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


 Delphia Randall
 1 years ago
 Views:
Transcription
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
Bullwhip Effect Measure When Supply Chain Demand is Forecasting
J. Basic. Appl. Sci. Res., (4)4743, 01 01, TexRoad Publicaio ISSN 0904304 Joural of Basic ad Applied Scieific Research www.exroad.com Bullwhip Effec Measure Whe Supply Chai emad is Forecasig Ayub Rahimzadeh
More informationA panel data approach for fashion sales forecasting
A pael daa approach for fashio sales forecasig Shuyu Re(shuyu_shara@live.c), TsaMig Choi, Na Liu Busiess Divisio, Isiue of Texiles ad Clohig, The Hog Kog Polyechic Uiversiy, Hug Hom, Kowloo, Hog Kog Absrac:
More informationThe Term Structure of Interest Rates
The Term Srucure of Ieres Raes Wha is i? The relaioship amog ieres raes over differe imehorizos, as viewed from oday, = 0. A cocep closely relaed o his: The Yield Curve Plos he effecive aual yield agais
More informationFORECASTING MODEL FOR AUTOMOBILE SALES IN THAILAND
FORECASTING MODEL FOR AUTOMOBILE SALES IN THAILAND by Wachareepor Chaimogkol Naioal Isiue of Developme Admiisraio, Bagkok, Thailad Email: wachare@as.ida.ac.h ad Chuaip Tasahi Kig Mogku's Isiue of Techology
More informationModeling the Nigerian Inflation Rates Using Periodogram and Fourier Series Analysis
CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 51 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis 1 Chukwuemeka O. Omekara, Emmauel J. Ekpeyog ad Michael P. Ekeree
More informationCHAPTER 22 ASSET BASED FINANCING: LEASE, HIRE PURCHASE AND PROJECT FINANCING
CHAPTER 22 ASSET BASED FINANCING: LEASE, HIRE PURCHASE AND PROJECT FINANCING Q.1 Defie a lease. How does i differ from a hire purchase ad isalme sale? Wha are he cash flow cosequeces of a lease? Illusrae.
More informationUNIT ROOTS Herman J. Bierens 1 Pennsylvania State University (October 30, 2007)
UNIT ROOTS Herma J. Bieres Pesylvaia Sae Uiversiy (Ocober 30, 2007). Iroducio I his chaper I will explai he wo mos frequely applied ypes of ui roo ess, amely he Augmeed DickeyFuller ess [see Fuller (996),
More informationREVISTA INVESTIGACION OPERACIONAL VOL. 31, No.2, 159170, 2010
REVISTA INVESTIGACION OPERACIONAL VOL. 3, No., 5970, 00 AN ALGORITHM TO OBTAIN AN OPTIMAL STRATEGY FOR THE MARKOV DECISION PROCESSES, WITH PROBABILITY DISTRIBUTION FOR THE PLANNING HORIZON. Gouliois E.
More informationModelling Time Series of Counts
Modellig ime Series of Cous Richard A. Davis Colorado Sae Uiversiy William Dusmuir Uiversiy of New Souh Wales Yig Wag Colorado Sae Uiversiy /3/00 Modellig ime Series of Cous wo ypes of Models for Poisso
More informationRanking of mutually exclusive investment projects how cash flow differences can solve the ranking problem
Chrisia Kalhoefer (Egyp) Ivesme Maageme ad Fiacial Iovaios, Volume 7, Issue 2, 2 Rakig of muually exclusive ivesme projecs how cash flow differeces ca solve he rakig problem bsrac The discussio abou he
More information1/22/2007 EECS 723 intro 2/3
1/22/2007 EES 723 iro 2/3 eraily, all elecrical egieers kow of liear sysems heory. Bu, i is helpful o firs review hese coceps o make sure ha we all udersad wha his heory is, why i works, ad how i is useful.
