Using Kalman Filter to Extract and Test for Common Stochastic Trends 1


 Delphia Randall
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1 Usig Kalma Filer o Exrac ad Tes for Commo Sochasic Treds Yoosoo Chag 2, Bibo Jiag 3 ad Joo Y. Park 4 Absrac This paper cosiders a sae space model wih iegraed lae variables. The model provides a effecive framework o specify, es ad exrac commo sochasic reds for a se of iegraed ime series. The model ca be readily esimaed by he sadard Kalma filer, whose asympoics are fully developed i he paper. I paricular, we esablish he cosisecy ad asympoic mixed ormaliy of he maximum likelihood esimaor, ad herefore, validae he use of coveioal mehods of iferece for our model. Moreover, we cosruc a race saisic, which ca be used o deermie he umber of commo sochasic reds i a sysem of iegraed ime series. I is show ha he limi disribuio of he saisic is sadard ormal. The es is very simple o impleme i pracical applicaios. Our simulaio sudy shows ha i behaves quie well i fiie samples. For a illusraio, we apply our mehodology o aalyze he commo sochasic red i various defaulfree ieres raes wih differe mauriies. Firs Draf: Ja 3, 26 This Versio: Sepember 24, 28 JEL Classificaio: C22, C5 Key words ad phrases: sae space model, Kalma filer, commo sochasic reds, maximum likelihood esimaio, asympoic heory. This versio is prepared for he preseaio a he Midwes Ecoomerics Coferece, Ocober 23, S. Louis. Chag ad Park graefully ackowledge he fiacial suppors from he NSF uder Gra No. SES45369/7352 ad SES5869, respecively. 2 Deparme of Ecoomics, Texas A&M Uiversiy 3 Deparme of Ecoomics, Rice Uiversiy 4 Deparme of Ecoomics, Texas A&M Uiversiy ad Sugkyukwa Uiversiy
2 . Iroducio The Kalma filer is he basic ool used i he sadard sae space models, which ypically deals wih dyamic ime series models ha ivolve uobserved variables. The applicaios of Kalma filer ca be foud i may fields icludig ecoomics ad fiace. The asympoic behavior of maximum likelihood ML esimaors based o he filer is well kow uder regular codiios, i.e., lieariy, Gaussiaiy, ad saioariy. If lieariy is violaed, he exeded Kalma filer is a sadard aleraive. Moreover, i is well kow ha he pseudoml esimaio performs well whe Gaussiaiy does o hold. To he bes of our kowledge, however, o research has bee doe o ivesigae he properies of he filer for he case ha saioariy is violaed. Oly very recely, Chag, Miller ad Park 27, which will be referred o as CMP hereafer, pioeered i developig a rigorous asympoic heory for he sae space models wih oe iegraed lae variable. Sice CMP allows for oly oe iegraed lae facor, i does o provide ay es for he umber of disic lae facors. This would ceraily be a impora limiaio i pracical applicaios. I may empirical aalysis, we see some srog evidece ha he commo sochasic reds i sysems cosisig of muliple iegraed ime series cao be explaied by a sigle facor. The presece of a sigle commo sochasic red would imply he presece of as may coiegraig relaios as oly oe e of he umber of iegraed ime series icluded i he sysem. This is highly ulikely, especially whe he uderlyig sysem is large ad ivolves may iegraed ime series, as is ofe he case i may pracical applicaios. The reader is referred o, e.g., Kim ad Nelso 999 for various models used i pracice ad previous empirical researches. I his paper, we exed CMP o allow for muliple lae facors, ad develop a es which ca be used o formally es for he umber of lae facors. Our framework is compleely geeral, excep ha we require he lae commo facors follow radom walks i a sric sese. Wihi his geeral framework, we show ha he ML esimaors of he parameers i he model are cosise ad asympoically mixed ormal. The sadard iferece based o he ML procedure is herefore valid. The covergece rae for he ML esimaors is as i he sadard model. However, we have a faser rae of covergece for he coefficie of lae commo sochasic reds alog he coiegraio space. This is i parallel o he covergece raes i oher ypes of coiegraed models. We also show ha a es based o a race saisic ca be applied i our model o es for he umber of commo sochasic reds, ad ha i has asympoically ormal disribuio. The es appears o be paricularly useful for a large sysem of iegraed ime series, which shares a relaively small umber of commo sochasic reds. The sae space modelig wih lae iegraed facors provides a aleraive way of aalyzig coiegraed sysems. I is i coras wih he coiegraig regressios cosidered by, for isace, Phillips 99 ad Park 992, ad also closely relaed o he error correcio formulaio used i Johase 988, 99 ad Ah ad Reisel 99. They all ca be used i modelig a sysem of coiegraed processes which share commo sochasic reds. The sae space model, however, is uique ad disiguishes iself from oher compeig models i ha i may allow for he commo sochasic reds o be modeled as pure radom walks. As we show i he paper, he sae space model wih commo
