FINANCIAL ECONOMETRIC MODELS Some contributions to the field

Transcription

1 FINANCIAL ECONOMETRIC MODELS Some conrbons o he feld João Ncola Inso Speror de Economa e Gesão/UTL, Ra do Qelhas 6, -78 Lsboa, Porgal, ncola@seg.l.p Absrac: Key words: For recen fnancal economerc models are dscssed. The frs ams o capre he volaly creaed by charss ; he second nends o model bonded random wals; he hrd nvolves a mechansm where he saonary s volaly-ndced, and he las one accommodaes nonsaonary dffson negraed sochasc processes ha can be made saonary by dfferencng. ARCH models, dffson processes, bonded random wal, volaly-ndced saonary, second order sochasc dfferenal eqaons.. INTRODUCTION. The objecve and scope of hs chaper Under he aspces of he Unversdade Técnca de Lsboa as par of commemoraon of 75 years of exsence, we were nved o prodce a paper ha reflecs some of or recen conrbons o he sae-of-he-ar on or feld. We have seleced for man conrbons n he fnancal economercs feld. Also, we brefly refer o some conrbons o he esmaon of sochasc dfferenal eqaons, alhogh he emphass of hs chaper s on specfcaon of fnancal economerc models. We gve he movaon behnd he models, and he more echncal deals wll be referred o he orgnal papers. The srcre of hs chaper s as follows. In secon. we refer some general properes of rerns and prces. In secon we menon a model ha ams o capre he volaly creaed by charss. Ths s done n a dscree-me seng n he conex of ARCH models; also a connosme verson s provded. In secon 3 we presen hree dffson processes,

2 João Ncola wh dfferen prposes. The frs one nends o model bonded random wals; he dea s o model saonary processes wh random wal behavor. In he second one we dscss processes where he saonary s volaly-ndced. Ths s applcable o every me seres where reverson effecs occr manly n perods of hgh volaly. In he las one, we focs on a second order sochasc dfferenal eqaon. Ths process accommodaes nonsaonary negraed sochasc processes ha can be made saonary by dfferencng. Also, he model sggess drecly modellng he (nsananeos) rerns, conrary o sal connos-me models n fnance, whch model he prces drecly.. Prces, rerns and sylzed facs An mporan sep n formng an economerc model consss n sdyng he man feares of he daa. In fnancal economercs wo of he mos mporan varables are prces and rerns (volaly s also fndamenal and we shall go bac o laer). Prces nclde, for example, soc prces, soc ndces, exchange raes and neres raes. If we collec daly daa, he prce s sally some ype of closng prce. I may be a bd prce, an as prce or an average. I may be eher he fnal ransacon prce of he day or he fnal qoaon. In dscree me analyss, researchers sally prefer worng wh rerns, whch can be defned by changes n he logarhms of prces (wh approprae adjsmens for any dvdend paymens). Le P be a represenave prce for a soc (or soc ndces, exchange rae, ec.). The rern r a me s defned as r log P log P. General properes (sylzed facs) are well nown for daly rerns observed over a few years of prces. The mos sgnfcan are: The (ncondonal) dsrbon of r s leporc and n some cases (for soc prces and ndces) asymmerc; The correlaon beween rerns s absen or very wea; The correlaons beween he magndes of rerns on nearby days are posve and sascally sgnfcan. These feares can be explaned by changes hrogh me n volaly. Volaly clserng s a ypcal phenomenon n fnancal me seres. As noed by Mandelbro [9], large changes end o be followed by large changes, of eher sgn, and small changes end o be followed by small changes. A measremen of hs fac s ha, whle rerns hemselves are ncorrelaed, absole rerns r or her sqares dsplay a posve, sgnfcan and slowly decayng aocorrelaon fncon: Corr r, r for τ rangng from a few mnes o several wees. Perods of hgh volaly lead o exreme vales (and hs o a leporc dsrbon). Fgre

