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Reprojecting Partially Observed Systems with Application to Interest Rate Diusions from January 5, 1992, to March 31, 1995 1

A. Ronald Gallant Department of Economics University of North Carolina Chapel Hill NC 27599-3305 USA

George Tauchen Department of Economics Duke University Durham NC 27708-0097 USA

First Draft: July 1996 This Version: August 1996

1 Supported by the National Science Foundation. The corresponding author is A. Ronald Gallant,

Department of Economics, University of North Carolina, CB# 3305, 6F Gardner Hall, Chapel Hill NC 27599-3305 USA; Phone 1-919-966-5338; FAX 1-919-966-4986; e-mail ron [email protected]. Updated versions of this paper are available by anonymous ftp at site ftp.econ.duke.edu in directory pub/arg/papers as le lv.psz, which is compressed PostScript.

Abstract We introduce reprojection as a general purpose technique for characterizing the observable dynamics of a partially observed nonlinear system. System parameters are estimated by method of moments wherein moments implied by the system are matched to moments implied by the transition density for observables that is determined by projecting the data onto its Hermite representation. Reprojection imposes the constraints implied by the system on the transition density and is accomplished by projecting a long simulation of the estimated system onto the Hermite representation. We utilize the technique to assess the dynamics of several diusion models for the short-term interest rate that have been proposed and compare them to a new model that has feedback from the interest rate into both the drift and diusion coecients of a volatility equation. This eort entails the development of new graphical diagnostics.

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1 Introduction We consider statistical methods for the analysis of dynamic nonlinear models that have unobserved variables. These models pervade science. Most often they arise from dynamical systems described by a system of deterministic or stochastic dierential equations in which the state vector is partially observed. For example, in epidemiology the SEIR model determines those susceptible, exposed, infected, and recovered from a disease whereas usually data are from case reports that report only those infected (Olsen and Schaer, 1990). Other examples are continuous and discrete time stochastic volatility models of speculative markets from nance (Ghysels, Harvey, and Renault, 1995), general equilibrium models from economics (Gennotte and Marsh, 1993); and compartment models from pharmacokinetics (Mallet, Mentre, Steimer, and Lokiec, 1988). Standard statistical methods, both classical and Bayesian, are usually not applicable in these situations either because it is not practicable to obtain the likelihood for the entire state vector or because the integration required to eliminate unobservables from the likelihood is infeasible. On a case-by-case basis, statistical methods are often available. However, our purpose here is to describe methods that are generally applicable. Although determining the likelihood of a nonlinear dynamical system that has unobserved variables is often infeasible, simulating the evolution of the state vector is often quite practicable. Our methods rely on this. Brie y, we project the observed data onto a Hermite series representation of the transition density of the observable process. Given a parameter setting for the system, we use simulation to compare the moments of the stationary density of the system to those of the projection using a comparison criterion that depends on the scores of the projection. A nonlinear optimizer is used to nd the parameter setting that minimizes the criterion. This is a method of moments (or minimum chi square) procedure that is as ecient in large samples as if maximum likelihood had been employed and is therefore termed ecient method of moments (EMM). Diagnostic tests are available to assess system adequacy as well as graphics that suggest reasons for failure. Subsequent reprojection of the estimated dynamical system on the Hermite series representation provides a facility for model elucidation that is as convenient as if a likelihood were available. 2

The use of method of moments together with simulation to estimate the parameters of dynamic models with unobserved variables is not new. Some of the earlier papers are Ingram and Lee (1991), Due and Singleton (1993), and Gourieroux, Monfort, and Renault (1993). The particular methods that we employ are due to Gallant and Tauchen (1996), who discuss the procedure, diagnostics, and asymptotics. The theoretical support for the projection that we employ is provided by Gallant and Long (1997), who show that it achieves the same eciency as maximum likelihood. We sketch the ideas from this literature with the intent of making this paper self contained. A forceful criticism of simulation based method of moments estimation has been that the method does not provide a representation of the observables in terms of their own past as does maximum likelihood based on a conditional density or do time series methods such as ARIMA, ARCH, or GARCH modeling (Jacquier, Polson, and Rossi, 1994). Therefore the methodology cannot be used for model elucidation by providing, for example, descriptions of the volatility of the observed process as a function of its own past. This has provided motivation for ad hoc methods on a case-by-case basis. The primary methodological contribution of this paper is to overcome this criticism. We introduce the notion of reprojection to get a representation of the observed process in terms of its own past that incorporates the dynamics implied by the nonlinear system under consideration. Once done, the methods of model elucidation introduced by Gallant, Rossi, and Tauchen (1992, 1993) may be applied. We employ the proposed methodology for estimation and diagnostic assessment of interest rate diusion models expressed as a partially observed system of stochastic dierential equations. We undertake this application because interest rate diusions of are considerable interest in their own right (At-Sahalia, 1996; Andersen and Lund, 1996a, 1996b; Hansen and Scheinkman, 1995; Lo, 1988; Melino, 1994) and we know of no other general purpose method for elucidating the dynamics of a discretely sampled system of stochastic dierential equations with a partially observed state.

