Itô Conditional Moment Generator and the Estimation of Short Rate Processes

It^o Conditional Moment Generator and the Estimation of Short Rate Processes1 Hao Zhou Mail Stop 91 Federal Reserve Board Washington, DC 20551 First Draft: January 1998 This Version: March 2003 1 ThispapergrowsoutofanessayofmyPhDdissertation,andispreviouslydistributedunderthe title \Jump-Di(cid:11)usion Term Structure and It^o Conditional Moment Generator." George Tauchen gave me valuable advice. I thank Ravi Bansal, David Bates, Tim Bollerslev, Peter Christo(cid:11)ersen, Sanjiv Das, Lars Hansen, Michael Hemler, Nour Meddahi, and Kenneth Singleton for their helpful suggestions. I am also grateful to the editor Ren(cid:19)e Garcia, the associate editor, and two anonymous referees for their constructive suggestions. Comments from the seminar participants at Duke, Brown, Virginia economics departments, SNDE 1999, FMA 1999, ES 2000 annual meetings, and Duke Risk Conference 2000 are greatly appreciated. The views presented here are solely those of the author and do not necessarily represent those of the Federal Reserve Board or its sta(cid:11). For questions and comments, please contact Hao Zhou, Trading Risk Analysis Section, Division of ResearchandStatistics, FederalReserveBoard,WashingtonDC20551USA;Phone1-202-452-3360; Fax 1-202-728-5887; e-mail hao.zhou@frb.gov.

Abstract This paper exploits the Ito^’s formula to derive the conditional moments vector for the class of interest rate models that allow for nonlinear volatility and flexible jump speci(cid:12)cations. Such a characterization of continuous-time processes by the Ito^ Conditional Moment Generator noticeably enlarges the admissible set beyond the a(cid:14)ne jump-di(cid:11)usion class. A simple GMM estimator can be constructed based on the analytical solution to the lower order moments, with natural diagnostics of the conditional mean, variance, skewness, and kurtosis. Monte Carlo evidence suggests that the proposed estimator has desirable (cid:12)nite sample properties, relative to the asymptotically e(cid:14)cient MLE. The empirical application singles out the nonlinear quadratic variance as the key feature of the U.S. short rate dynamics. Keywords: Ito^ Conditional Moment Generator, Short Term Interest Rate, Jump-Di(cid:11)usion Process, Quadratic Variance, Generalized Method of Moments, Monte Carlo Study. JEL classi(cid:12)cation: C51, C52, G12

1 Introduction In modeling the short-term interest rate, researchers face the challenge of accommodating all relevant features in a single model speci(cid:12)cation. Those features, include but are not limited to, (1) short-term persistence, (2) long-run mean reversion, (3) nonlinear state-dependence in volatility, (4) non-Gaussian features in skewness and kurtosis. The celebrated CIR model (Cox et al., 1985) and its various extensions, although appealing in their general equilibrium nature and closed-form solution, have di(cid:14)culty in (cid:12)tting all these features simultaneously for the US interest rate data (Brown and Dybvig, 1986).1 Rigorous speci(cid:12)cation tests using historical data tend to reject the square-root model (A¨(cid:16)t-Sahalia, 1996b; Conley et al., 1997; Gallant and Tauchen, 1998). Although having an inherent advantage in (cid:12)tting features (1) and (2) as indicated in the literature, the CIR-type model fails to capture the rich volatility dynamics and the nonlinear non-Gaussian features. Consequently, e(cid:11)ortstomodifythesquare-rootmodellargelyconcentrateonmoreflexible speci(cid:12)cations ofthevolatilitydynamics. ItisclearthattheCIR modelisjustonespecialcase of so-called linear CEV (constant elasticity of volatility) speci(cid:12)cation, where the elasticity equals one half. Recent comparative studies (Chan et al., 1992;Conley et al., 1997;Tauchen, 1997; Christo(cid:11)ersen and Diebold, 2000) found that an elasticity around one and one-half is more desirable. Alternatively, one can estimate the volatility function nonparametrically (A¨(cid:16)t-Sahalia, 1996a; Stanton, 1997; Jiang and Knight, 1997; Jiang, 1998; Bandi, 2002; Bandi andPhillips, 2003). The empirical (cid:12)ndings along this line suggest that the square-root model (cid:12)ts reasonably well for the medium range of interest rates, but the estimated nonlinearity at both the high and low ends is neither accurate nor conclusive. A pertinent approach is to introduce an unobserved stochastic volatility factor into the di(cid:11)usion function, which (cid:12)nds considerable support in empirical studies (Andersen and Lund, 1996, 1997).2 The jumpdi(cid:11)usion approach to interest-rate modeling (and bond pricing exercise) is of more recent 1The bivariate extensions of CIR speci(cid:12)cation (Gibbons and Ramaswamy, 1993; Chen and Scott, 1993; Pearson and Sun, 1994) also meet with poor empirical performance. Du(cid:14)e and Singleton (1997) found favorable evidence for a two-factor CIR model with serially-correlated error structure. Dai and Singleton (2000) estimated more flexible three-factor a(cid:14)ne speci(cid:12)cations similar to Chen (1996) and Balduzzi et al. (1996) for interest rate swap data after 1987. 2Thispaperfocusesonthemaximumflexibilityintheunivariatesetting,andtheextensiontomultivariate or stochastic volatility is deferred to future research. 1

