(since y Xb a scalar and therefore equals its transpose)

Transcription

1 Backgroud: Prvoul w lard rort of tmator wh w orv d data {w } draw from om dtruto P th tmator ar fucto of th data w ow w wll orv data {w } { } from om dtruto P whr w thk of a th outcom/ddt varal ad a th covarat/ddt varal W wll trtd th codtoal ma P Y d μ P d Though th fuctoal form of μp ca qut gral w tud hr th ca whr th cfc lar fuctoal form whr μ P P P d d Hr w add a cotat to to clud th trct trm W charactrz tru P a th mmzr of S lat quar: arg m Y P R P [ d d ] Aumg th radom vctor d uch that - ad Y a radom calar varal wth frt momt ft Th: Tru P uqul gv [ ] Y P d P d d P d B Aa rcl: Y Th th Ordar Lat Squar tmator of th tru P whr d d { } { } P Y ar orvd OLS tmator: Suo w orv W Y d from om dtruto P ro: / / S Y thod of Drvato: Lar Algra: Fd th t aromato of th mag of fdg t mallt Fd orthogoal rojcto of oto th ac ad th colum of R Pr oj df or orth roj [ m ] r m d data ad th" t" aro P f vrtl th OLS Calculu: mz SSR/ um of rdcto rror SSR c a calar ad thrfor qual t trao SSR FOC : c calar f vrtl th Projcto ad Ahlator atrc P ad P P ad A k m ad th" t" aro P OLS m Prort: a P P ad ar mmtrc ad dott c P Sc rojctg oto th colum ac of gv ou th am thg d Sc vcto m orthogoal to th ac P W ca alwa rght a vctor R a th rojcto oto orthogoal uac Follow from a f P Fttd jut th orthogoal rojcto of oto th colum ac of g A matr rtur th lar comato of that th rojcto of a vctor oto colum ac of : A A h P A PA A 3 Th tutv c for a A lv colum ac of Trac -k j SSR Trac P th rojcto oto th colum ac of ad th rojcto oto th ac that orthogoal to m P dmott P P 3 PA A P P P P P P P A P A A PA A A

2 SSR SR ad Samlg rror: Rlatoh to Tru rror trm SSR Sum of quard OLS rdual OLS tmat of SSR var of rror trm 5 SR Std rr of th rgro t of td dv of rror trm Samlg rror A amlg rror th rojcto of th rror matr oto th colum ac of wh?? How do w thk aout th dffrc tw th amlg rror ad th tru rror? What do t ma to rojct a radom vctor oto a ac? odl A odl a t of rtrcto o th jog dtruto of th ddt ad ddt varal- a modl a t of jot dtruto atfg a t of aumto W wll 3 modl ach of whch mak a t of aumto aout th jot dtruto of : Clacal Rgro Aumto 5 wth Gaua rror: Aumto 6 : Gralzd Lat Squar - Rla Codtoal Homokdatct ad o Sral Corrlato Rla Aumto a ad 3: Rla vrthg d BC mmtr ad dmotc 5 - th dgr of frdom Th da that aramtr d to tmatd for otag th rdual vctor ud to calculat or cfcall ha to atf th k ormal quato: -

3 5 Aumto to Clacal Rgro odl : Lart 6 : Rlatoh tw ddt var ad rgror lar Y or k k Strct ogt: Rgror ar trctl ogou t rror trm ar radom wth codtoal ad ucodtoal ma k Y k k k Y Y tuto: O avrag our modl mak th rght gu of! Hr th ma codtoal o rgror for all orvato othr word f w tak th jot dtruto of th radom varal f ad codr th codtoal dtruto f gral th codtoal ma a olar fucto of Th trct ogt aumto a that th fucto a cotat of valu ot that th aumto alo ot too rtrctv f w allow for a cotat trm th rgror mlcato of trct ogt: a Ucodtoal a : ach Rgror orthogoal to th rror trm for all orvato: j k jk jk jk jk jk jk jk c c ach rgror ad ach rror trm ar ucorrlatd: Th follow from a ad c th aumto a that ach of th rgror larl ucorrlatd wth th rror trm Cov jk jk jk 3 o ultcollart: Rgror ar larl ddt from ach othr Rak wth roalt ot: Th allow u to a that vrtl! a Homokdatct: Codtoal cod momt a cotat 7 Var > for Var o Sral Corrlato tw Orvato: Cov j j for j j Th follow from c th codtoal var-cov matr dagoal th off-dag lmt ar ot: ad 5 th matr of codtoal cod momt Var hrcal roortoal to th dtt matr For Gaua odl: For uro of hoth ttg w oft add th followg dtrutoal aumto 5 ormalt of th rror Trm: Y ot: Th aumto ar aout th TR aml W mo o d aumto f a radom aml { } d acro orvato th our aumto ca rtatd for gl orvato Y k k or am a for c j j for j o k k k k 3 Rak wth roalt Sam > ad Cov j j Th mlcato of d aml that th jot dtruto of do ot dd o So th ucodtoal cod momt cotat acro w gt ucodtoal homokdatct wth d ad th fuctoal form of th codtoal cod momt am acro! Howvr th valu of th codtoal cod momt am acro do ot follow o aumto codtoal homokdatct rma rtrctv for radom aml 6 Th ot a rtrctv a aumto a t m cau ma o-lar rlatoh ca larzd g gg 7 Homokdatct a that th codtoal varac of th rror trm a cotat codtoal cod momt cotat trct ogt

