This question is about showing that the variance of the wls estimator is smaller than that of the ols estimator when there is heteroskedasticity:

111 the notes. we claimed that (Hint: start with the fact that :31:1(2'1 E)2 > 0.) Then substitute X12 for 21-. \"Then will equality hold? 7.1 An Example We begin with a simple illustrative example where we can directly show all of the consequences of heteroskedasticity. Example 7.1. Suppose Y; = BX;te;, i =1, 2, ..., N. such that E[ex] = 0, E[e?(x] = 02X?, and Elec,|x] =0 for all i # j, i = 1, 2, ..., N. We also assume that _, X? + 0. (Since we have only one regressor, we will use X; to represent the ith observation of variable X. We use x to represent all of the observations of X .) The OLS estimator for B, in this example is Bols = EM XY . Ci Xi ( B, Xi te;) = Bit which is unbiased for B1: E[ Bols | x] = Bit = B1.\fexample. Since 5'3?\" is a sample mean of a random variable with variance 02: its variance is A 2 var [Kids] I U N ' It can be shown [:see exercises) that Therefore varLgim] <_i varle>