origin (Baz and Das, 1996; Das, 1998), and its general equilibrium formulation is explored by Ahn and Thompson (1988).3 Theinnovationofthispaperistogeneratetheparametricconditionalmomentsonlyusing the Ito^’s formula and to construct a computationally e(cid:14)cient GMM estimator. Maximum Likelihood Estimation (MLE) is available only for a very restricted class of jump-di(cid:11)usion models (Lo, 1988). Our method di(cid:11)ers with the in(cid:12)nitesimal generator of Hansen and Scheinkman (1995) (GMM) in that it fully exploits the conditional information, does not relyonsimulationsasdoDu(cid:14)eandSingleton(1993)(SMM),usesmodel-dependentmoments instead of data-dependent moments (Gallant and Tauchen, 1996) (EMM), generalizes to an arbitrary number of moments rather than only to conditional mean and variance (Fisher and Gilles, 1996) (QML), and has reliable small sample properties in comparison with the nonparametric approach (A¨(cid:16)t-Sahalia, 1996a) (NP). As shown below, our method reduces a complicated task of solving a stochastic di(cid:11)erential equation (SDE) to a simple matrix solution of an ordinary di(cid:11)erential equation (ODE) system. The computational burden is reduced to a minimum of elementary algebra.4 Another important advantage is that the characterization of short rate processes by the Ito^’s approach allows for nonlinear volatility and semiparametric jump speci(cid:12)cations. Within the univariate paradigm, nonlinearity is indispensable to successful modeling of the U.S. short term interest rate. In the literature, the most closely related method is to identify the stochastic di(cid:11)erential equations with an orthogonal series representation (Hansen et al., 1998), which is attributed to the generalized eigenvalue-eigenfunction technique (Wong, 1964). We further justify the aforementioned methodology by a numerical exercise and illustrate by an empirical application. Monte Carlo evidence suggests that the (cid:12)nite sample e(cid:14)ciency of the proposed GMM estimator is comparable to the asymptotically e(cid:14)cient MLE, the 3Recently there is a growing literature on jump-di(cid:11)usion interest rate modeling (see Chacko and Das, 1999;Johannes, 1999;Piazzesi,2000,among many others), which ranges from short-ratedynamics to (cid:12)xedincome derivatives, from market-implied jumps to macroeconomic announcements, and from parametric to nonparametric speci(cid:12)cations. 4Alternatively, an equivalent spectral method of moments is developed by exploiting the closed-form conditionalcharacteristicfunctionsforthea(cid:14)nejump-di(cid:11)usionmodel(ChackoandViceira,1999;Singleton, 2001;JiangandKnight,2002;Carrascoetal.,2002). However,theselectionofspectralmomentsremainsasa di(cid:14)cultchallenge,whereasintheclassicalmethodofmoments,anaturalchoiceis thelower-ordermoments. Moreover, a strategy to derive moments using the Ito^’s formula alone and not relying on the characteristic function or moment generating function may be more desirable for certain non-standard processes,e.g., the quadratic variance model discussed in this paper. 2

sampling t-statistics of individual parameter is not far away from the normal reference distribution, and the GMM test of over-identifying restrictions has a typical upward bias but with reasonable magnitude. When applied to the U.S. short term interest rate from 1954 to 2002, both the square-root model and the restricted CEV model are rejected outright. Adding jumps shows some improvement but only the quadratic variance model cannot be rejected atthe1%signi(cid:12)cance level. U-shapedvolatility andnonlinear higher ordermoments seem to be the main challenges of (cid:12)tting the U.S. short rate dynamics, in additional to the well-known linear mean persistence. The remainder of this paper is organized as follows: Section 2 derives the conditional moments for an admissible class of processes including the square-root, the restricted CEV, the jump-di(cid:11)usion, and the quadratic variance; Section 3 builds an easy-to-implement GMM estimator and provides some (cid:12)nite sample evidence; Section 4 applies the estimating procedure to the four models mentioned above and contrasts the speci(cid:12)cation di(cid:11)erences using the conditional moment pro(cid:12)les; and Section 5 concludes. 2 Ito^ Conditional Moment Generator This section outlines a strategy to derive the conditional moments simultaneously for certain continuous-time processes, relying only on the Ito^’s formula and the speci(cid:12)cations of drift, di(cid:11)usion, and jump functions. The resulting characterization not only nests the popular a(cid:14)nejump-di(cid:11)usionclass, butalsofeaturesnonlinearquadraticvarianceandsemiparametric flexible jumps. 2.1 A General Characterization of Admissible Processes Supposethattheevolution ofthestatevariable(i.e., theshortrate)isgoverned byareducedform jump-di(cid:11)usion process dr = (cid:22) dt+(cid:27) dW +J dN((cid:26) t); (1) t t t t t t where W is a standard Brownian motion, N((cid:26) t) is a Poisson driving process with an intent t sity function (cid:26) , and J is the jump size with distribution (cid:5)(J ). Note that both the jump t t t rate and jump size are allowed to be state-dependent but conditionally independent of each 3

other and with respect to the Brownian motion. Process (1) must satisfy certain regularity conditions and the critical ones are: (a) both (cid:22) and (cid:27) are Lipschitz continuous, (b) (cid:26) and t t t (cid:5)(J ) are F measurable. t t − The strategy is to solve all the conditional moments up to the K’th order simultaneously, by (cid:12)rst applying the Generalized It^o’s lemma (Merton, 1971; Lo, 1988) to each rk for k = T 1;2;(cid:1)(cid:1)(cid:1);K, and then take the conditional expectation " Z (cid:18) (cid:19) # T 1 E (r k ) = r k +E (cid:22) kr k−1 + (cid:27) 2 k(k −1)r k−2 +(cid:26) E [(r +J ) k −r k ] du (2) t T t t u u u u u J u u u t 2 Interchanging the expectation and integration operators, and taking the derivative with respect to time T, we arrive at a di(cid:11)erential equation system " # dE (rk) 1 Xk (cid:16) (cid:17) t s = E (cid:22) kr k−1 + (cid:27) 2 k(k −1)r k−2 +(cid:26) k r k−i E (J i ) : (3) t s s s s s i s J s ds 2 i=1 with boundary condition E (rk) = rk. The following proposition characterizes the class of t t t jump-di(cid:11)usion processes that sustain a closed-form solution to equation (3), Proposition 1 (Characterization) The su(cid:14)cient condition for the K-dimensional ordinary di(cid:11)erential equation system (3) to have a (cid:12)rst order linear solution, is to restrict the drift, di(cid:11)usion, and jump functions in the following forms (1) (cid:22) = (cid:20)((cid:18) −r ); and t t q (2) (cid:27) = (cid:27) +(cid:27) r +(cid:27) r2; and t 0 1 t 2 t P (3) (cid:26) E (Jk) = k J r j . t J t j=0 kj t Many linear or nonlinear restrictions need to be imposed to ensure existence and identi(cid:12)cation, for example, the sign constraints on (cid:20);(cid:18);(cid:27) ;(cid:27) ;(cid:27) and the zero constraints on some 0 1 2 J . The proof only involves a straightforward veri(cid:12)cation, hence omitted.5 kj For the admissible process under Proposition 1, the K-vector of its conditional moments E (R ) = [E (r );E (r2);(cid:1)(cid:1)(cid:1);E (rK)]0 is characterized by a linear di(cid:11)erential equation syst s t s t s t s tem, dE (R ) t s = A((cid:12))E (R )+g((cid:12)); (4) t s ds 5ThenecessaryconditionalfortheK-dimhensionalordinarydi(cid:11)erentialequa P tionsy (cid:0) st (cid:1) em(3)tohaivea(cid:12)rst order linear solution, is to require the term (cid:22) s kr s k−1+ 1 2 (cid:27) s 2k(k−1)r s k−2+(cid:26) s k i=1 k i r s k−iE J(J s i) to be a k’th order polynomial of r s, which is trivial and not as informative as the su(cid:14)cient condition. 4