More informationReaction Rates. Example. Chemical Kinetics. Chemical Kinetics Chapter 12. Example Concentration Data. Page 1
Page Chemical Kieics Chaper O decomposiio i a isec O decomposiio caalyzed by MO Chemical Kieics I is o eough o udersad he soichiomery ad hermodyamics of a reacio; we also mus udersad he facors ha gover
More informationA Strategy for Trading the S&P 500 Futures Market
62 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 A Sraegy for Tradig he S&P 500 Fuures Marke Edward Olszewski * Absrac A sysem for radig he S&P 500 fuures marke is proposed. The sysem
More informationStudies in sport sciences have addressed a wide
REVIEW ARTICLE TRENDS i Spor Scieces 014; 1(1: 195. ISSN 999590 The eed o repor effec size esimaes revisied. A overview of some recommeded measures of effec size MACIEJ TOMCZAK 1, EWA TOMCZAK Rece years
More informationA formulation for measuring the bullwhip effect with spreadsheets Una formulación para medir el efecto bullwhip con hojas de cálculo
irecció y rgaizació 48 (01) 933 9 www.revisadyo.com A formulaio for measurig he bullwhip effec wih spreadshees Ua formulació para medir el efeco bullwhip co hojas de cálculo Javier ParraPea 1, Josefa
More informationCOLLECTIVE RISK MODEL IN NONLIFE INSURANCE
Ecoomic Horizos, May  Augus 203, Volume 5, Number 2, 6775 Faculy of Ecoomics, Uiversiy of Kragujevac UDC: 33 eissn 2279232 www. ekfak.kg.ac.rs Review paper UDC: 005.334:368.025.6 ; 347.426.6 doi: 0.5937/ekohor30263D
More informationHilbert Transform Relations
BULGARIAN ACADEMY OF SCIENCES CYBERNEICS AND INFORMAION ECHNOLOGIES Volume 5, No Sofia 5 Hilber rasform Relaios Each coiuous problem (differeial equaio) has may discree approximaios (differece equaios)
More informationWhy we use compounding and discounting approaches
Comoudig, Discouig, ad ubiased Growh Raes Near Deb s school i Souher Colorado. A examle of slow growh. Coyrigh 00004, Gary R. Evas. May be used for orofi isrucioal uroses oly wihou ermissio of he auhor.
More informationIntroduction to Statistical Analysis of Time Series Richard A. Davis Department of Statistics
Iroduio o Saisial Aalysis of Time Series Rihard A. Davis Deparme of Saisis Oulie Modelig obeives i ime series Geeral feaures of eologial/eviromeal ime series Compoes of a ime series Frequey domai aalysishe
More information4. Levered and Unlevered Cost of Capital. Tax Shield. Capital Structure
4. Levered ad levered Cos Capial. ax hield. Capial rucure. Levered ad levered Cos Capial Levered compay ad CAP he cos equiy is equal o he reur expeced by sockholders. he cos equiy ca be compued usi he
More informationResearch Article Dynamic Pricing of a Web Service in an Advance Selling Environment
Hidawi Publishig Corporaio Mahemaical Problems i Egieerig Volume 215, Aricle ID 783149, 21 pages hp://dx.doi.org/1.1155/215/783149 Research Aricle Dyamic Pricig of a Web Service i a Advace Sellig Evirome
More informationUNDERWRITING AND EXTRA RISKS IN LIFE INSURANCE Katarína Sakálová
The process of uderwriig UNDERWRITING AND EXTRA RISKS IN LIFE INSURANCE Kaaría Sakálová Uderwriig is he process by which a life isurace compay decides which people o accep for isurace ad o wha erms Life
More informationExchange Rates, Risk Premia, and Inflation Indexed Bond Yields. Richard Clarida Columbia University, NBER, and PIMCO. and
Exchage Raes, Risk Premia, ad Iflaio Idexed Bod Yields by Richard Clarida Columbia Uiversiy, NBER, ad PIMCO ad Shaowe Luo Columbia Uiversiy Jue 14, 2014 I. Iroducio Drawig o ad exedig Clarida (2012; 2013)
More informationTesting Linearity in Cointegrating Relations with an Application to Purchasing Power Parity
