Question

sunien er sitgle linear regeson mode %, are nenrandom Zn other words, var(ulx) eloes rutderend on X Homos kadati) V ua uun are mctually indepea Previde a matlenstia proof fhr the egation ECa-)-62 ·nsi8-- assumptions?-V. lt 2. 2 ? n-2

Many thanks in advance.

Answer 1

The unbiased estimator of the error variance ơ can be written as UNBIASEDNESS OF Ơ2 Under Assumptions E.1 through E.4, ớ2 is unbiased for σ2: E(G21X-r' for all σ2 > 0 PROOF: Write u = y-xp = y-XXX)-X'y = My = Mu, where M = 1,-XXX)-X', and the last equality follows because MX-0. Because M is symmetric and idempotent, E.4 Because u'Mu is a scalar, it equals its trace. Therefore, The last equality follows from tr(M) = trln)-trXXX)-1X1 = n-tr(XX)-XX-n- tr (lk + 1-n-(k + 1)= n-k-1. Therefore,

note that above proof for multiple regression , you can put k = 1 for simple linear regression

to get the result