Ecoometrica, Vol. 7, No. 1 (Jauary, 22), MAXIMUM LIKELIHOODESTIMATION OF DISCRETEL SAMPLED DIFFUSIONS: A CLOSED-FORM APPROXIMATION APPROACH By acie Aït-Sahalia 1 Whe a cotiuous-time diffusio is

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Ecoometrica, Vol. 7, No. 1 (Jauary, 22), MAXIMUM LIKELIHOODESTIMATION OF DISCRETEL SAMPLED DIFFUSIONS: A CLOSED-FORM APPROXIMATION APPROACH By acie Aït-Sahalia 1 Whe a cotiuous-time diffusio is observed oly at discrete dates, i most cases the trasitio distributio ad hece the likelihood fuctio of the observatios is ot explicitly computable. Usig Hermite polyomials, I costruct a explicit sequece of closed-form fuctios ad show that it coverges to the true (but ukow) likelihood fuctio. I documet that the approximatio is very accurate ad prove that maximizig the sequece results i a estimator that coverges to the true maximum likelihood estimator ad shares its asymptotic properties. Mote Carlo evidece reveals that this method outperforms other approximatio schemes i situatios relevat for fiacial models. Keywords: Maximum-likelihood estimatio, cotiuous-time diffusio, discrete samplig, trasitio desity, Hermite expasio. 1 itroductio Cosider a cotiuous-time parametric diffusio (1.1) dx t = X t dt + X t dw t where X t is the state variable, W t a stadard Browia motio, ad are kow fuctios, ad a ukow parameter vector i a ope bouded set R K. Diffusio processes are widely used i fiacial models, for istace to represet the stochastic dyamics of asset returs, exchage rates, iterest rates, macroecoomic factors, etc. While the model is writte i cotiuous time, the available data are always sampled discretely i time. Igorig the differece ca result i icosistet estimators (see, e.g., Merto (198) ad Melio (1994)). A umber of ecoometric methods have bee recetly developed to estimate the parameters of (1.1), without requirig that a cotiuous record of observatios be available. Some of these methods are based o simulatios (Gouriéroux, Mofort, ad Reault (1993), Gallat ad Tauche (1996)), others o the geeralized method of momets (Hase ad Scheikma (1995), Duffie ad Gly (1997), Kessler ad Sorese 1 I am grateful to David Bates, Reé Carmoa, Freddy Delbae, Ro Gallat, Lars Hase, Bjarke Jese, Per Myklad, Peter C. B. Phillips, Rolf Poulse, Peter Robiso, Chris Rogers, Agel Serrat, Chris Sims, George Tauche, ad i particular a co-editor ad three aoymous referees for very helpful commets ad suggestios. Robert Kimmel ad Erst Schaumburg provided excellet research assistace. This research was supported by a Alfred P. Sloa Research Fellowship ad by the NSF uder Grat SBR Mathematica code to calculate the closed-form desity sequece ca be foud at yacie. 223 224 yacie aït-sahalia (1999)), oparametric desity-matchig (Aït-Sahalia (1996a, 1996b)), oparametric regressio for approximate momets (Stato (1997)), or are Bayesia (Eraker (1997) ad Joes (1997)). As i most cotexts, provided oe trusts the parametric specificatio (1.1), maximum-likelihood is the method of choice. The major caveat i the preset cotext is that the likelihood fuctio for discrete observatios geerated by (1.1) caot be determied explicitly for most models. Let p X x x deote the coditioal desity of X t+ = x give X t = x iduced by the model (1.1), also called the trasitio fuctio. Assume that we observe the process at dates t = i i =, where is fixed. 2 Bayes rule combied with the Markovia ature of (1.1), which the discrete data iherit, imply that the log-likelihood fuctio has the simple form (1.2) l l { p X X i X i 1 i=1 For some of the rare exceptios where p X is available i closed-form, see Wog (1964); i fiace, the models of Black ad Scholes (1973), Vasicek (1977), Cox, Igersoll, ad Ross (1985), ad Cox (1975) all rely o the kow closed-form expressios. If samplig of the process were cotiuous, the situatio would be simpler. First, the likelihood fuctio for a cotiuous record ca be obtaied by meas of a classical absolutely cotiuous chage of measure (see, e.g., Basawa ad Prakasa Rao (198)). 