Consider the following population model

= 1+

wherehas mean 0 and variance2. Note that we do not have the intercept here (it is not necessary

and adding it in does not change anything). Define1to be the least squares estimator of1. Let1be some other (hypothetical) estimator of1thatmay not be the least squares estimator. We want to show that

(1):1/(1)^2 1/(1)^2

so let us do it step-by-step.

a) Show that 1/=1(1)^2 ==1(1)^2 +=1(11)^2.

• Hint1:Noticethat( )^2=( + )^2

b) Argue that (a) implies (1). Also, what does (1) mean?

3': = 31x1- + 8t where at- has mean 0 and variance 02. Note that we do not have the intercept here (it is not necessary and adding it in does not change anything). Define [)1 to be the least squares estimator of 31- Let 51 be some other (hypothetical) estimator of 31 that may not be the least squares estimator. We want to show that TI. TI. 1 1 .. (1): 52m b19602 s 320;- taxi-)2, i=1 i=1 so let us do it step-by-step. 1 ~ 1 1 2 ~ a) Show that;2?=1(yi b13502 = ; 1:104 1913502 + ;Z?=1xi (b1 b1)2- - Hint 1: Notice that (yl- Elsa-)2 = (yi blxi + blxi 51x02. b) Argue that (a) implies (1). Also, what does (1) mean