CARDIFF BUSINESS SCHOOL WORKING PAPER SERIES

Transcription

1 CARDIFF BUSINESS SCHOOL WORKING PAPER SERIES Cadiff Ecoomic Wokig Pape Woo K Wog A Uique Othogoal Vaiace Decompoitio E008/0 Cadiff Buie School Cadiff Uiveity Colum Dive Cadiff CF0 3EU Uited Kigdom t: +44 (0) f: +44 (0) wwwcadiffacuk/cab ISSN Apil 008 Thi wokig pape i poduced fo dicuio pupoe oly Thee wokig pape ae epected to be publihed i due coue i evied fom ad hould ot be quoted o cited without the autho witte pemiio Cadiff Ecoomic Wokig Pape ae available olie fom: Equiie: EcoWP@cadiffacuk

2 A UNIQUE ORTHOGONAL VARIANCE DECOMPOSITION WOON K WONG INVESTMENT MANAGEMENT RESEARCH UNIT CARDIFF BUSINESS SCHOOL 3 Mach 008 Cadiff Uiveity Abecoway Buildig Colum Dive Cadiff CF0 3EU Tel: ; Fa: ; wogwk3@cadiffacuk

3 A UNIQUE ORTHOGONAL VARIANCE DECOMPOSITION ABSTRACT Let e ad Σ be epectively the vecto of hock ad it vaiace covaiace mati i a liea ytem of equatio i educed fom Thi aticle how that a uique othogoal vaiace decompoitio ca be obtaied if we impoe a etictio that maimize the tace of A a poitive defiite mati uch that Az = e whee z i vecto of ucoelated hock with uit vaiace Such a etictio i meaigful i that it aociate the laget poible weight fo each elemet i e with it coepodig elemet i z It tu out that A = Σ the quae oot of Σ KEYWORDS: Vaiace decompoitio Choleky decompoitio uique othogoal decompoitio ad quae oot mati JEL Claificatio: C0 INTRODUCTION VARIANCE DECOMPOSITION IS OFTEN CARRIED OUT i a ecoometic aalyi Fo eample Shoock (98) coide the iue of iequality decompoitio by facto compoet I a tuctual VAR ytem ecoomic theoy i ofte employed i ode to cotuct the tuctual hock that ae ucoelated with each othe; ee fo itace Sim (986) Beake (986) ad Blachad ad Quah (989) Howeve it i well kow that the vaiace decompoitio fo a igle equatio ytem i ot uique I the cae of a tuctual VAR aalyi the electio of odeig i Choleky decompoitio i geeally ad hoc ad covicig idetifyig aumptio ae had to come by Thi aticle popoe a uique othogoal vaiace decompoitio that ca be applied to both igle a well a multiple equatio ytem Thi i evideced fom the emak by Hamilto (994 p 335) that if thee wee compellig idetifyig aumptio fo uch a ytem the fiece debate amog macoecoomit would have bee ettled log ago!

4 Let e ad Σ be epectively the vecto of hock ad it vaiace covaiace mati i the equatio ytem i educed fom Let A be a decompoitio mati uch that ad Az = e whee z i vecto of ucoelated hock with uit vaiace It i how that if we etict A to be poitive defiite ad it tace be maimized a uique decompoitio mati give by A A = Σ A = Σ i obtaied While A beig poitive defiite i a vey geeal coditio the maimizatio of it tace i ituitively appealig Fo the ake of agumet let coide z a the udelyig tuctual hock The the highe i the tace of A the le i ay the i-th compoet hock i e ca be liealy eplaied by the othe compoet of tuctual hock i z I thi ee the tace of A meaue the etet fo each idividual compoet of e which may be egaded a obevable to be eplaied by it ow coepodig tuctual hock compoet Thi i paticulaly meaigful if thee i o ecoomic theoy available to idetify the ytem Thi pape i ogaized a follow Sectio povide the motivatio ad poof of a uique othogoal decompoitio fo a imple two-vaiable ytem Sectio 3 geealize the eult to the -vaiable cae Numeical eample ae give i the Sectio 4 ad Sectio 5 coclude with ome emak THE TWO-VARIABLE CASE Motivatio Without lo of geeality let u coide the followig imple cae: y = e + e whee e = e e ) i obevable eially ucoelated with 3 ( σ σ va( e ) = Σ = σ σ I a tuctual VAR aalyi e may be egaded a dive by udelyig ucoelated tuctual hock ε = ε ε ) i the followig mae ( () e = γ ε + γ ε () e = γ ε + γ ε If we ae iteeted i detemiig the cotibutio of a compoet towad the total vaiatio the the vaiace of the coepodig tuctual hock ca be abitaily et to oe; ee Remak 3 3 If e i ot obevable they ae omally etimated a the eidual of a VAR i educed fom

