Generalized Least Squares

• Consider

y = X + u; Euu0 = :

Show that GLS is BLUE.

• Consider the model

We know that the statistical properties of the OLS estimator of improve as N increases. So consider doubling the sample size by just using every observation twice. Derive the statistical properties of an estimator that uses each observation twice.

• Consider a population with a joint density of (y;X): f (y;X):

Now consider a sample of this population, where observation i is sampled with known probability p(X_i). Such a method is called stratied sampling, and it is used to oversample people with certain characteristics (e.g., race).

1. a) Given your sample, suggest an estimator of

_y = Ey

of the form

;

i.e., what are good choices of? Show that your estimator is unbiased and derive its variance.

1. b) Consider the true model

y_i = X_i + u_i; u_i iid(0;²):

How should you use the information about sampling probabilities in p(X_i) in a GLS framework to weight observations and get a more ecient estimator of than the OLS estimator?

1

4) Consider the process

1. Find the autocovariance function for
2. Let

zt zt1 = ut

where the process for u_t is the same as above. Write the process for z_t as an ARMA process.