3 sochasic reds specified as pure radom walks is o compaible wih a fiie order error correcio model ECM or vecor auoregressio VAR. Therefore, he esig procedure ha are based o a fiie order ECM or VAR is o applicable for he sae space models we cosider i he paper. The res of he paper is orgaized as follows. I Secio 2, we iroduce our sae space model ad oulie he Kalma filerig echique used o esimae he model. Some prelimiary resuls are also icluded i his secio. Secio 3 ad 4 prese he mai heoreical fidigs. I Secio 3, we esablish he cosisecy ad asympoic mixed ormaliy of he ML esimaors. Theories abou he deermiaio of umber of commo sochasic reds are preseed i Secio 4. I paricular, we iroduce ad aalyze a race saisic o es for he umber of commo sochasic reds i Secio 4. Secio 5 preses some simulaio resuls for he fiie sample performace of our esimaors ad es saisic. A empirical illusraio follows i Secio 6. Here we use our mehodology o ivesigae a sysem of ieres raes wih differe mauriies. Secio 7 cocludes he paper. Mahemaical proofs are give i Appedix The Model ad Prelimiary Resuls We cosider he sae space model give by uder he followig assumpios: SSM: y is a pdimesioal observable ime series, SSM2: x is a qdimesioal vecor of lae variable, y = A x + u x = x + v SSM3: A is a p q marix of ukow parameers of rak q, where q p, SSM4: u ad v are p ad qdimesioal idepede, ideically disribued iid errors ha are ormal wih mea zero ad variace Λ ad ideiy marix I q, respecively, ad idepede of each oher, ad SSM5: x is idepede of u ad v, ad assumed o be give. Our model ca be used o exrac commo sochasic reds i ime series y. Noice ha lae variable x is defied as a vecor of radom walks, our model provides a aural way o decompose a coiegraed ime series io a permae ad rasiory compoes. The parameer A ad he lae commo sochasic reds x are o globally ideified i our model. Obviously, he observable ime series y have he same likelihood uder joi rasformaio A A H ad x H x 2 for ay qdimesioal orhogoal marix H. They are ideified oly up o he equivalece class defied by he rasformaio i 2. However, boh of A ad x are locally ideified. Ideed, we may easily see ha, for ay qdimesioal orhogoal marix H, A H is
4 o i he eighborhood of ay p q marix A of rak q defied by he Euclidea or ay equivale orm i he vecor space of p q marices. Of course, x is ideified locally if A is. I he subseque developme of our heory, we will o impose ay exra resricios o globally ideify A ad x. This does o seem o be ecessary for mos poeially useful applicaios of our model, for which we would be primarily ieresed i fidig ou he dimesio of commo sochasic reds ad exracig radom walks represeig hem. All he resuls i he paper for A ad x should herefore be ierpreed as applyig o a member of he equivale class give by he rasformaio i 2. To ease he exposiio of he paper, we firs assume ha q, i.e., he dimesio of x ad rak of A, is kow o explai how o exrac x ad o develop he asympoic heory for he ML esimaio of A. A es for q based o a race saisic will he be iroduced ad discussed laer. Throughou he paper, we will maily look a he simple model give by. This is purely for exposiioal coveiece. Our subseque resuls exed rivially o a more geeral class of sae space models wih measureme equaio give by y = A x + m Π k y k + u, 3 k= i place of he oe i. The iclusio of he lagged differeces of y i 3 oly iroduces more parameers associaed wih he observable saioary compoes of he model, ad would o affec our asympoic heory i ay impora maer. I our subseque developme of he heory, we will meio explicily wha modificaios are eeded o accommodae he geeral model i 3. I all cases, he ecessary modificaios are obvious ad sraighforward. The model defied i ca be esimaed by he usual Kalma filer. Le F be he σfield geeraed by y,..., y, ad for z = x or y, we deoe by z s he codiioal expecaio of z give F s ad by Ω s ad Σ s he codiioal variaces of x ad y give F s, respecively. The Kalma filer cosiss of he predicio ad updaig seps. For he predicio sep, we uilize he relaioships ad x = x, y = Ax, Ω = Ω + I q, Σ = AΩ A + Λ. O he oher had, he updaig sep relies o he relaioships x = x + Ω A Σ y y, Ω = Ω Ω A Σ AΩ. The ML esimaio mehod is used i esimaig he ukow parameers. 3
5 For may uses of Kalma filer, he primary goal is o calculae a forecas ad also he codiioal variace of he observed ime series y as a fucio of previous observaios. However, i he case ha he value of he uobserved variable is of ieres for is ow sake, smoohig echique is ofe used, deoed x = Ex F. The smoohed series x is esimaed codiioally o all of he iformaio i he sample  o jus he iformaio up o ime. The followig is he key equaio for smoohig: x = x + Ω Ω + x + x +. This procedure works recursively by sarig from =. Sarig value x ogeher wih series x, x +, Ω ad Ω + are achieved i he esimaio procedure. The reader is referred o Hamilo 994 or Kim ad Nelso 999 for more deails of his echique. Oe hig is clear ha smoohig is implemeed afer he model parameers are esimaed, herefore his procedure has o effec o he parameer esimaes. For ay give values of A ad Λ, here exis seady sae values of Ω ad Σ, which we deoe by Ω ad Σ. 4 Lemma 2. The seady sae values Ω ad Σ exis ad are give by Ω = 2 I q + I q + 4A Λ A /2, Σ = 2 AI q + I q + 4A Λ A /2 A + Λ for p q marix A ad p p marix Λ. We will se Ω = Ω I q 4 for he res of he paper, so ha Ω akes is seady sae value Ω for all. Of course, Σ also becomes ime ivaria ad akes is seady sae value Σ uder his coveio. 5 The followig lemma specifies x more explicily as a fucio of he observed ime series y ad he iiial value x. To simplify he exposiio, we le y =. Lemma 2.2 We have x = A Λ A A Λ y I q Ω k A Λ A A Λ y k + I q Ω x for all 2. k= The resul of Lemma 2.2 is give eirely by he predicio ad updaig seps of Kalma filer. I paricular, i holds eve uder misspecificaio of our model i. 5 Though we do o show explicily i he paper, Ω always coverges i our experimes o he seady sae value Ω as icreases, regardless of he sarig values.
