What regression does. = β + β x : The conditional mean of y given x is β + β x 1 2

Transcription

1 Sectn Smple Regressn What regressn es Relatnshp between varables Often n ecnmcs we beleve that there s a (perhaps causal) relatnshp between tw varables Usually mre than tw, but that s eferre t anther ay We call ths the ecnmc mel Example: Grae n Ecn 0 vs number f rm-mates takng Ecn 0 Functnal frm Is the relatnshp lnear? y = β + β x x s calle the regressr The lnear frm s a natural frst assumptn, unless thery rejects t β s slpe, whch etermnes whether relatnshp between x an y s pstve r negatve β s ntercept r cnstant term, whch etermnes where the lnear relatnshp ntersects the y axs Is t plausble that ths s an exact, etermnstc relatnshp? Data (almst) never ft exactly alng lne Why? Measurement errr (ncrrect efntn r msmeasurement) Other varables that affect y Relatnshp s nt purely lnear Relatnshp may be fferent fr fferent bservatns S the ecnmc mel must be mele as etermnng the expecte value f y E y x ( ) = β + β x : The cntnal mean f y gven x s β + β x te that ths says nthng abut ther aspects f the strbutn Hw es a change n x affect the varance f y? (We assume that t es nt) Hw es a change n x affect the mean, r the 75 th percentle, r any ther aspect f the strbutn f y? (If y s assume t be nrmal, then everythng abut the strbutn epens nly n mean an varance) Other regressn technques (n partcular, quantle regressn) allw us t examne the mpact f x n aspects f the strbutn f y ther than the mean ~ 5 ~

2 Ang an errr term fr a stchastc relatnshp gves us the actual value f y: y = β + β x + e Errr term e captures all f the abve prblems Errr term s cnsere t be a ranm varable an s nt bserve rectly Varance f e s σ, whch s the cntnal varance f y gven x, the varance f the cntnal strbutn f y gven x The smplest, but nt usually val, assumptn s that the cntnal varance s the same fr all bservatns n ur sample (hmskeastcty) E ( y x ) β =, whch means that the expecte value f y ncreases by β x unts when x ncreases by ne unt Des t matter whch varable s n the left-han se? At ne level, n: x = ( y β e), s β β x = γ + γ y+ v, where γ, γ =, v = e β β β Fr purpses f mst estmatrs, yes: We shall see that a crtcally mprtant assumptn s that the errr term s nepenent f the regressrs r exgenus varables Are the errrs shcks t y fr gven x r shcks t x fr gven y? It mght nt seem lke there s much fference, but the assumptn s crucal t val estmatn Exgenety: x s exgenus wth respect t y f shcks t y nt affect x, e, y es nt cause x Where the ata cme frm? Sample an ppulatn We bserve a sample f bservatns n y an x Depenng n cntext these samples may be Drawn frm a larger ppulatn, such as census ata r surveys Generate by a specfc ata-generatng prcess (DGP) as n tmeseres bservatns We usually wul lke t assume that the bservatns n ur sample are statstcally nepenent, r at least uncrrelate: ( j) cv y, y = 0, j We wll assume ntally (fr a few weeks) that the values f x are chsen as n an experment: they are nt ranm ~ 6 ~

