Ch.2 The simple regression model. The Simple Regression Model. 2.1 Definition of the model. A Simple Assumption for u. The Concept of Error Term

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1 Chapter Ch The smple regresso model The Smple Regresso Model = + + u Defto of the smple regresso model Dervg the OLS estmates 3 Mechacs of OLS 4 Uts of measuremet & fuctoal form 5 Epected values & varaces of OLSE 6 Regresso through the org Specal Lecture Specal Lecture Defto of the model Equato (), = + + u, defes the Smple Regresso model I the model, we tpcall refer to as the Depedet Varable as the Idepedet Varable s as parameters, ad u as the error term Specal Lecture 3 Specal Lecture 4 The Cocept of Error Term u represets factors other tha that affect If the other factors u are held fed, so that u =, the = E : eld = + fertlzer + u (3) u cludes lad qualt, rafall, etc E : wage = + educ + u (4) u cludes eperece, ablt, teure, etc A Smple Assumpto for u The average value of u, the error term, the populato s That s, E(u) = Ths s ot a restrctve assumpto, sce we ca alwas use to ormalze E(u) to To draw ceters parbus coclusos about how affects, we have to hold all other factors ( u) fed Specal Lecture 5 Specal Lecture 6 Smple Regresso Model

2 Chapter Zero Codtoal Mea We eed to make a crucal assumpto about how u ad are related We wat t to be the case that kowg somethg about does ot gve us a formato about u, so that the are completel urelated That s, that E(u ) = E(u) = (5&6), whch mples E( ) = + (PRF) (8) Specal Lecture 7 E( ) as a lear fucto of, where for a the dstrbuto of s cetered about E( ) f() E( ) = + Specal Lecture 8 Dervg the OLSE Basc dea of regresso s to estmate the populato parameters from a sample Let (, ): =,, } deote a radom sample of sze from the populato For each observato ths sample, t wll be the case that = + + u (9) Specal Lecture 9 Populato regresso le, sample data pots ad the assocated error terms 4 3 } u } u 3 u 3 4 E( ) = + u 4 Specal Lecture Dervg OLSE usg MM To derve the OLS estmates, we eed to realze that our ma assumpto of E(u ) = E(u) = also mples that Cov(,u) = E(u) = Because Cov(X,Y) = E(XY) E(X)E(Y) (B7) Now we prepare restrctos to estmate s E(u) = () E(u) = () Cot Dervg OLSE usg MM Sce u =, we ca rewrte; E(u) = E( ) = () E(u) = E[( )] = (3) These are called momet restrctos The approach to estmato mples mposg the populato momet restrctos o the sample momets It meas, a sample estmator of E(X), the mea of a populato dstrbuto, s smpl the arthmetc mea of the sample Specal Lecture Specal Lecture Smple Regresso Model

3 Chapter More Dervato of OLS We wat to choose values of the parameters that wll esure that the sample versos of our momet restrctos are true The sample versos are as follows: ˆ ˆ (4) ˆ ˆ (5) Specal Lecture 3 Cot More Dervato of OLS Gve the defto of a sample mea, ad propertes of summato, we ca rewrte the frst codto as follows ˆ ˆ (6) or ˆ ˆ (7) So the OLS estmated slope s ˆ (9) Specal Lecture 4 Summar of OLS slope estmate The slope estmate s the sample covarace betwee ad dvded b the sample varace of If ad are postvel (egatvel) correlated, the slope wll be postve (egatve) eeds to var our sample See (8) & Fgure (3) More OLS Itutvel, OLS s fttg a le through the sample pots such that the sum of squared resduals s as small as possble, hece the term s called least squares The resdual, û, s a estmate of the error term, u, ad s the dfferece betwee the ftted le (sample regresso fucto) ad the sample pot Specal Lecture 5 Specal Lecture 6 Sample regresso le, sample data pots ad the assocated estmated error terms 4 3 } û } û 3 û û ˆ ˆ ˆ Specal Lecture 7 Alterate approach to dervato Gve the tutve dea of fttg a le, we ca set up a formal mmzato problem ˆ ˆ () The frst order codtos, whch are the almost same as (4) & (5), ˆ ˆ, ˆ ˆ Specal Lecture 8 Smple Regresso Model 3