6 I follows from Lemma 2. ha Ω > I q, ad herefore, < Ω < I q. As a cosequece, we have < I q Ω < I q, ad herefore, he magiude of he erm I q Ω x is geomerically decliig as. I implies ha he effec of x o x dilues ou as, as log as x is fixed ad fiie a.s. Therefore, we may se x = 5 wihou affecig our asympoic resuls. Le Ω be he value of Ω defied wih he rue values A ad Λ of A ad Λ. If we deoe by x he value of x uder model, we may deduce from Lemma 2.2 ad smoohig echique ha 5 Proposiio 2.3 We have x = x + Ω I q Ω k A Λ A A Λ u k I q Ω k= k= k v k for all 2, ad for all. x = x + I q Ω k= k x +k +k where Proposiio 2.3 implies i paricular ha x x = Ω a b, a = I q Ω k A Λ A A Λ u k ad b = I q Ω k v k. k= Uder he assumpio ha u ad v are iid radom sequeces, he ime series a ad b become he saioary firsorder VAR processes give by a = I q Ω a + A Λ A A Λ u, b = I q Ω b + v respecively, sice < I q Ω < I q. Clearly, every compoe of x or x is coiegraed wih he correspodig compoe of x wih ui coiegraig coefficie. The sochasic reds i x may herefore be ideified ad represeed by hose i x or x. I seems worh oig ha he resuls i Proposiio 2.3 do o rely o he iid assumpio of u ad v. I paricular, our resuls here imply ha we may exrac he commo sochasic red i y usig he predicig ad smoohig seps of Kalma filer, as log as u ad v are k=
7 geeral saioary processes. Apparely, we eed o kow he rue parameer values o obai x or x. The rue parameer values are ypically ukow ad have o be esimaed. I mos pracical applicaios, we should herefore use he parameer esimaes o compue x or x. I is raher clear ha he esimaes of x ad x based o he esimaed parameer values are close o x ad x, respecively, if we use he cosise parameer esimaes. Oce we obai x, we may decompose ime series y io he permae ad rasiory PT compoes. If we deoe hem as y P ad y T, respecively, hey are give by y P = A x ad y T = y A x. 6 The permae compoe y P is I, whereas he rasiory compoe y T is I. Noe ha he permae compoe y P is predicable, while he rasiory compoe y T is a marigale differece sequece mds ad upredicable. The Kalma filer has exacly he same predicio ad updaig seps for he measureme equaio 3, if we le m y = Ax + Π k y k. i place of y = Ax. Therefore, i is clear ha Lemma 2. ad Proposiio 2.3 hold for his geeral model wihou ay modificaio. Moreover, Lemma 2.2 coiues o be valid if we oly replace y wih y m k= Π k y k. The heory of Kalma filer for he geeral model is hus followed immediaely. 3. Asympoics for Maximum Likelihood Esimaio I his secio, we cosider he maximum likelihood esimaio of our model. I paricular, we esablish he cosisecy ad asympoic Gaussiaiy of he maximum likelihood esimaor uder ormaliy. Because he iegraed process is ivolved, he usual asympoic heory for ML esimaio of sae space models give by, for isace, Caies 988, does o apply. CMP develops a geeral asympoic heory of ML esimaio, which allows for he presece of osaioary ime series. They obai he asympoics of ML esimaors of he parameers i heir model, where he umber of lae variable is resriced o oe. I his paper, we derive he asympoic properies of he ML esimaors of he parameers i he sae space model ha has muliple sochasic lae variables. I developig our asympoic heory, we will frequely refer o he resuls obaied previously i CMP. We le θ be a κdimesioal parameer vecor ad defie k= ε = y y o be he predicio error wih codiioal mea zero ad variace marix Σ. Uder ormaliy, he loglikelihood fucio of y,..., y is give by l θ = 2 log de Σ r Σ 2 ε ε 6
8 igorig he uimpora cosa erm. Here, Σ ad ε are i geeral give as fucios of θ. Le s θ ad H θ be he score vecor ad Hessia marix, i.e., s θ = l θ θ ad Afer applyig some algegra, we may deduce ha s θ = vec Σ vec Σ + vec Σ vec 2 θ 2 θ ad H θ = Iκ vec Σ 2 2 θ θ + 2 I κ + vec Σ 2 θ vec Σ 2 θ + ε θ Σ ε vec Σ vec Σ ε ε Σ Σ vec Σ θ vec Σ Σ Σ θ H θ = 2 l θ θ θ. Σ ε ε Σ Σ 2 vec Σ θ θ Σ Σ ε ε + Σ ε ε θ I ε Σ 2 θ θ ε ε θ ε + Σ Σ ε θ Σ ε, 7 vec Σ θ ε θ ε Σ Σ vec Σ θ as give i CMP. Here ad elsewhere i he paper, vec A deoes he colum vecor obaied by sackig he rows of marix A. Deoed by ˆθ he maximum likelihood esimaor of θ, he rue value of which is se as θ. As i he sadard saioary model, he asympoics of ˆθ i our model ca be obaied from he firs order Taylor expasio of he score vecor, which is give by s ˆθ = s θ + H θ ˆθ θ, 7 where θ lies i he lie segme coecig ˆθ ad θ. Assumig ha ˆθ is a ierior soluio, we have s ˆθ = immediaely. Therefore, i is ow clear from 7 ha we may wrie ν T ˆθ θ = ν T H θ T ν ν T s θ 8 for appropriaely defied κdimesioal square marices ν ad T, which are iroduced here respecively for he ecessary ormalizaio ad roaio. Upo appropriae choice of he ormalizaio marix sequece ν ad roaio marix T, we will show ha