3 We wll a ranm regressrs sn an scver that they n t change thngs much as lng as x s nepenent f e Gals f regressn True regressn lne: actual relatnshp n ppulatn r DGP True β an f (e x) Sample f bservatns cmes frm rawng ranm realzatns f e frm f (e x) an plttng pnts apprprately abve an belw the true regressn lne We want t fn an estmate regressn lne that cmes as clse t the true regressn lne as pssble, base n the bserve sample f y an x pars: Estmate values f parameters β an β Estmate prpertes f prbablty strbutn f errr term e Make nferences abut the abve estmates Use the estmates t make cntnal frecasts f y Determne the statstcal relablty f these frecasts Summarzng assumptns f smple regressn mel Assumptn #0: (Implct an unstate) The mel as specfe apples t all unts n the ppulatn an therefre all unts n the sample All unts n the ppulatn uner cnseratn have the same frm f the relatnshp, the same ceffcents, an errr terms wth the same prpertes If the Unte States an Mal are n the ppulatn, they really have the same parameters? Ths assumptn unerles everythng we n ecnmetrcs, an thus t must always be cnsere very carefully n chsng a specfcatn an a sample, an n ecng fr what ppulatn the results carry mplcatns SR: y = β + β x + e SR: E( e ) = 0, s E( y) = β + β x te that f x s ranm, we make these cntnal expectatns E( e x) = 0 E ( y x) = β + βx var e =σ = var y SR3: ( ) ( ) If x s ranm, ths becmes var ( e x) =σ = var ( y x) We shul (an wll) cnser the mre general case n whch varance vares acrss bservatns: heterskeastcty cv e, e = cv y, y = 0 SR4: ( j) ( j) Ths, t, can be relaxe: autcrrelatn ~ 7 ~

4 SR5: x s nn-ranm an takes n at least tw values We wll allw ranm x later an see that E( e x ) = 0 mples that e must be uncrrelate wth x e~ 0, σ SR6: (ptnal) ( ) Ths s cnvenent, but nt crtcal snce the law f large numbers assures that fr a we varety f strbutns f e, ur estmatrs cnverge t nrmal as the sample gets large Example: Assess the valty f these assumptns fr 0 rm-mate mel Strateges fr btanng regressn estmatrs What s an estmatr? A rule (frmula) fr calculatng an estmate f a parameter (β, β, r σ ) base n the sample values y, x Estmatrs are ften ente by ^ ver the varable beng estmate: An estmatr f β mght be ente ˆβ Hw mght we estmate the β ceffcents f the smple regressn mel? Three strateges: Meth f least-squares Meth f maxmum lkelh Meth f mments All three strateges wth the SR assumptns lea t the same estmatr rule: the rnary least-squares regressn estmatr: (b, b, s ) Meth f least squares Estmatn strategy: Make sum f square y-evatns ( resuals ) f bserve values frm the estmate regressn lne as small as pssble Gven ceffcent estmates b, b, resuals are efne as e y b bx Or eˆ = y yˆ, wth y b + bx Why nt mnmze the sum f the resuals? ˆ ~ 8 ~ ˆ We n t want sum f resuals t be large negatve number: Mnmze sum f resuals by havng all resuals nfntely negatve Many alternatve lnes that make sum f resuals zer (whch s esrable) because pstves an negatves cancel ut Why use square rather than abslute value t eal wth cancellatn f pstves an negatves? Square functn s cntnuusly fferentable; abslute value functn s nt Least-squares estmatn s much easer than least-absluteevatn estmatn

5 Prmnence f Gaussan (nrmal) strbutn n nature an statstcal thery fcuses us n varance, whch s expectatn f square Least-abslute-evatn estmatn s ccasnally ne (specal case f quantle regressn), but nt cmmn Least-abslute-evatn regressn gves less mprtance t large utlers than least-squares because squarng gves large emphass t resuals wth large abslute value Tens t raw the regressn lne twar these pnts t elmnate large square resuals Least-squares crtern functn: ˆ ( ) S = e = y b b x = = Least-squares estmatrs s the slutn t mns Snce S s a cntnuusly fferentable functn f the estmate parameters, we can fferentate an set the partal ervatves equal t zer t get the leastsquares nrmal equatns: S = ( y b bx)( x) = 0, b = yx + b x + b x = 0 = = = S = ( y b bx) = 0 b = y b b x = 0 = = y b bx = 0 b = y bx te that the b cntn assures that the regressn lne passes thrugh the pnt ( x, y ) b, b Substtutng the secn cntn nt the frst ve by : yx ( y bx ) x bx ( yx yx) b ( x x ) x x ( x x) + + = 0 + = 0 b ( )( ) yx yx y y x x σˆ = = = XY σˆ X The b estmatr s the sample cvarance f x an y ve by the sample varance f x What happens f x s cnstant acrss all bservatns n ur sample? Denmnatr s zer an we can t calculate b ~ 9 ~