Tesig Lieariy i Coiegraig Relaios wih a Applicaio o Purchasig Power Pariy Seug Hyu Hog Deparme of Ecoomics Cocordia Uiversiy ad Peer C. B. Phillips Cowles Foudaio, Yale Uiversiy Uiversiy of Aucklad ad
More informationDAMPING AND ENERGY DISSIPATION
DAMPING AND ENERGY DISSIPATION Liear Viscous Dampig Is A Propery Of The Compuer Model Ad Is No A Propery Of A Real Srucure 19.1. INTRODUCTION I srucural egieerig, viscous, velociydepede dampig is very
More informationCircularity and the Undervaluation of Privatised Companies
CMPO Workig Paper Series No. 1/39 Circulariy ad he Udervaluaio of Privaised Compaies Paul Grou 1 ad a Zalewska 2 1 Leverhulme Cere for Marke ad Public Orgaisaio, Uiversiy of Brisol 2 Limburg Isiue of Fiacial
More informationCombining Adaptive Filtering and IF Flows to Detect DDoS Attacks within a Router
KSII RANSAIONS ON INERNE AN INFORMAION SYSEMS VOL. 4, NO. 3, Jue 2 428 opyrigh c 2 KSII ombiig Adapive Filerig ad IF Flows o eec os Aacks wihi a Rouer Ruoyu Ya,2, Qighua Zheg ad Haifei Li 3 eparme of ompuer
More informationTesting Linearity in Cointegrating Relations With an Application to Purchasing Power Parity
Tesig Lieariy i Coiegraig Relaios Wih a Applicaio o Purchasig Power Pariy Seug Hyu HONG Korea Isiue of Public Fiace KIPF), Sogpaku, Seoul, Souh Korea 38774 Peer C. B. PHILLIPS Yale Uiversiy, New Have,
More informationA Queuing Model of the Ndesign Multiskill Call Center with Impatient Customers
Ieraioal Joural of u ad e ervice, ciece ad Techology Vol.8, o., pp. hp://dx.doi.org/./ijuess..8.. A Queuig Model of he desig Muliskill Call Ceer wih Impaie Cusomers Chuya Li, ad Deua Yue Yasha Uiversiy,
More informationDistributed Containment Control with Multiple Dynamic Leaders for DoubleIntegrator Dynamics Using Only Position Measurements
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 553 Disribued Coaime Corol wih Muliple Dyamic Leaders for DoubleIegraor Dyamics Usig Oly Posiio Measuremes Jiazhe Li, Wei Re, Member, IEEE,
More informationTHE FOREIGN EXCHANGE EXPOSURE OF CHINESE BANKS
Workig Paper 07/2008 Jue 2008 THE FOREIGN ECHANGE EPOSURE OF CHINESE BANKS Prepared by Eric Wog, Jim Wog ad Phyllis Leug 1 Research Deparme Absrac Usig he Capial Marke Approach ad equiyprice daa of 14
More informationAPPLICATIONS OF GEOMETRIC
APPLICATIONS OF GEOMETRIC SEQUENCES AND SERIES TO FINANCIAL MATHS The mos powerful force i he world is compoud ieres (Alber Eisei) Page of 52 Fiacial Mahs Coes Loas ad ivesmes  erms ad examples... 3 Derivaio
More informationDeterminants of Public and Private Investment An Empirical Study of Pakistan
eraioal Joural of Busiess ad Social Sciece Vol. 3 No. 4 [Special ssue  February 2012] Deermias of Public ad Privae vesme A Empirical Sudy of Pakisa Rabia Saghir 1 Azra Kha 2 Absrac This paper aalyses
More informationRanking Optimization with Constraints
Rakig Opimizaio wih Cosrais Fagzhao Wu, Ju Xu, Hag Li, Xi Jiag Tsighua Naioal Laboraory for Iformaio Sciece ad Techology, Deparme of Elecroic Egieerig, Tsighua Uiversiy, Beijig, Chia Noah s Ark Lab, Huawei
More informationOn Motion of Robot Endeffector Using The Curvature Theory of Timelike Ruled Surfaces With Timelike Ruling
O Moio of obo Edeffecor Usig he Curvaure heory of imelike uled Surfaces Wih imelike ulig Cumali Ekici¹, Yasi Ülüürk¹, Musafa Dede¹ B. S. yuh² ¹ Eskişehir Osmagazi Uiversiy Deparme of Mahemaics, 6480UKEY
More informationA Heavy Traffic Approach to Modeling Large Life Insurance Portfolios
A Heavy Traffic Approach o Modelig Large Life Isurace Porfolios Jose Blache ad Hery Lam Absrac We explore a ew framework o approximae life isurace risk processes i he sceario of pleiful policyholders,
More informationHYPERBOLIC DISCOUNTING IS RATIONAL: VALUING THE FAR FUTURE WITH UNCERTAIN DISCOUNT RATES. J. Doyne Farmer and John Geanakoplos.