3 Mar-86 Jn-87 Ag-88 Nov-89 Feb-9 May-9 Ag-93 Oc-94 Jan-96 Apr-97 Jl-98 Oc-99 Dec- Mar- Jn-3 Sep-4 Nov-5 Fnancal economerc models 3 shows a ypcal me seres of rerns. Any economerc model for rerns shold capre hese general feares of fnancal me seres daa. The sascal feares of prces are no so obvos. In general, mos of he seres conan a clear rend (e.g. soc prces when observed over several years), ohers shows no parclar endency o ncrease or decrease (e.g. exchange raes). Shocs o a seres end o dsplay a hgh degree of perssence. For example, he Federal Fnds Rae experenced a srong pwards srge n 973 and remaned a he hgh level for nearly wo years. Also, he volaly of neres raes seems o be perssen. We wll resme some of hese feares n chaper Fgre. Mcrosof daly rerns from 986 o 6. DISCRETE-TIME MODELS. The ARCH famly In a semnal paper Engle [3] nrodced he so called aoregressve condonal heerosedascy model. These models have proven o be exremely sefl n modellng fnancal me seres. Also, hey have been sed n several applcaons (forecasng volaly, CAPM, VaR, ec.). The ARCH() s he smples example of an ARCH process. One assmes ha he dsrbon of he rern for perod, gven pas nformaon, s r F ~ D, where D s he condonal dsrbon, s he condonal mean and

4 4 João Ncola r, (, ) s he condonal varance. A large error n perod (ha s a hgh vale for r ) mples a hgh vale for he condonal varance n he nex perod. Generally, s a wea componen of he model snce s dffcl o predc he rern r based on a F -mensrable sochasc process. In many cases s a posve consan. Ths, eher a large posve or a large negave rern n perod mples hgher han average volaly n he nex perod; conversely, rerns close o he mean mply lower han average volaly. The erm aoregressve (from ARCH) comes from he fac ha he sqared errors follow an aoregressve process. In fac, from where r we have v, v and snce v s a marngale dfference (by consrcon, assmng E v ) one concldes ha s an aoregressve process of order one. There are a grea nmber of ARCH specfcaons and many of hem have her own acronyms, sch GARCH, EGARCH, MARCH, AARCH, ec.. One more ARCH model he Trend-GARCH.. Movaon In recen lerare a nmber of heerogeneos agen models have been developed based on he new paradgm of behavoral economcs, behavoral fnance and bonded raonaly (see [7] for a srvey on hs sbjec). Bascally, mos models n fnance dsngsh beween sophscaed raders and echncal raders or charss. Sophscaed raders, sch as fndamenalss or raonal arbragers end o psh prces n he drecons of he raonal expecaon fndamenal vale and hs ac as a sablsng force. Charss base her decsons manly on sascs generaed by mare acvy sch as pas prces and volme. Techncal analyss do no aemp o measre he nrnsc vale of a secry; nsead hey loo for paerns and ndcaors on soc chars ha wll deermne a soc's fre performance. Ths, here s he belef ha secres move n very predcable rends and paerns. As De Long e al. [] recognse, hs acvy can lm he wllngness of fndamenalss o ae posons agans nose raders (charss). In fac, f

