Lecture-3: Simple Regression Model Properties-II

Transcription

1 Lectre-3: Smple Regreon Model Properte-II

2 In Toda Cla Recap Smple regreon model etmaton Ga-Markov Theorem Hand calclaton of regreon etmate

3 3 So the OLS etmated lope 0 provded that n n n

4 Smmar of OLS lope etmate 4 The lope etmate the ample covarance between and dvded b the ample varance of If and are potvel correlated, the lope wll be potve If and are negatvel correlated, the lope wll be negatve Onl need to var n or ample

5 More OLS Inttvel, OLS fttng a lne throgh the ample pont ch that the m of qared redal a mall a poble, hence the term leat qare The redal, û, an etmate of the error term,, and the dfference between the ftted lne (ample regreon fncton) and the ample pont 5

6 Smple Regreon Fncton (SRF) }. û. 3 û { û 4. { 0 }. û 3 4

7 PRF and SRF Relatonhp Oberve the relatonhp between error term (), and the redal ( ) 7

8 Algebrac Properte of OLS The m of the OLS redal zero Th, the ample average of the OLS redal zero a well The ample covarance between the regreor and the OLS redal zero The OLS regreon lne alwa goe throgh the mean of the ample 8

9 Algebrac Properte (prece) 9 n 0 and th, n n 0 n 0 0

10 Algebrac Properte () Then SST SSE SSR 0 We can thnk of each obervaton a beng made p of an eplaned part,and an neplaned part, We then defne the followng : the totalm of qare (SST) the eplaned m of qare (SSE) the redal m of qare (SSR)

11 SST = SSE + SSR Algebrac Properte (3)

12 Proof that SST = SSE + SSR 0 we know that and SSE SSR

13 Goodne-of-Ft () How do we thnk abot how well or ample regreon lne ft or ample data? 3 Can compte the fracton of the total m of qare (SST) that eplaned b the model, call th the R- qared of regreon R = SSE/SST = SSR/SST r = ( ) ( )

14 Unbaedne of OLS Ame the poplaton model lnear n parameter a = Ame we can e a random ample of ze n, {(, ): =,,, n}, from the poplaton model. Th we can wrte the ample model = Ame E( ) = 0 and th E( ) = 0 Ame there varaton n the 4

15 Unbaedne of OLS () In order to thnk abot nbaedne, we need to rewrte or etmator n term of the poplaton parameter Start wth a mple rewrte of the formla a 5, where

16 6 Unbaedne of OLS (cont) 0 0 0

17 7 Unbaedne of OLS (cont) th and, the nmerator can be rewrtten a o, 0,

18 8 Unbaedne of OLS (cont) then, o that, let E d E d d

19 Unbaedne Smmar The OLS etmate of and 0 are nbaed Proof of nbaedne depend on or 4 ampton f an ampton fal, then OLS not necearl nbaed Remember nbaedne a decrpton of the etmator n a gven ample we ma be near or far from the tre parameter 9

20 Varance of the OLS Etmator Now we know that the amplng dtrbton of or etmate centered arond the tre parameter Want to thnk abot how pread ot th dtrbton Mch eaer to thnk abot th varance nder an addtonal ampton, o Ame Var( ) = (Homokedatct) 0

21 Varance of OLS (cont) Var( ) = E( )-[E( )] E( ) = 0, o = E( ) = E( ) = Var() Th alo the ncondtonal varance, called the error varance, the qare root of the error varance called the tandard devaton of the error Can a: E( )= 0 + and Var( ) =

22 Homokedatct Graphcal lltraton of homokedatct The varablt of the noberved nflence doe not dependent on the vale of the eplanator varable

23 Heterokedatct 3 An eample for heterokedatct: Wage and edcaton The varance of the noberved determnant of wage ncreae wth the level of edcaton

24 4 Varance of OLS (cont) Var d d Var d d Var d Var Var

25 Varance of OLS Smmar The larger the error varance,, the larger the varance of the lope etmate The larger the varablt n the, the maller the varance of the lope etmate A a relt, a larger ample ze hold decreae the varance of the lope etmate Problem that the error varance nknown 5

26 Etmatng the Error Varance We don t know what the error varance,,, becae we don t oberve the error, 6 What we oberve are the redal, û We can e the redal to form an etmate of the error varance

27 7 Error Varance Etmate (cont) / etmator of an nbaed Then, n SSR n Wh n-?

28 Error Varance Etmate (cont) 8 recall f e / that d Standard error of we bttte for then we have the tandard error of, the regreon

29 Ga-Markov Ampton () 9 Standard ampton for the lnear regreon model Ampton SLR. (Lnear n parameter) In the poplaton, the relatonhp between and lnear Ampton SLR. (Random amplng) The data a random ample drawn from the poplaton Each data pont therefore follow the poplaton eqaton

30 Ga-Markov Ampton () 30 Ampton for the lnear regreon model (cont.) Ampton SLR.3 (Sample varaton n eplanator varable) Ampton SLR.4 (Zero condtonal mean) The vale of the eplanator varable are not all the ame (otherwe t wold be mpoble to td how dfferent vale of the eplanator varable lead to dfferent vale of the dependent varable) Ampton SLR.5 (Homokedatct) The vale of the eplanator varable mt contan no nformaton abot the mean of the noberved factor The vale of the eplanator varable mt contan no nformaton abot the varablt of the noberved factor

31 Smple Regreon Etmaton 3 Term Coeffcent of Intercept coeffcent Coeffcent of Determnaton (r ) Standard error (β ) Standard error (β 0 ) Formlae 0 n n e / r = ( ) e(β 0 )=qrt ( σ n ( ) ) ( ) where, σ = n t-tat Coeffcent/tandard error