( ) ( )) ' j, k. These restrictions in turn imply a corresponding set of sample moment conditions:

Transcription

1 esng he Random Walk Hypohess If changes n a sees P ae uncoelaed, hen he followng escons hold: va + va ( cov, 0 k 0 whee P P. k hese escons n un mply a coespondng se of sample momen condons: g µ + µ (,, k, l 0 kl,,. ( ( k k( ( µ ( µ ρ µ ρ ρ µ µ ρ l l hese momen condons lead o an esmao fo he auocoelaon of : ( µ ( µ k ˆ k. / ρ ( + µ * Le µ. Unde he assumpon ha he ae saonay and egodc, and also assumng * *. E 0 * * *. E k 0 3. E * * 4 *3 * 4. E 0, k, gven esuls n Hansen (98, s possble o show ha he asympoc dsbuon of ˆ ρ ˆ ρ... ˆ ρ s gven by ( ' k l a ˆ ρ ~ N 0, I. hs dsbuon s deved usng Hansen s (98 devaon of he asympoc vaancecovaance max of he esmao, DS 0 0 ' D 0. Fo hs poblem,

2 D Noe ha because hs max s dagonal, we only have o woy abou he (3,3, (3,4 and (4,4 elemens of he vaance-covaance max. I s easy o show ha hese elemens ae equal o 4, 0and 4, especvely. heefoe, he vaance-covaance max fo he esmaos s educed o , whch gves us he desed esul. Snce any lnea combnaon of nomals s nomal, he followng moe geneal esul s ue: ˆ ρ ~ ( 0, N I N N N N ( ' ˆ ~ 0, D N DD ρ M N N N M M ( ( ' ' M ˆ ˆ Dρ DD Dρ ~ χ. M M M M hs equaon conans ehe exac o appoxmae epesenaons of many of he ecen sascs used n he leaue o es fo seal coelaon. Fo example, he mulpeod sample auocoelaon, he commonly used vaance ao sasc, he Box-Pece sasc, among ohes, fall no hs class. able povdes some examples of he weghs D and he coespondng sascs.

3 ABLE able povdes examples of a geneal class of sascs fo esng whehe a sees P P s uncoelaed. Specfcally, hs class can be epesened as lnea combnaons of conssen auocoelaon esmaos: ˆ ρ Dˆ ρ D whee ˆ ρ ˆ µ (... Dk Dl ( P P ˆ µ ( P P ˆ µ ( P ˆ P µ ( P P hs class ncludes many of he sascs cuenly employed n he fnances and macoeconomcs leaue; ncludng he vaance-ao sasc and mulpeod auocoelaon, among ohes. ˆ β Sasc Auocoelaon P P ˆ µ P P ˆ µ ( ( ( P ˆ P µ Vaance Rao ( P ˆ P µ ( P ˆ P µ Vˆ ˆ ρ Repesenaon Dˆ ρ mn(, ˆ ρ ( ( ˆ ρ ypcal Weghs D D D mn, / D / Q ˆ β, egadeesh (99 Regesson ( P P ˆ µ ( P P ˆ µ ( P ˆ P µ Box-Pece ( ˆ ( P ˆ P µ P P µ P P ˆ µ ˆ ρ / D / D I( n eqn. ρ ˆ 3

4 Examples Consde one pacula sasc; namely, he mulpeod auocoelaon looked a n Fama- Fench (988 and Huznga (987. Specfcally, ( p ˆ( ˆ p µ p p µ ( ˆ P P µ ( ˆ ( ˆ 0 + µ 0 + µ ( ˆ 0 + µ ˆ β (. ( Auocoelaon epesenaon o see he beakdown of he above esmao n ems of conssen esmaos of he ˆ ρ, consde he followng easonng: auocoelaons,. hen he numeao of ˆ β ( esmaes ( hen he numeao of cov( +,,cov( +,,cov(, and cov(,, mplyng one ˆ ρ (, and one ρ and so foh. Suppose ; ˆ ρ. Now suppose 3 ; cov,, mplyng one β esmaes ρ, wo In geneal, suppose ; ˆ β esmaes he fs ode covaance once, second ode covaance wce,...h ode covaance mes, (+h ode covaance mes,., (-h ode covaance wce, and h ode covaance once. he esul ha ˆ β s appoxmaely a lnea combnaon of ˆk ρ fo dffeen k hen follows. In geneal, ˆ β Asympoc Dsbuon hen he numeao of ˆ ρ mn(, /. Above we showed ha he sascs could be wen as lnea combnaons of conssen esmaos of dffeen auocoelaons. We can use hs esul o explcly deve he sascal elaon beween hese sascs. Noe ha, unde he null hypohess of a andom 4