5 Fnancal economerc models 5 nose raders oday are pessmss and he prce s low, a fndamenals wh a shor me horzon byng hs asse can sffer a loss f nose raders become even more pessmsc. Conversely, a fndamenals sellng an asse shor when he prce s hgh can lose money f nose raders become more bllsh n he near fre. "Nose raders hs creae her own space. [...] Arbrage does no elmnae he effec of nose becase nose self creaes rs" (De Long e al., []). As a conseqence, echncal raders or charss, sch as feedbac raders and rend exrapolaors end o psh prces away from he fndamenal and hs ac as a desablsng force, creang volaly. Based on hese deas, Ncola [6] proposed an economerc model, n a dscree and connos-me seng, based on a echncal radng rle o measre and capre he ncrease of volaly creaed by charss... The Trend-GARCH In order o derve he model we now focs more closely on a by-sell rle sed by charss. One of he mos wdely sed echncal rles s based on he movng average rle. Accordng o hs rle, by and sell sgnals are generaed by wo movng averages of he prce level: a long-perod average and a shor-perod average. A ypcal movng average radng rle prescrbes a by (sell) when he shor-perod movng average crosses he long-perod movng average from below (above) (.e. when he orgnal me seres s rsng (fallng) relavely fas). As can be seen, he movng average rle s essenally a rend followng sysem becase when prces are rsng (fallng), he shor-perod average ends o have larger (lower) vales han he longperod average, sgnallng a long (shor) poson. Hence, he hgher he dfference beween hese wo movng averages, he sronger he sgnal o by or sell wold be and, a he same me, he more charss deec he by or sell sgnals. As a conseqence, a movemen n he prce and n he volaly ms, n prncple, be expeced, whenever a rend s spposed o be naed. How o ncorporae hs nformaon n he specfcaon of he condonal varance s explaned below. To smplfy, we assme (as ohers) ha he shor-perod movng average s js he crren (or laes) mare prce and he long-perod one s an exponenally weghed movng average (EWMA), whch s also an adapve expecaon of he mare prce. In hs formlaon, he excess demand fncon of nose raders can be gven as a fncon of log S m q f log S m, f x () where S denoes he mare prce and average, represened here as an EWMA, m s he long-perod movng

6 6 João Ncola m m log S,. The dervave of f (see eqaon ()) s posve as, he hgher he qany log S m, he sronger he sgnal o by wold be. Conversely, he lower he qany log S m, he sronger he sgnal o sell wold be. Based on hese deas and n Baer [7], Ncola [6] proposes he followng model, nspred by he GARCH(,) specfcaon: r m m, log S log S, m,,, () where, r s he log rern, s he condonal mean, s assmed o be a seqence of..d. random varables wh E and Var. The condonal varance ncorporaes a measre of charss radng acvy, hrogh he erm log S m. We presen some properes of hs model. Sppose ha. Ths, S log S log S log S r. (3) On he oher hand, he EWMA process has he followng solon m m log S. (4) Combnng eqaons (3) and (4), and assmng m, we have, afer some smplfcaons, log S m r log S r If he seqence r dsplays very wea dependence, one can assme, ha s r. In hs case, we have

7 Fnancal economerc models 7 log S m. The model nvolves explcly he dea of he movng average rle, whch we ncorporae sng eqaon m m logs. Ths movng average represenaon grealy faclaes he esmaon of he model and he sdy of he saonary condons. The expresson r can be ndersood as a rend componen, whch approxmaely measres rend esmaes n echncal radng models. When he mos recen rerns have he same sgnal, ha s, when log S m r s hgh, charss see a general drecon of he prce (ha s, a rend) whch s generally classfed as an prend or downrend. In hese cases, charss ncrease her acvy n he mare, byng and sellng and hs ncreasng volaly. On he oher hand, when he rend s classfed as rangebond, prce swngs bac and forh for some perods, and as conseqence, he qany r s low (he posve rerns end o compensae he negave ones). In hs case, here s mch less rade acvy by charss, and he volaly assocaed wh hem s low. I can be proved nder he condons,,,, and s a seqence of..d. random varables wh E and and Var ha he process s covarance-saonary f and only f. Condons for he exsence of a nqe src saonary solon are also sded n Ncola [6]. The saonary maes sense becase prends or downrends canno perss over me. To assess he mean draon of a rend componen, cold be neresng o calclae he speed of adjsmen arond zero. The hgher he parameer he lower he speed of reverson. A sefl ndcaor of he speed of adjsmen s he so-called half-lve ndcaor, whch, n or case, s gven by he expresson log / / log. Esmaon of model () s sraghforward. One can se he psedo maxmm lelhood based on he normal dsrbon (for example). A nll hypohess of neres s wheher he erm log S m eners n he specfcaon of he condonal varance, ha s, H:. Under hs hypohess, s no denfed, ha s, he lelhood fncon does no depend on and he asympoc nformaon marx s snglar. One smple approach consss of consderng Daves's bond when q parameers are denfed only nder alernave hypohess (see Ncola, [6]). An emprcal llsraon s provded n Ncola [6]. Also, when he lengh of he dscree-me nervals beween observaons goes o zero, s shown ha, n