9 8 ML: ν T s θ d N as for some N, ML2: ν T H θ T ν d M > a.s. as for some M, ad ML3: There exiss a sequece of iverible ormalizaio marices µ such ha µ ν a.s., ad such ha sup µ T H θ H θ T µ p, θ Θ where Θ = {θ µ T θ θ } is a sequece of shrikig eighborhoods of θ. As show by Park ad Phillips 2 i heir sudy of he oliear regressio wih iegraed ime series, codiios MLML3 above are sufficie o derive he asympoics for ˆθ. I fac, uder codiios MLML3, we may deduce from 8 ad coiuous mappig heorem ha ν T ˆθ θ = ν T H θ T ν ν T s θ + o p d M N 9 as. I paricular, ML3 esures ha s ˆθ = wih probabiliy approachig o oe ad ν T H θ H θ T ν p as. This was show by Wooldridge 994 for he asympoic aalysis of exremum esimaors i models icludig osaioary ime series. To obai he limi disribuio of s θ, we firs le ε, / θ ε ad / θ vecσ be defied respecively as ε, / θ ε ad / θ vecσ evaluaed a he rue parameer value θ of θ. The we have s θ = vec Σ Σ Σ 2 θ vec ε ε Σ As show i CMP, ε θ Σ ε. ε ε Σ d N, I + KΣ Σ as, where K is he commuaio marix, ad ε ε Σ ad Noe i paricular ha ε θ Σ ε are asympoically idepede. 2 ε = y y = A x x + u, ad as a cosequece ε, F is a marigale differece sequece ad / θ ε is a predicable sequece wih respec o he filraio F.
10 If our model were saioary, he limi disribuio would herefore be easily derivable from, 2 ad ε ε θ Σ ε ε d N, plim θ Σ θ, 3 which ca be readily obaied by employig he sadard marigale CLT. Of course, asympoics i 3 does o hold for our osaioary model wih iegraed lae variables. As we will show below i Lemma 3., he mulivariae process / θ ε is give by a mixure of saioary ad osaioary processes. Our subseque asympoic aalysis will herefore be focused o solvig he complexiy caused by his mixure of saioariy ad osaioariy. Now we look a our model more specifically. The parameer θ for our model is give by θ = veca, vλ, 4 wih he rue value θ = veca, vλ. Here ad elsewhere i he paper, va deoes he subvecor of veca wih all subdiagoal elemes of A elimiaed. Therefore, va vecorizes oly he oreduda elemes of A. We may relae veca ad va by DvA = veca, where D is he duplicaio marix. See, e.g., Magus ad Neudecker 988, pp The dimesio of θ is give by κ = pq + pp + /2, sice i paricular here are oly pp + /2 umber of oreduda elemes i Λ. For our model, we may easily deduce from Lemma 2.2 ad Proposiio 2.3 ha Lemma 3. We have ε veca = I p Λ A A Λ A A x + a u, v ad ε vecλ = b u, v, where a u, v ad b u, v are saioary liear processes drive by u ad v. Accordig o Lemma 3., ε ε θ = veca, ε vλ is a marix ime series cosisig of a mixure of iegraed ad saioary processes sice a u, v ad b u, v are saioary liear processes drive by u ad v. Noice ha P = I p Λ A A Λ A A 5 is a p qdimesioal oorhogoal projecio o he space orhogoal o A alog Λ A. Naurally, we have A P =. Cosequely, A I q aihilaes he commo sochasic reds i ε / veca, ad herefore A I q ε / veca becomes saioary. Ulike ε / veca, i is raher clear from Lemma 3. ha ε / vecλ is eirely saioary. 9
11 I order o effecively deal wih he sigulariy of he marix P i 5, we follow CMP ad iroduce he ecessary roaio. Le B be a p p q marix saisfyig he codiios B Λ A = ad B Λ B = I p q. 6 Noe ha if raka = q = p, such a B does o exis. I he followig discussio we will focus o he case where q < p. I is easy o deduce ha P = I p Λ A A Λ A A = Λ B B, 7 sice P is a projecio marix such ha A P = P Λ A =. Now he κdimesioal roaio marix T is defied as T = T N, T S, 8 where T N ad T S are marices of dimesios κ κ ad κ κ 2 wih κ = p qq ad κ 2 = q 2 + pp + /2, which are give by T N = B I q ad T S = A A Λ A /2 I q I pp+/2 respecively. I follows immediaely from Lemma 3., 6 ad 7 ha T N ε θ = B ε I q = B x + c N u, v 9 veca ad T S ε θ = A Λ A /2 A I ε q veca ε vλ = c S u, v 2 for some saioary liear processes c N u, v ad c S u, v drive by u ad v. Moreover, we ca easily ge he iverse of he roaio marix T as B T Λ I q = A Λ A /2 A Λ I q 2 I pp+/2 from our defiiio of T give above i 8. Before derivig he mai asympoic resuls for he ML esimaor ˆθ of θ, we eed o esablish wo lemmas, which will be preseed subsequely. They are sraighforward exesios of Lemmas 3.3 ad 3.4 i CMP.