3 Secod, whe the samplig iterval goes to zero, expasios for the trasitio fuctio i small time are available i the statistical literature (see, e.g., Azecott (1981)). Dacuha-Castelle ad Flores-Zmirou (1986) calculate expressios for the trasitio fuctio i terms of fuctioals of a Browia Bridge. With discrete-time samplig, the available methods to compute the likelihood fuctio ivolve either solvig umerically the Fokker- Plack-Kolmogorov partial differetial equatio (see, e.g., Lo (1988)) or simulatig a large umber of sample paths alog which the process is sampled very fiely (see Pederse (1995) ad Sata-Clara (1995)). Neither method produces a closed-form expressio to be maximized over : the criterio fuctio takes either the form of a implicit solutio to a partial differetial equatio, or a sum over the outcome of the simulatios. By cotrast, I costruct a closed-form sequece p J X of approximatios to the trasitio desity, hece from (1.2) a sequece l J of approximatios to the loglikelihood fuctio l. I also provide empirical evidece that J = 2 or 3 is amply adequate for models that are relevat i fiace. 4 Sice the expressio 2 See Sectio 3.1 for extesios to the cases where the samplig iterval is time-varyig ad eve possibly radom. 3 Note that the cotiuous-observatio likelihood is oly defied if the diffusio fuctio is kow. 4 I additio, Jese ad Poulse (1999) have recetly completed a compariso of the method of this paper agaist four alteratives: a discrete Euler approximatio of the cotiuous-time model maximum likelihood estimatio 225 Notes: This figure reports the average uiform absolute error of various desity approximatio techiques applied to the Vasicek, Cox-Igersoll-Ross ad Black-Scholes models. Euler refers to the discrete-time, cotiuous-state, first-order Gaussia approximatio scheme for the trasitio desity give i equatio (5.4); Biomial Tree refers to the discrete-time, discrete-state (two) approximatio; Simulatios refers to a implemetatio of Pederse (1995) s simulated-likelihood method; PDE refers to the umerical solutio of the Fokker-Plack-Kolmogorov partial differetial equatio satisfied by the trasitio desity, usig the Crak- Nicolso algorithm. For implemetatio details o the differet methods cosidered, see Jese ad Poulse (1999). Figure 1. Accuracy ad speed of differet approximatio methods for p X. to be maximized is explicit, the effort ivolved is miimal, idetical to a stadard maximum-likelihood problem with a kow likelihood fuctio. Examples are cotaied i a compaio paper (Aït-Sahalia (1999)), which provides, for differet models, the correspodig expressio of p J X. Besides makig maximum-likelihood estimatio feasible, these closed-form approximatios have other applicatios i fiacial ecoometrics. For istace, they could be used for derivative pricig, for idirect iferece (see Gouriéroux, Mofort, ad Reault (1993)), which i its simplest versio uses a Euler approximatio as istrumetal model, or for Bayesia iferece basically wheever a expressio for the trasitio desity is required. The paper is orgaized as follows. Sectio 2 describes the sequece of desity approximatios ad proves its covergece. Sectio 3 studies the properties of the resultig maximum-likelihood estimator. I Sectio 4, I show how to calculate i closed-form the coefficiets of the approximatio ad readers primarily iterested i applyig these results to a specific model ca go there directly. l J (1.1), a biomial tree approximatio, the umerical solutio of the PDE, ad simulatio-based methods, all i the cotext of various specificatios ad parameter values that are relevat for iterest rate ad stock retur models. To give a idea of the relative accuracy ad speed of these approximatios, Figure 1 summarizes their mai results. As is clear from the figure, the approximatio of the trasitio fuctio derived here provides a degree of accuracy ad speed that is umatched by ay of the other methods. 226 yacie aït-sahalia Sectio 5 gives the results of Mote Carlo simulatios. Sectio 6 cocludes. All proofs are i the Appedix. 