8 8 João Ncola some condons, he dscree-me process converges n dsrbon o he solon of he dffson process d c d dw,,, d d dw,,,. 3. CONTINUOUS-TIME MODELS 3. A bonded random wal process 3.. Movaon Some economc and fnancal me seres can behave js le a random wal (RW) (wh some volaly paerns) b de o economc reasons hey are bonded processes (n probably, for nsance) and even saonary processes. As dscssed n Ncola [] (and references heren) hs can be he case, for example, of neres raes, real exchange raes, some nomnal exchange raes and nemploymen raes among ohers seres. To bld a model wh sch feares s necessary o allow RW behavor drng mos of he me b force mean reversons whenever he processes ry o escape from some nerval. The am s o desgn a model ha can generae pahs wh he followng feares: as long as he process s n he nerval of moderae vales, he process bascally loos le a RW b here are reverson effecs owards he nerval of moderae vales whenever he process reaches some hgh or low vales. As we wll see, hese processes can adm - relyng on he parameers - saonary dsrbons, so we come o an neresng conclson: processes ha are almos ndsngshable from he RW process can be, n effec, saonary wh saonary dsrbons. 3.. The model If a process s a random wal, he fncon E x (where ) ms be zero (for all x ). On he oher hand, f a process s bonded (n probably) and mean-reverng o (say), he fncon E x ms be posve f x s below and negave f x s above.

9 Fnancal economerc models 9 Now consder a process ha s bonded b behaves le a RW. Wha nd of fncon shold E x be? As he process behaves le a RW, () ms be zero n some nerval and, snce he process s bonded, () ms be posve (negave) when x s ''low'' (''hgh''). Moreover we expec ha: () E x s a monoonc fncon whch, assocaed wh (), means ha he reverson effec shold be srong f x s far from he nerval of reverson and shold be wea n he oppose case; (v) E x s dfferenable (on he sae space of ) n order o assre a smooh effec of reverson. To sasfy ()-(v) we assme x x E x e e e wh,,. Le x x s fx a x e e e. Wh or assmpon abo E x we have he bonded random wal process (BRW) n dscree-me: e e e,. where are he nsances a whch he process s observed, (... T ), s he nerval beween observaons,, and are parameers dependng on and,,,... s a seqence of..d. random varables wh E and Var. I can be proved (see []) ha he seqence formed as a sep fncon from, ha s f, converges wealy (.e. n dsrbon) as o he solon o he sochasc dfferenal eqaons (SDE) : d e e e d dw, c. (5) where c s a consan and W s a sandard Wener process ( ). The case a x (for all x ) leads o he Wener process (whch can be ndersood as he random wal process n connos-me). I s sll obvos ha a, so ms behave js le a Wener process when crosses. However, s possble, by selecng adeqae vales for, and o have a Wener process behavor over a large nerval cenred on (ha s, sch ha a x over a large nerval cenred on ). Neverheless, whenever escapes from some levels here wll always be reverson effecs owards he. A possble drawbac of model (5) s ha he dffson coeffcen s consan. In he exchange rae framewor and nder a arge zone regme, we shold observe a volaly of shape '' '' wh respec o x (maxmm volaly a he cenral rae) (see [8]). On he oher hand, nder a free floang regme, s common o observe a ''smle'' volaly (see [8]). For boh possbles, we allow he volaly o be of shape '' '' or '' ' '' by assmng a specfcaon le exp x. c