4 Chapter 3 Propertes of OLS Algebrac Propertes of OLS The sum of the OLS resduals s zero Thus, the sample average of the OLS resduals s zero as well ad thus, (3) Cot Algebrac Propertes The sample covarace betwee the regressors ad the OLS resduals s zero (3) 3 The OLS regresso le alwas goes through the mea of the sample ˆ ˆ Specal Lecture 9 Specal Lecture Cot Algebrac Propertes We ca thk of each observato as beg made up of a eplaed part, ad a ueplaed part, ˆ (3) The we defe the followg : ˆ SST SSE (33) (34) SSR (35) The, SST SSE SSR (36) Specal Lecture Goodess-of-Ft It s useful we thk about how well the sample regresso le fts sample data From (36), SSE SSR R (38) SST SST R dcates the fracto of the sample varato that s eplaed b the model Specal Lecture 4 Measuremet Uts & Fucto Form If we use the model * = * + **+ u* stead of = + + u, we get ˆ * ˆ ˆ * c c ad ˆ d where * = c ad * = d Smlarl, ˆ * ˆ where * = l ad * = l Specal Lecture 3 Specal Lecture 4 Smple Regresso Model 4

5 Chapter 5 Meas & Varace of OLSE Now, we vew ˆ as estmators for the parameters that appears the populato, whch meas propertes of the dstrbutos of over dfferet ˆ radom samples from the populato Ubasedess of OLS Ubased estmator: A estmator whose epected value (or mea of ts samplg dstrbuto) equals the populato value (regardless of the populato value) Cot Ubasedess of OLS Assumpto for ubasedess Lear parameters as = + + u Radom samplg (, ): =,,, }, Thus, = + + u 3 Sample varato the, thus ( ) 4 Zero codtoal mea, E(u ) = Specal Lecture 5 Specal Lecture 6 Cot Ubasedess of OLS I order to thk about ubasedess, we eed to rewrte our estmator terms of the populato parameter ˆ u ( ) ( ) the ˆ E E( u ) ( ) * we ca also get E( ˆ ) the same wa (49),(5) (53) Ubasedess Summar The OLS estmates of ad are ubased Proof of ubasedess depeds o our 4 assumptos f a assumpto fals, the OLS s ot ecessarl ubased Remember ubasedess s a descrpto of the estmator a gve sample our estmate ma be ear or far from the true parameter Specal Lecture 7 Specal Lecture 8 Varaces of the OLS Estmators Now we kow that the samplg dstrbuto of our estmate s cetered aroud the true parameter We wat to thk about how spread out ths dstrbuto s It s much easer to thk about ths varace uder a addtoal assumpto, so assume 5 Var(u ) = (Homoskedastct) Cot Varace of OLSE s also the ucodtoal varace, called the error varace, sce Var(u ) = E(u ) - [E(u )] E(u ) =, so = E(u ) = E(u ) = Var(u) Ad, the square root of the error varace, s called the stadard devato of the error The we ca sa E( )= + ad Var( ) = Specal Lecture 9 Specal Lecture 3 Smple Regresso Model 5

6 Chapter Homoskedastc Case Heteroskedastc Case f( ) f( ) E( ) = + E( ) = + 3 Specal Lecture 3 Specal Lecture 3 Cot Varace of OLSE Var ˆ ) ( ) ( (57 ) The larger the error varace,, the larger the varace of the slope estmate The larger the varablt the, the smaller the varace of the slope estmate As a result, a larger sample sze should decrease the varace of the slope estmate Specal Lecture 33 Estmatg the Error Varace We do t kow what s the error varace,, because we do t observe the errors, u What we observe are ol the resduals, û, ot the errors, u So we ca use the resduals to form a estmate of the error varace Specal Lecture 34 Cot Error Varace Estmate u ˆ ˆ ˆ ˆ The, a ubased estmator of ˆ u ˆ ˆ (6) s Cot Error Varace Estmate ˆ ˆ recall that sd se ˆ Stadard error of the regresso ˆ If we substtute ˆ for, the we have the stadard error of ˆ, ˆ ( ) Var( ˆ) Specal Lecture 35 Specal Lecture 36 Smple Regresso Model 6

7 Chapter 6 Regresso through the Org Now, cosder the model wthout a tercept: ~ ~ (63) Solvg the FOC to the mmzato problem, OLS estmated slope s ~ (66) * Recall that a tercept ca alwas ormalze E(u) to the model wth Specal Lecture 37 Smple Regresso Model 7