12 Lemma 3.2 If we le U r, V r, W r = for r,, he i follows ha r Σ ε, r T N U r, V r, W r d U, V, W ε r θ, T S ε θ Σ as, where U, V, ad W are possibly degeerae Browia moios such ha V ad W are idepede of U, ad such ha V rσ V r dr is of full rak a.s. ad We may readily esablish from Lemma 3.2 he joi asympoics of T N T S ε θ Σ ε d ε V rdur, 22 ε θ Σ ε d W, 23 where we deoe W simply as W. This coveio will be made for he res of he paper. Because of he idepedece of V ad U, he limiig disribuio i 22 is mixed ormal. O he oher had, he idepedece of W ad U reders he wo limi disribuios i 22 ad 23 o be idepede. Clearly, we have W = d N, varw, where Moreover, if we defie varw = plim T S ε ε θ Σ θ Z = 2 T S vecσ Σ Σ θ vec T S. ε ε Σ, he i follows ha Z Z, where Z = d N, varz wih varz = 2 T S vecσ Σ Σ θ vecσ θ T S. As oed earlier, Z is also idepede of U, V ad W iroduced i Lemma 3.2. Now we are ready o derive he limi disribuio for he ML esimaor ˆθ of θ defied i 4. They are give by 9 wih he roaio marix T i 8 ad he sequece of ormalizaio marix ν = diag I κ, I κ2, as we sae below as a heorem.
13 Theorem 3.3 All hree codiios i MLML3 are saisfied for our model. I paricular, ML ad ML2 hold, respecively, wih N = V rdur Z W 2 ad M = V rσ V r dr varw + varz i oaios iroduced before. Theorem 3.3 is compleely aalogous o Theorem 3.5 i CMP. I paricular, Theorem 3.3 shows ha he resuls i Theorem 3.5 of CMP exeds well o he mulidimesioal case, hough he proof is much more ivolved o deal wih he mulidimesioaliy of he commo sochasic red. As i CMP, we le ad Q = V rσ V r R S V rdur = varw + varz W Z, 24 where R ad S are κ 2 , ad pp + /2dimesioal, respecively. Noe ha Q has a mixed ormal disribuio, whereas R ad S are joily ormal ad idepede of Q. Now we may easily deduce from Theorem 3.3 ha vˆλ vλ d S, ad B Λ I q vec Â d Q 25 A Λ A /2 A Λ I q vecâ veca d R, 26 similarly as i CMP. I paricular, i follows immediaely from 25 ad 26 ha vec Â veca d A A Λ A /2 I q R, which has a degeerae ormal disribuio, if q < p. From Theorem 3.3 ad he subseque remarks, we kow ha he ML esimaors Â ad ˆΛ coverge a he sadard rae, ad have ormal limi disribuios. However, i he case where q < p he limi disribuio of Â is degeerae. I he direcio of B Λ, i has a rae of covergece ad a mixed ormal limi disribuio. The ormal ad mixed ormal asympoic disribuios of ML esimaors validae he coveioal iferece for
14 hypohesis esig i such sae space models where muliple iegraed lae variables are icluded. As discussed i CMP, he asympoic resuls for he ML esimaor for our model also hold, a leas qualiaively, for more geeral models, such as he ype of he models icludig lagged erms i measureme equaios. Eve for he case where ime series cosiss o oly sochasic iegraed reds, bu deermiisic liear ime red, afer some proper roaio of he ime series, see, e.g., Park 992, our asympoic heories are applicable for he roaed ime series. The roaio simply separaes ou he compoe domiaed by a deermiisic liear ime red ad he compoe represeed as a purely sochasic iegraed process. 4. Deermiaio of Number of Commo Treds I he asympoic aalysis of he ML esimaor for our model defied i, we assume ha he umber of commo sochasic reds i y is kow o be q. This of course is equivale o assumig ha he umber of coiegraig relaioships i he pdimesioal ime series y is kow o be p q. From our aalysis i he previous secio, we may ideed readily deduce ha B Λ y = B Λ u ad varb Λ u = I p q. I is herefore clearly see ha Λ B is he marix of p q coiegraig vecors, which yield coiegraig errors wih ideiy covariace marix. However, he umber of commo sochasic reds or he coiegraig relaioships is ypically ukow i empirical sudies. I his secio, we will develop a es based o a race saisic for esig he umber of commo sochasic reds, ad explai how we may use he es o deermie he dimesioaliy of he lae iegraed processes i our model. Needless o say, esig for he umber of commo sochasic reds is equivale o esig for he umber of coiegraig relaioships. Therefore, a leas cocepually, we may use