4 Clacal Rgro aumto 5: Prort of OLS tmator Ft Saml Prort of OLS tmator 8 a Uad: Udr 3 ot: Strct ogt CRTCAL rovg th athg hort of t wll ot uffc For aml t ot ough to aum or orthogoalt Sc mot tm r modl do ot atf trct ogt ad at mot th orthogoalt codto th OLS tmator ot uad Varac of OLS : Udr 5 Var c Gau-arkov Th: Udr 5 th OLS tmator ffct th cla of lar ad uad tmator That for a uad that lar Var Var OLS th matr Var Var a d matr ag: For a rgro coff th varac of th OLS t o largr tha that of a othr lar uad tmator Var Var d a Var Var a for a k vctor a [ ] for a vctor wth th lmtad o w aov alo tru Var Var for k kk OLS kk BLU Bt Lar Uad tmator: Gau-arkov a that c OLS lar ad uad th aov roof amog all othr lar ad uad tmator of th OLS tmator ffct th that t codtoal varac matr Var OLS th mallt amog th lar uad tmator That wh OLS calld BLU kk ot: Th do ot ma that OLS tmator th mot ffct amog all lar tmator! wh? d Covarac of OLS tmator ad rror 9 : Udr Co var OLS whr or rcl th ma that vr radom RV ucorrlatd wth vr RV : Cov j j j for k ad j Or outr roduct form: Cov [ ] 8 a Var k k trct ogt [ ] [ ] [ ] [ ] [ ] a [ ] Var Var Var Var c a co DD DD DD Var Var c DD d 5 c lar o w wrt a C for om matr C Lt D C A C D A whr A C D A D A D D D D D D D D D c uad D D c uad D c trct ogt th alo ma that D for a o zro D D D A D A rort of OLS Samlg rror Var Var Var D A D A Var D A c D A fucto of 9 D A D A cvar cod hom DD AD DA AA ad w kow DA D c D ad AA d A ad Cov A A A A A c A Alv m roj oto OLS OLS

5 Ft Saml Prort of Rcall SSR OLS OLS a Uad of : Udr rovdd > o that wll-dfd mlcato: B law of tratd ctato tmat of Var : Sc th tmat of a atural tmat of Var / Var Ug th aov rult o th varac of ad th followg dtruto of w ca ow tt hoth aout OLS rgro coffct aumg ormalt th rror trm Ft Saml Hoth Ttg Gaua Rgro udr ormalt aumto: Prlmar: Dtruto of amlg rror ad Sc ormal alo ad 6 W kow from ft aml rort of OLS tmator that ad Var So amlg rror ad Ttg Hoth aout dvdual OLS Rgro Coffct Suo H : : z ull Udr k f w do ot kow th tru oulato varac th w ca coduct hoth ttg ug aml S ad th t- dtruto Udr th ull t S t k Trac ad Trac Show rvou From Trac oral corr for c m m m j j j j j j j P Trac P Trac BA Trac AB c Trac Trac Trac Trac P k Sc /- th roof amout to howg that : Trac ad Trac Th th tadard rror of th OLS tmat that gt t out STATA Rcall Df: Lt Z hav a ormal dtruto wth ma ad varac Lt V hav a ch-quar dtruto wth v dgr of frdom Suo that Z ad V ar ddt Th v t v V Z T Hr Z Z / / Z ad χ ot: c A rak A th dmott matr rojcto A a mmtrc ad χ Fact: f Hr rak trac - Fact: f A dmott th raka traca