5 walk, s possble o show ha he dffeen conssen esmaos of he h ode ˆ ρ and ˆ ρ k, ae asympocally pefecly coelaed. Gven ha mos of auocoelaon he ecen sascs can be wen as lnea combnaons of he ˆ ρ 's fo dffeen s, s saghfowad o calculae he on confdence neval of any wo sascs. Fo example, ˆ β, and he peod consde a compason beween he peod auocoelaon, vaance ao, V ˆ (. Usng he epesenaons as lnea combnaons of auocoelaon, we can we hese sascs especvely n he followng fom: ˆ β ( ˆ ρ + ˆ ρ ˆ ρ ˆ ˆ ρ + ρ Vˆ + (( ˆ ρ( + ( ˆ ρ( ˆ ρ (... ˆ ˆ + + ρ ( + ρ (. Usng he epesenaons gven above, we can we down he weghs ha ˆ β place on he auocoelaons: Auocoelaons : ˆ ρ ˆ ρ... ˆ ρ ˆ ρ ˆ ρ... ˆ ρ ˆ ρ ˆ β weghs : ˆ + V ( weghs : and Vˆ Usng hs nfomaon, s possble o calculae he covaance beween ˆ β Vˆ Specfcally, cov( ˆ β, Vˆ (, yeldng he on asympoc dsbuon: and. + ˆ β a 0 3 N,. Vˆ ( 0 ( 4 ( 6 ˆ β and Vˆ, s equal o Fom above, he coelaon coeffcen beween he sascs, appoxmaely 75%. heefoe, ove 50% of he vaaon n he -monh auocoelaon can be explaned by he -monh vaance ao. In conas, he coelaon beween ˆ β and Vˆ s only 35%. hee afoemenoned esuls concenng he asympoc dsbuon of ˆ Fs, he asympoc vaance of ˆ ρ equals one fo all and. ˆ ρ and ˆl ρ fo l esmaos (e.g. ˆ ρ and ˆ ρ ( k fo all on dsbuon calculaons exemely easy. ae asympocally uncoelaed. hd, any wo h ρ ae of pacula mpoance. Second, any wo esmaos (e.g. ode auocoelaon ae asympocally pefecly coelaed. hese hee esuls make 5

6 on es Acoss Holdng Peods Wh espec o he Fama and Fench (988 mulpeod auocoelaons, he ypcal ' elemens of asympoc vaance-covaance max of he ˆ β s ae (, + s k + ˆ β 3 k va β k s( k, k + + k 3k whee s k, [( l mn( k, l]. l he coelaon beween any wo esmaos can be expessed as ρ k 3 ( s( k, + ( + ( + k k Fo close o k, he egesson coeffcens ae hghly coelaed. Fo example, consde he esmaos ˆ β ˆ 48 and β 60. he coelaon s.9; heefoe, ove 80% of he vaaon n 60 ˆβ can be explaned by 48. ˆβ A popula sasc o es mulple escons n fnance has been he Wald sasc. As an example, consde he sasc fo a on es of β βk 0 (denoe ˆ β : 3 ˆ β 3 ˆ k k β ˆ ˆ k + 6 ρ kββ k χ. k + + ( + ( k + ρ k he evdence a peods s clealy no ndependen of evdence a k peods and he on es eflecs hs dependence. Fo examples, consde he esmaes of he 48 and 60 monh seal coelaon coeffcens of he NYSE equal weghed pofolo, ˆ β and ˆ β If he economecan wee o gnoe he coelaon ρ 48,60.9 and ea he wo esmaos ndependenly, he value of he sasc would be 80% hghe han he ( ˆ ˆ β48, β 60 sasc whch akes accoun of he coelaon. Noe ha he 60 monh esmao has 48 monhs of daa n common wh he 48 monh esmao. Gven ha ˆ β 48.36, wha should we expec 60 ˆβ o equal unde he null See Fama-Fench (988, able. 6

7 hypohess? Indvdually, he expeced value of 60 ˆβ equals zeo; howeve, hs wll no be he case when ˆ β In fac, ˆ ˆ ( β48.36, β s 6.3 mes geae han ˆ ˆ ( β48 36, β Smply saed, he seal coelaon paen n he esmaes.e. ˆ β.36, ˆ β 0.0 s no conssen wh he coelaon paen n he esmaos. ( Mulvaae ess he esuls above ook each asse sepaaely. Wha s he asympoc dsbuon beween a a b b ˆ ρ, ˆ, ˆ and ˆ k ρl ρk ρ l, fo any asse a and b and kh and lh ode auocoelaon? he devave max agan s gong o be dagonal, so we can concenae on he elemens of he vaance-covaance max of he momen condons elaed o he auocoelaon esmaos, yeldng a 0 cov 0 a a ab cov 0 0 a a ab a, cov ab b b b 4 0 cov ab b b b whch yelds he followng vaance-covaance max fo vecos ˆa ρ and ˆ ρ b, I, whee has ypcal elemen ρ ab on he off-dagonals and on he dagonals (.e. he coelaon max wh squaed elemens. Small Sample Popees Noe ha n pacce wh peod euns and obsevaons, we have only / ndependen obsevaons. hee s some effcency gan by usng ovelappng euns bu s no of a mulple- fold magnude. he sascs dscussed so fa assume fxed and le,. Howeve, fo lage, hese sascs seem o possess poo small sample popees. In fac, n many nsances n he empcal leaue (boh wh acual and smulaed daa, eseaches le gow wh. We can deve an alenave asympoc heoy whch les / δ, a consan beween 0 and, wh, he queson s whch asympoc heoy (fxed vesus / δ povdes a bee appoxmaon n small samples. Consde he peod eun, x( R 0. 7