he exisig es such as Johase 998, 99 o deermie p q or q, i.e., he umber of coiegraig vecors or he umber of commo sochasic reds. However, usig he mehods based o a fiie order VAR or ECM as Johase s approach has wo impora shorcomigs i our coex. Firs, as we will show subsequely, our model cao be represeed as ay fiie order vecor auoregressio or error correcio model. Ay fiie order VAR or ECM is herefore icosise wih our model. Secod, our model is poeially more useful for a large sysem of ime series which share a few commo sochasic reds. For such sysems, VAR or ECM formulaios ofe become oo flexible, allowig oo may parameers. I paricular, i is impossible o use log VAR s or ECM s, ryig o fi a ifiie order VAR or ECM. 3 Proposiio 4. We have y = B Λ B y C k y k + ε, 27 k=
15 4 where C k = A I q Ω k A Λ A A Λ. Proposiio 4. makes clear he differece bewee our model ad he coveioal ECM. From 27, we may immediaely see ha y is geeraed as VAR, which i paricular implies ha he our model is o represeable as a fiieorder VAR. Moreover, we have rak deficiecies i he shorru coefficies C k, as well as i error correcio erm B Λ B. Noe ha C k are of rak q ad Λ B C k = for all k =, 2,.... I he coveioal ECM, here is o such rak resricio imposed o he shorru coefficies. As a cosequece, Johase s approach, based o fiie order ECM s, is o applicable i our model. This is also rue for he geeral measureme equaio 3. Ideed, i is easy o see ha Proposiio 4. coiues o hold i his case oly wih y replaced by y m k= Π k y k. Clearly, i order o es he umber of commo sochasic reds i our framework, a ew esig mehod is eeded. Now we cosider he ull hypohesis which will be esed agais H : rak A = q, H : rak A > q. To deermie he umber of commo reds, we es H sequeially sarig from q =. If H is o rejeced for q =, he we coclude ha here exiss a sigle commo red. If, o he oher had, H is rejeced i favor of q >, he we es for he ull hypohesis wih q = 2. We may coiue his procedure uil H is o rejeced. The umber of commo reds is he deermied as he value of q, for which H is o rejeced for he firs ime. Our procedure here is i coras wih ha of Johase, which ess for he umber of coiegraig relaioships i a reversed order. I his approach, he ull hypohesis of o coiegraio i.e., p = q is firs esed agais oe coiegraig relaioship i.e., q = p, which will he be esed agai wo coiegraig relaioships i.e., q = p 2 if he ull hypohesis of o coiegraio is rejeced i favor of he aleraive hypohesis. To deermie he umber of coiegraig relaioships, we mus coiue he es uil he ull hypohesis is o rejeced. This is he same as our procedure. I our framework, which seems more useful o aalyze relaively large dimesioal sysems sharig a few commo sochasic reds, we believe ha our approach is more desirable. As q ges large, he esimaio of our model becomes compuaioally quie burdesome. wih where The es will be based o he saisic defied as τ = /ˆω r ˆB ˆΛ y y ˆΛ ˆB p q ˆω 2 = 2p q vec I p q ˆB ˆΛ ˆB ˆΛ avarˆλ ˆ ˆΛ ˆB ˆΛ ˆB vec I p q, avarˆλ ˆ is a cosise esimae of he asympoic variace of ˆΛ. 28
16 5 Uder he ull hypohesis H, we have ˆB ˆΛ y y ˆΛ ˆB ˆB ˆΛ u u ˆΛ B Λ Λ Λ B = I p q, due i paricular o he fac ha B Λ A =. O he oher had, we have uder he aleraive hypohesis H ˆB ˆΛ y y ˆΛ ˆB ˆB ˆΛ A 2 x x ˆB A ˆΛ ˆB p, sice ˆB ˆΛ A does o vaish as ges large. Therefore, cosise is he es which rejecs he ull hypohesis H i favor of he aleraive hypohesis H whe he value of he saisic τ is large. The followig heorem esablishes he limi ull disribuio of τ. Theorem 4.2 Uder he ull hypohesis, we have τ d N, as. As show i Theorem 4.2, he limi disribuio of τ is sadard ormal. I does o iclude ay uisace parameers. Therefore, he implemeaio of he es is ruly simple. For he es saisic τ, we may use ay cosise esimae of he asympoic variace of ˆΛ. The mos aural choice would be o use he egaive hessia marix. To be more precise, we defie ˆT S H ˆθ ˆT S = H H H2 S =, H 2 H22 where ˆT S is he marix defied similarly as T S wih A replaced by Â ad he pariio of H S is made coformably wih ˆT S. The a cosise esimae of he asympoic variace of vˆλ is give by H 22 = H 22 H H H 2 2. The correspodig asympoic variace of ˆΛ is give by D H 22 D, where D is he duplicaio marix. A similar approach is possible whe he measureme equaio is give more geerally as i 3. I his case, we may simply modify he es saisic τ by replacig y wih y m ˆΠ k y k, k=