6 Ttg Lar Jot Hoth aout OLS Rgro Coffct Wald Tt W ca gralz th aov to tt ull hoth that mo a rtrcto o ot ol a gl dvdual coffct ut a lar comato of thm wrtt a a tm of lar quato Suo H : R # r r# r whr R ad r ar kow ad cfd th hoth #r th dmo or r or # of quato #r < Ad RakR #r w mo R to hav full row rak to mak ur that thr ar o rdudat quato ad that quato ar cott wth ach othr W rjct/do t rjct ull ad o th followg ft aml dtruto of Wald Stattc 3 Udr th ull f ull wr tru th Wald tattc F#r- dtrutd ot: Th alo th Lagrag ult Stat R r F [ R R ] R r /# r R r RVar R R r /# r F # r Dco Rul: Rjct for larg valu of W! Altratvl w ca u th followg mor covt ro of F ug th lklhood-rato rcl F SSR SSR R SSR U U /# r / 3 ot o t v F Stattc: A F ad t tt ar quvalt wh ttg hoth aout dvdual coffct: Sc hoth aout dvdual coffct ar lar hoth th t-tt of H : ca wrtt a [ [ ] ] R r whr R vctor w kth lmt ad r F t T F gral B f ull that a t of dvdual rgro coffct qual crta valu th F rfrrd to T f th z g lvl ach of th t-tt alha th ovrall z ot alha F tt a lklhood rato tt ad LR tt hav crta dral rort S t Fall w hav to how that ad / ar ddtl dtrutd Rcall: For Y Jotl ormal f CovY Y ddt Sc A ad ad ar jotl ormal codtoal o Alo th ar ucorrlatd o ad ar ddtl dtrutd codtoal o Sc Z ad ar fucto of ad thrfor th ar alo ddt 3 U / u Rcall th F tattc Df: Lt U ChSqu RV V ChSqv U ad V ddt RV Th F F u v V / v w/# r W ca wrt W whrw R r R R R r/ # r ad q q / wch-q#r udr th ull: Udr ull R r R R R R # r R R R Rcall f a m-dmoal radom vctor wth m μ Σmm th μ Σ μ χ m So w Ch-Sq#r q Ch-Sq- a how aov 3 w ad q ddt am raog aov: w a fucto of ad q a fucto of ad ddt w ad q ddt ot: Th drvato of th F-Rato ad o th Wald-Prcl ad thrfor oft calld th Wald Stattc cau t ad o OLY th urtrctd tmator whch ot cotrad to atf th rtrcto of th ull hoth S 7 PS3 Q 7

7 Rlato to amum Lklhood wth ormall dtrutd rror L tmator of 5 : Suo Aumto 5 hold Th th L of th OLS tmator ad L of SSR OLS/L tmator of UVU c t rach CRLB ad uad 6 : Udr aumto 6 th OLS tmator UVU that a othr uad ut ot carl lar tmator ha largr codtoal varac th matr ot: OLS tmator of do ot atta th CRLB c Var / 7 ad ot uad thrfor ot UVU 5 Udr aumto 6 / / / / Pr / : : SSR P P lug cl L FOC wrt fd Hold FOC wrt a π π π π ψ [ ] T ψ ad lt 6 Rcall th CR qualt: Suo that T a ral valud tattc ad dot th d vctor of artal drvatv ad uo that ogular ad th codto of th aov thorm hold Th for all Θ ψ ψ ψ ψ d f T Var whr k Hr Var Var Var rak χ χ χ