2 a sequece of expasios of the trasitio fuctio To uderstad the costructio of the sequece of approximatios to p X, the followig aalogy may be helpful. Cosider a stadardized sum of radom variables to which the Cetral Limit Theorem (CLT) apply. Ofte, oe is willig to approximate the actual sample size by ifiity ad use the N 1 limitig distributio for the properly stadardized trasformatio of the data. If ot, higher order terms of the limitig distributio (for example the classical Edgeworth expasio based o Hermite polyomials) ca be calculated to improve the small sample performace of the approximatio. The basic idea of this paper is to create a aalogy betwee this situatio ad that of approximatig the trasitio desity of a diffusio. Thik of the samplig iterval as playig the role of the sample size i the CLT. If we properly stadardize the data, the we ca fid out the limitig distributio of the stadardized data as teds to (by aalogy with what happes i the CLT whe teds to ifiity). Properly stadardizig the data i the CLT meas summig them ad dividig by 1/2 ; here it will ivolve trasformig the origial diffusio X ito aother oe, which I call Z below. I both cases, the appropriate stadardizatio makes N 1 the leadig term. I will the refie this N 1 approximatio by correctig for the fact that is ot (just as i practical applicatios of the CLT is ot ifiity), i.e., by computig the higher order terms. As i the CLT case, it is atural to cosider higher order terms based o Hermite polyomials, which are orthogoal with respect to the leadig N 1 term. But i what sese does such a expasio coverge? I the CLT case, the covergece is uderstood to mea that the series with a fixed umber of corrective terms (i.e., fixed J ) coverges whe the sample size goes to ifiity. I fact, for a fixed, the Edgeworth expasio will typically diverge as more ad more corrective terms are added, uless the desity of each of these radom variables was close to a Normal desity to start with. I will make this statemet precise later, usig the criterio of Cramér (1925): the desity p z to be expaded aroud a N 1 must have tails sufficietly thi for exp z 2 /2 p z 2 to be itegrable. The poit however is that the desity p X caot i geeral be approximated for fixed aroud a Normal desity, because the distributio of the diffusio X is i geeral too far from that of a Normal. For istace, if X follows a geometric Browia motio, the right tail of the correspodig log-ormal desity p X is too large for its Hermite expasio to coverge. Ideed, that tail is of order x 1 exp l 2 x as x teds to +. Similarly, the expasio of ay N v desity aroud a N 1 diverges if v 2, ad hece the class of trasitio desities p X for which straight Hermite expasios coverge i the sese of addig more terms (J icreases with fixed) is quite limited. maximum likelihood estimatio 227 To obtai evertheless a expasio that coverges as more correctio terms are added while remais fixed, I will show that the trasformatio of the diffusio process X ito Z i fact guaratees (ulike the CLT situatio) that the resultig variable Z has a desity p Z that belogs to the class of desities for which the Hermite series coverges as more polyomial terms are added. I will the costruct a coverget Hermite series for p Z. Sice Z is a kow trasformatio of X, I will be able to revert the trasformatio from X to Z ad by the Jacobia formula obtai a expasio for the desity of X. As a result of trasformig Z back ito X, which i geeral is a oliear trasformatio (uless x is idepedet of the state variable x), the leadig term of the expasio for the desity p X will be a deformed, or stretched, Normal desity rather tha the N 1 leadig term of the expasio for p Z. The rest of this sectio makes this basic ituitio rigorous. I particular, Theorem 1 will prove that such a expasio coverges uiformly to the ukow p X. 2 1 Assumptios ad First Trasformatio X I start by makig fairly geeral assumptios o the fuctios ad. I particular, I do ot assume that ad satisfy the typical growth coditios at ifiity, or do I restrict attetio to statioary diffusios oly. Let D X = x x deote the domai of the diffusio X. I will cosider the two cases