5 Oe geeal iteet i to fid out the cotibutio of ay ε i the vaiatio of y Fom the above ytem of equatio the equied vaiace cotibutio i γ + γ ) va( ) ( ε But ε i uobevable ad γ ae ukow I the abece of ecoomic theoy thi pape popoe to decompoe the vaiatio i y = e + e by coideig a poitive defiite decompoitio mati uch that a A = a a a e = Az which ca be witte a (3) e = az + a z (4) e = az + a z z = ( z z ) i a eially ucoelated adom vecto with idetity covaiace mati We may egad the adom vaiable z a uit tuctual hock aociated with ε The vaiace cotibutio of iteet ca the be egaded a the vaiatio cotibuted by z which i implie ( a + a Now to obtai A we make ue of the fact that A A = Σ which ) (5) a + a = σ (6) a + a = σ (7) a a + aa = σ Sice thee ae oly thee equatio available to olve fo fou ukow i A the decompoitio mati i ot uique which i a well kow fact Oe appoach i to apply the Choleky decompoitio fo which we may chooe to etict a to be zeo aumig that z doe ot cotibute towad e i (3) Sice A i poitive defiite both a ad a ae poitive The etictig a = 0 implie that a attai it laget poible value the tadad deviatio of e ad z cotibute to the vaiatio i y by ( a + a) Alteatively oe could vay a o that a = 0 Thi i equivalet to uig the othe odeig choice i the Choleky decompoitio eultig i the aumptio that z doe ot cotibute towad e at all We have the oppoite effect i thi cae: the

6 vaiace cotibutio of z i imply a with the magitude of a educed ad a attai it maimum value which i the tadad deviatio of e The above eample illutate why the electio of odeig i the Choleky decompoitio could affect the outcome datically Howeve it doe offe hit o how we could decompoe a vaiace if thee i o ecoomic theoy available to idetify the ytem Fit chage i a elemet of A ay a othe elemet of A o that the elatiohip A A caue coepodig chage i all = Σ i maitaied Secod i the abece of ecoomic theoy it i meaigful to chooe a value of a uch that the tace of A a + a ae maimized Doig o i equivalet to the ytem beig idetified with each e i aociated with maimum weight give to it coepodig z i ; the laget poible aociatio betwee each e i ad it udelyig tuctual hock Ideed thi pape how that whe the choice of elemet of A i eticted to maimizig the tace of the mati A i uiquely detemied to be Σ Othogoal Decompoitio fo the Cae We hall ow poceed to obtai the uique othogoal vaiace decompoitio fo the two vaiable ytem coideed above Fit we aume that the decompoitio mati A i poitive defiite The ettig the deivative of t(a) the tace of A to zeo lead u to A = Σ The poof i completed by howig that the ecod deivative of t(a) at the tuig poit i egative Now ice t ( A) = a + a diffeetiatig t(a) with epect to a give (8) d t( A) = da + da Sice Σ i cotat diffeetiate (5) (7) with epect to a yield (9) a da + a 0 = (0) a da + a da 0 = () a da + da a + a + a da 0 Fom (9) we have () da = a a =

7 Fom (0) the deivative of a ca be witte a (3) da = aa da Note that A i poitive defiite o a ad a ae poitive Subtitute the eult of () ad (3) ito () we have a ( aa aa ) (4) da = = a a a ( a a a a ) ice by vitue of poitive defiitee of A a a a a 0 The eult i (4) above > eable u to obtai da i (3) i tem of a ad a which i (5) da = a a Now ubtitutig () ad (5) ito (8) we have at a tuig poit whe d t( A) = 0 a a + a a 0 = which implie a = a That i A A = A = Σ Sice both A ad Σ ae eal ymmetic ad poitive defiite To pove that A = Σ i uique A = Σ i a maimum tuig poit we eed to how that d t( A) = d a + d a i egative whe a = a Fit diffeetiate () with epect to a we have 3 d a = a + a a da = a a a < 0 Sice a = a diffeetiate (3) with epect to a yield 3 d a = a + a a da = a a a < 0 REMARK : Maimizig the tace of A lead u to the ymmetical etictio which i the equied additioal equatio to idetify the fou ukow i (5) (7) REMARK : The ymmetical etictio a = a i ot the ame a etictig the γ ad γ i () ad () to be the ame If we aume that va( ε i ) = va( ei ) the the tuctual decompoitio ca be epeed a mati with i-th diagoal ety equal to va( e i ) Σ Λ ε = e whee Λ i a diagoal