6 Ths s ur frst encunter wth the prblem f cllnearty: f x s a cnstant then x s a lnear cmbnatn f the ther regressr the cnstant ne that s multple by b Cllnearty (r multcllnearty) wll be mre f a prblem n multple regressn If t s extreme (r perfect), t means that we can t calculate the slpe estmates The abve equatns are the rnary least-squares (OLS) ceffcent estmatrs Meth f maxmum lkelh Cnser the jnt prbablty ensty functn f y an x, f (y, x β, β ) The functn s wrtten s cntnal n the ceffcents β t make explct that the jnt strbutn f y an x are affecte by the parameters Ths functn measures the prbablty ensty f any partcular cmbnatn f y an x values, whch can be lsely thught f as hw prbable that utcme s, gven the parameter values Fr a gven set f parameters, sme bservatns f y an x are less lkely than thers Fr example, f β = 0 an β < 0, then t s less lkely that we wul see bservatns where y > 0 when x > 0, than bservatns wth y < 0 The ea f maxmum-lkelh estmatn s t chse a set f parameters that makes the lkelh f bservng the sample that we actually have as hgh as pssble The lkelh functn s just the jnt ensty functn turne n ts hea: ( β β ) ( β β ) L, x, y f x, y, If the bservatns are nepenent ranm raws frm entcal prbablty strbutns (they are IID), then the verall sample ensty (lkelh) functn s the pruct f the ensty (lkelh) functn f the nvual bservatns: (,,,,,, β, β ) = (, β, β ) f x y x y x y f x y n n = n L β, β x, y, x, y,, x, y = L β, β x, y ( ) ( ) n n = If the cntnal prbablty strbutn f e cntnal n x s Gaussan (nrmal) wth mean zer an varance σ : (,, ) (,, ) f x y β β = L β β x y = e πσ y β βx σ ( ) Because f the expnental functn, Gaussan lkelh functns are usually manpulate n lgs ~ 0 ~

7 te that because the lg functn s mntnc, maxmzng the lg-lkelh functn s equvalent t maxmzng the lkelh functn tself Fr an nvual bservatn: ln L = ln( πσ ) ( y β β x ) Aggregatng ver the sample: = ( β β ) = ( β β ) ln L, x, y ln L, x, y = = ln( πσ ) ( y β βx) = σ = ln( πσ ) ( y β βx) σ = ~ ~ σ The nly part f ths expressn that epens n β r n the sample s the fnal summatn Because f the negatve sgn, maxmzng the lkelh functn (wth respect t β) s equvalent t mnmzng the summatn But ths summatn s just the sum f square resuals that we mnmze n OLS Thus, OLS s MLE f the strbutn f e cntnal n x s Gaussan wth mean zer an cnstant varance σ, an f the bservatns are IID Meth f mments Anther general strategy fr btanng estmatrs s t set estmates f selecte ppulatn mments equal t ther sample cunterparts Ths s calle the meth f mments In rer t emply the meth f mments, we have t make sme specfc assumptns abut the ppulatn/dgp mments Assume E( e ) 0, = Ths means that the ppulatn/dgp mean f the errr term s zer Crrespnng t ths assumptn abut the ppulatn mean f e s the sample mean cntn eˆ = 0 Thus we set the sample mean t the value we have assume fr the ppulatn mean Assume cv ( xe, ) = 0, whch s equvalent t ( ) E x E( x) e = 0 Crrespnng t ths assumptn abut the ppulatn cvarance between the regressr an the errr term s the sample cvarance cntn: ( x ) ˆ x e = 0 Agan, we set the sample mment t the zer value that we have assume fr the ppulatn mment

