s row Endogeney Is he erm gven o he suaon when one or more of he regressors n he model are correlaed wh he error erm such ha E( u 0 The 3 man causes of endogeney are: Measuremen error n he rgh hand sde varables Smulaney ( way causaly beween deenden and rgh hand sde varables Omed varables In racce he soluons o hese roblems and he roeres of he resulng esmaon echnques can only be esablshed for large samles (asymocally In he absence of endogeney OLS wll roduce a conssen esmaor of he rue arameer values Gven ( y + ( u + u The behavour of hs esmaor as he samle sze ges larger can be deermned by akng he robably lms u lm( lm + ( Snce where BB he h of hen hs samle average wll no go o zero as he samle sze ges larger ( converges o a fne value, say QBB so lm lm Q However
u u should gve an average value of zero as he samle sze ges larger (ends o nfny f he Gauss Markov assumon abou he varables and he resdual beng uncorrelaed s rue. So u lm lm u 0 Hence n ( lm( and OLS sad o be a conssen esmaor of If hs assumon s volaed hen OLS wll be nconssen (effecvely he resdual erm becomes a funcon of so ha y b + u( hence dy/d b +du/d so s dffcul o dsngush beween he conrbuon of he wo channels of nfluence of
y y + > BeB Errors n Varables One common suaon where he assumon of no endogeney may be volaed s when one or more of he varables s measured wh error Smles o llusrae hs usng he varable model before generlsng Deenden Varable Measured wh Error True: Observe: rue y y b + u ( rue + e ( e deenden varable measured wh error e e s a random resdual erm jus lke u, so e ~( 0, σbeb and errors n measuremen are assumed o be uncorrelaed wh oher resduals Sub. ( no ( y - e b + u y b + u + e y b + v where v u + e (3 Ok o esmae (3 by OLS, snce E(u E(e 0 Cov(,u Cov(,e 0 (nohng o sugges values of he varable correlaed wh meas. error n deenden varable So OLS esmaes are unbased n hs case Bu sandard errors are larger han would be n absence of meas. error Var(v Var( u + e Var(u + Var(e σbub σbeb σbub he resdual varance n resence of measuremen error n deenden varable now also conans an addonal conrbuon from error n y varable, σ and so, F values smaller han should be leadng o Tye II error (acce false null Measuremen Error n Elanaory Varable True: rue rue b + u (
erm y y y and Observe: rue + w ( e rhs var. measured wh error (w sub. ( no ( rue rue rue b(-w + u b - bw + u b + v (3 where now v - bw + u (so resdual erm agan consss of comonens OLS on (3 gves y rue rue ( b + u b u + (4 Ineresed n he asymoc roeres of hs esmaor so ake robably lm. Assumng u w are ndeenden (so ha level of gves no ndcaon abou he sze/sgn of he error n measuremen hen lm u lm ( + w u 0 so nd n (4 vanshes bu lm lm ( + w lm + lm u lm lm ( + w lm σ σ + σ w 0 so lm( σ b σ + σ w b
y row observaon. or σ w lm( b σ + σ w b σ + w σ If b > 0 hen b < b If b < 0 hen b > b so ha measuremen error n he rgh hand sde varable means ha he OLS esmaes wll be nconssen and suffer from aenuaon bas (closer o zero n absolue values The rao σ w σ + σ w σ σ + σ w s called he relably rao (he rooron of varaon n he unobserved rue varable ha s accouned for by he varaon n he observed varable and he rao σ w σ s called he nose o sgnal rao An ncrease n relably (fall n nose-sgnal means a fall n measuremen error and he OLS esmaes are closer o her rue values. Generalsng o he k varable model True: Observe: rue rue rue + u ( + w ( Where now w s an BBk mar of measuremen errors and he of w corresonds h o he measuremen errors on all he varables assocaed wh he h Conssency of he OLS esmaor deends on he behavour of lm( lm y