8 REMARK 3: I meauig the vaiace cotibutio of each compoet it make o diffeece which fom of decompoitio Σ / z = e o Σ Λ ε = e i ued 3 THE GENERAL -VARIABLE CASE 3 Othogoal Decompoitio fo the Cae The eult i the above two-vaiable cae ca be geealized ito the geeal multivaiate -vecto cae Geeally peakig ice Σ i ymmetic it ha ( + ) / ditict elemet Howeve A ha ukow paamete to be idetified To detemie the ytem it i thu eceay to impoe ( ) / etictio Simila to the twovaiable cae maimizig the tace of a poitive defiite mati A lead u to the ame ymmetical etictio fo A which povide the additioal ( ) / etictio We hall ow fomally tate the eult i Theoem below THEOREM : Let e be a eially ucoelated adom -vecto with va(e ) i ymmetical ad poitive defiite Let A be a poitive defiite mati uch that A A = Σ ad va( z ) = I The maimizig the tace of A lead u to uique poitive defiite quae oot of Σ = Σ which Az = e A = Σ which i the PROOF: The poof compie thee tep Fit we demotate that a ymmetical A give ie to a zeo deivative of t(a) It i the how that the ecod deivative of t(a) i egative whe A i ymmetical Fially we pove that poit Thoughout the poof the idetity of A A = Σ i epeatedly ued A = Σ i the global maimum To pove that ymmety of A implie a zeo deivative of t(a) we diffeetiate A A = Σ ad obtai (6) A da + da A = 0 Sice A i poitive defiite (7) t ( da ) = t( A da A ) A eit Left multiply (6) by Sice t ( da) = t( da ) ad t ( AB) = t( BA) (7) ca be ewitte a (8) t ( da) = t( da A A ) A ad take tace

9 Now A = A o A A = I Theefoe (8) ca hold oly whe d t( A) = t( da) = 0 To how that it i a maimum tuig poit we eed to pove that d t( A) < 0 whe A i ymmetical The ecod ode diffeetial of (9) A d A + d A A + ( da) = 0 A = Σ ca be witte a Left multiply (9) by A ad take tace t d A + t A d A A + t A da = (0) ( ) ( ) ( ( ) ) 0 Becaue t ( A d A A) = t( d A) ad poitive defiitee of A implie that A eit we ca wite (0) a d t( A) = t( d A) = t which ule A i a ull mati i egative We have demotated i the above that veify that it i a global maimum let coditio A A = Σ implie that () Σ D + DΣ + DD = 0 Left multiply () by Σ ad take tace () t ( D) + t( Σ DD ) = 0 ( A ( da) ) = t( da A A da) A = Σ + A = Σ i a maimum tuig poit To D whee D i ay abitay mati The /4 /4 But t ( Σ DD ) = t( D Σ Σ D) 0 with equality if ad oly if D = 0 i which cae A = Σ Fo the othe cae of t ( Σ DD ) > 0 () implie that t ( D) < 0 Sice A = Σ + D we have t ( Σ ) > t( A) QED 3 Aymptotic Ditibutio of ˆΣ Hee we deive the aymptotic ditibutio of ˆΣ whe it i etimated fom a ealvalued -vecto ample of ( e K e T ) We aume that e t i ditibuted IID N ( 0 Σ) whee Σ i a poitive defiite ymmetic mati ˆΣ i etimated by takig quae oot T t of Σˆ which i give by Σˆ = T = ( et e)( et e) whee e i the ample mea of e t

