YI – bx1)2 + “-Ridgeb 2•

f. Using “-Ridge = 0.5, repeat (c) and (d). Find the value of feRidge.

g. Using “-Ridge = 3, repeat (c) and (d). Find the value of ffiRidge.

e. Find the value of ffiRidge.