8 Trt ad Hoth Ttg udr Gaua rror:

9 : Gralzd Lat Squar GLS Rlag aumto a ad Codtoal Homokdatct ad o Sr al Corrlato w orv whch w aum to atf lart trct ogt ad o multcollat ut What t a: f rror ot codtoall homokdatc th valu of th dagoal lmt of ar ot th am f thr corrlato rror tw orvato th off-dagoal lmt of ar o- Coquc of Rlag Aumto a Codtoal Homokdat ct ad o Sral Corrlato: A Gau arkov Thorm o r hold for th OLS tmator : BLU om othr tmator B T-tt for lar rtrcto o a gl coffct ad F-tt jot lar rtrcto o r vald: t-rato ad Wald tattc/f-rato o r t ad f dtrutd C OLS tll uad c uad do ot rqur aumto a ad ol aumto 3 GLS odl Aumto: Aumto 3 from Lart Strct ogt ad o ultcollart Aumto c: V for om V kow mmtrc otv dft o-gular/vrtl matr W factor out from vr lmt for covc ak ot car 3 GLS tmator ad BLU: f th matr fucto VV - kow th a clvr traformato of th data w ca ota a modl that atf Gaua odl aumto ad th OLS tmator for th modl GLS wll BLU For a mmtrc otv dft matr V thr t ogular matr C t V - C C 8 Lt C C C atf aumto : Lart: From aumto for our modl ca r-wrtt a: Strct ogt: ad cota am f C C Aumto 3 o ultcollart: C ogular Rak Rak wth roalt C C C C CV C C V C CVC Codtoal Homokdatct: GLS tmator jut th OLS tmator ald to th traformd modl! C C C C C C C C V GLS [ ] [ ] V 9 Ft Saml Prort of GLS tmator a Uad: Udr GLS aumto 3 GLS ro for th Varac: Udr aumto Var GLS V c ffcc of GLS Gau-arkov Ald: Udr aumto th GLS ffct that th codtoal varac of a uad tmator that lar gratr tha or qual to Var th matr ot o th lmtg atur of GLS: Th c ft aml rort of GLS rt o th trct ogt aumto a wll a th kowldg of V what V a a fucto of th data A w ll tm-r cott whr rror ar oft rall corrlatd trct ogt too trog ad lmt th uful of th GLS rocdur So thr OLS or GLS hav th c ft-aml rort uch a uad Howvr gv that th rgror ar r-dtrmd wakr aumto tha trct ogt th th OLS tmator whch gor th ral corrlato th rror wll hav om good larg aml rort uch a cotc ad amtotc ormalt GLS wll ot hav th rort udr th wakr aumto GLS V mmtrc V orthogoall dagoalzal ctral thorm V Λ a matr of orthoormal gvctor orthogoal ad Λ matr of gvalu 8 orthogoal V Λ c V V V Λ V Λ / / / Λ Λ C C whr C Λ ot: Th alo ml that V C C C V C CV C CVC 9 Proof ar am a for ut ald to th traformd data

10 5 3 Scal Ca: A quar Htr tct Corrc dtoal ht Wghtd Lat S okda to: Co rocdatct Var of var wth o corr rror g W orv wth { } d whr ad o ral corrlato aumto -3 hold Codtoal htrokdatct: [ ] om for fucto kow Var Varac of var wth! Th V ad C matrc ar gv : Th w ota th traformd tm dvd vrthg / / / / V C o C C V V f w kow th fucto th w ca mlmt th aov chm ad rform OLS o th traformd modl ractc w do t kow uch fucto o w thr hav to aum a fuctoal form or tmat th ukow fucto ot: th modl orvato wth hghr codtoal varac gt a lowr wght ad vc vra B Srall Corrlatd rror g S o ow w hav Pra-Whto Corrcto u wth ad aumto a hold ut < j j Cov th form of corrlato calld AR Th V ad C matrc ar gv : C Radom ffct odl Ca alo trrtd a GLS S latr rou Rgro v Poold Rgro t u codtoal homokdatct hold th ru oold rgro thr wa th OLS offct ot tmat ar th am HW of 7 C V G Rug OLS rgro grou am a oolg rgro ug dumm varal th dffrc what aumto ou mak aou th codtoal varac f ou aum that th codtoal varac var grou th ru arat rgro f ou aum that th codtoal varac th am acro gro c S