HYPERBOLIC DISCOUNTING IS RATIONAL: VALUING THE FAR FUTURE WITH UNCERTAIN DISCOUNT RATES By J. Doye Farmer ad Joh Geaakoplos Augus 2009 COWLES FOUNDATION DISCUSSION PAPER NO. 1719 COWLES FOUNDATION FOR
More informationKyoungjae Kim * and Ingoo Han. Abstract
Simulaeous opimizaio mehod of feaure rasformaio ad weighig for arificial eural eworks usig geeic algorihm : Applicaio o Korea sock marke Kyougjae Kim * ad Igoo Ha Absrac I his paper, we propose a ew hybrid
More information3. Cost of equity. Cost of Debt. WACC.
Corporae Fiace [090345] 3. Cos o equiy. Cos o Deb. WACC. Cash lows Forecass Cash lows or equiyholders ad debors Cash lows or equiyholders Ecoomic Value Value o capial (equiy ad deb)  radiioal approach
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationTesting the Weak Form of Efficient Market Hypothesis: Empirical Evidence from Jordan
Ieraioal Busiess ad Maageme Vol. 9, No., 4, pp. 9 DOI:.968/554 ISSN 984X [Pri] ISSN 9848 [Olie] www.cscaada.e www.cscaada.org Tesig he Wea Form of Efficie Mare Hypohesis: Empirical Evidece from Jorda
More informationHanna Putkuri. Housing loan rate margins in Finland
Haa Pukuri Housig loa rae margis i Filad Bak of Filad Research Discussio Papers 0 200 Suome Pakki Bak of Filad PO Box 60 FI000 HESINKI Filad +358 0 83 hp://www.bof.fi Email: Research@bof.fi Bak of Filad
More informationPERFORMANCE COMPARISON OF TIME SERIES DATA USING PREDICTIVE DATA MINING TECHNIQUES
, pp.5766. Available olie a hp://www.bioifo.i/coes.php?id=32 PERFORMANCE COMPARISON OF TIME SERIES DATA USING PREDICTIVE DATA MINING TECHNIQUES SAIGAL S. 1 * AND MEHROTRA D. 2 1Deparme of Compuer Sciece,
More informationAn Approach for Measurement of the Fair Value of Insurance Contracts by Sam Gutterman, David Rogers, Larry Rubin, David Scheinerman
A Approach for Measureme of he Fair Value of Isurace Coracs by Sam Guerma, David Rogers, Larry Rubi, David Scheierma Absrac The paper explores developmes hrough 2006 i he applicaio of markecosise coceps
More informationMechanical Vibrations Chapter 4
Mechaical Vibraios Chaper 4 Peer Aviabile Mechaical Egieerig Deparme Uiversiy of Massachuses Lowell 22.457 Mechaical Vibraios  Chaper 4 1 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory Impulse Exciaio
More informationStock price prediction using a fusion model of wavelet, fuzzy logic and ANN
0 Ieraioal Coferece o Ebusiess, Maageme ad Ecoomics IPEDR Vol.5 (0 (0 IACSIT Press, Sigapore Sock price predicio usig a fusio model of wavele, fuzzy logic ad Nassim Homayoui + ad Ali Amiri Compuer egieerig
More informationGranger Causality Analysis in Irregular Time Series
Grager Causaliy Aalysis i Irregular Time Series Mohammad Taha Bahadori Ya Liu Absrac Learig emporal causal srucures bewee ime series is oe of he key ools for aalyzig ime series daa. I may realworld applicaios,
More informationImplementation of Lean Manufacturing Through Learning Curve Modelling for Labour Forecast
Ieraioal Joural of Mechaical & Mecharoics Egieerig IJMMEIJENS Vol:09 No:0 35 Absrac I his paper, a implemeaio of lea maufacurig hrough learig curve modellig for labour forecas is discussed. Firs, various
More informationCapital Budgeting: a Tax Shields Mirage?