10 By the Cetal Limit Theoem it i etablihed that Σˆ ha a aymptotic omal ditibutio give by Σ ˆ d (3) T ( vech vechσ) N (0 V ) whee vech i a opeato that tack ditict elemet of a ymmetic mati ito a vecto (the tackig ule give by Magu ad Neudecke (999 p 49) ae adopted hee) Let D be the duplicatio mati uch that D vech( Σ) = vec( Σ) ad Mooe-Peoe ivee of + D be the + D that evee the opeatio that i D vec( Σ) = vech( Σ) The vaiace covaiace mati i (3) ca be witte a + + (4) V = D ( Σ Σ)( D ) Now we tate the ecod eult of thi aticle ad povide it poof below THEOREM : Give a eal-valued -vecto IID Gauia ample of e K e ) with zeo mea ad va( e t ) ( T = Σ a poitive defiite ymmetic mati The etimato of obtaied by takig quae oot of the maimum likelihood etimato Σˆ i aymptotically ditibuted a vechσ ˆ / d (5) T ( vechσ ) N(0 V ) whee (6) ( ) ( ) V / = D ( I Σ ) D D ( Σ Σ) D D ( I Σ ) D Σ PROOF: Sice Σ i a cotiuou mati fuctio of Σ applyig the delta method to (3) yield the eult of (5) So we ut eed to pove (6); deive the Jacobia mati of Σ ad obtai the equied vaiace covaiace mati We begi thi by coideig the diffeetial of Σ Σ = Σ which i give by (7) Σ dσ + dσ Σ = dσ Applyig the vec opeato to (7) we have (8) ( I Σ ) vec( dσ ) + ( Σ I) vec( dσ ) = vec( dσ)

11 Let K be the pemutatio mati uch the tem Σ / I i (8) ca be witte a K ( I Σ ) K fom (8) Due to ymmety of Σ K vec( (9) ( I + K )( I Σ ) d vecσ = d vecσ dσ ) = d vecσ Thu we have Now + + K = D D D D D ) D + I whee the Mooe-Peoe ivee i = ( Left multiply (9) by D ad implify the duplicatio matice (30) D ( I Σ ) D d vechσ = D D d vechσ Note that the dimeio of D i ( + ) / ad ha full colum ak o the ivee of ( D ( I Σ ) ) eit Theefoe the equied diffeetial D epeed a d vechσ = ( D ( I Σ ) D ) D D d vechσ d vechσ i (30) ca be which by the idetificatio theoem fo mati fuctio (Magu ad Neudecke (999 p 96)) yield the equied Jacobia mati Σ = D ( I Σ ) D D D (3) ( ) By the delta method V = Σ V ( Σ ) Subtitutig (3) fo the Jacobia mati / ad implifyig the duplicatio matice we aive at (6) i Theoem QED REMARK 4: Theoem alo hold if e t i o-gauia but ditibuted IID(0 Σ ) with zeo fouth ode cumulat 4 NUMERICAL EXAMPLES I thi ectio we povide thee umeical eample baed o the wok of Campbell ad Amme (993) They ue a VAR model to decompoe the ece tock etu ( e t+ ) ece 0-yea bod etu ( b t+ ) ad uepected yield pead iovatio ( t+ ) ito chage i epectatio of futue tock divided iflatio hot-tem eal iteet ate

12 ad chage i epectatio of futue ece tock ad bod etu The thee vaiable of iteet ca be witte a e t + = ed t + e t + e t + b t+ = b t+ b t+ b t+ t+ = t+ + t+ + t+ whee the ubcipt d ad tad fo divided eal iteet ate iflatio ad ece etu epectively So fo itace e d t+ ca be itepeted a the ew about futue divided fo the uepected ece tock etu b t+ efe to the ew o futue ece bod etu wheea t+ i the ew about futue iflatio fo the uepected yield pead iovatio A diffeet odeig i the Choleky decompoitio yield vatly diffeet eult Campbell ad Amme epot all 6 vaiace-covaiace tem (which ae tadadized to um to equal oe) ad R tatitic fom imple egeio of the vaiable of iteet o each of thei coepodig compoet Thee tatitic ae povided i Table I II ad III I each table we alo povide the quae oot decompoitio mati ( Σ ) the aociated vaiace cotibutio (vc) the Chi-quaed tatitic fo tetig equality of vaiace cotibutio a well a the vaiace cotibutio obtaied uig diffeet odeig choice i the Choleky decompoitio The Chi-quaed tet ae cotucted a follow Fit let c be the um of the -th colum elemet of Σ Sice the vaiace of vaiable of iteet i tadadized the othogoal vaiace cotibutio fom compoet i imply give by vc = c To cay out the hypothei tetig of vc = vc k k we make ue of the fact that equality implie c = ± c ad calculate k (3) ( c Cq = ( c c ) k + c ) k va( c va( c c ) k + c ) k if c if c c k c k 0 < 0