10 João Ncola Dependng on he we wll have volaly of '' '' or '' '' form. Narally, leads o consan volaly. Ths specfcaon, wh, can also be approprae for neres raes. We propose, herefore, / d e e e d e dw, c. (6) Some properes are sded n []. Under some condons boh solons are saonary (wh nown saonary denses). To apprecae he dfferences beween he Wener process (he nbonded RW) and he bonded RW, we smlae one rajecory for boh processes n he perod, wh. We consdered,, and 4. The pahs are presened n fgre. In he neghborhood of he fncon a x s (approxmaely) zero, so behaves as a Wener process (or a random wal n connos-me). In effec, f a x, we have d dw (or W ). We draw wo arbrary lnes o show ha he bonded random wal afer crossng hese lnes ends o move oward he nerval of moderae vales. / Bonded Random Wal Wener (Random Wal) Fgre. Bonded Randow Wal vs. Wener Process 3. Processes wh volaly-ndced saonary 3.. Movaon Shor-erm neres rae processes have shown a leas wo man facs. Frsly, he mean-reverng effec s very wea (see, for example, Chan e al. [9] or Band [5]). In fac, he saonary of shor-erm neres rae processes s qe dbos. The sal n roo ess do no clearly eher rejec or accep he hypohess of saonary. Snce neres rae processes are

11 Fnancal economerc models bonded by a lower (zero) and pper (fne) vale a pre n roo hypohess seems mpossble snce a n roo process goes o or wh probably one as me goes o. Some ahors have addressed hs qeson. The sse s how o reconcle an apparen absence of meanreverng effecs wh he fac ha he neres rae s a bonded (and possbly saonary) process. Whle Aï-Sahala [] and Ncola [] sggess ha saonary can be drf-ndced, Conley e al. [] (CHLS, henceforh) sgges ha saonary s prmarly volaly-ndced. In fac, has been observed ha hgher volaly perods are assocaed wh mean reverson effecs. Ths, he CHLS hypohess s ha hgher volaly njecs saonary n he daa. The second (well nown) fac s ha he volaly of neres raes s manly level dependen and hghly perssen. The hgher (lower) he neres rae s he hgher (lower) he volaly. The volaly perssence can hs be parally arbed o he level of perssence of he neres rae. The hypohess of CHLS s neresng snce volaly-ndced saonary can explan marngale behavor (fac one), level volaly perssence (fac wo), and mean-reverson. To llsrae hese deas and show how volaly can njec saonary we presen n fgre 3 a smlaed pah from he SDE: I s worh menonng ha he Eler scheme d dw (7) Y Y Y,..d. N, Fgre 3. Smlaed pah from he SDE (7) canno be sed snce Y explodes as (see [4, 7]). For a mehod o smlae, see Ncola [4]. Snce he SDE (7) has zero drf, we cold

12 João Ncola expec random wal behavor. Neverheless, fgre 3 shows ha he smlaed rajecory of exhbs reverson effecs owards zero, whch s assred solely by he srcre of he dffson coeffcen. I s he volaly ha ndces saonary. In he neghborhood of zero he volaly s low so he process ends o spend more me n hs nerval. If here s a shoc, he process moves away from zero and he volaly ncreases (snce he dffson coeffcen s x ) whch, n rn, ncreases he probably ha crosses zero agan. The process can reach exreme peas n a very shor me b qcly rerns o he neghborhood of zero. I can be proved, n fac, ha s a saonary process. Ths, s a saonary local marngale b no a marngale snce E converges o he saonary mean as and s no eqal o as wold be reqred f was a marngale. 3.. A defnon of volaly-ndced saonary To or nowledge, CHLS were he frs o dscss volaly-ndced saonary (VIS) deas. Rcher [7] generalzes he defnon of CHLS. Bascally, her defnon saes ha he saonary process (solon of he sochasc dfferenal eqaon (SDE) d a d b dw ) has VIS a bondares l and r f lm s x and lm s x x l x r where s s he scale densy, s x exp z x a / b d ( z san arbrary vale). There s one dsadvanage n sng hs defnon. As shown n [5], he VIS defnon of CHLS and Rcher does no clearly denfy he sorce of saonary. I can be proved ha her defnon does no exclde meanreverson effecs and hs saonary can also be drf-ndced. A smple and a more precse defnon s gven n Ncola [5]. Consder he followng SDEs d dy a a Y d d b dw. dw We say ha a saonary process has VIS f he assocaed process Y does no possess a saonary dsrbon (acally, hs corresponds o wha defned n Ncola [5] as VIS of ype ). The non s smple: alhogh he process Y has he same drf as ha of he process, Y s nonsaonary (by defnon) whereas s saonary. The sbson of for b x ransforms a nonsaonary process Y no a saonary