where A((cid:1)) is a K (cid:2) K lower-triangular matrix and g((cid:1)) is a K (cid:2) 1 vector. Both A((cid:1)) and g((cid:1)) are nonlinear functions of the parameter vector (cid:12) = [(cid:20);(cid:18);(cid:27) ;(cid:27) ;(cid:27) ;J ;(cid:1)(cid:1)(cid:1);J ]0, 0 1 2 10 KK de(cid:12)ned by the underlying the jump-di(cid:11)usion process (1). Since the coe(cid:14)cients of such a non-homogeneous linear (cid:12)rst-order di(cid:11)erential equation do not depend on time, one obtains the following closed-form solution, (cid:16) (cid:17) E (R ) = e (T−t)A((cid:12)) R +A −1 ((cid:12)) e (T−t)A((cid:12)) −I g((cid:12)); (5) t T t where I is the K (cid:2)K identity matrix and e((cid:1)) denotes the matrix exponential. There are some advantages in using the \Ito^ Transformation" to generate the conditional moments. From the perspective of richer dynamics, although the drift function has to be restricted as linear, the di(cid:11)usion function can be nonlinear, and the jump function only requires the speci(cid:12)cation of its moments. More detailed examples are examined in the next subsection to illustrate the enhanced flexibility of such an Ito^ characterization. From the perspective of easier implementation, the calculation of moments in a typical matrix programming language remains a one-line code as equation (5), and the computation of each entry of A((cid:1)) and g((cid:1)) in equation (3) does not require di(cid:11)erentiation; whereas using the conditional moment generating function involves messy high order derivatives. Once computed, a moment-based estimator (like GMM) is readily available, while a likelihood-based method requires the Fourier inversion of the characteristic function. It is also possible to apply the Ito^ transformation to processes that lack analytical solution to the moment generating function. The major disadvantage of relying on a potentially limited set of moments, is the possible loss of estimation e(cid:14)ciency relative to MLE. To address this concern, the next section designs a GMM estimator and quanti(cid:12)es its adequate (cid:12)nite sample performance. 2.2 Leading Empirical Examples To illustrate the applicability of the proposed methodology, here we present several speci(cid:12)cations that are useful to model the short term interest rate. Only the solutions to the (cid:12)rst four moments are spelled out, as the higher order moments are trivial extensions. 5

2.2.1 Flexible Jump-Di(cid:11)usion Process We start with a simple jump-di(cid:11)usion process p dr = (cid:20)((cid:18)−r )dt+(cid:27) r dW +J dN((cid:26) t) (6) t t t t t t where (cid:26) = (cid:26) and J is speci(cid:12)ed by its four moments. Although the di(cid:11)usion part of this t t model is a(cid:14)ne, the state variable may not be a(cid:14)ne if the jump-size moments are statedependent as in Proposition 1. The solution to its (cid:12)rst four conditional moments in the form of equation (5), can be characterized by the matrix A((cid:12)) 2 3 −(cid:20)+(cid:26)E(J) 0 0 0 6 7 6 P 7 6 2(cid:20)(cid:18)+(cid:27)2 −2(cid:20)+(cid:26) 2 (2)E(Ji) 0 0 7 6 i=1 i 7 6 P 7 6 0 3(cid:20)(cid:18)+3(cid:27)2 −3(cid:20)+(cid:26) 3 (3)E(Ji) 0 7 4 i=1 i 5 P 0 0 4(cid:20)(cid:18) +6(cid:27)2 −4(cid:20)+(cid:26) 4 (4)E(Ji) i=1 i and the vector g((cid:12)) 2 3 (cid:20)(cid:18) 6 7 6 7 6 0 7 6 7 6 7 6 0 7 4 5 0 If we specialize to the case of uniform distribution (J (cid:24) U[ar ;br ]), the moments of the t t t jump-size are, respectively, E(J) = (b2−a2) r , E(J2) = (b3−a3) r2, E(J3) = (b4−a4) r3, and 2(b−a) t 3(b−a) t 4(b−a) t E(J4) = (b5−a5) r4. 5(b−a) t Two important points are worth noting here. First, the model is not a(cid:14)ne as the conditional variance is not linear but quadratic in the state variable, which is qualitatively similar to the quadratic variance di(cid:11)usion model discussed next. Second, the particular state-dependence of the jump-size rules out the possibility of negative interest rate, under the mild restriction that −1 (cid:20) a < b < +1. Negative short rate level is di(cid:14)cult to dealt with for certain a(cid:14)ne speci(cid:12)cations and is conceptually problematic in a nominal economic environment. 2.2.2 Quadratic Variance Di(cid:11)usion Model An important alternative to the a(cid:14)ne variance model is the \quadratic variance" process de(cid:12)ned as q dr = (cid:20)((cid:18)−r )dt+ (cid:27)2 −(cid:27)2r +(cid:27)2r2dW (7) t t 0 1 t 2 t t 6

