Consider the following population model ???????? = ????1???????? + ???????? where ???????? has mean 0 and variance ????2. Note that we do not have the

Consider the following population model ???????? = ????1???????? + ???????? where ???????? has mean 0 and variance ????2. 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 ????1. Let ????1 be some other (hypothetical) estimator of ????1 that . 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???????? (????1 −????1)^2 . b) Argue that (a) implies (1). Also, what does (1) mean? 3′: = 31×1- + 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- – nt 1: Notice that (yl- — Elsa-)2 = (yi — blxi + blxi — 51×02. b) Argue that (a) implies (1). Also, what does (1) mean?