where D X = + ad D X = +. The latter case is ofte relevat i fiace, whe cosiderig models for asset prices or omial iterest rates. I additio, the fuctio is ofte specified i fiacial models i such a way that lim x + x = ad ad/or violate the liear growth coditios ear the boudaries. For these reasos, I will devise a set of assumptios where growth coditios (without costrait o the sig of the drift fuctio ear the boudaries) are replaced by assumptios o the sig of the drift ear the boudaries (without restrictio o the growth of the coefficiets). The assumptios are: Assumptio 1 (Smoothess of the Coefficiets): The fuctios x ad x are ifiitely differetiable i x, ad three times cotiuously differetiable i, for all x D X ad. Assumptio 2 (No-Degeeracy of the Diffusio): 1. If D X = +, there exists a costat c such that x c for all x D X ad. 2. If D X = + lim x + x = is possible, but the there exist costats such that x x for all x ad. Whether or ot lim x + x =, is a odegeerate o +, that is: for each , there exists a costat c such that x c for all x + ad. The first step I employ towards costructig the sequece of approximatios to p X cosists i stadardizig the diffusio fuctio of X, i.e., trasformig X 228 yacie aït-sahalia ito defied as 5 (2.1) X X = du/ u where ay primitive of the fuctio 1/ may be selected, i.e., the costat of itegratio is irrelevat. Because od X, the fuctio is icreasig ad ivertible for all. It maps D X ito D = y ȳ, the domai of, where y lim x x + x ad ȳ lim x x x. For example, if D X = + ad x = x, the = 1 X 1 if 1 (so D = +, = l X if = 1 (so D = + ad = 1 X 1 if 1 (so D =. For a give model uder cosideratio, assume that the parameter space is restricted i such a way that D is idepedet of i. This restrictio o is iessetial, but it helps keep the otatio simple. By applyig Itô s Lemma, has uit diffusio, that is (2.2) d t = t dt + dw t where y = 1 y 1 y 1 2 x 1 y Assumptio 3 (Boudary Behavior): For all y ad its derivatives with respect to y ad have at most polyomial growth 6 ear the boudaries ad lim y y + or y y + where is the potetial, i.e., y 2 y + y / y /2. 1. Left Boudary: If y =, there exist costats such that for all y ad y y where either 1 ad , or = 1 ad 1. Ify =, there exist costats E ad K such that for all y E ad y Ky. 2. Right Boudary: If ȳ =+, there exist costats E ad K such that for all y E ad y Ky. Ifȳ =, there exist costats such that for all y ad y y where either 1 ad or = 1 ad 1/2. Note that is ot restricted from goig to ear the boudaries. Assumptio 3 is formulated i terms of the fuctio for reasos of coveiece, but the restrictio it imposes o the origial fuctios ad follows from (2.1). Assumptio 3 oly restricts how large ca grow if it has the wrog sig, meaig that is positive ear y ad egative ear y: the liear growth is the maximum possible rate. But if has the right sig, the process is beig pulled 5 The same trasformatio, sometimes referred to as the Lamperti trasform, has bee used, for istace, by Flores (1999). 6 Defie a ifiitely differetiable fuctio f as havig at most polyomial growth if there exists a iteger p such that y p f y is bouded above i a eighborhood of ifiity. If p = 1 f is said to have at most liear growth, ad if p = 2 at most quadratic growth. Near, polyomial growth meas that y +p f y is bouded. maximum likelihood estimatio 229 back away from the boudaries ad I do ot restrict how fast mea-reversio occurs (up to a arbitrary large polyomial rate for techical reasos). The costraits o the behavior of the fuctio are essetially the best possible for the followig reasos. If has the wrog sig ear a ifiity boudary, ad grows faster tha liearly, the explodes (i.e., ca reach the ifiity boudary) i fiite time. Near a zero boudary, say y =, if there exist ad 1 such that y ky i a eighborhood of +, the becomes attaiable. The behavior of the diffusio implied by the assumptios made is fully characterized by the followig propositio, where T if t t D = y ȳ deotes the exit time from D : Propositio 1: Uder Assumptios 1 3, (2.2) admits a weak solutio t t, uique i probability law, for every distributio of its iitial value. 