Suppose the Xi are IID N (0, 1). Let i = 0.025(X4 i − 3X2 i ) and Yi =

Xi + i. An investigator does not know how the data were generated, and runs a regression of Y on X.

(a) Show that R2 is about 0.97. (This is hard.)

(b) Do the OLS assumptions hold?

(c) Should the investigator trust the usual regression formulas for standard errors?

Hints. Part

(a) can be done by calculus—see the end notes to chapter 5 for the moments of the normal distribution—but it gets a little intricate.

A computer simulation may be easier. Assume there is a large sample, e.g., n = 500.