Generalized Least Squares
1) Consider
y = X¯ + u; Euu0 = :
Show that GLS is BLUE.
2) 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.
3) 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(Xi). Such a method is called strati…ed sampling, and it is used to oversample people with certain characteristics (e.g., race).
a) Given your sample, suggest an estimator of
¹y = Ey
of the form
;
i.e., what are good choices of ? Show that your estimator is unbiased and derive its variance.
b) Consider the true model
yi = Xi¯ + ui; ui » iid(0;¾2):
How should you use the information about sampling probabilities in p(Xi) in a GLS framework to weight observations and get a more e¢cient estimator of ¯ than the OLS estimator?
1
4) Consider the process
a) Find the autocovariance function for
b) Let
zt ¡ µzt¡1 = ut
where the process for ut is the same as above. Write the process for zt as an ARMA process.