13 Fnancal economerc models 3 process. Ths, he saonary of can only be arbed o he role of he dffson coeffcen (volaly) and n hs case we have n fac a pre VIS process. The followng s a smple creron o denfy VIS, n he case of sae space,. We say ha a saonary process wh bondares l and r has VIS f lm xa x or lm xa x. x 3... An example: Modellng he Fed fnds Rae wh VIS Processes wh VIS are poenally applcable o neres rae me-seres snce, as has been acnowledged, reverson effecs (owards a cenral measre of he dsrbon) occr manly n perods of hgh volaly. To exemplfy a VIS process monhly samplng of he Fed fnds rae beween Janary 96 and December was consdered. As dscssed n Ncola [5], here s emprcal evdence ha sppors he specfcaon x d exp / / dw (8) where logr and r represens he Fed fnds rae. The sae space of r s, and s,. Tha s, can assme any vale n R. Ths ransformaon preserves he sae space of r, snce r exp. By Iô's formla, eqaon (8) mples a VIS specfcaon for neres raes dr r e log r d r e / / log r dw I can be proved ha s an ergodc process wh saonary densy p x m m x x dx e x.e. logr N,/. By he connos mappng heorem, r exp s an ergodc process. Frhermore, has a log-normal saonary densy. There s some emprcal evdence ha sppors he above models. I s based on for facs:. The emprcal margnal dsrbon of logr maches he (margnal) dsrbon ha s mplc n model (8).. The resls of Dcey-Fller ess are compable wh a zero drf fncon for, as specfed n model (8). 3. Nonparamerc esmaes of a x and b x do no rejec specfcaon (8).

14 4 João Ncola 4. Paramerc esmaon of model (8) operforms common onefacor models n erms of accracy and parsmony. The esmaon of SDE (8) s dffcl snce he ranson (or condonal) denses of reqred o consrc he exac lelhood fncon are nnown. Several esmaon approaches have been proposed nder hese crcmsances (see Ncola [] for a bref srvey). To esmae he parameers of eqaon (8) we consdered he smlaed maxmm lelhood esmaor sggesed n Ncola [] (wh N and S ). The mehod proposed by Aï-Sahala [3] wh J (Aï-Sahala's noaon for he order of expanson of he densy approxmaon) gves smlar resls. The approxmaon of he densy based on J s oo complcaed o mplemen ( nvolves dozens of nrcae expressons ha are dffcl o evalae). The proposed model compares exremely favorably wh oher proposed one-facor connos-me models. In able we compare he proposed model wh oher relevan models for neres raes. Only he proposed mehod was esmaed by s. Remanng nformaon was obaned from able VI of Aï-Sahala []. For comparson prposes he proposed model was esmaed sng he same mehod appled o he oher models (we consdered he densy approxmaon proposed by Aï-Sahala [3] wh J, n he perod Janary-63 o December-98). Table 5 ndcaes ha he proposed model operforms he ohers n erms of accracy and parsmony. Table. Log-Lelhood of some Paramerc Models, Models Loglelhooh Nº Parameers dr r d dw dr r d r dw / dr r r d r dw dr r d r dw r 3 4r 3 / dr r d r dw dr r e log r / / log r d r e dw A second order sochasc dfferenal eqaon In economcs and fnance many sochasc processes can be seen as negraed sochasc processes n he sense ha he crren observaon behaves as he cmlaon of all pas perrbaons. In a dscree-me framewor he concep of negraon and dfferenaon of a sochasc

