In some applications, such as regressing annual income on the number of years of education, the variance of y tends to be larger at higher values of x.

Consider the model E(yi) = ????xi, assuming var(yi) = xi????2 for unknown ????2.

a. Show that the generalized least squares estimator minimizes ∑

i

(yi −

????xi)

2∕xi (i.e., giving more weight to observations with smaller xi) and has ????̂

GLS = ȳ∕x̄, with var(????̂

GLS) = ????2∕(

∑

i xi).

b. Show that the ordinary least squares estimator is ????̂ = (

∑

i xiyi)∕(

∑

i x2 i ) and has var(????̂) = ????2(

∑

i x3 i )∕(

∑

i x2 i )

2.

c. Show that var(????̂) ≥ var(????̂

GLS).