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77
Likelihood Inference for Discretely Observed NonLinear Diffusions
 Econometrica
, 1998
"... This paper is concerned with the Bayesian estimation of nonlinear stochastic differential equations when only discrete observations are available. The estimation is carried out using a tuned MCMC method, in particular a blocked MetropolisHastings algorithm, by introducing auxiliary points and usin ..."
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Cited by 155 (18 self)
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This paper is concerned with the Bayesian estimation of nonlinear stochastic differential equations when only discrete observations are available. The estimation is carried out using a tuned MCMC method, in particular a blocked MetropolisHastings algorithm, by introducing auxiliary points and using the EulerMaruyama 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.
Numerical Techniques for Maximum Likelihood Estimation of ContinuousTime Diffusion Processes
 JOURNAL OF BUSINESS AND ECONOMIC STATISTICS
, 2001
"... 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, sinc ..."
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Cited by 87 (0 self)
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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 simulationbased 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. shortterm 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 10,000fold 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. shortterm interest rate.
Estimating Equations Based on Eigenfunctions for a Discretely Observed Diffusion Process
, 1995
"... : A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the g ..."
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Cited by 58 (13 self)
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: A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discretetime observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study. Keywords: generator, optimal estimating function, stochastic differential equation, quasilikelihood. 1 Introduction Martingale estimating functions have turned out to give good and relatively simple estimators for discretely observed diffusion models, for which the likelihood function is only explicitly known in special cases. These estimators have th...
Stochastic volatility models as hidden Markov models and statistical applications
 Bernoulli
, 2000
"... This paper deals with the ®xed sampling interval case for stochastic volatility models. We consider a twodimensional diffusion process (Yt, Vt), where only (Yt) is observed at n discrete times with regular sampling interval Ä. The unobserved coordinate (Vt) is ergodic and rules the diffusion coef®c ..."
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Cited by 45 (5 self)
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This paper deals with the ®xed sampling interval case for stochastic volatility models. We consider a twodimensional diffusion process (Yt, Vt), where only (Yt) is observed at n discrete times with regular sampling interval Ä. The unobserved coordinate (Vt) is ergodic and rules the diffusion coef®cient (volatility) of (Yt). We study the ergodicity and mixing properties of the observations (YiÄ). For this purpose, we ®rst present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (Vt). When the stochastic differential equation of (Vt) depends on unknown parameters, we derive momenttype estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n 1=2. Examples of models coming from ®nance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.
Bayesian inference for nonlinear multivariate diffusion models observed with error
 Computational Statistics and Data Analysis
, 2008
"... Diffusion processes governed by stochastic differential equations (SDEs) are a well established tool for modelling continuous time data from a wide range of areas. Consequently, techniques have been developed to estimate diffusion parameters from partial and discrete observations. Likelihood based i ..."
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Cited by 32 (7 self)
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Diffusion processes governed by stochastic differential equations (SDEs) are a well established tool for modelling continuous time data from a wide range of areas. Consequently, techniques have been developed to estimate diffusion parameters from partial and discrete observations. Likelihood based inference can be problematic as closed form transition densities are rarely available. One widely used solution involves the introduction of latent data points between every pair of observations to allow an EulerMaruyama approximation of the true transition densities to become accurate. In recent literature, Markov chain Monte Carlo (MCMC) methods have been used to sample the posterior distribution of latent data and model parameters; however, naive schemes suffer from a mixing problem that worsens with the degree of augmentation. In this paper, we explore an MCMC scheme whose performance is not adversely affected by the number of latent values. We illustrate the methodology by estimating parameters governing an autoregulatory gene network, using partial and discrete data that is subject to measurement error.
Bayesian sequential inference for nonlinear multivariate diffusions
 Statistics and Computing
, 2006
"... In this paper, we adapt recently developed simulationbased sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m −1 latent data points between every pair of observations. Sequential ..."
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Cited by 28 (3 self)
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In this paper, we adapt recently developed simulationbased sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m −1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters online. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. shortterm interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. shortterm interest rate. 1
Estimating Functions for Discretely Sampled DiffusionType Models. Chapter of the Handbook of financial econometrics, AitSahalia and Hansen eds. http://home.uchicago.edu/ lhansen/handbook.htm Birgé
 in Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics
, 2004
"... Estimating functions provide a general framework for finding estimators and studying their properties in many different kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obta ..."
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Cited by 26 (9 self)
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Estimating functions provide a general framework for finding estimators and studying their properties in many different kinds of statistical models, including stochastic process models. An estimating function is a function of the data as well as of the parameter to be estimated. An estimator is obtained by equating the estimating function to zero and solving the resulting
Parametric Inference for Diffusion Processes Observed At Discrete Points in Time: A Survey
"... This paper is a survey of existing estimation techniques for stationary and ergodic diffusion processes observed at discrete points in time. The reader is introduced to the following techniques: (i) estimating functions with special emphasis on martingale estimating functions and socalled simple es ..."
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Cited by 24 (2 self)
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This paper is a survey of existing estimation techniques for stationary and ergodic diffusion processes observed at discrete points in time. The reader is introduced to the following techniques: (i) estimating functions with special emphasis on martingale estimating functions and socalled simple estimating functions; (ii) analytical and numerical approximations of the likelihood which can in principle be made arbitrarily accurate; (iii) Bayesian analysis and MCMC methods; and (iv) indirect inference and EMM which both introduce auxiliary (but wrong) models and correct for the implied bias by simulation
Estimation of continuoustime markov processes sampled at random time intervals
, 2004
"... We introduce a family of generalizedmethodofmoments estimators of the parameters of a continuoustime Markov process observed at random time intervals. The results include strong consistency, asymptotic normality, and a characterization of standard errors. Sampling is at an arrival intensity that ..."
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Cited by 21 (0 self)
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We introduce a family of generalizedmethodofmoments estimators of the parameters of a continuoustime Markov process observed at random time intervals. The results include strong consistency, asymptotic normality, and a characterization of standard errors. Sampling is at an arrival intensity that is allowed to depend on the underlying Markov process and on the parameter vector to be estimated. We focus on financial applications, including tickbased sampling, allowing for jump diffusions, regimeswitching diffusions, and reflected
LikelihoodBased Specification Analysis of ContinuousTime Models of the ShortTerm Interest Rate
 JOURNAL OF FINANCIAL ECONOMICS
, 2003
"... An extensive collection of continuoustime models of the shortterm interest rate are evaluated over data sets that have appeared previously in the literature. The analysis, which uses the simulated maximum likelihood procedure proposed by Durham and Gallant (1999), provides new insights regardin ..."
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Cited by 18 (0 self)
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An extensive collection of continuoustime models of the shortterm interest rate are evaluated over data sets that have appeared previously in the literature. The analysis, which uses the simulated maximum likelihood procedure proposed by Durham and Gallant (1999), provides new insights regarding several previously unresolved questions. For single factor models, I find that the volatility rather than the drift is the critical component in model specification. Allowing for additional flexibility beyond a constant term in the drift provides negligible benefit. While constant drift would appear to imply that the short rate is nonstationary, in fact stationarity is volatilityinduced. The simple constant elasticity of volatility model fits weekly observations of the threemonth Treasury bill rate remarkably well but is easily rejected when compared to more flexible volatility specifications over daily data. The methodology of Durham and Gallant can also be used to estimate stochastic volatility models. While adding the latent volatility component provides a large improvement in the likelihood for the physical process, it does little to improve bondpricing performance.