8 Pluggng the expressn fr the resual nt the sample mment expressns abve: ( ) y b b x = 0, b = y bx Ths s the same as the ntercept estmate equatn fr the least-squares estmatr abve ( x x)( y b bx) = 0, ( x x)( y y + bx bx) = 0, ( x x)( y y) b ( x x)( x x) = 0, ( x x)( y y) b = x x ( ) Ths s exactly the same equatn as fr the OLS estmatr Thus, f we assume that E( e ) = 0, an cv ( xe, ) = 0 n the ppulatn, then the OLS estmatr can be erve by the meth f mments as well (te that bth f these mment cntns fllw frm the extene assumptn SR that E(e x) = 0) Evaluatng alternatve estmatrs (nt mprtant fr cmparsn here snce all three are same, but are they any g?) Desrable crtera Unbaseness: estmatr s n average equal t the true value E ( β ˆ ) = β Small varance: estmatr s usually clse t ts expecte value var ( β ˆ) = E ( β ˆ Eβˆ) Small RMSE can balance varance wth bas: RMSE = MSE MSE E ( β β ˆ ) We wll talk abut BLUE estmatrs as mnmum varance wthn the class f unbase estmatrs Samplng strbutn f OLS estmatrs b an b are ranm varables: they are functns f the ranm varables y an e We can thnk f the prbablty strbutn f b as ccurrng ver repeate ranm samples frm the unerlyng ppulatn r DGP ~ ~

9 In many (mst) cases, we cannt erve the strbutn f an estmatr theretcally, but must rely n Mnte Carl smulatn t estmate t (See belw) Because OLS estmatr (uner ur assumptns) s lnear, we can erve ts strbutn We can wrte the OLS slpe estmatr as b = = = = = = = = ( y y)( x x) =β + ( x x) ( β +β x + e y)( x x) ( x x) ( β +β x + e ( β +βx) )( x x) ( x x) ( β ( x x) + e)( x x) = ( x x) ( ) ( x x) e x x The thr step uses the prperty y = β + β x, snce the expecte value f e s zer Fr nw, we are assumng that x s nn-ranm, as n a cntrlle experment If x s fxe, then the nly part f the frmula abve that s ranm s e The frmula shws that the slpe estmate s lnear n e Ths means that f e s Gaussan, then the slpe estmate wll als be Gaussan Even f e s nt Gaussan, the slpe estmate wll cnverge t a Gaussan strbutn as lng as sme mest assumptns abut ts strbutn are satsfe Because all the x varables are nn-ranm, they can cme utse when we take expectatns, s ( x x) e ( x x) E( e) = = E( b) = β + E =β + =β ( x x ) ( x x ) = = What abut the varance f b? ~ 3 ~

10 We wll the etals f the analytcal wrk n matrx frm because t s easer var b = E b β ( ) ( ) = E = σ = = ( x x) ( x x) E( e ) = = ( x x) HGL equatns 4 an 6 prve frmulas fr varance f b an the cvarance between the ceffcents: var ( b ) =σ ( b b ) = = x ( ) x x cv, =σ < 0 ( x x) = x te that the cvarance between the slpe an ntercept estmatrs s negatve: verestmatng ne wll ten t cause us t unerestmate the ther What etermnes the varance f b? Smaller varance f errr mre precse estmatrs Larger number f bservatns mre precse estmatrs Mre spersn f bservatns arun mean mre precse estmatrs What we knw abut the verall prbablty strbutn f b? If assumptn SR6 s satsfe an e s nrmal, then b s als nrmal because t s a lnear functn f the e varables an lnear functns f nrmally strbute varables are als nrmally strbute If assumptn SR6 s nt satsfe, then b cnverges t a nrmal strbutn as prve sme weak cntns n the strbutn f e are satsfe These expressns are the true varance/cvarance f the estmate ceffcent vectr Hwever, because we nt knw σ, t s nt f practcal use t us We ~ 4 ~

