Lecture 3: Simple Omitted Variables Problem

Transcription

1 Lectre Notes o Advaced Ecoometrcs Taash Yamao Fall Semester 5 Lectre 3: Smple Omtted Varales Prolem I ths lectre, we epad the smple lear regresso model to mltvarate ler model. The ma motvato to clde more tha oe varale to a regresso model s that t s ver dffclt to detf the effect of oe depedet varale,, o the depedet varale,, wthot cosderg the other factors. For stace, we ca measre the effect of schoolg more precsel f we compare people who have the eactl the same characterstcs, ecept the schoolg. There have ee ma stdes o tws. If we fd a dfferece wage rates etwee the two detcal people ecept the schoolg, the we ca coclde that the dfferece s de to the dfferece the schoolg. Ufortatel, socal scece, t s mpossle to fd eve two persos eactl the same ecept oe factor. Eve tws are dfferet ma was. The prolem of omttg mportat varales regresso aalses s called the omtted varales prolem, ad ths s the core of ma prolems ecoometrcs. I start ths lectre wth the smplest case of the omtted varales prolem. Omtted Varales Prolem: a smple case Sppose that the tre mode shold e:. 3- Bt sppose that we have formato ol o t ot so that we ca ol estmate a smple varate model wth t ot. Accordg to the least sqared approach from the prevos lectre, we ow that the least sqared estmators are ˆ 3- ˆ 3-3 However, the tre model s 3-. Ths, we ca replace 3-3 sg the tre model 3-,

2 ˆ As we dd efore, we ca smplf the rght had sde of the eqato: ˆ Ths, tag the epectato of oth sdes of the eqato, the last term ecomes zero der assmptos SLR -4, whle the secod term remas: E 3-4 Ths dcates that the least sqared estmate of, wll e ased f the secod term of 3-4 s ot zero,.e., ad are ot correlated. Note, frther, that the last part of the secod term s smpl a least sqared estmator of a smple varate regresso model etwee ad : e E δ Ths, 3-4 ca e wrtte as

3 E ˆ δ Therefore the as s δ. Ths, the sgs of adδ determe the drecto of the as pward or dowward. If the two parameters, ad δ, have the same sgs oth postve or oth egatve, the the drecto of as s postve. Bt f the parameters have the opposte sgs, the the drecto of as s egatve. For stace, let s sa that we wat to estmate the effect of agrcltral credt o farm prodctvt t do ot have formato o the edcato of farmers. Ths, the error term of a varate model cldes edcato as a omtted varale: farm_prod credt edc Becase edc s correlated wth farm_prod ad credt postvel, postve. The drecto of a potetal as s postve. adδ are oth The magtde of the as depeds o the asolte vales of ad δ. If the correlatos etwee ad ad etwee ad are strog, the the sze of the as ecomes gger. O the other had, f oe of the two correlatos s wea the sze of the as ecomes small. Eample 3-: Idvdal No-farm Icome Ugada I rral areas of Ugada, o-farm come provdes mch eeded cash to farm hoseholds. Most edcated me ad wome rral areas have opporttes to hold reglar jos, mag a costat mothl wage throghot a ear. Other people who are ot fortate eogh to hold reglar jos are lel to ear o-farm come from small self-emploed sesses sch as mag asets or tradg goods. Sppose that we are terested the geder dffereces o-farm come ad wat to test a hpothess that wome mae less o-farm come tha me. Bt a smple comparso of o-farm come etwee me ad wome does ot provde a relale test for ths hpothess f characterstcs of me ad wome are ot smlar. Oe major factor o-farm come s edcato. If me are etter edcated tha wome ad me mae more o-farm come tha wome, we ca ot e sre f the hgh o-farm come s a reslt of geder or edcato. For eample, let s se the data from Ugada. The data are collected FASID collaorato wth Maerere Uverst Ugada 3. The data come from 94 3

4 hoseholds. Amog them, we fd 648 people who eared some come from o-farm actvtes. Tale dcates average o-farm come US$, average schoolg ears, age, ad oservatos for me ad wome. Tale. Idvdal No-farm Icome Ugada tale female, cmea ocus mea ed mea age age f%8.f row; female meaocus meaed meaage Nage Total As o ca see Tale, me are slghtl etter edcated ad mae more o-farm come. To compare o-farm come etwee me ad wome whle holdg edcato levels costat, we have created categores for edcato: o edcato categor, -4 ears of schoolg, 5-7 ears of schoolg, 8- ears of schoolg 3, ad more tha ears of schoolg 4. The we have calclated average o-farm come for each categor for me ad wome separatel. Althogh wome mae less o-farm come tha me geeral, t-tests dcate that the dfferece s statstcall sgfcat at the 5 percet level ol for Categor 3. Tale. No-farm Icome Geder ad Edcato. tale edcat female, cmea ocus f%8.f row; female edcat Dfferece p-vale Total Now, let s cosder whch drecto the as wold e f we estmate the o-farm come eqato wth a female dmm t ot edcato. Edcato the depedet varale s postvel correlated wth edcato the omtted varale, ad edcato the omtted varale s egatvel correlated wth the female dmm the depedet varale. Ths, accordg to the theor aove, the drecto of the as shold e egatve. lo-farm come female edc. reg loc female Sorce SS df MS Nmer of os 648 4

5 F, Model Pro > F. Resdal R-sqared Adj R-sqared.84 Total Root MSE.484 loc Coef. Std. Err. t P> t [95% Cof. Iterval] female _cos reg loc female ed Sorce SS df MS Nmer of os F, Model Pro > F. Resdal R-sqared Adj R-sqared.894 Total Root MSE.3556 loc Coef. Std. Err. t P> t [95% Cof. Iterval] female ed _cos The reslts the frst model dcate that wome mae aot 63 percet less o-farm come tha me, t the reslts the secod model dcate that the dfferece s aot 55 percet. Ths, the reslts dcate that the estmated coeffcet was ased dowward whe ed was ot clded. Of corse, there are proal ma other varales that shold e clded ths model to estmate the geder dfferece more precsel. We wll come ac to ths model later ths corse. Ed of Eample Mltvarate Lear Regresso Model: Illstrato It s clear that we eed to clde more tha oe varale regresso models to tae to accot ma factors. How ma varales shold e clded? We wll std how we shold select varales ased o derlg ecoomc theor ad emprcal aalses later. For ow, let s cosder a geeral case wth depedet varales pls a tercept. A mltvarate ler regresso model wth -depedet varales s.. for,,. 5

6 6 where,, ad are oservato s, ad. As the varate model, let s start wth the least sqared resdals approach, whch s called the Ordar Least Sqared OLS regresso. Least Sqared Resdals Approach As the varate regresso, we wat to fd a set of estmators, j, that mmze the sm of sqared resdals: m F.O.C. w.r.t. w.r.t. 3- w.r.t. Whe there s ol oe depedet varale ths, we cold solve the frst order codtos easl. Bt whe there are depedet varales, solvg the frst order codtos for varales wold e complcated. Matr maes t easer.