11 Larg Saml Thor of OLS tma tor : rvo u cto aum to of ad ar farl trct W how hr that rlag ma of th aumto OLS whl do ot hav th ft aml rort wll hav c amtotc rort Bac Tm Sr Coct: of radom varal z Stochatc roc : a quc { } Ralzato/Saml Path of a tochatc roc: a ralzato of { } z quc of ral umr Tm Sr: f th quc of rv dd tm th tochatc roc calld a tm r w oft wll u tm r to rfr to th ralzato ad th tochatc roc ml a: { z } tru ma of ach of th rv th quc d for rgodc Statoart: Th fudamtal rolm tm-r aal that w orv th ralzato of th roc ol oc w gt ol aml ad orvato of ralzd { z }! dall w would lk to orv htor ma tm ovr to ota mor aml ut clarl th ot fal But f ach of th z com from th am dtruto tatoart th w ca vw ach ralzato of z a ralzato from th am dtruto Furthrmor f th roc ot too rtt rgodct th ach { } lm t of { z } othr lmt th ca th tm avrag ovr th lmt of { z } Dfg Statoar ad rogodc Proc Strctl Statoar Proc: A tochatc roc { } z wll cota om formato ot avalal from th wll cott for th ml ma! trctl tatoar f th jot dtruto of z z z z dd ol o for a gv ft tgr r ad a t of ucrt r - - r - ut ot r g jot dtruto of z z 5 am a z z 6 what mattr th rlatv oto th quc ot th aolut oto! { } Wakl Statoar Proc: A tochatc roc z wakl or covarac tatoar f: z do ot dd o ad Covz z -j t ft ad dd ol o j ut ot o g Covz z 5 Covz z 6 rg z ad to rgodc f for a two oudd fucto: f : R R g : R L R odc Proc: A tatoar roc { } lm f z z z z f z z g z k g l k z l Hurtcall a tatoar roc rgodc rgodc tatoart f t amtotcall ddt a rv or radom vctor otod far aart th quc ar almot ddtl dtrutd rgodc tatoart mortat dvlog larg aml thor cau of th rgodc thorm rgodc Thorm: Lt { } a tatoar ad rgodc roc wth z Th z z μ a z μ da: Th gralz olmogorov LL cau rgodc thorm allow for ral ddc whra ol rul t out d aumto rovdd that th ral ddc daar th -ru tatoar rgodc amtotcall larg aml mlcato: A momt of a tatoar ad rgodc roc f t ad ft cottl tmatd th aml momt Vctor Proc artgal artgl Dffrc artgal Vctor Proc: A vctor roc { g } calld a martgal f g g g g for ot: Th codtog t g g oft ca lld th formato t ad { g } calld a martgal c t formato t t ow at valu artgal Dffrc Squ g wth calld a marggal dffrc quc md or c: A vctor roc { } g martgal dffrc f th ctato codtoal o t at valu alo : g g g mortat Prort: A martgal dffrc quc ha o ral corrlato Covg g j { } g for rgodc Statoar artgal Dffrc CLT: Lt a vctor martgal dffrc quc that tatoar ad rgodc wth g g 3 ad lt g g D Th g g Σ OLS rocdur mortat coomtrc cau t ha good amtotc rort for a cla of modl dffrt from that ar uful coomc W rt a modl hr wth th wdt rag of coomc alcato t rl o o dtrutoal aumto ormalt rror trm ot car ad th trct ogt aumto rlac a much wakr aumto that th ar rdtrmd z olmogorov Scod Strog Law of Larg umr: Lt { } d wth z μ { z } { f z } Th z z μ a Sc f rgodc tatoar th alo rgodc tatoar for a maural fucto f

12 Aumto : To tud larg aml rort of OLS w rla aumto of furthr ad mo om add l aumto Th da that udr th fo llowg t of l tr ct aumto though w ca t a c ft aml rort w ca clam c amtotc rort Th DGP 5 quc of RV that gratd our ft aml/data atf th followg: Lart: for whr a vctor of laator varal uorvd rror trm k k rgodc Statoart: lmt th quc ar amtotcall ddt o ral ddc daar Th dmoal vctor tochatc roc { } jotl tatoar ad rgodc Th aumto aga allow for ral ddc to daar a aml gt larg ad allow u to vok rgodc thorm ad rgodc tatoar CLT ot: d aml ar trvall rgodc tatoar Prdtrmd Rgror / Cotmoraou Orthogoalt Codto: All th rgror ar rdtrmd th that th ar orthogoal to th cotmoraou rror trm: k k [ ] g whr g ot: Th wakr tha trct ogt Rak Codto: o multcollart th lmt matr ogular ad hc ft Dot Σ artgal Dffrc wth Ft Scod omt: g a martgal dffrc quc wth ft cod momt { g } a martgal dffrc quc o g wth g g - g - g for > o ral corrlato g Th matr of cro momt g g ogular P o g g g g S A var g rgodc Dffrc CLT ot: T h aumto trogr tha 3 c g md g W d t to drv amtotc ormalt of OLS tmator ot: Th aumto hard to trrt Oft w trrt a uffct codto: th rror trm rall ucorrlatd ad alo ucorrlatd wth th currt ad at rgror ot: S a matr of fourth momt ot: Th aumto alo ml that th rror trm ar ot rall corrlatd Add l : Ft fourth momt for Rgror: [ k j ] t ad ft for all kj For cott tmato of S Codtoal Homokdatct: > { } 3 Sc g tatoar th matr of cro momt do ot dd o Alo w mlctl aum that all th cro momt t ad ar ft Commt aout aumto: d trvall rgodc tatoar: f{ } d w hav radom aml th trvall atf rgodc tatoart Though trval th a mortat cal ca of rgodc tatoart! Prdtrmd v Strctl ogou rgror: Prdtrmd rgror ar ot rqurd to trctl ogou Th a wakr aumto tha Strct ogt ml that all th rgror ar orthogoal to all th rror trm currt at ad futur o th othr had rdtrmd rgror rtrct ol th cotmoraou rlatoh tw th rror trm ad th rgror Rak Codto a o ultcollart th lmt: Sc ft lm S Σ wth roalt th rgodc thorm whr S So for uffctl larg S ogular ad quvaltl rak for uffctl larg S a matr of th momt: Sc g w ca rwrt S th jth lmt j j whch volv th momt So for cott tmato of S w d a addtoal aumto that th th momt t A uffct codto for g to a md: A uffct codto that ar to trrt : rror trm rall ucorrlatd ad ucorrlatd wth th currt ad at rgror B f { } tatoar th { } alo tatoar ucodtoall homokdatc that do ot dd o But th do ot ml rror ar codtoall homokdatc thrfor w d a addtoal aumto 7 g η f whr η d ad η Th Εη f η f ddc do ot dd o ut Ε η f Ε η f Ε η f ddc dd o c var acro! 5 Data Gratg Proc DGP th tochatc roc / quc of radom varal that gratd th ft aml Y d Thrfor f w cf th DGP th jot dtruto of th ft aml Y d ca dtrmd! ft-aml thor whr th aml z fd ad ft w dfd a modl a a t of jot dtruto of Y d larg aml thor a modl tatd a a t of DGP that atf a t of aumto