11 nee an estmatr fr σ n rer t calculate a stanar errr f the ceffcents: an estmate f ther stanar evatn The requre estmate n the classcal case s s eˆ = We ve by because ths s the number f egrees f freem n ur regressn Degrees f freem are a very mprtant ssue n ecnmetrcs It refers t hw many ata pnts are avalable n excess f the mnmum number requre t estmate the mel In ths case, t takes mnmally tw pnts t efne a lne, s the smallest pssble number f bservatns fr whch we can ft a bvarate regressn s Any bservatns beyn make t (generally) mpssble t ft a lne perfectly thrugh all bservatns Thus, s the number f egrees f freem n the sample We always ve sums f square resuals by the number f egrees f freem n rer t get unbase varance estmates Fr example, n calculatng the sample varance, we use s = ( z z) because there are egrees f = freem left after usng ne t calculate the mean Here, we have tw ceffcents t estmate, nt just ne, s we ve by The stanar errr f each ceffcent s the square rt f the crrespnng agnal element f that estmate cvarance matrx te that the HGL text uses an alternatve frmula base n σ ˆ = eˆ = Ths estmatr fr σ s base because there are nly egrees f freem n the resuals are use up n estmatng the β parameters Intructn t Stata Stata wrks n a ataset (ta fle) Stata cmmans: Enter at prmpt Chse frm menu/wnws In large samples they are equvalent Enter nt a fle fr batch executn The Stata screen ~ 5 ~

12 Results wnw Cmman wnw Varables wnw Revew wnw Prpertes wnw Lg fles Set ne up s stuents can see t later Openng a ata set Shw ata etr/brwser Cmmans t statstcal analyss summarze reg Graphcs cmmans Use menus t get see all ptns wthut rememberng hw t type Sample analyss: Ree Ecn 0 graes Depenent varable gpnts Shw summary statstcs Pnt ut screte strbutn: Is ths a prblem? Regressn n sngle varable: hsgpa Interpretng ceffcents (nte that ntercept s autmatcally nclue: nnt ptn) Pnt ut stanar errr, t statstc, p value, cnfent lmts te mssng bservatns Shw utreg usng graeregs, se Alternatve: regress n rr Shw hw utreg as clumns utreg usng graeregs, se merge Calculate precte values wth prect prect gpahat Graph actual an precte vs rr Dsplay hypthetcal precte values wth margns margns, at(rr=(5 4 3 )) Transfrmatn: satc00 = satv00 + satm00 Regress n satc00 Cmpare t hsgpa regressn Regressn n ummy varable Regress n female Interpretatn f ceffcents Categry mean prectns: margns female Multple regressn emnstratn Reg gpnts rr satv00 satm00 female ~ 6 ~

13 Shw utreg wth multple varables utreg usng graeregs, se merge A takng t regressn an nterpret Use margns t slate prectns f hypthetcal nvual varables wth thers at means margns, at(rr = (5 4 3 )) atmeans margnsplt Mnte Carl meths Base n HGL Appenx G Hw we evaluate an estmatr such as OLS? Uner smple assumptns, we can smetmes calculate the estmatr s theretcal prbablty strbutn We can ften calculate the theretcal strbutn t whch the estmatr cnverges n large samples even when we cannt calculate the small-sample strbutn In general (an, n partcular, when we cannt calculate the true strbutn), we can smulate the mel ver thusans f samples t estmate ts strbutn The estmatn f the prbablty strbutn f an estmatr thrugh smulatn s calle Mnte Carl smulatn an s an ncreasngly mprtant tl n ecnmetrcs Cnser smple Mnte Carl example: (MC Class Demta) Let s suppse that we are wrkng wth a gven, fxe = 57 We have fxe, gven values f the x varable fr all 57 bservatns Usng HGL s ex9-3ta wth avertsng varable as x We assume that the true ppulatn values f β an β are 0 an 3 Clse t estmate values fr regressn f sales n avertsng The true errr term s IID nrmal wth varance 009 (stanar evatn 03) T use Mnte Carl t smulate the strbutn f the OLS estmatrs, we generate M replcatns f the samplng experment: M sets f 57 IID (0, 009) smulate bservatns n e usng ranm number generatr (We wul generate sample values fr x f t were nt beng taken as fxe) Calculate the M sets f 57 values f y fr each bservatn as β + β x + e wth knwn values f the parameters an x an smulate values f e Run M regressns fr the M smulate samples, keepng the estmate values f nterest (presumably ˆβ an ˆβ, but pssbly als ther values) Lk at strbutn f the estmatrs ver M replcatns t apprxmate the actual strbutn Mean ~ 7 ~