13 The vaiace tem i (3) ae calculated uig (6) i Theoem with T = 44 4 Ude the ull hypothei that the two vaiace cotibutio ae equal Cq i ditibuted a Chi-quaed with oe degee of feedom < Iet Table I > Fom Table I we ca ee that coelatio betwee the thee compoet that eplai ece tock etu ae low A a eult aw vaiace i Σ R tatitic ad vaiace cotibutio (vc) ae of coitet magitude Ecept fo the fit ad thid electio of odeig Choleky method yield elatively imila eult too Tet o equality of quae oot vaiace cotibutio eveal that they ae igificatly diffeet fom each othe < Iet Table II > Net we look at the vaiace decompoitio fo ece bod etu give i Table II It ca be ee that both the iflatio ( b ) ad ece etu ( b ) compoet have lage vaiace ad that coelatio betwee the thee compoet ae faily high Ecept fo the Choleky decompoitio the fit thee meaue of vaiace cotibutio ae coitet with each othe Though vc ugget b ha the laget vaiace cotibutio wheea the aw vaiace i Σ ugget b cotibute the mot the Chi-quaed tet eveal that the diffeece i iigificat The tet howeve cofim that the eal iteet compoet ( b ) ha the leat cotibutio to the vaiatio i ece bod etu Fo the Choleky decompoitio vaiatio i the meaue of cotibutio i huge fo diffeet odeig Fo eample electig the ecod odeig yield 007 ad 0775 fo b ad b epectively wheea the fouth electio choice yield hugely cotatig eult of 0754 ad 074 fo b ad b epectively < Iet Table III > Table III povide the vaiace decompoitio fo uepected yield pead iovatio Fit it i oticeable that both the iflatio ( ) ad eal iteet ate ( ) compoet have lage vaiace ad ae highly egatively coelated (coelatio equal -099) Depite thei lage vaiace imple egeio of uepected yield pead iovatio 4 Campbell ad Amme (993) ue mothly data fom Jauay 95 to Febuay 987 a total of 44 moth

14 o each of both compoet yield vey low R tatitic of 007 ad 0003 Squae oot decompoitio eveal that the cotibutio of ad to the total vaiatio ae ad 089 epectively While thee figue ae moe eible tha the tatitic oe poit meit futhe dicuio That i i pite of the fact that va( ) va( ) ad that va( ) i much lage tha va( ) vc ( vc of ) i the mallet at 077 Howeve a moe caeful aalyi eveal that it i a plauible outcome Fit a oted above coelatio betwee ad cov( ) i egative wheea cov( ) i poitive The eultig R i highly egative Secod Σ implie that while the aociated uit-vaiace ucoelated tuctual hock z iovate with a impact coefficiet a high a 778 thi effect i geatly educed by it oppoite effect o with a egative impact coefficiet of -5 Moeove egative cov( ) implie that the vaiace cotibutio by i futhe educed albeit by a mall egative impact coefficiet of -007 o Though va( ) i low le tha 3% of va( ) + va( ) it vaiace cotibutio accodig to the quae oot method i elatively high Thi ca be eplaied by the high egative coelatio betwee ad which implie that thei hock ae i oppoite diectio ad the et effect become much malle Alo it i oted that the coelatio betwee ad the othe two compoet ae mall Ideed vc = 0 89 i coitet with the coepodig elatively high R tatitic of 035 The Chi-quaed tet led futhe cedibility to the quae oot decompoitio appoach Though coideably lage tha vc eem vc the tet eveal a iigificat diffeece a outcome coitet with thei imila-ize vaiace ad lage coelatio The tet betwee ad vc howeve cofim that eplai vaiatio i the yield pead iovatio igificatly moe tha Ulike the eult i Table I the Choleky method geeate vatly diffeet figue of vaiace cotibutio if diffeet electio of odeig ae ued Thi i due to the fact that the compoet i Table I ae elatively ucoelated Fially we emak that becaue the umeical eample coideed have oly thee vaiable above aalye vc