Theoreical ad Applied Ecoomics Volume XVIII (211), No. 3(556), pp. 314 Capial Budgeig: a Tax Shields Mirage? Vicor DRAGOTĂ Buchares Academy of Ecoomic Sudies vicor.dragoa@fi.ase.ro Lucia ŢÂŢU Buchares
More informationTHE IMPACT OF FINANCING POLICY ON THE COMPANY S VALUE
THE IMPACT OF FINANCING POLICY ON THE COMPANY S ALUE Pirea Marile Wes Uiversiy of Timişoara, Faculy of Ecoomics ad Busiess Admiisraio Boțoc Claudiu Wes Uiversiy of Timişoara, Faculy of Ecoomics ad Busiess
More informationON THE RISKNEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS WITH NUMERICAL METHODS IN VIEW ABSTRACT KEYWORDS 1. INTRODUCTION
ON THE RISKNEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS WITH NUMERICAL METHODS IN VIEW BY DANIEL BAUER, DANIELA BERGMANN AND RÜDIGER KIESEL ABSTRACT I rece years, markecosise valuaio approaches have
More informationA simple SSDefficiency test
A simple SSDefficiecy es Bogda Grechuk Deparme of Mahemaics, Uiversiy of Leiceser, UK Absrac A liear programmig SSDefficiecy es capable of ideifyig a domiaig porfolio is proposed. I has T + variables
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationIDENTIFICATION OF MARKET POWER IN BILATERAL OLIGOPOLY: THE BRAZILIAN WHOLESALE MARKET OF UHT MILK 1. Abstract
IDENTIFICATION OF MARKET POWER IN BILATERAL OLIGOPOLY: THE BRAZILIAN WHOLESALE MARKET OF UHT MILK 1 Paulo Robero Scalco Marcelo Jose Braga 3 Absrac The aim of his sudy was o es he hypohesis of marke power
More informationTESTING FOR SERIAL INDEPENDENCE OF GENERALIZED ERRORS
TESTING FOR SERIAL INDEPENDENCE OF GENERALIED ERRORS aichao Du Idiaa Uiversiy April 4, 2008 Absrac I his paper, we develop a Neymaype smooh es for he serial depedece of uobservable geeralized errors.
More information1. Introduction  1 
The Housig Bubble ad a New Approach o Accouig for Housig i a CPI W. Erwi iewer (Uiversiy of Briish Columbia), Alice O. Nakamura (Uiversiy of Albera) ad Leoard I. Nakamura (Philadelphia Federal Reserve
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationChapter 4 Return and Risk
Chaper 4 Reur ad Risk The objecives of his chaper are o eable you o:! Udersad ad calculae reurs as a measure of ecoomic efficiecy! Udersad he relaioships bewee prese value ad IRR ad YTM! Udersad how obai
More informationManaging Learning and Turnover in Employee Staffing*
Maagig Learig ad Turover i Employee Saffig* YogPi Zhou Uiversiy of Washigo Busiess School Coauhor: Noah Gas, Wharo School, UPe * Suppored by Wharo Fiacial Isiuios Ceer ad he Sloa Foudaio Call Ceer Operaios
More informationFinancial Data Mining Using Genetic Algorithms Technique: Application to KOSPI 200
Fiacial Daa Miig Usig Geeic Algorihms Techique: Applicaio o KOSPI 200 Kyugshik Shi *, Kyougjae Kim * ad Igoo Ha Absrac This sudy ieds o mie reasoable radig rules usig geeic algorihms for Korea Sock Price
More informationM O N A S H U N I V E R S I T Y
ISSN 44077X ISBN 0 736 06 M O N A S H U N I V E R S I T Y AUSTRALIA Forecasig Sales of Slow ad Fas Movig Iveories Ralph Syder Workig Paper 7/99 Jue 999 DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS
More informationConvergence of Binomial Large Investor Models and General Correlated Random Walks
Covergece of Biomial Large Ivesor Models ad Geeral Correlaed Radom Walks vorgeleg vo Maser of Sciece i Mahemaics, DiplomWirschafsmahemaiker Urs M. Gruber gebore i Georgsmariehüe. Vo der Fakulä II Mahemaik
More informationGeneral Bounds for Arithmetic Asian Option Prices