+ + + + u w w w (( ( ( ( lm lm( Snce, w and u are ndeenden can gnore cross-roducs, so + w w ( ( lm lm( [QBB + ΛBwB ] - QBB - [QBB + ΛBwB ] -- ΛBwB whch s a mure of all he arameers n he model B. Can show ha hs general resul also holds when he ms-measured s a dummy varable B. In he smle varable model can oban a bound for he rue value of lm(/ lm( δ < < where δ s he coeffcen on y n a reverse regresson of on y
are endogenous Insrumenal Varable Esmaon Snce he roblem arses because he error erm s effecvely correlaed wh he (msmeasured varable hs causes endogeney bas. The soluon s o relace he varable of concern wh oher varable(s ha are correlaed wh he varable of concern bu uncorrelaed wh he resdual erm. Gven y + u ( where : ] [ and BB he se of kbb varables Consder an BBL mar : ] [ Such ha z. lm lm W z u u. lm lm 0 re-mully ( by y + u ( wh Var( u E( uu σ ( suggess can use GLS esmaor n ( ( ( ( y (3 GLS wh varance/covarance mar ( ( ( (4 Var σ
In he (jus denfed case where LK (he same number of nsrumens as here are endogenous varables hen (3 smlfes o y ( and (4 becomes ( ( ( Var σ Can show ha hs esmaor s conssen snce y lm lm lm( + u lm lm lm( so lm( and s conssen oe ha because s effecvely a measure of he correlaon beween he nsrumen and he endogenous varable hen f hs s low, he varance of he esmaor wll be large. Hence esmaes ycally have larger sandard errors when comared wh OLS esmaes. oe. Conssency says nohng abou he erformance of esmaors n small samles
Two Sage Leas Squares If an nsrumen(s ess hen he model s sad o be jus(over denfed If no nsrumen can be found he model s sad o be undenfed. An equvalen way o ge he esmaor whch can hel llumnae he denfcaon ssue s based on he followng. Gven ( ( ( y Le - BzB ( So ( y I follows ha - BzB ( γ SLS ( y so he esmaor can be obaned by a wo sage leas squares rocedure:. Regress he endogenous varable on he se of nsrumens (n racce hs means regressng any endogenous varable on all he eogenous varables n he model and any nsrumens. Subsue he redced values of for n he model (redced values of eogenous varables are dencal o he rue values and regress y on The frs sage of hs rocess wll hel llumnae wheher he chosen nsrumens are n fac sgnfcan redcors of he endogenous varable(s In hs case Var( σ ( ( whch can be wren as
and so > gven han Var( σ ( ( ( σ - [ BzB BzB] So Var( SLS σ ( wh a conssen esmae of σ s ( y by u SLS usls SLS SLS ( y SLS SLS ( y SLS ( y In he resence of heeroskedascy (and endogeneey he heeroskedasc robus varance esmaor s gven by Var( robus SLS ( S sls ( where S sls u sls - vald only asymocally Effcency of he Esmaor Can show ha SLS s he mos effcen of all esmaors usng lnear nsrumens However he ssue hen arses as o whch nsrumens are he mos effcen o use for SLS. Snce Var( sls σ [ BzB BzB] - σ - [ BzB] I s ossble o comare esmaors based on dfferen ses of nsrumens (and hence dfferen marces Consder BB BB ha LBB (a leas one more nsrumen n BB LBB BB - so ha BzB BB(BB BB BB and - BzB BB(BB BB BB
Var( and Var( sls σ [ BzB] - sls σ [ BzB] - If BzB > BzB Then Var( sls < Var( sls Davdson & MacKnnon ( 9 show ha s a osve sem-defne mar and so BzB - BzB [ BzB - BzB ] ncreasng he number of nsrumens leads o an ncrease n he asymoc effcency of he esmaor Bu Can also show ha n small samles ncreasng he number of nsrumens leads o an ncrease n small samle bas (In fne samle models, he eeced value of he jus denfed esmaor does no es and n over-denfed models he eeced value s based because of he z u correlaon beween and u, u s unlkely o dsaear n fne samles Weak Insrumens I s also rue ha he weaker he correlaon beween nsrumen and endogenous varable, he larger wll be he bas of even n large samles In he smle varable model wh a sngle nsrumen z Then Gven y + u
