In this paper, we characterize, interpret, and test the over-identifying restrictions imposed in affine models of the term-structure. "We begin by showing, using the classification scheme proposed by Dai, Liu, and Singleton [10] for general affine diffusions, that the family of N-factor models can be classified into N + 1 non-nested sub-families of models. For each subfamily, we derive a canonical model with the property that every admissible member of this family is equivalent to or a nested special case of our canonical model. Second, using our classification scheme and canonical models, we show that many of the three-factor models in the literature impose potentially strong over-identifying restrictions, and we completely characterize these restrictions. Finally, we compute simulated-method-of-moments estimates for several members of the sub-family of three-factor models that nest the "benchmark" model of Chen [8], and test the over-identifying restrictions on the joint distribution...

This paper is concerned with the Bayesian estimation of non-linear stochastic differential equations when only discrete observations are available. The estimation is carried out using a tuned MCMC method, in particular a blocked Metropolis-Hastings algorithm, by introducing auxiliary points and using the Euler-Maruyama discretisation scheme. Techniques for computing the likelihood function, the marginal likelihood and diagnostic measures (all based on the MCMC output) are presented. Examples using simulated and real data are presented and discussed in detail.

This article presents a technique for nonparametrically estimating continuous-time di#usion processes which are observed at discrete intervals. We illustrate the methodology by using daily three and six month Treasury Bill data, from January 1965 to July 1995, to estimate the drift and di#usion of the short rate, and the market price of interest rate risk. While the estimated di#usion is similar to that estimated by Chan, Karolyi, Longsta# and Sanders (1992), there is evidence of substantial nonlinearity in the drift. This is close to zero for low and medium interest rates, but mean reversion increases sharply at higher interest rates.

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 diffusion 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 diffusion coefficients of a volatility equation. This effort entails the development of new graphical diagnostics.

Efficient Method of Moments (EMM) is used to fit the standard stochastic volatility model and various extensions to several daily financial time series. EMM matches to the score of a model determined by data analysis called the score generator. Discrepancies reveal characteristics of data that stochastic volatility models cannot approximate. The two score generators employed here are "Semiparametric ARCH" and "Nonlinear Nonparametric". With the first, the standard model is rejected, although some extensions are accepted. With the second, all versions are rejected. The extensions required for an adequate fit are so elaborate that nonparametric specifications are probably more convenient. Corresponding author: George Tauchen, Duke University, Department of Economics, Social Science Building, Box 90097, Durham NC 27708-0097 USA, phone 1-919-660-1812, FAX 1-919-684-8974, e-mail get@tauchen.econ.duke.edu. 0 1 Introduction The stochastic volatility model has been proposed as a descripti...

This paper studies the empirical performance of jump-diffusion models that allow for stochastic volatility and correlated jumps affecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the in-sample estimation period. This contrasts previous findings where stochastic volatility paths are found to be too smooth relative to the option implied dynamics. While the models perform well during the high volatility estimation period, they tend to overprice long dated contracts out-of-sample. This evidence points towards a too simplistic specification of the mean dynamics of volatility.

This paper proposes a new method for estimation of parameters in diffusion processes from

We examine the empirical evidence on the expectations hypothesis of the term structure of interest rates in the United States, the United Kingdom, and Germany using the Campbell-Shiller (1991) regressions and a vector-autoregressive methodology. We argue that anomalies in the U.S. term structure, documented by Campbell and Shiller (1991), may be due to a generalized peso problem in which a high-interest rate regime occuued less frequently in the sample of U.S. data than was rationally anticipated. We formalize this idea as a regime-switching model of short-term interest rates estimated with data from seven countries. Technically, this model extends recent research on regime-switching models with state-dependent transitions to a cross-sectional setting. Use of the small sample distributions generated by the regime-switching model for inference considerably weakens the evidence against the expectations hypothesis, but it remains somewhat implausible that our data-generating process produced the U.S. data. However, a model that combines moderate time-variation in term premiums with peso-problem effects is largely consistent with term structure

Stochastic differential equations often provide a convenient way to describe the dynamics of economic and financial data, and a great deal of effort has been expended searching for efficient ways to estimate models based on them. Maximum likelihood is typically the estimator of choice; however, since the transition density is generally unknown, one is forced to approximate it. The simulation-based approach suggested by Pedersen (1995) has great theoretical appeal, but previously available implementations have been computationally costly. We examine a variety of numerical techniques designed to improve the performance of this approach. Synthetic data generated by a CIR model with parameters calibrated to match monthly observations of the U.S. short-term interest rate are used as a test case. Since the likelihood function of this process is known, the quality of the approximations can be easily evaluated. On data sets with 1000 observations, we are able to approximate the maximum likelihood estimator with negligible error in well under one minute. This represents something on the order of a 10,000-fold reduction in computational effort as compared to implementations without these enhancements. With other parameter settings designed to stress the methodology, performance remains strong. These ideas are easily generalized to multivariate settings and (with some additional work) to latent variable models. To illustrate, we estimate a simple stochastic volatility model of the U.S. short-term interest rate.

anonymous referees and seminar participants at Stanford University and the 1999 Econometric Society