The Uiversiy of Ediburgh Geeral Bouds for Arihmeic Asia Opio Prices Colombia FX Opio Marke Applicaio MSc Disseraio Sude: Saiago Sozizky s1200811 Supervisor: Dr. Soirios Sabais Augus 16 h 2013 School of
More informationAbstract. 1. Introduction. 1.1 Notation. 1.2 Parameters
1 Mdels, Predici, ad Esimai f Oubreaks f Ifecius Disease Peer J. Csa James P. Duyak Mjdeh Mhashemi {pjcsa@mire.rg, jduyak@mire.rg, mjdeh@mire.rg} he MIRE Crprai 202 Burlig Rad Bedfrd, MA 01730 1420 Absrac
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationIntroduction to Hypothesis Testing
Iroducio o Hyohei Teig Iroducio o Hyohei Teig Scieific Mehod. Sae a reearch hyohei or oe a queio.. Gaher daa or evidece (obervaioal or eerimeal) o awer he queio. 3. Summarize daa ad e he hyohei. 4. Draw
More informationFEBRUARY 2015 STOXX CALCULATION GUIDE
FEBRUARY 2015 STOXX CALCULATION GUIDE STOXX CALCULATION GUIDE CONTENTS 2/23 6.2. INDICES IN EUR, USD AND OTHER CURRENCIES 10 1. INTRODUCTION TO THE STOXX INDEX GUIDES 3 2. CHANGES TO THE GUIDE BOOK 4 2.1.
More informationThe Norwegian Shareholder Tax Reconsidered
The Norwegia Shareholder Tax Recosidered Absrac I a aricle i Ieraioal Tax ad Public Fiace, Peer Birch Sørese (5) gives a ideph accou of he ew Norwegia Shareholder Tax, which allows he shareholders a deducio
More informationTeaching Bond Valuation: A Differential Approach Demonstrating Duration and Convexity
JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier 2008 3 Teachig Bod aluaio: A Differeial Approach Demosraig Duraio ad ovexi TeWah Hah, David Lage ABSTRAT A radiioal bod pricig scheme used i iroducor
More informationA GLOSSARY OF MAIN TERMS
he aedix o his glossary gives he mai aggregae umber formulae used for cosumer rice (CI) uroses ad also exlais he ierrelaioshis bewee hem. Acquisiios aroach Addiiviy Aggregae Aggregaio Axiomaic, or es aroach
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationAPPLIED STATISTICS. Economic statistics
APPLIED STATISTICS Ecoomic saisics Reu Kaul ad Sajoy Roy Chowdhury Reader, Deparme of Saisics, Lady Shri Ram College for Wome Lajpa Nagar, New Delhi 0024 04Ja2007 (Revised 20Nov2007) CONTENTS Time
More informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101115. Macroeconomericians
More informationCAPITAL BUDGETING. Capital Budgeting Techniques
CAPITAL BUDGETING Capial Budgeig Techiques Capial budgeig refers o he process we use o make decisios cocerig ivesmes i he logerm asses of he firm. The geeral idea is ha he capial, or logerm fuds, raised
More informationhttp://www.ejournalofscience.org Monitoring of Network Traffic based on Queuing Theory
VOL., NO., November ISSN XXXXXXXX ARN Joural of Sciece a Techology  ARN Jourals. All righs reserve. hp://www.ejouralofsciece.org Moiorig of Newor Traffic base o Queuig Theory S. Saha Ray,. Sahoo Naioal
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More information14 Protecting Private Information in Online Social Networks
4 roecig rivae Iormaio i Olie Social eworks Jiamig He ad Wesley W. Chu Compuer Sciece Deparme Uiversiy o Calioria USA {jmhekwwc}@cs.ucla.edu Absrac. Because persoal iormaio ca be ierred rom associaios
More informationChinese Stock Price and Volatility Predictions with Multiple Technical Indicators
Joural of Iellige Learig Sysems ad Applicaios, 2011, 3, 209219 doi:10.4236/jilsa.2011.34024 Published Olie November 2011 (hp://www.scirp.org/joural/jilsa) 209 Chiese Sock Price ad Volailiy Predicios wih
More information12. Spur Gear Design and selection. Standard proportions. Forces on spur gear teeth. Forces on spur gear teeth. Specifications for standard gear teeth