7 The boudaries of D are uattaiable, i the sese that Prob T = = 1. Fially, if + is a right boudary, the it is atural if, ear + y Ky ad etrace if y Ky for some 1. If is a left boudary, the it is atural if, ear y K y ad etrace if y K y for some 1. If is a boudary (either right or left), the it is etrace. 8 Note also that Assumptio 3 either requires or implies that the process is statioary. Whe both boudaries of the domai D are etrace boudaries, the the process is ecessarily statioary with commo ucoditioal (margial) desity for all t { y / ȳ { v (2.3) y exp 2 u du exp 2 u du dv y provided that the iitial radom variable is itself distributed with desity (2.3) (see, e.g., Karli ad Taylor (1981)). Whe at least oe of the boudaries is atural, statioarity is either precluded or implied i that the (oly) possible cadidate for statioary desity,, may or may ot be itegrable ear 7 A weak solutio to (2.2) i the iterval D is a pair W, a probability space ad a filtratio, such that W satisfies the stochastic itegral equatio that uderlies the stochastic differetial equatio (2.2). For a formal defiitio, see, e.g., Karatzas ad Shreve (1991, Defiitio 5.5.2). Uiqueess i law meas that two solutios would have idetical fiite-dimesioal distributios, i.e., i particular the same observable implicatios for ay discrete-time data. From the perspective of statistical iferece from discrete observatios, this is therefore the appropriate cocept of uiqueess. 8 Natural boudaries ca either be reached i fiite time, or ca the diffusio be started or escape from there. Etrace boudaries, such as, caot be reached startig from a iterior poit i D = +, but it is possible for to begi there. I that case, the process moves quickly away from ad ever returs there. Typically, ecoomic cosideratios require the boudaries to be uattaiable; however, they say little about how the process would behave if it were to start at the boudary, or whether that is eve possible, ad hece it is sesible to allow both types of boudary behavior. 23 yacie aït-sahalia the boudaries. 9 Next, I show that the diffusio admits a smooth trasitio desity: Propositio 2: Uder Assumptios 1 3, admits a trasitio desity p y y that is cotiuously differetiable i , ifiitely differetiable i y D ad y D, ad three times cotiuously differetiable i. Furthermore, there exists such that for every, there exist positive costats C i i= 4, ad D such that for every ad y y D 2 : p y y C 1/2 e 3 y y 2 /8 e C 1 y y y +C 2 y y +C 3 y +C 4 y 2 (2.4) p y y / y (2.5) D 1/2 e 3 y y 2 /8 P y y e C 1 y y y +C 2 y y +C 3 y +C 4 y 2 where P is a polyomial of fiite order i y y, with coefficiets uiformly bouded i. Fially, if ear the right boudary + ad ear the left boudary (either or ), the =+. The ext result shows that these properties essetially exted to the diffusio X of origial iterest. Corollary 1: Uder Assumptios 1 3, (1.1) admits a weak solutio X t t, uique i probability law, for every distributio of its iitial value X. The boudaries of D X are uattaiable, i the sese that Prob T X = = 1 where T X if t X t D X. I additio, X admits a trasitio desity p X x x which is cotiuously differetiable i , ifiitely differetiable i x D X ad x D X, ad three times cotiuously differetiable i. 2 2 Secod Trasformatio Z The boud (2.4) implies that the tails of p have a Gaussia-like upper boud. I light of the discussio at the begiig of Sectio 2 about the requiremets for covergece of a Hermite series, this is a big step forward. However, while, thaks to its uit diffusio = 1, is closer to a Normal variable tha X is, it is ot practical to expad p. This is due to the fact that p gets peaked aroud the coditioal value y whe gets small. Ad a Dirac mass is ot a particularly appealig leadig term for a expasio. For that reaso, I perform a further trasformatio. For give , ad y R, defie the pseudoormalized icremet of as (2.6) Z 1/2 y 9 For istace, both a Orstei-Uhlebeck process, where y = y, ad a Browia motio, where y =, satisfy th
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