15 Fnancal economerc models 5 process plays an essenal role n modern economercs analyss. For nsance, he sochasc process y ;,,,... where y y (..d.n, ) s an example of an negraed process. Noce ha y can be wren as y y, or y y x, (9) where x. One way o deal wh sch processes s o se a dfferenced-daa model (for example, y, n he prevos example). Dfferencng has been sed mosly o solve non-saonary problems vewed as n roos alhogh, hsorcally, dfferenced-daa models arose early n economercs as a procedre o remove common rends beween dependen and ndependen varables. In emprcal fnance, mos wor on negraed dffson processes s relaed o sochasc volaly models (see for example, Genon-Caalo and Laredo [4]) and realzed volaly (see for example, Andersen e al. [4] and Barndorff-Nelsen and Sheppard [6]). However, negraed and dfferenaed dffson processes n he same sense as negraed and dfferenaed dscree-me processes are almos absen n appled economercs analyss. One of he reasons why connos-me dfferenaed processes have no been consdered n appled economercs s, perhaps, relaed o he dffcles n nerpreng he 'dfferenaed' process. In fac, f Z s a dffson process drven by a Brownan moon, hen all sample fncons are of nbonded varaon and nowhere dfferenable,.e. dz / d does no exs wh probably one (nless some smoohng effec of he measremen nsrmen s nrodced). One way o model negraed and dfferenaed dffson processes and overcome he dffcles assocaed wh he nondfferenably of he Brownan moon s hrogh he represenaon dy d d a d b dw () where a and b are he nfnesmal coeffcens (respecvely, he drf and he dffson coeffcen), W s a (sandard) Wener process (or Brownan moon) and s (by hypohess) a saonary process. In hs model, Y s a dfferenable process, by consrcon. I represens he negraed process, Y Y d ()

16 6 João Ncola (noe he analogy wh he correspondng expresson n a dscree-me seng, y y x, eqaon (9)) and dy / d s he saonary dfferenaed process (whch can be consdered he eqvalen concep o he frs dfferences seqence n dscree-me analyss). If represens he connosly componded rern or log rern of an asse, he frs eqaon n sysem () shold be rewren as d logy d. Ncola [3] arges ha () can be a sefl model n emprcal fnance for a leas wo reasons. Frs, he model accommodaes nonsaonary negraed sochasc processes ( Y ) ha can be made saonary by dfferencng. Sch ransformaon canno be done n common nvarae dffson processes sed n fnance (becase all sample pahs from nvarae dffson processes are nowhere dfferenable wh probably one). Ye, many processes n economcs and fnance (e.g. soc prces and nomnal exchange raes) behave as he cmlaon of all pas perrbaons (bascally n he same sense as n roo processes n a dscree framewor). Second, n he conex of soc prces or exchange raes, he model sggess drecly modellng he (nsananeos) rerns, conrary o sal connos-me models n fnance, whch drecly model he prces. General properes for rerns (sylzed facs) are well nown and docmened (for example, rerns are generally saonary n mean, he dsrbon s no normal, he aocorrelaons are wea and he correlaons beween he magnde of rerns are posve and sascally sgnfcan, ec.). One advanage of drecly modellng he rerns ( ) s ha hese general properes are easer o specfy n a model le () han n a dffson nvarae process for he prces. In fac, several neresng models can be obaned by selecng a x and b x appropraely. For example, he choce a x x and b x leads o an negraed process Y whose rerns,, have an asymmerc leporc saonary dsrbon (see he example below). Ths specfcaon can be appropraed n fnancal me seres daa. Bbby and Sørensen [8] had already noced ha a smlar process o () cold be a good model for soc prces. We observe ha he model defned n eqaon () can be wren as a second order SDE, d dy / d a d b dw. These nds of eqaons are common n engneerng. For nsance, s sal for engneers o model mechancal vbraons or charge on a capacor or condenser sbmed o whe nose excaon hrogh a second order sochasc dfferenal eqaon. Inegraed dffsons le Y n eqaon () arse narally when only observaons of a rnnng negral of he process are avalable. For nsance, hs can occr when a realzaon of he process s observed afer passage hrogh an elecronc fler. Anoher example s provded by ce-core daa on oxygen soopes sed o nvesgae paleoemperares (see Dlevsen and Sørensen []).