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2 Ecient Method of Moments

2.1 Projection

M Let fytg1 t=?1 ; yt 2 < ; be a discrete stationary time series. The stationary distribution of a contiguous subsequence yt?L; : : :; yt is presumed to have a density p(y?L ; : : : ; y0) de ned over : (1 ? cos(100u))=100 j100uj < =2: For  with nonnegative integer elements, let z = z11


zMM and jj = PMk=1 k ; similarly for x . Consider the density (z; x)]2(z) (3) hK (zjx) = R [P[PK(u; K x)]2(u) du formed from the polynomial Kx Kz  X  X a x z PK (z; x) = =0 =0

where (z) = (2)?M=2e?z0 z=2 . The product [PK (z; x)]2(z) is a Hermite polynomial of degree Kz in z with positivity enforced whose coecients are polynomials of degree Kx in x. The shape of the innovation density hK (ztjxt?1) varies with xt?1 which permits hK (ztjxt?1) to exhibit general conditional shape heterogeneity. By putting selected elements of the matrix A = [a ] to zero PK (z; x) can be made to depend on only Lp  L lags from x: In applications where M is large the coecients a  corresponding to monomials z that 5

represent high order interactions can be set to zero with little eect on the adequacy of approximations. Let Iz = 0 indicate that no interaction coecients are set to zero, Iz = 1 indicate that coecients corresponding to interactions z of order larger than Kz ? 1 are set to zero, and so on; similarly for x and Ix. The change of variables yt = Rxt?1 zt + xt?1 to obtain the density fP [R?1t?1 (yt ? xt?1 )R; xt?1]g2[R?1 xt?1 (yt ? xt?1 )] : (4) fK (ytjxt?1; ) = K j xdet( 1 = 2 Rxt?1 )j [PK (u; xt?1)]2(u) du completes the description of the SNP density. The vector  contains the coecients A = [a ] of the Hermite polynomial, the coecients [b0; B ] of the location function, and the coecients [0; P ] of the scale function. To achieve identi cation, the coecient a0;0 is set to 1. The tuning parameters are Lu; Lr ; Lp; Kz ; Iz ; Kx; and Ix; which determine K and the dimension pK of : Some characteristics of fK (ytjxt?1; ) may be noted. If Kz ; Kx; and Lr are put to zero, then fK (ytjxt?1; ) de nes a Gaussian vector autoregression. If Kx and Lr are put to zero, then fK (ytjxt?1; ) de nes a non-Gaussian vector autoregression model with homogeneous innovations. If Kz and Kx are put to zero, then fK (ytjxt?1; ) de nes a Gaussian ARCH model. If Kx is put to zero, then fK (ytjxt?1; ) de nes a non-Gaussian ARCH model with homogeneous innovations. If Kz > 0; Kx > 0; Lp > 0; Lu > 0; and Lr > 0; then fK (ytjxt?1; ) de nes general nonlinear process with heterogeneous innovations. How best to select the tuning parameters Lu ; Lr ; Lp; Kz ; Iz ; Kx; and Ix is an open question. A strategy found to work well is to move upward along an expansion path using the BIC criterion BIC = sn(~) + (1=2)(pK =n) log(n); n X sn () = ? n1 log[fK (~ytjy~t?L; : : :; y~t?1; )]; t=0 (Schwarz, 1978) to guide the search, models with small values of BIC being preferred. The expansion path has a tree structure. Rather than examining the full tree, the strategy is to expand rst in Lu with Lr = Lp = Kz = Kx = 0 until BIC turns upward. Next, expand Lr with Lp = Kz = Kx = 0; then expand Kz with Kx = 0; and lastly Lp and Kx . It is useful to expand in Kz ; Lp and Kx at a few intermediate values of Lr because it sometimes happens that the smallest value of BIC lies elsewhere within the tree. 6

When the estimated transition density fK (ytjxt?1; ~n) is used in connection with the estimator described in Subsection 2.2, it is essential that fK (ytjxt?1; ~n) not represent an explosive process, as discussed in detail by Tauchen (1995). With persistent processes, such as our application in Section 3, this can happen. We examine long simulations from candidate estimates fK (ytjxt?1; ~n) and exclude from consideration those that are explosive.

2.2 Estimation We now suppose that a dynamical system de nes the density p(y?L; : : : ; y0j) for observables of Subsection 2.1, where  2