No sign restrictions are imposed in the GMM estimation procedure, but are adopted here in line with the actual result to highlight some nice properties|non-zero volatility when rate approaches zero, high volatility when rate is high, and comparable scale of the local variance parameter (as in the square-root model). For this quadratic variance model, the conditional moments are characterized by the equation (5) in terms of 2 3 −(cid:20) 0 0 0 6 7 6 7 6 2(cid:20)(cid:18) −(cid:27)2 −2(cid:20)+(cid:27)2 0 0 7 6 1 2 7 A((cid:12)) = 6 7 6 3(cid:27)2 3(cid:20)(cid:18)−3(cid:27)2 −3(cid:20)+3(cid:27)2 0 7 4 0 1 2 5 0 6(cid:27)2 4(cid:20)(cid:18)−6(cid:27)2 −4(cid:20)+6(cid:27)2 0 1 2 and 2 3 (cid:20)(cid:18) 6 7 6 7 6 (cid:27)2 7 6 0 7 g((cid:12)) = 6 7 6 0 7 4 5 0 Note that the solution structure is similar for both the jump-di(cid:11)usion and the quadratic variance models, and that the only di(cid:11)erence is in each entry. This feature makes the numerical calculation of the moments straightforward and fast. The quadratic variance model has several important advantages. First, the model is not a(cid:14)ne hence its moment generating function or characteristic function may not be easy to derive. Then the It^o conditional moment generator may be the only choice among all the non-simulation-based methods to calculate the moments. Second, there is a great deal of debate about whether the volatility is linear or nonlinear, e.g., the \U" shaped volatility pattern reported by A¨(cid:16)t-Sahalia (1996a). Here we can provide a simple parametric nonlinear alternativeandafeasibleGMMestimatorwithconditionalmomentbaseddiagnostics. Third, the quadratic variance speci(cid:12)cation seems to nest several famous short rate models, namely, log-linear ((cid:27) = (cid:27) = 0), Ornstein-Uhlenbeck ((cid:27) = (cid:27) = 0), and square-root ((cid:27) = (cid:27) = 0 0 1 1 2 0 2 and reversing the sign of (cid:27)2). Of course, the obvious disadvantage is that the bond pricing 1 solution is not easily obtained except for using Monte Carlo simulation. Nevertheless, the empirical evidence of Section 4 seems to suggest that the quadratic variance function is indispensable in modeling the univariate short rate dynamics. 7

2.2.3 Cubic or Transformable CEV Model Some models are not directly solvable by the It^o conditional moment generator, but can be \reduced" to the tractable cases by appropriate transformations. For a detailed discussion on the reducibility technique, see Chapter 4 of Kloeden and Platen (1992). Consider the following nonlinear drift and constant-elasticity-of-volatility (CEV) speci(cid:12)cation dr = (cid:20)((cid:18)r 2γ−1 −r )dt+(cid:27)r γ dW ; (8) t t t t t where under appropriate parameter restrictions r 2 (0;+1) and has a positive starting t value. Note that the cross restriction on parameter γ between drift and di(cid:11)usion is required for the reducibility, and may prove to be empirically too restrictive relative to the standard linear drift CEV model. Marsh and Rosenfeld (1983) (cid:12)rst proposed such a modeling strategy and estimated with maximum likelihood for distinct values of γ = 0;0:5;1. Eom (1997) studied the distributional properties and the optimal GMM instruments for γ 2 [0;1). Ahn and Gao (1999) examined the term structure implications for the case of γ = 1:5 and estimated with GMM. The adopted GMM estimators were based on time discretization and approximate (cid:12)rst and second moments. Using the transformation x = r(cid:11), which is a state-preserving transformation when r 2 t t t (0;+1) and γ 2 [0;1) or γ 2 (1;+1), one arrives at the familiar square-root model6 p dx = a(b−x )dt+c x dW : (9) t t t t The above transformation can be characterized by the following proposition Proposition 2 (Transformation) The mappings between the CEV process (8) and the square-root model (9) are (cid:11) = 2(1−γ) a = 2(1−γ)(cid:20) (10) (1−2γ)(cid:27)2 b = (cid:18) + 2(cid:20) c = 2(1−γ)(cid:27) TheproofisastraightforwardapplicationoftheIto^’slemma, andisavailablefromtheauthor upon request. The solution to conditional moment of the transformed process x is a special t 6Whenγ =1thesquare-rootprocess(8)isreducedtothelog-normalprocessdr t =(cid:20)((cid:18)−1)r t dt+(cid:27)r t dW t, and the parameters (cid:20) and (cid:18) are not separately identi(cid:12)able. 8

case of thejump-di(cid:11)usion process (6)without jumps (letting(cid:26) = 0would besu(cid:14)cient). The t fourth parameter γ in the nonlinear drift CEV model (8) is identi(cid:12)ed through the nonlinear but monotonic transformation x = r(cid:11), given that r 2 (0;+1). t t t 3 Estimation Strategy and Monte Carlo Evidence Deriving the conditional moment restrictions (5) only achieves half of the task for estimating the underlying continuous-time model. The other half rests on designing an appropriate estimator with desirable large and small sample properties. The purpose of the this section is to outline an easy-to-implement GMM estimator based on the moment condition solution (5), and to assure the readers that the estimator performs reasonably well for the benchmark CIR model under empirically plausible scenarios. 3.1 The GMM Estimator The condition moments solution (5) can be spelled out as a vector-auto-regressive (VAR) formula 2 3 2 32 3 2 3 E (r ) d 0 0 0 r d 6 t t+1 7 6 11 76 t 7 6 01 7 6 7 6 76 7 6 7 6 E (r2 ) 7 6 d d 0 0 76 r2 7 6 d 7 E t [h t+1 ((cid:12))] = 6 6 t t+1 7 7 −6 6 21 22 7 7 6 6 t 7 7 −6 6 02 7 7 = 0 (11) 6 E (r3 ) 7 6 d d d 0 76 r3 7 6 d 7 4 t t+1 5 4 31 32 33 54 t 5 4 03 5 E (r4 ) d d d d r4 d t t+1 41 42 43 44 t 04 which is a recursive simultaneous equation system and its unrestricted version can be estimatedbytheordinaryleastsquare(OLS).Toformageneralizedmethodofmoments(GMM) estimator, a natural choice of instruments is the constant one and the lagged variables, hence the moment condition vector (with a total of fourteen equations) 2 3 (E(r )−r )(1;r )0 6 t+1 t+1 t 7 6 7 6 (E(r2 )−r2 )(1;r ;r2)0 7 f t ((cid:12)) (cid:17) 6 6 t+1 t+1 t t 7 7 (12) 6 (E(r3 )−r3 )(1;r ;r2;r3)0 7 4 t+1 t+1 t t t 5 (E(r4 )−r4 )(1;r ;r2;r3;r4)0 t+1 t+1 t t t t By construction E[f ((cid:12) )] = 0, and the corresponding GMM or minimum chi-square estit 0 mator is de(cid:12)ned by (cid:12) ^ = argming ((cid:12))0Wg ((cid:12)), where g ((cid:12)) refers to the sample mean T T T T 9