13 3 Prort of th OLS tmator of ad : 6 A Cotc of for : Udr aumto lm k B Amtotc ormalt of : Udr 5 Rcall: Σ ad g g S Avar g k D Σ S Σ whr A var Σ SΣ C Cott tmat of Avar : Suo thr avalal a cott tmator Ŝ of S Th udr aumto Avar cottl tmatd A var S S S whr S D Cotc of tmato of varac of tru rror cott: Udr aumto P rovdd t ad ft OLS OLS rdual for orvato ot o Cott tmato of S: How do w ota cott tmator Ŝ of S kk kk from th aml? o U OLS Rdual for rror: Sc w do t orv gg w do t orv tru rror w u a cott tmator for om cott tmator of Th Cott tmato of S: Suo th coffct tmat ud for calculatg th rdual for Ŝ cott for ad uo S g g t ad ft Th udr aumto ad 6 Proof mlar to that for D Hoth Ttg Amtotc dtruto ad 6 mlcato udr Codtoal Homokdatct 6 S cott for S 6 Proof: A Y S g S g P S Σ rgodc Thorm c rgodc tatoar covrgc a whchml ro P S Σ CT w kowσ t P g g rgodc Thorm c g rgodc Statoar aga covrgc actuall a P Slutk B D D g S Statoar rgodc CLT Σ S Σ Slutk C From aov th S P S S Σ S Σ Slutk D P P t uffc to how that c / o Slutk [ ] [ ] g S P P P P g ad S c ad g g

14 5 Larg aml rort of WLS tmator ot o Bt Lar Prdctor: How do w trrt OLS f all w hav rgodc tatoart? otvato: Our thor aout ft ad larg aml rort rl o th lart aumto a wll a othr What f ralt o of th aumto ar atfd ct rgodc tatoart o for aml f w ol hav d aml ad w go ahad ad al OLS to th aml What t that w tmat? OLS tr : OLS gv u th t lar rdctor or th lat quar rojcto th t wa to larl com th laator varal to rdct th ddt varal Th: Suo w orv ad kow t jot dtruto th t rdctor of that t mmz S 7 f for a fucto f aumg Y ad f ft [ ] [ ] Sc ca hghl olar f w rtrct th rdctor to g a lar fucto of th w ca how that th t rdctor OLS! Proof: Th th am a our frt drvato of th OLS tmator Suo w wat to fd th t mmz S lar rdctor/aromato of ad o Thu w fd * that atf th orthogoalt codto th lar comato of uch that th ctd dtac mallt orthogoal trm * * * [ ] - Th: Th lat quar rojcto L th t lar rdctor of that t mmz Pf: Th: OLS Cottl tmat th Projcto Coffct ow uo w hav a aml of z draw from a rgodc tatoar tochatc roc { } wth th jot dtruto whch do ot dd o cau of tatoart dtcal to that of th aov aml Th cottl tmat * 7 [ ] [ ] [ ] [ f f f f f ] [ f ] f [ f ] f [ ] [ ] f