Question

Consider the model y = a + bX + e. Show that the least squares estimator for b is unbiased and consistent. You can assume that the 5 standard disturbance term assumptions are true. For each step explain why it is true.

Answer 1

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we have the model & Y= at bxte. of bis givers by the least square estimate 6- cxi-2) (Yi-y) according to the model, Y - atbxte . Eyir =[a+b x,te] - ali = 1 sabiexi + rei yana + b + ē s Y = a +bx+ Cho Xi- + = + a + b+ e − c a+bK+ %) = b[xi -& J+ (ei-ė) Hence =(xi-x) (Yi-5) = + (x; -=)[b[vi-16]+(@i-ė)] moram 5b(x1=87) * (< Cxi-x) Cei-e)] - = b &-)'+ EC8-F) ei - ; -5) ē = b (ki-P)ci-) ei =é[mã- zi] Ee (mi-meJ=c ther 6 = E(Xi) Yi-V), b=&i=8) (xi-x)e; (xi – x)2 =(xi-532 ē <Xi- & {(xi-x) ei {(xi - x)2 = b + {(xi)ei <(x-)2 or by (xi - x) (ei-e) - 0

thes (6) - E[b+ F(xi-T)ei 1 {(xi-xy +E[6] + E ( 263-) ei bis constant ( -X of linear model, x's are finced in they are not By the assumption random variables, thens is not sandom, they are xi ť < xi-x) constant = 0 another assumption logo destes buen Fleidzo 0 - EC6")= béois if 6 is unbiased Sel= cm3) ) = 4+ (<5 -x) <(xi - Elaleiks.so] XL 2 X =(xi-x) was I undes least square assumptton of Red . y=0 sample Elei/xi] o sample I by another assumptes thens (6²) = bto =b → is unbiased for b

consistency of last square estimator we have 6 = b + (xi-x) ei = 6 + 16 =(xi-x) ei In {(xi-x)2 multiply and divide by / in the 2nd term. of = 6+ b =(Xi-8) (ei-e) → Elli -1)? using low of large number we have, § Elli-x) Ceizē) — P > cou (xi, ei) we know that ē=0 One of the assumption disturbance team them ei-(e;- e) ToR by $ $&1x)2 P, Varcxi) by method of moments converges in pro. M us. ちそれ → EC) to 6 -»)? LUCx). Varcx) >o from the assumption that Ele/x]-o and E[e]-o, e fx are indep Box Exi, eit - H et ** itao --xeito ic, coucxi, ei)=o lie is indep of Efes Efxicits Heftilitai] ther b-b+ b < Cxi-x) (ei-ė) 10 € (xi-) Numerafos = <61-) Cei-e) - Pyo Denomianoton by var (X) This naxi-x Xeie) Co. In z(xi-x² Bybto - ² Pb & ba is consistent fos b o