14 Varance/stanar evatn/stanar errr Quantles fr use n nference Demnstrate usng Stata Setup ata x s alreay n MC Class Demta Create fle prgram lstest g e=rnrmal(0, 03) g y=0 + 3*x+e reg y x rp e y en La t nt memry: run lstest Run smulatn wth 5000 replcatns smulate b=_b[x], reps(5000): lstest Shw summary stats, hstgram, centles (5, 975) Hw g s the OLS estmatr? Is OLS the best estmatr? Uner what cntns? Uner classcal regressn assumptns SR SR5 (but nt necessarly SR6) the Gauss- Markv Therem shws that the OLS estmatr s BLUE Any ther estmatr that s unbase an lnear n e has hgher varance than b te that (5, 0) s an estmatr wth zer varance, but t s base n the general case Vlatn f any f the SR SR5 assumptns usually means that there s a better estmatr Least-squares regressn mel n matrx ntatn (Frm Grffths, Hll, an Juge, Sectn 54) We can wrte the th bservatn f the bvarate lnear regressn mel as y = β + β x + e Arrangng the bservatns vertcally gves us such equatns: y = β + β x + e, y =β +β x + e, y x e =β +β + ~ 8 ~

15 Ths s a system f lnear equatns that can be cnvenently rewrtten n matrx frm There s n real nee fr the matrx representatn wth nly ne regressr because the equatns are smple, but when we a regressrs the matrx ntatn s mre useful Let y be an clumn vectr: y y y = y Let X be an matrx: x x X = x β s a clumn vectr f ceffcents: β β = β An e s an vectr f the errr terms: e e e = e Then y = Xβ + e expresses the system f equatns very cmpactly (Wrte ut matrces an shw hw multplcatn wrks fr sngle bservatn) In matrx ntatn, eˆ = y Xb s the vectr f resuals Summng squares f the elements f a clumn vectr n matrx ntatn s just the nner pruct: eˆ ˆˆ =, = ee where prme entes matrx transpse Thus we want t mnmze ths expressn fr least squares ee ˆˆ = ( y Xb) ( y Xb) = ( y bx )( y Xb) = yy bxy + bxxb Dfferentatng wth respect t the ceffcent vectr an settng t zer yels Xy + XXb = 0, r XXb = Xy Pre-multplyng by the nverse f X X yels the OLS ceffcent frmula: ( ) b = XX Xy (Ths s ne f the few frmulas that yu nee t memrze) ~ 9 ~

16 te symmetry between matrx frmula an scalar frmula X y s the sum f the crss pruct f the tw varables an X X s the sum f squares f the regressr The frmer s n the numeratr (an nt nverte) an the latter s n the enmnatr (an nverte) In matrx ntatn, we can express ur estmatr n terms f e as ( ) ( XX ) X ( Xβ e) ( ) β ( ) β ( XX ) Xe b= XX Xy = + = XX XX + XX Xe = + When x s nn-stchastc, the cvarance matrx f the ceffcent estmatr s als easy t cmpute uner the OLS assumptns Cvarance matrces: The cvarance f a vectr ranm varable s a matrx wth varances n the agnal an cvarances n the ffagnals Fr an M vectr ranm varable z, the cvarance matrx s t the fllwng uter pruct: (( )( ) ) E ( z Ez) E( z Ez)( z Ez) E( z Ez)( zm Ez) E ( z Ez )( z Ez ) E ( z Ez ) E ( z Ez )( z Ez ) cv( z) = E z Ez z Ez M = E z Ez zm Ez E z Ez zm Ez E zm Ez ( )( ) ( )( ) ( ) In ur regressn mel, f e s IID wth mean zer an varance σ, then cv e = E ee =σ I, wth I beng the rer- entty Ee = 0 an ( ) ( ) matrx We can then cmpute the cvarance matrx f the (unbase) estmatr as cv = E ( ( XX ) Xe )(( XX ) Xe ) ( b) = E ( b β)( b β) = = E ( XX ) XeeX ( XX ) ( XX ) X E ( ee ) X( XX ) ( XX ) XX ( XX ) ( XX ) =σ =σ What happens t var ( ) b as gets large? Summatns n X X have atnal terms, s they get larger Ths means that nverse ~ 30 ~

