Spring 1998 John Rust
Economics 551b 37 Hillhouse, Rm. 27

MIDTERM EXAM (first installment)

(Due: Monday, February 2, 1998)

QUESTION 1 Suppose the random variables are multivariate normal, where the dimension of is and the dimension of is . Show that

where is the vector of least squares coefficients:

In other words, you have shown that when the random variables are normally distributed the best nonlinear predictor and the best linear predictor coincide.

Hint: Show this result in several steps, following the path below:

A.

(Step 1) Show that if are any random variables where is finite and nonsingular, and is finite, then we can write

where is a random variable satisfying where is a vector of 0s.

B.

(Step 2) Use the result in part A to show that can also be written as

(hint) note that if is the random variable in the vector we can write

C.

Show that the above result implies that when or , we have two equivalent expressions for the OLS coefficients :

D.

Now, fill in the details of Greene's exposition of the marginal and conditional distributions of the multivariate normal in section 3.10.1 of his book, and show that if has a joint multivariate normal distribution, then the conditional density f(y|X) (i.e. the conditional density of given that ) is normally distributed, , where

and

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