You travel to the planet Estimation as part of an economics expedition. Families are organized as pairs of individuals identified as alphas and betas. You are interested in their wage distributions.

Let Xa,Xb denote the wages of a family pair. Let ¹a, ¹b, ¾2

a, ¾2 b and ¾ab denote the means, variances, and covariance of the alpha and beta wages within a family. You gather a random sample of n families and obtain the data set {(Xa1,Xb1) , (Xa2,Xb2) , ..., (Xan,Xbn)}. We assume that the family pairs (Xai ,Xbi )

aremutually independent and identically distributed across i .

(a) Write down an estimator bµ for µ Æ ¹b ¡¹a, the wage difference between betas and alphas.

(b) Find var[Xb ¡Xa]. Write it in terms of the parameters given above.

(c) Find the asymptotic distribution for p

n

¡bµ¡µ

¢

as n!1. Write it in terms of the parameters given above.

(d) Construct an estimator of the asymptotic variance. Be explicit.