15 baed o vaiace covaiace mati ae feaible Howeve i pactice we ofte have moe tha thee vaiable to aalyze Ideed vaiable imply that thee ae ( +) / vaiace covaiace tem to coide Woe till fo the Choleky decompoitio thee ae! electio of odeig 4 CONCLUSION WITH SOME REMARKS Thi aticle popoe a othogoal vaiace decompoitio that maimize the tace of a poitive defiite decompoitio mati Whe thee i o ecoomic theoy oe ca ely upo to decompoe the hock the tace of A ha meaigful itepetatio: a lage tace mea a highe aociatio betwee obevable hock ad thei coepodig tuctual hock It tu out that uch a decompoitio mati i uique ad equal to the quae oot of the vaiace covaiace mati Limitig ditibutio of the etimato of quae oot decompoitio mati i deived ad umeical eample ae povided to illutate it uefule Though thi aticle coide the imple IID eal-valued -vecto cae it eult ca be eadily eteded to fo eample the VARMA cae of Mittik ad Zadozy (993) Fom the umeical eample we ca ee that a diffeet electio of odeig i the Choleky method could yield a vatly diffeet outcome Theefoe the quae oot decompoitio i a ueful alteative fo compaio I paticula whe thee ae may vaiable i the ytem the popoed method i able to povide a cocie aalyi Fially ice multiple egeio ad VAR model ae ubiquitou i mot ocial ciece tudie thi aticle popoe a imple mea of vaiace decompoitio that i ituitive ad equie o pio ifomatio REFERENCE BERNANKE B (986): Alteative eplaatio of moey-icome coelatio Caegie-Rochete Cofeece Seie o Public Policy BLANCHARD O J AND D QUAH (989): The dyamic effect of aggegate demad ad upply ditubace Ameica Ecoomic Review CAMPBELL J Y AND J AMMER (993): What move the tock ad bod maket? A vaiace decompoitio fo log-tem aet etu Joual of Fiace

16 HAMILTON J D (994): Time Seie Aalyi New Jeey: Piceto MAGNUS J R AND H NEUDECKER (999): Mati diffeetial calculu with applicatio i tatitic ad ecoometic New Yok: Joh Wiley MITTNIK S AND P A ZADROZNY (993): Aymptotic ditibutio of impule epoe tep epoe ad vaiace decompoitio of etimated liea dyamic model Ecoometica SIMS C (986): Ae foecatig model uable fo policy aalyi? Fedeal Reeve Bak of Mieapoli Quately Review 3 6 SHORROCKS A F (98): Iequality decompoitio by facto compoet Ecoometica 50 93

17 TABLE I VARIANCE DECOMPOSITION FOR EXCESS STOCK RETURNS e d e e Σ Pael A e d e e Simple egeio R Σ Pael B e d e e Σ decompoitio vc Vaiace decompoitio uig Choleky Method Pael C d d d d d d Ece tock etu ae decompoed ito e d e ad e which ae epectively ew about futue divided eal iteet ate ad ece tock etu Data i Pael A i obtaied fom Campbell ad Amme (993 Table III) The figue ae vaiace covaiace mati (coelatio i bold italic fot) ad R tatitic obtaied fom imple egeio of ece tock etu o each compoet I Pael B Σ i the quae oot decompoitio mati vc i vaiace cotibutio ad tet tatitic fo equality of vc ae i bold italic fot Ude the ull hypothei the tet tatitic i ditibuted a Chi-quaed with degee of feedom; 384 ad 6635 ae citical value at 5% ad % igificace level epectively Pael C povide the vaiace cotibutio uig Choleky method with diffeet odeig

18 TABLE II VARIANCE DECOMPOSITION FOR EXCESS BOND RETURNS b b b Σ Pael A b b b Simple egeio R Σ Pael B b b b Σ decompoitio vc Vaiace decompoitio uig Choleky Method Pael C Ece bod etu ae decompoed ito b b ad b which ae epectively ew about futue iflatio eal iteet ate ad ece bod etu Data fom Pael A i obtaied fom Campbell ad Amme (993 Table IV) Deciptio fo figue i Pael A B ad C ae imila to Table I The 5% ad % citical value of Chi-quaed tatitic ude ull hypothei ae 384 ad 6635 epectively

19 TABLE III VARIANCE DECOMPOSITION FOR YIELD SPREAD INNOVATIONS Σ Pael A Simple egeio R Σ Pael B Σ decompoitio vc Vaiace decompoitio uig Choleky method Pael C Ece bod etu ae decompoed ito ad which ae epectively ew about futue iflatio eal iteet ate ad ece bod etu Data fom Pael A i obtaied fom Campbell ad Amme (993 Table VIII) Deciptio fo figue i Pael A B ad C ae imila to Table I The 5% ad % citical value of Chi-quaed tatitic ude ull hypothei ae 384 ad 6635 epectively