. Spur Gear Desig ad selecio Objecives Apply priciples leared i Chaper 11 o acual desig ad selecio of spur gear sysems. Calculae forces o eeh of spur gears, icludig impac forces associaed wih velociy ad
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationNo. 16. Closed Formula for Options with Discrete Dividends and its Derivatives. Carlos Veiga, Uwe Wystup. October 2008
Cere for Pracical Quaiaive Fiace No. 16 Closed Formula for Opios wih Discree Divideds ad is Derivaives Carlos Veiga, Uwe Wysup Ocober 2008 Auhors: Prof. Dr. Uwe Wysup Carlos Veiga Frakfur School of Frakfur
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationDerivative Securities: Lecture 7 Further applications of BlackScholes and Arbitrage Pricing Theory. Sources: J. Hull Avellaneda and Laurence
Deivaive ecuiies: Lecue 7 uhe applicaios o Blackcholes ad Abiage Picig heoy ouces: J. Hull Avellaeda ad Lauece Black s omula omeimes is easie o hik i ems o owad pices. Recallig ha i Blackcholes imilaly
More informationA New Hybrid Network Traffic Prediction Method
This full ex paper was peer reviewed a he direcio of IEEE Couicaios Sociey subjec aer expers for publicaio i he IEEE Globeco proceedigs. A New Hybrid Nework Traffic Predicio Mehod Li Xiag, XiaoHu Ge,
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationFakultet for informasjonsteknologi, Institutt for matematiske fag
Page 1 of 5 NTNU Noregs eknisknaurviskaplege universie Fakule for informasjonseknologi, maemaikk og elekroeknikk Insiu for maemaiske fag  English Conac during exam: John Tyssedal 73593534/41645376 Exam
More informationImproving Survivability through Traffic Engineering in MPLS Networks
Improvig Survivabiiy hrough Traffic Egieerig i MPLS Neworks Mia Ami, KiHo Ho, George Pavou, ad Michae Howarh Cere for Commuicaio Sysems Research, Uiversiy of Surrey, UK Emai:{M.Ami, K.Ho, G.Pavou, M.Howarh}@eim.surrey.ac.uk
More informationThe effect of the increase in the monetary base on Japan s economy at zero interest rates: an empirical analysis 1
The effec of he icrease i he moeary base o Japa s ecoomy a zero ieres raes: a empirical aalysis 1 Takeshi Kimura, Hiroshi Kobayashi, Ju Muraaga ad Hiroshi Ugai, 2 Bak of Japa Absrac I his paper, we quaify
More informationRevisions to Nonfarm Payroll Employment: 1964 to 2011
Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm
More informationOptimal Combination of International and Intertemporal Diversification of Disaster Risk: Role of Government. Tao YE, Muneta YOKOMATSU and Norio OKADA
京 都 大 学 防 災 研 究 所 年 報 第 5 号 B 平 成 9 年 4 月 Auals of Disas. Prev. Res. Is., Kyoo Uiv., No. 5 B, 27 Opimal Combiaio of Ieraioal a Ieremporal Diversificaio of Disaser Risk: Role of Goverme Tao YE, Muea YOKOMATSUaNorio
More informationOutline. Numerical Analysis Boundary Value Problems & PDE. Exam. Boundary Value Problems. Boundary Value Problems. Solution to BVProblems
Oulie Numericl Alysis oudry Vlue Prolems & PDE Lecure 5 Jeff Prker oudry Vlue Prolems Sooig Meod Fiie Differece Meod ollocio Fiie Eleme Fll, Pril Differeil Equios Recp of ove Exm You will o e le o rig
More informationPredicting Indian Stock Market Using Artificial Neural Network Model. Abstract
Predicig Idia Sock Marke Usig Arificial Neural Nework Model Absrac The sudy has aemped o predic he moveme of sock marke price (S&P CNX Nify) by usig ANN model. Seve years hisorical daa from 1 s Jauary
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More information