17 Fnancal economerc models 7 To llsrae connos-me negraed processes we presen n fgre 4 wo smlaed ndependen pahs of Y Y d where s governed by he sochasc dfferenal eqaon d. d..5 dw ( s also represened n fgre 4). All pahs are composed of observaons defned n he nerval,. I s neresng o observe ha Y dsplays all he feares of an negraed process (wh a posve drf, snce E. ): absence of mean reverson, shocs are perssen, mean and varance depend on me, ec. On he oher hand, he ncondonal dsrbon of (rern) s asymmerc and leporc A: Inegrae Process - Y me A: Dfferencaed Process me B: Inegrae Process - Y me B: Dfferencaed Process me Fgre 4 Smlaon of wo ndependen pahs from a second order SDE Esmaon of second order sochasc dfferenal eqaons rases new challenges for wo man reasons. On he one hand, only he negraed process Y s observable a nsans,,,... and hs n model () s a laen non-observable process. In fac, for a fxed samplng nerval, s mpossble o oban he vale of a me from he observaon Y whch represens he negral Y d. On he oher hand, he esmaon of model () canno n prncple be based on he observaons Y,,,... snce he condonal dsrbon of Y s generally nnown, even f ha of s nown. An excepon s he case where follows an Orsen-Uhlenbec process, whch s analyzed n Gloer [6]. However, wh dscree-me observaons Y,,,... (o smplfy we se he noaon, where ), and gven ha

18 8 João Ncola Y Y d d d, we can oban a measre of a nsan sng he formla: Y Y ~. () Narally, he accracy of () as a proxy for depends on he magnde of. Regardless of he magnde of we have n or sample, we shold base or esmaon procedres on he sample ~,,,... snce s no observable. Paramerc and sem-paramerc esmaon of negraed dffsons s analyzed n Gloer [5, 6] and Dlevsen and Sørensen []. In Ncola [3] s spposed ha boh nfnesmal coeffcens a and b, are nnown. Non-paramerc esmaors for he nfnesmal coeffcens a and b are proposed. The analyss reveals ha he sandard esmaors based on he sample ~,,,... are nconssen even f we allow he sep of dscrezaon o go o zero asympocally. Inrodcng slgh modfcaons o hese esmaors we provde conssen esmaors. See also []. ACKNOWLEDGEMENTS I wold le o han Tom Knder for helpfl commens. Ths research was sppored by he Fndação para a Cênca e a Tecnologa (FCT) and by POCTI. REFERENCES. Aï-Sahala, Y. (996), Tesng Connos-Tme Models of he Spo Ineres Rae, The Revew of Fnancal Sdes 9, Aï-Sahala, Y. (999), Transon Denses for Ineres Rae and Oher Nonlnear Dffsons, The Jornal of Fnance LIV, Aï-Sahala, Y. (), Maxmm Lelhood Esmaon of Dscreely Sampled Dffsons: a Closed-Form Approxmaon Approach, Economerca 7, Andersen T. & T. Bollerslev & F. Debold & P. Labys () The Dsrbon of Exchange Rae Volaly. Jornal of he Amercan Sascal Assocaon 96, Band, F. (), Shor-Term Ineres Rae Dynamcs: A Spaal Approach, Jornal of Fnancal Economcs 65, Barndorff-Nelsen, O. & N. Sheppard () Economerc Analyss of Realzed Volaly and s se n Esmang Sochasc Volaly Models. Jornal of he Royal Sascal Socey B 64, 53-8.

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