8 Fo smplcy, we ae gong o esc ouselves o nonovelappng obsevaons and he vaance-ao sasc. When consuced usng he nonovelappng obsevaons, he sasc s / ( (... ( x ˆ µ + x ˆ µ + + x ˆ µ R whee R R ˆ, ˆ, µ µ R and whee s assumed ha s an nege. In he usual fxed asympoc eamen, unde he null hypohess he numeao and denomnao convege n pobably o va( R, so ha ( convege n pobably o one. Lo and MacKnlay (988, 989 sudy he fxed asympoc dsbuon of (. hey assume ha { R } ae..d. wh a nomal dsbuon and show ha: d ( ( N(0, (. I uns ou ha hs appoxmaon woks well only fo small. he poo pefomance of he fxed asympoc appoxmaon when s lage n elaon o ndcaes a need fo an alenave appoach n whch s explcly ecognzed as lage. In he nonovelappng case.e., (, such an alenave appoxmaon s n fac easy o develop. o smplfy exposon, assume (unealscally, noe ha s no necessay ha µ s known o equal zeo so ha R ε and he ems nvolvng ˆµ can be dopped. In hs case, when s an nege ( can be ewen as ( ( ε + ( ε... ( ε / R he numbe of nonovelappng obsevaons used o consuc ( If δ as, emans fxed a. (Noe ha as nceases wll no eman an nege: fo hs o δ fomally o apply, ake he lm along he sequence δ,,,3,..., whee δ s an nege. hs esuls n a smple lmng dsbuon fo (. Because as, each of N 0, vaaes. Upon he paal sums above conveges n dsbuon o ndependen ˆ R dvdng by (whch s conssen fo, one fnds decly ha hs sasc has a lmng ch-squaed dsbuon: d ( δχ. δ 8

9 hs smple esul hghlghs hee key feaues ha ypcally dsngush he δ asympoc appoxmaon fom he fxed appoxmaons. Fs, hs vaance-ao sasc s no conssen, bu ahe has a nondegeneae lmng dsbuon whou fs scalng by. Second, hs lmng dsbuon s no he nomal dsbuon, bu s nonnomal hee, a χ, dvded by s degees of feedom. hd, hs esul holds even f hee s δ nonnomaly and heeoskedascy (as long as he aveage condonal heeoskedascy ends o a consan vaance. 9

10 ABLE Empcal Dsbuon of Maxmum Mulpeod Auocoelaons Acoss Reun Hozons: [-6,8,0] yeas (70 obs. able epos he empcal dsbuon of he peod seal coelaon esmaos lages absolue devaon fom zeo acoss hozons {,4,...7,96,0 }. Denoe hs sasc ˆ β *, whee * s he peod n whch he lages devaon occus. he dsbuon s compued usng 5000 eplcaons of daa geneaed fom an..d. nomally dsbued andom vaable (n whch he mean and vaance ae chosen o mach he equal-weghed sock eun ndex. Also povded ae he empcal dsbuons of ˆ β ˆ * and β * (when * { } 36,48,60. A. Fequency Dsbuon fo * y. y. Y. 3 y. 4 y. 5 y. 6 y. 8 y. 0 Pecenage 3% 7.% 8.86% 8.4% 8.76% 3.46% 7.3% 33.06% B. Empcal Dsbuon of ˆ β * Empcal CDF Values Sasc Mean β * ˆ β * ˆ β ( * 3, 4,5, * { } 0

11 ABLE 3 on es Sascs fo Pofolo Reuns ( Reun Hozon (yea-6,8,0 able 3 epos F ess (denoe ˆ β fo whehe he seal coelaon esmaes of -6, and 0 yea euns on 9 dffeen pofolos (.e. ndex, 0 sze and 7 ndusy ae each only sgnfcanly dffeen fom zeo. Columns 5 and 6 epo heeoskedascy conssen ( ˆ β sascs. Columns 7 and 8 povde he max ˆ β sasc aken fom he acual daa acoss he -6,8 and 0 yeas hozons fo each pofolo. he ess ae pefomed on ovelappng monhly euns fo he peod he empcal p value s he p value fo he sasc geneaed fom each pofolo s empcal dsbuon unde he null hypohess. hs dsbuon was compued fom 5000 eplcaons usng daa ndependenly dawn wh eplacemen fom he sample dsbuon of each pofolo s euns.