1 Problem Set 3 Problem 1 If a vector x of random variables has a normal distribution with mean (vector) and variance (matrix) , then S = (x )0 1 (x...

Answer to all 3 parts of problem 1? (PDF attached)

Problem 1 If a vector x of random variables has a normal distribution with mean (vector)

and variance (matrix) , then S = (x????)0????1(x????) has a chi-squared (2) distribution

with parameter (degrees of freedom) equal to the number of variables in X. If you know

eigen-vectors and -values you can show this yourself. Use this fact to nd an asymptotic

test of the equality of the male and female regression coecients in the wage regressions

estimated separately on the male and female samples. We do not make the assumption that

the variances of the errors in the two regressions are equal as was done in the Oaxaca paper.

(i) Derive the variance of the dierence of the male and female regression coecients. Do

we have to worry about their covariance? Why (not)?

(ii) Assume that the CLR assumptions hold in the male and female populations and that

the variance of the errors in the two populations is known. Suggest a test statistic

that under the null hypothesis of equal coecients has a 2 distribution. What is the

parameter, i.e. the degrees of freedom of this distribution? Hint: Use the fact that the

sum or dierence of (vectors of) random variables that have the normal distribution

also have the normal distribution.

(iii) If the variances of the errors are unknown and potentially unequal, suggest a test that

in large samples has a 2 distribution if the null hypothesis is correct. Show that the

test statistic has the same distribution under the null if the CLR assumptions do not

hold, but we assume that the independent variables and the errors are uncorrelated.

Hint: you can cite results in the lecture notes.