P of the moment conditions, g ((cid:12)) (cid:17) 1=T T−1 f ((cid:12)), and W denotes the asymptotic co- T t=1 t variance matrix of g ((cid:12) ) (Hansen, 1982). An iterative estimator of W is adopted here; and T 0 since the error is not serially correlated, only the heteroscedasticity need to be accounted for. Under standard regularity conditions, the minimized value of the objective function (normalized by the sample size) is asymptotically distributed a chi-square random variable, which allows for an omnibus test of the overidentifying restrictions. Moreover inference regarding individual parameters is readily available from the standard formula of the asymptotic variance-covariance matrix, (@f ((cid:12))=@(cid:12)0W@f ((cid:12))=@(cid:12))=T. t t 3.2 Considerations for Identi(cid:12)cation and E(cid:14)ciency Identi(cid:12)cation, or globalidenti(cid:12)cation, is equivalent to theassumption that theGMM estimatorachieves aunique minimum at some(cid:12) 2 B, where B is acompact set. Intheunrestricted 0 recursive VAR model (11), the total number of identi(cid:12)able parameters is fourteen, which can be easily veri(cid:12)ed by the standard order and rank conditions. Since the underlying jumpdi(cid:11)usion model is nonlinear, the identi(cid:12)cation issue becomes more complicated|on the one hand, the restricted nonlinear dynamics may not be able to identify as many as fourteen parameters; on the other hand a nonlinear structure usually helps to identity more parameters than a linear structure. There is not much theoretical guidance in literature on how to verify the identi(cid:12)cation condition in a nonlinear model before the model is actually estimated. However, there is a su(cid:14)cient condition|plim(@g =@(cid:12)0W @g =@(cid:12))=T being nonsingular| T T T that can be numerically veri(cid:12)ed with the estimation result from a given sample data set. It is equivalent to the more primitive condition of local identi(cid:12)cation that the gradient is of full column rank and the Hessian is negative de(cid:12)nite. In practice, all the empirical examples seem not to violate this su(cid:14)cient condition, except a variation of the jump-di(cid:11)usion model where both the jump-rate and jump-size parameters are state-independent constants. Following Hansen (1985) and Hansen et al. (1988), the conditional moment restriction E [h ((cid:12))] = 0 indicated by equation (11) implies an e(cid:14)cient choice of instruments as t t+1 E [@h ((cid:12) )=@(cid:12)]Var [h ((cid:12) )]−1. In theory such a choice of instruments should be ideal, t t+1 0 t t+1 0 but in practice other considerations may favor the natural choice of (12). First, the optimal instruments involve unknown true distribution parameter (cid:12) , which has to be approximated 0 in the GMM estimation procedure. Second, to calculate the optimal instruments one needs 10

to solve for eight lower order moments if one uses only four lower order moments in the estimation, which is trivial analytically using the Ito^ approach but may be numerically unstable for real world data sets. Further, there is a logical inconsistency|one has the knowledge of eight order moments but does not use it in the moment condition restriction. Meddahi and Renault (1997) proposed an interesting treatment that reduces the information of the third and fourth conditional moments to the unconditional skewness and kurtosis, and achieves the e(cid:14)cient estimates of conditional mean and variance. The GMM estimator implemented here explicitly incorporates the conditional third and fourth moments and is conceptually related to their e(cid:14)cient estimation of the (cid:12)rst two conditional moments. The relative e(cid:14)ciency of the proposed GMM estimator can be judged in a Monte Carlo setting, against the asymptotically e(cid:14)cient MLE (which is theoretically superior to GMM for a given set of moment restrictions with optimal instruments). 3.3 Monte Carlo Evidence To assess the (cid:12)nite sample performance of the proposed GMM estimator, a limited Monte p Carlo study is conducted for the benchmark square-root model dr = (cid:20)((cid:18)−r )dt+(cid:27) r dW , t t t t in comparisons with the MLE estimates reported by Durham and Gallant (2002). There are six scenarios chosen in their paper, with varying degrees of persistence and volatility, and a (cid:12)xed long-run mean of 6 percent. I adopted the exact same setup with 1000 observations in each random sample and a total of 512 Monte Carlo replications. To avoid the discretization bias, I simulate the square-root model from the exact non-central Chi-square distribution (cid:18) (cid:19) q (cid:16) p (cid:17) f(r jr ;(cid:20);(cid:18);(cid:27)) = ce −u−v v 2 I 2 uv ; (13) t+(cid:1) t q u where q = 2(cid:20)(cid:18)=(cid:27)2−1; c = 2(cid:20)=(cid:27)2(1−e−(cid:20)(cid:1)), u = cr e−(cid:20)(cid:1), v = cr , and I ((cid:1)) is a modi(cid:12)ed t t+(cid:1) q Bessel function of the (cid:12)rst kind with a fractional order q (Oliver, 1972). A composite method of generating random number (Devroye, 1986) is adopted here after transforming the above density function into, X1 yj+(cid:21)−1e−y uje−u X1 f(y) = (cid:1) = Gamma(yjj +(cid:21);1)(cid:1)Poisson(jju) (14) Γ(j +(cid:21)) j! j=0 j=0 with y = v and (cid:21) = q + 1. In practice, one (cid:12)rst draws a random number j from the Poisson(jju)distribution; then draws another random number y from the Gamma(yjj+(cid:21);1) 11