17 where ˆΠ k is he ML esimae of Π k for k =,..., m. This is show i he proof of Theorem 4.2. Moreover, i is clear ha ˆΠ k is asympoically idepede of ˆΛ. Therefore, we may jus igore he blocks of he hessia marix correspodig o ˆΠ k ad proceed as above, whe we compue he cosise esimae of he asympoic variace of vec ˆΛ. 5. Simulaios 6 5 Desiies of Â 2 Desiies of ˆΛ Desiies of Roaed Â.5 Desiies of raios Figure : Desiies of MLE ad raios, =5 I his secio, we perform a se of simulaios o ivesigae he fiie sample properies of he ML esimaes. We look a a specific model of 3 observable ime series wih 2 commo sochasic reds. I order o saisfy Assumpio SSM4, he error erms u ad v are geeraed as follows: u = ε ε 2 + ε 3 ε 2 ad v = ε4 ε 5 where ε ε 5 are idepede ad radomly draw from N,. The idepedece bewee u ad v herefore follows ad he covariace marix of v is a ideiy marix. Λ, he covariace marix of u, ca be easily derived as well. We prese Λ ad he arbirarily seleced rue value of A as follows: A = ad Λ =,
18 The iiial value of he sae variable x is se o be. I his way, Assumpio SSM5 is also saisfied ad Theorem 3.3 is readily applicable o his model. Accordig o he heorem, he ML esimaors Â ad ˆΛ coverge o Gaussia disribuios a he sadard rae. However, he asympoic disribuio of Â is degeeraed ormal. Â coverges o a mixed ormal disribuio a a faser rae i he direcio of B Λ, he coiegraig space of observable ime series. However, i he direcios orhogoal o he coiegraig space, i coverges a sadard rae ad has a asympoically ormal disribuio. I he simulaio, he samples of size 5 are draw 2 imes o esimae he ML esimaors. The saisics based o hese esimaors are also derived. I esimaig Λ, we esimae he cholesky riagle of Λ isead. I ha way, we oly esimae he oreduda parameers ad also esure he esimaed covariace marix o be posiive defiie. To choose he marix B i he roaio marix, we firs regress a radomly picked 3 dimesioal vecor o Λ A, ad he ormalize he residual, such ha he ormalized residual e saisfies he codiio e Λ e =. We ake e as our B. I is clear ha B saisfies he cosrais specified i 6. The simulaio resuls are summarized i Figure. The disribuios of he ML esimaors are ceered wih heir rue values. The fiie sample behavior of he ML esimaors are as expeced. The disribuios of Â ad ˆΛ are symmeric ad well ceered as prediced by heir asympoic heories. The lef boom of preses he disribuios of roaed Â. The solid curves represe he disribuios of Â o he direcio of B Λ ad he res of he curves represe he disribuios of Â o he direcios orhogoal o B Λ. As expeced, he solid curves are seeper sice he roaed Â coverges a a faser rae o he direcios defied by he coiegraig space. Disribuios of raios are preseed a he righ boom of. I oal 3 curves are ivolved, 2 are he raios of he ML esimaors of he ukow parameers ad oe represes sadard ormal disribuio. The hiree curves seem o be very much overlapped, which is cosise wih he asympoic heory. 6. A Empirical Illusraio The opic of deermiig he relaioship amog he yields o defaulfree securiies ha differ oly i heir erms o mauriy has log bee a opic of cocer for ecoomiss. Mos researches are coduced i he framework of srucural models. Srucural models focus o explaiig ad esig he erm premium. Depedig o he assumpio of he drivig diffusio processes, srucural models ca be divided io oefacor ad mulifacor models. Oefacor models, like oefacor imehomogeeous models of Vasicek, CoxIgersollRoss, Doha, ad he Expoeial Vasicek model, model he isaaeous spo ieres rae via oe drivig diffusio process. Mulifacor models, such as he wofacor model i Hall ad Whie 994, assume ha ieres raes are affeced by muliple correlaed diffusio processes. Accordig o Jamshidia ad Zhu 997, mulifacor models ca explai much more variaios i hisorical yield curves ha oefacor models do. However, how may facors should be icluded o icrease he explaaory power of models wihou causig he overfiig problem is o clear. This problem becomes eve less clear whe we cosider a group of ieres raes alogeher i oe model. We expec some reds 7