17 matrx gets smaller an varance ecreases: mre bservatns mples mre accurate estmatrs te that varance als ncreases as the varance f the errr term ges up Mre mprecse ft mples less precse ceffcent estmates Our estmate cvarance matrx f the ceffcents s then s ( ) XX The (, ) element f ths matrx s s ˆ e = = ( x x) ( x x) = = Ths s the frmula we calculate n class fr the scalar system Thus, t summarze, when the classcal assumptns hl an e s nrmally strbute, ( σ ( XX ) ) b~ β, Asympttc prpertes f OLS bvarate regressn estmatr (Base n S&W, Chapter 7 t cvere n class Sprng 04) Cnvergence n prbablty (prbablty lmts) Assume that S, S,, S, s a sequence f ranm varables In practce, they are gng t be estmatrs base n,,, bservatns p S μ f an nly f lm Pr S μ δ = 0 fr any δ > 0 Thus, fr any small value f δ, we can make the prbablty that S s further frm μ than δ arbtrarly small by chsng large enugh p If S μ, then we can wrte plm S = μ Ths means that the entre prbablty strbutn f S cnverges n the value μ as gets large Estmatrs that cnverge n prbablty t the true parameter value are calle cnsstent estmatrs Cnvergence n strbutn If the sequence f ranm varables {S } has cumulatve prbablty strbutns F, F,, F,, then S lm F t = F t, fr all t at whch F s cntnuus S f an nly f ( ) ( ) ~ 3 ~ If a sequence f ranm varables cnverges n strbutn t the nrmal strbutn, t s calle asympttcally nrmal

18 Prpertes f prbablty lmts an cnvergence n strbutn Prbablty lmts are very frgvng: Slutsky s Therem states that plm (S + R ) = plm S + plm R plm (S R ) = plm S plm R plm (S / R ) = plm S / plm R The cntnuus-mappng therem gves us Fr cntnuus functns g, plm g(s ) = g(plm S ) An f S g S g S S, then ( ) ( ) Further, we can cmbne prbablty lmts an cnvergence n strbutn t get If plm a = a an S, then a S Central lmt therems S as a ± S a± S S / a S/ a These are very useful snce t means that asympttcally we can treat any cnsstent estmatr as a cnstant equal t the true value There s a varety wth slghtly fferent cntns Basc result: If {S } s a sequence f estmatrs f μ, then fr a we varety f unerlyng strbutns, ( S ) ( ) the unerlyng statstc μ 0, σ, where σ s the varance f Applyng asympttc thery t the OLS mel Uner the mre general cntns than the nes that we have typcally assume (nclung, specfcally, the fnte kurtss assumptn, but nt the hmskeastcty assumptn r the assumptn f fxe regressrs), the OLS estmatr satsfes the cntns fr cnsstency an asympttc nrmalty var ( x E( x) ) e β 0, Ths s general case wth var ( x ) heterskeastcty Wth hmskeastcty, the varable reuces t the usual frmula: σ ( b β) 0, var ( x ) ( b ) plm σ ˆ =σ, as prven n Sectn 73 b b t b β = se ( b ) Chce fr t statstc: ( ) 0, ~ 3 ~

19 If hmskeastc, nrmal errr term, then exact strbutn s t If heterskeastc r nn-nrmal errr (wth fnte 4 th mment), then exact strbutn s unknwn, but asympttc strbutn s nrmal Whch s mre reasnable fr any gven applcatn? Lnearty an nnlnearty The OLS estmatr s a lnear estmatr because b s lnear n e (whch s because y s lnear n β), nt because y s lnear n x OLS can easly hanle nnlnear relatnshps between y an x lny = β + β x y = β + β x etc Dummy (ncatr) varables take the value zer r ne Example: MALE = f male an 0 f female y = β + β MALE + e Fr females, E[ y MALE ] =β Fr males, E[ y MALE ] = β + β Thus, β s the fference between the expecte value f males an females ~ 33 ~