distribution; and (cid:12)nally calculates the target state variable r = y=c. See Zhou (2001) for t+(cid:1) implementation detail. Table 1 compares the parameter estimates of the proposed GMM estimator in this paper with those of the MLE estimator, under the six scenarios (a-f) in Durham and Gallant (2002). In terms of bias, only the mean-reversion parameter (cid:20) has a sizable upward bias when the persistence level is high (scenario a, b, and d)|about 10% of the parameter value for MLE and about 20% for GMM; while for the less persistent scenarios (c, e, and f), the bias is noticeably reduced. This is a classical case of (cid:12)nite sample bias in estimating the AR(1) coe(cid:14)cient for near-unit-root processes. For the long-run mean parameter (cid:18) and the local variance parameter (cid:27), MLE has negligible positive bias and GMM has negligible negative biases. In terms of relative e(cid:14)ciency, the proposed GMM estimator is remarkably close to the asymptotically e(cid:14)cient MLE. The root-mean-squared-error of GMM is no more 10% larger than that of MLE for most parameters in the persistent cases (scenarios a, b, and d), and is practically indistinguishable for most parameters in the less persistent cases (scenarios c, e, and f). Figure 1 reports the GMM test of the overidentifying restrictions, which exhibits a typical over-rejection bias but with a reasonable size comparing with the reference level. The sampling distribution of the t-test statistics is graphed in Figure 2, and indicates that the (cid:12)nite sample distortion is rather small comparing with the reference Normal(0,1) distribution. 4 Empirical Application In this section, the It^o moment generator and the related GMM estimator are applied to the empirical U.S. interest rate data. The weekly 3-month t-bill rate from January 1954 to July 2002, totaling 2504 observations, is obtained from the Federal Reserve Bank of St. Louis public website. The time series plot is given in Figure 3 and the summary statistics are reported in Table 2. The short rate exhibits the typical features found in literature|high persistence (auto-regressive coe(cid:14)cients close to one), high volatility (standard deviation 277 basis points), moderately high skewness (1.14) and kurtosis (4.87). I will focus on the estimation result of the four empirical examples (including the benchmark CIR model) presented in Section 2, and illustrate how to use the conditional moment functions to further 12

compare di(cid:11)erent model speci(cid:12)cations. 4.1 Estimation Result The GMM estimator designed in Section 3 is applied to the four candidate models discussed in Section 2: square-root, restricted CEV, jump-di(cid:11)usion, and quadratic variance.7 The results are summarized in Table 3. The standard square-root model is strongly rejected by the GMM speci(cid:12)cation test, with achi-square(df=11)of49.36. Thelong-runmeanparameter (0.0497)isabout50basispoints lower than the sample average (0.0542), the mean-reversion parameter is very low (0.0020) and imprecisely estimated with standard error 0.0013, and the local variance parameter is also lower (0.0062which implies a unconditional standard deviation 0.0219versus the sample standard deviation 0.0277). The nonlinear drift CEV model is also strongly rejected with a chi-square (df=10) of 48.57. Although most parameters are accurately estimated, the model also has di(cid:14)culty in nailing down the mean reversion parameter (cid:20) (0.0017 with standard error 0.0010). The restrictedCEVmodelaccuratelyestimateselasticity parameteras0.4825withstandarderror 0.0028, which con(cid:12)rms the empirical (cid:12)nding by Eom (1997). All other parameter estimates are close to and/or slightly lower than the square-root estimates.8 The jump-di(cid:11)usion model is implemented here with a constant jump-rate (cid:26) and a uniformjump-size (−ar ;ar ). The symmetry restriction onjump size is to ensure identi(cid:12)cation. t t The result predicts roughly two jumps per year; with a state-dependent jump-size of plus or minus 119 basis points at the sample average (0.0542), plus or minus 13 basis points at sample minimum (0.0058), and plus or minus 368 basis points at sample maximum (0.1676). Such a jump pattern is more realistic than the constant jump-size distribution, and can rule 7Aspointedoutbyareferee,onecouldestimateacomprehensivemodelnestingbothtime-varyingjumps and conditional quadratic variance. I found out that such a speci(cid:12)cation is not empirically identi(cid:12)able by the GMM estimator. My intuition is that the particular jump and di(cid:11)usion speci(cid:12)cations adopted here produce the similar quadratic conditional variance. Therefore they are substituting for each other instead ofbeing complementary. This canbe easilyseenfromthe diagnosticconditionalmomentgraphsinthe next subsection. 8Thisresultdi(cid:11)ersfromthetypicalempirical(cid:12)ndingforthelineardriftCEVmodel,inthattheelasticity coe(cid:14)cient there is found to be in the range of 1.0-1.5 (Chan et al., 1992; Conley et al., 1997; Tauchen, 1997; Christo(cid:11)ersen and Diebold, 2000), possibly because that the nonlinear drift CEV model imposes an unrealistic restriction across the drift and di(cid:11)usion functions. 13