19 are commo o differe ieres raes sice uder defaulfree assumpio, he purchase of a logerm asse is equivale o ha of a sequece of shorerm asses. Therefore, he umber of reds mus be much smaller ha he umber of he ieres raes uder cosideraio. I his secio, we focus o esig ad exracig he commo uobservable sochasic reds of 9 defaulfree ieres raes. Our aalysis is based o he sae space model described i Secio 2. The ieres raes used here are Secodary Marke Rae for Treasury Bills wih mauriies 3 ad 6 mohs ad Treasury Cosa Mauriy Rae of Treasury Bods wih mauriies, 2, 3, 5, 7,, 3 years. The daa is obaied from Federal Reserve Bak a S. Louis, ad he selecio of daa is based o availabiliy. The daa is mohly ad ragig from February s, 977 o February s, /77 2/82 2/87 2/92 2/97 2/2 Figure 2: DefaulFree Ieres Raes wih Differe Mauriies A salie feaure of ieres raes is srog persisece. The persisece ca be easily see from Figure 2. Alhough srog persisece does o ecessarily imply a presece of ui roo, i is rue ha i may empirical sudies, ui roo ess fail o rejec he ull hypohesis ha ieres raes have a ui roo. For example Nelso ad Plosser 982 ivesigae a se of macroecoomic variables by usig DickeyFuller ype ess ha are developed i Dickey ad Fuller 979, 98, ad coclude ha may macroecoomic ime series, icludig bod yield, are beer characerized as havig a radom walk compoe ha as saioary wih drif or red saioary. I his paper, we revisi his problem by coducig he augmeed Dickey Fuller es o he 9 ieres rae series. The umber of lags i he auoregressio fucio is seleced by Akaike Iformaio Crieria ad Bayesia Iformaio Crieria. The auoregressio has a cosa erm bu o a ime red sice o ecoomic heory suggess ha omial ieres raes should exhibi a deermiisic ime red ad if a ieres were o be described by a saioary process, surely i would have a posiive mea. The augmeed Dickey Fuller ess o he ie ieres raes fail o rejec ui roo hypoheses a 95% sigificace level. This gives us a reaso o believe ha
20 ieres raes are drive by some osaioary sochasic processes. The deails of he esig resuls are available upo reques. We use he model defied i o exrac he commo reds of he 9 ieres raes. Based o he asympoic aalysis i he previous secios, he model ca be esimaed by he ordiary Kalma filer ad he ML esimaes of model parameers are cosise ad asympoically Gaussia. Moreover, if he umber of reds is kow, he exraced reds oly differ from he rue reds by a saioary compoe. We apply he sequeial es iroduced i Secio 4 i esig he umber of commo reds. 9 H : k = q H : k > q where q < 9 is he umber of reds uder he ull. The es saisic is give i 28. Before calculaig he es saisic uder he ull hypohesis, we firs eed o esimae he model wih he umber of reds assumed i he ull. We shall always sar from q = ad sop whe we fail o rejec he ull. The likelihood fucio is formalized i he sadard way as for geeral saioary sae space models. The iiial covariace marix of he sae variable is se o be Ω I q, where Ω is he seady sae value derived i Lemma 2.: Ω = 2 I q + I q + 4A Λ A /2. The selecio of he iiial value of he sae variable is o clear. Alhough i should o maer asympoically as discussed i Lemma 2.2, i migh sill play a role i fiie sample esimaio. Kim ad Nelso 999 sugges droppig some of he iiial observaios whe evaluaig he likelihood fucio. This approach causes iformaio loss as we ca see. Aoher approach is o rea x as aoher model parameer ad esimae. This approach is applaudable whe he umber of reds is o big, i.e. he dimesio of x is low. I his applicaio, we use he secod approach. I esimaig he covariace marix of he error erm i he measureme equaio, i order o esure ha he esimaed covariace marix is posiive defiie ad o esimae oly oreduda parameers we esimae is Cholesky decomposiio isead. We repor he model esimaors for q = i Table. I order o ge ˆB i he race saisic, we firs regress a radomly geeraed 9 8 marix o ˆΛ Â, ad he ormalize he obaied error erm so ha for he ormalized error e, e ˆΛ e =. I is clear ha he ormalized error saisfies he codiios i 6 ad herefore ca be used as ˆB. The race saisic ad is variace are ow ready o be calculaed. I our case, τ = Accordig o he es saisic, we fail o rejec he ull hypohesis of q =, i.e., he ieres raes uder cosideraio are drive by a sigle sochasic process. To verify he es resul, we also esimae he model uder he assumpio of q = 2, ad exrac he correspodig commo reds. The maximum likelihood esimaes for q = 2
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