Session 12

Exercise Session 12, December 1st ,2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua Chen (Cissy), Natalia Guseva

Problem 1 OLS estimator in …nite sample. Recall b = (X 0 X) 1. Find the sampling error. (Hint: b

1

X 0Y

)

2. Show OLS estimator is unbiased. (Hint: E(b j X) = ) 3. Find the variance of b for given X: Problem 2 Let ( ; F; P) be a probability space. There exists two random variables X and Y . If we can observe X,what we can Rsay about Y ? One way is to 2 use mean square error, M SE = E(Y g(X))2 = fY (!) g(X(!))g dP (!). Show that g (X) = E(Y j X) is the best solution. Problem 3 Variable X is normally distributed with mean

and variance

2

.

1. Assume 2 = 80: The observed value of the sample mean X of a random sample of size 20 is 81:2. Find a 95% con…dence interval for . 2. Assume mately.

2

= 9:Find n such that P r(X

1<

< X + 1) = 0:9 approxi-

Problem 4 Show for any two random variables X and Y , V arX = E(V ar(X j Y )) + V ar(E(X j Y )): Problem 5 Suppose the distribution of Y conditional on X = x is N (x; x2 ) and the marginal distribution of X is uniform (0; 1):Find EY; V arY; and Cov(X; Y ):

1