out the negative interest rates, which is quite troublesome in a nominal economic environment. Nevertheless, the model is rejected at the p-value 0.0002, and the parameter (cid:18) is unconvincingly large (0.0995). Thequadraticvariancemodelperformsthebest, andisnotrejectedatthe1%signi(cid:12)cance level (p-value 0.0121). All the parameter estimates are highly signi(cid:12)cant. The estimates of the drift parameters fall between the square-root model (similarly the CEV model) and the jump-di(cid:11)usion model. The parameter estimates of the di(cid:11)usion function guarantee that (a) instant variance does not admit negative value, (b) minimum volatility is achieved at a positive short rate level, and (c) volatility increases more when short rate level is high than when short rate level is low (see the conditional moment graphs bellow). Such a result from a parametric perspective seems to be con(cid:12)rm the (cid:12)nding of A¨(cid:16)t-Sahalia (1996a) from a nonparametric perspective. 4.2 Conditional Moment Graphs The conditional moment vector (11) not only serves as the basis for constructing a GMM estimator, but also provides intuitive diagnostics in conditional mean, volatility, skewness, and kurtosis. The conditional mean and conditional variance in discrete sampling intervals, are equivalent to the drift and volatility functions in instant times for the pure di(cid:11)usion processes, but more general in covering also the jump-di(cid:11)usion processes. The conditional skewness and kurtosis provide natural assessment on how much the implied transitional density deviates from the conditional normality. Higher order conditional moments are especially informative about the jump impact when the time horizon is longer than zero, but the instantaneous higher order moments cannot provide any new information than the instantaneous drift and volatility. Figure 4 plots the conditional mean (top panel) and conditional variance (bottom panel). It is clear that the square-root model has the least persistence in level. Although the restricted CEV model has a potential nonlinear drift, the estimated mean function is mostly linear andclosetothesquare-rootmodel. Ontheotherhand, thejump-di(cid:11)usionmodelisthe most persistent case, suggesting an observational equivalence between occasional jumps and near unit-root in interest rate processes. Our preferred quadratic variance model has a linear mean function with a moderate persistence among the four models. Turning to the condi- 14

tional volatility, both the square-root model and the restricted CEV model produce nearly identical linear volatility pro(cid:12)les, underpinning the clear rejection by the GMM speci(cid:12)cation tests. Jump-di(cid:11)usion process provides a slightly nonlinear quadratic variance function, due to the state-dependent jump-size speci(cid:12)cation (J (cid:24) U[−ar ;ar ]) that di(cid:11)ers from the t t t standard a(cid:14)ne jump-di(cid:11)usion models. Of course the most dramatic result comes from the U-shaped quadratic variance model, which partially con(cid:12)rms the nonparametric (cid:12)nding of the nonlinear volatility by A¨(cid:16)t-Sahalia (1996a) and the parametric (cid:12)nding of CEV elasticity between 1.0 and 1.5 (Chan et al., 1992; Conley et al., 1997; Tauchen, 1997; Christo(cid:11)ersen and Diebold, 2000). The post-war U.S. history suggests that the interest rate volatility is certainly high when the short rate level is high, but the volatility is also none trivial when the rate is close to zero. Therefore a nonlinear dependence of short rate volatility on its level may be better captured by a quadratic variance model than by a standard a(cid:14)ne model. Figure 5 depicts the conditional skewness and kurtosis functions and o(cid:11)ers some assessment of the departures from the conditional normality. From the top panel we can see that both the square-root and the nonlinear CEV model give a virtually same hyperbolic skewness function|shooting up at the lower end and approaching zero at the higher end. The jump-di(cid:11)usion process has a similar pro(cid:12)le but a uniformly higher skewness once the short rate level reaches above 2 percent. The quadratic volatility model is unique in presenting a nonlinear increasing skewness function that approaches -0.1 at the low end and +0.1 at the high end. Turning to the bottom panel, again, both the square-root and the nonlinear CEV models give a virtually same hyperbolic kurtosis function|shooting up at the lower end and approaching three at the higher end. Note that the jump-di(cid:11)usion model gives an extraordinarily high kurtosis, ranging from 9 at the lower end to 44 at the higher end (outside and above the picture range). Usually introducing jumps helps to increase the model skewness and kurtosis, but an unusually high kurtosis of 9-44 must be caused by the restrictive jump speci(cid:12)cation (constant jump rate (cid:26) = (cid:26) and uniform jump size J (cid:24) U[−ar ;ar ]), which is t t t t imposed to identify all the model parameters. The preferred quadratic model has a nonlinear V-shaped kurtosis function for the short rate level between 0 and 8 percent and then mostly a constant. In short, the quadratic variance model produces unique nonlinear conditional skewness and kurtosis, which are dramatically di(cid:11)erent from all other candidate models. 15

5 Conclusion This paper proposes an It^o’s approach to generate the conditional moments for continuous time Markov processes and gives a characterization of the class of admissible models. The resulting conditional moment vector forms the basis of a natural GMM estimator. Monte Carlo evidence suggests that such a moment generator and the related estimator behave reasonably well for a benchmark square-root model. When applied to the empirical U.S. short ratedata,theproceduresinglesoutthequadraticvariancemodelastheonlyunrejectedspeci(cid:12)cation at the one percent level. The benchmark square-root model, the state-dependent jump-di(cid:11)usion process, and the nonlinear drift CEV model all fail in the GMM tests of overidentifying restrictions. Further diagnostics suggests that the U-shaped conditional variance and non trivial conditional skewness and kurtosis are important in modeling the short rate dynamics in the univariate setting. One important extension is to estimate a multivariate asset return model with possible quadratic volatility components. 16

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Table 1: Monte Carlo Experiment This table compares the (cid:12)nite sample performance of the GMM estimator proposed in this paper with that of the MLE estimator provided by Durham and Gallant (2002). (cid:1) stands for the discrete sampling interval and df for the degree of freedom of the implied non-central chi-square distribution. The random sample size is chosen as 1000 and the number of Monte Carlo replicates is 512. Here we report the mean bias and root-mean-squared-error. True Value MLE GMM MLE GMM Mean Bias Mean Bias Root-MSE Root-MSE Scenario (a), (cid:1) = 1=12, df = 5:33 (cid:20) = 0:50 0.0489 0.0828 0.1344 0.1473 (cid:18) = 0:06 0.0006 -0.0027 0.0080 0.0086 (cid:27) = 0:15 0.0002 -0.0028 0.0034 0.0046 Scenario (b), (cid:1) = 1=12, df = 2:48 (cid:20) = 0:50 0.0597 0.1036 0.1413 0.1697 (cid:18) = 0:06 -0.0003 -0.0052 0.0114 0.0118 (cid:27) = 0:22 0.0001 -0.0045 0.0054 0.0075 Scenario (c), (cid:1) = 1=12, df = 133:33 (cid:20) = 0:50 0.0438 -0.0042 0.1299 0.1221 (cid:18) = 0:06 0.0001 -0.0000 0.0016 0.0017 (cid:27) = 0:03 0.0002 -0.0003 0.0007 0.0008 Scenario (d), (cid:1) = 1=12, df = 4:27 (cid:20) = 0:40 0.0458 0.0892 0.1210 0.1446 (cid:18) = 0:06 0.0008 -0.0046 0.0102 0.0102 (cid:27) = 0:15 0.0001 -0.0029 0.0035 0.0048 Scenario (e), (cid:1) = 1=12, df = 53:33 (cid:20) = 5:00 0.0151 0.0169 0.4630 0.4580 (cid:18) = 0:06 0.0001 -0.0002 0.0008 0.0009 (cid:27) = 0:15 0.0000 -0.0040 0.0043 0.0059 Scenario (f), (cid:1) = 2, df = 53:33 (cid:20) = 0:50 0.0013 0.0283 0.0430 0.0483 (cid:18) = 0:06 0.0004 -0.0022 0.0018 0.0029 (cid:27) = 0:15 0.0004 -0.0041 0.0056 0.0066 23