( z y y z zu + z zu + z + s s zu z where sbzub s he samle covarance beween z and u sbzb s he samle covarance beween z and As he samle covarance becomes a conssen esmae of he oulaon covarance and so Cov( zu lm( + Cov( z Corr( zu * s. e.( u lm( + Corr( z * s. e.( So even f he correlaon beween z and u s small and esmae can be a long way off he rue value f he correlaon beween z and s small Comarng hs wh he OLS esmaor Cov( u Corr( u * s. e.( u lm( OLS + + Var( s. e.( so beer o use OLS raher han f corr(zu/corr(z > corr(u
BB ( BB from : δbb + ] + whch sage ] sage How o so a weak nsrumen a The raw correlaon coeffcen s low b Even beer f we regress he endogenous varable on he se of eogenous varables and he nsrumen(s, hen we would eec he nsrumen(s o have a non-zero effec afer neng ou he conrbuon of he eogenous varables c Canno use he R b snce hs could be hgh urely because of he eogenous varables and no he nsrumens so could look a he aral R nes ou he effec of he eogenous varables, assumng [ BB BB and [BB : BB and s obaned from he regresson ( ( γ + v d Look a he F value n a es of δbb 0 n he s of he SLS esmaon BB BBδBB e As a rule of humb, F values below 0 sugges ha here may be a roblem wh weak nsrumens [ Can show ha when here s a sngle endogenous varable hen lm( + lm( OLS F s where F s he F value from he of he SLS esmaon rocess. If F0 hen he relave sze of he dscreancy beween he and OLS esmaes s aromaely :9, small enough o refer over OLS ]
, Tesng for Endogeney Hausman Tes Under he null hyohess of no endogeney boh OLS and wll gve conssen esmaes of he rue coeffcen values, bu OLS wll be he mos effcen If endogeney s resen hen only s conssen Hence H0: lm( 0 OLS Le q OLS q Then ~ (0, s. e.( q and H q Var(qq ~ χ B(kB where k no. of rgh hand sde varables n he model (hough he es can equally be aled o a sub-se of varables n he model Hausman shows ha (asymocally Var(q Var( Var( Whch usng a common (conssen esmae of σ s SLS ( y SLS ( y SLS u SLS u SLS OLS namely hen H can be esmaed as H q[( ( s ] q ~ χ ( k and f χ > hen rejec he null ha here s no endogeney resen χ crcal
An alernave (asymoc equvalen aroach s gven by he Durbn-Wu-Hausman form of he es. Regress he endogenous varable on he se of nsrumens and save he redced value and he resduals. Include eher he redced values and he resduals as an addonal regressor(s one for each endogenous varable n he orgnal model 3. Use a (F es for he sgnfcance of hese addonal erms 4. If sgnfcan hen rejec he null of no endogeney (nuvely f regress endogenous varable on a se of nsrumens hen he resulng redced value should be uncorrelaed wh he deenden varable n he orgnal model Tesng Over-Idenfyng Resrcons In general can es he assumon ha he nsrumen s uncorrelaed wh he resdual, bu f here s more han nsrumen can es wheher he addonal nsrumens are correlaed wh he srucural form resdual Tes s a varan of he Hausman es n ha can comare nsrumens wh jus jus over based on he full se of based on a subse of nsrumens ha jus denfy he model. nsrumens are nvald hen s a conssen esmaor by assumon, bu If he addonal lm( over So could n rncle aly H q Var(qq ~ χ B(kB
Where now over q jus and f χ > χ crcal hen rejec he null ha he over-denfyng resrcons (era nsrumens are vald n he sense of beng uncorrelaed wh he srucural form resduals However hs can nvolve many dfferen varaons (f he model s over-denfed whch nsrumen s used for he jus-denfed case? An asymoc equvalen varaon of hs es ha avods hs roblem s based on he followng. esmae he srucural form by /SLS usng all ossble nsrumens and save he resduals, u Regress on all he eogenous varables n he sysem 3 Under he null ha all nsrumens are uncorrelaed wh he resduals u hen can show ha u *R ~χ Where q no. of over-denfyng resrcons (LBB KBB Bq Agan f χ > χ crcal hen rejec he null ha he over-denfyng resrcons (some (bu whch? of era nsrumens are vald