Table 2: Summary Statistics of Three Month T-Bill Rates The following table summarizes the weekly U.S. 3-month t-bill rates from January 1954 to July 2002 with a total of 2504 observations. The data is obtained from the public website of the Federal Reserve Bank of St. Louis. Moments and Quantiles jth Order Autocorrelations Mean 0.0542 (cid:26) 1.0000 0 Std. Dev. 0.0277 (cid:26) 0.9964 1 Skewness 1.1414 (cid:26) 0.9912 2 Kurtosis 4.8728 (cid:26) 0.9856 3 Minimum 0.0058 (cid:26) 0.9798 4 5%-qntl. 0.0171 (cid:26) 0.9734 5 25%-qntl. 0.0347 (cid:26) 0.9667 6 Medium 0.0504 (cid:26) 0.9600 7 75%-qntl. 0.0689 (cid:26) 0.9537 8 95%-qntl. 0.1045 (cid:26) 0.9477 9 Maximum 0.1676 (cid:26) 0.9418 10 24

Table 3: Empirical Estimation Results This table presents the main empirical results of the four model speci(cid:12)cations discussed in Section 2 and estimated by the GMM estimator outlined in Section 3. Square-Root Nonlinear CEV Jump-Di(cid:11)usion Quadratic Variance (cid:20) = 0.0020 0.0017 0.0005 0.0010 (0.0013) (0.0010) (0.0001) (0.0002) (cid:18) = 0.0497 0.0458 0.0995 0.0669 (0.0131) (0.0128) (0.0186) (0.0016) (cid:27) = 0.0062 0.0059 0.0031 (0.0002) (0.0002) (0.0012) γ = 0.4825 (0.0028) (cid:26) = 0.0381 (0.0025) a = 0.2196 (0.0265) (cid:27) = 0.0015 0 (0.0002) (cid:27) = 0.0097 1 (0.0005) (cid:27) = 0.0412 2 (0.0006) Chi-Square = 49.3617 48.5714 31.6703 21.1222 d.o.f = 11 10 9 9 p-value = 0.0000 0.0000 0.0002 0.0121 25

100 80 Rejection Curve 60 40 Reference Curve 20 0 0 20 40 60 80 100 Nominal Level of Test noitcejeR fo egatnecreP Scenario (a) 100 Rejection Curve 80 60 40 Reference Curve 20 0 0 20 40 60 80 100 Nominal Level of Test noitcejeR fo egatnecreP Scenario (b) 100 Rejection Curve 80 60 40 Reference Curve 20 0 0 20 40 60 80 100 Nominal Level of Test noitcejeR fo egatnecreP Scenario (c) 100 80 Rejection Curve 60 40 Reference Curve 20 0 0 20 40 60 80 100 Nominal Level of Test noitcejeR fo egatnecreP Scenario (d) 100 Rejection Curve 80 60 40 Reference Curve 20 0 0 20 40 60 80 100 Nominal Level of Test noitcejeR fo egatnecreP Scenario (e) 100 Rejection Curve 80 60 40 Reference Curve 20 0 0 20 40 60 80 100 Nominal Level of Test noitcejeR fo egatnecreP Scenario (f) Figure 1: GMM Speci(cid:12)cation Test of Overidentifying Restrictions. 26

Scenario (a): k Scenario (a): q Scenario (a): s 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −5 Scenar 0 io (b): k 5 −5 Scenar 0 io (b): q 5 −5 Scenar 0 io (b): s 5 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −5 Scenar 0 io (c): k 5 −5 Scenar 0 io (c): q 5 −5 Scenar 0 io (c): s 5 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −5 Scenar 0 io (d): k 5 −5 Scenar 0 io (d): q 5 −5 Scenar 0 io (d): s 5 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −5 Scenar 0 io (e): k 5 −5 Scenar 0 io (e): q 5 −5 Scenar 0 io (e): s 5 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −5 Scena 0 rio (f): k 5 −5 Scena 0 rio (f): q 5 −5 Scenar 0 io (f): s 5 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −5 0 5 −5 0 5 −5 0 5 Figure 2: \- -" Normal (0,1) reference density; \|" t-test statistics. 27

US 3−Month Treasury Bill Rate: 1954−2002 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1960 1970 1980 1990 2000 Figure 3: Time Series Plot of the Short Term Interest Rate. 28

x 10 −4 Conditional Mean 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 Zero as Reference Square−Root Nonlinear CEV −3 Jump−Diffusion Quadratic Variance −3.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Short Rate Level x 10 −5 Conditional Variance 6 Square−Root Nonlinear CEV Jump−Diffusion 5 Quadratic Variance 4 3 2 1 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Short Rate Level Figure 4: Conditional Mean and Variance. 29

Conditional Skewness 0.25 0.2 0.15 0.1 0.05 0 −0.05 Zero as Reference Square−Root −0.1 Nonlinear CEV Jump−Diffusion Quadratic Variance 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Short Rate Level Conditional Kurtosis Three as Reference Square−Root Nonlinear CEV 3.04 Jump−Diffusion (9−44) Quadratic Variance 3.03 3.02 3.01 3 2.99 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Short Rate Level Figure 5: Conditional Skewness and Kurtosis. 30