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128
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 115 (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 75 (15 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 58 (7 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.
On asymptotics of estimating functions
 Brazilian Journal of Probability and Statistics
, 1999
"... The asymptotic theory of estimators obtained from estimating functions is reviewed and some new results on the multivariate parameter case are presented. Specifically, results about existence of consistent estimators and about asymptotic normality of these are given. First a very general stochastic ..."
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Cited by 50 (6 self)
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The asymptotic theory of estimators obtained from estimating functions is reviewed and some new results on the multivariate parameter case are presented. Specifically, results about existence of consistent estimators and about asymptotic normality of these are given. First a very general stochastic process setting is considered. Then it is demonstrated how more specific conditions for existence of√ nconsistent and asymptotically normal estimators can be given for martingale estimating functions in the case of observations of a Markov process. Key words: asymptotic normality, consistency, diffusion processes, estimating equations, likelihood inference, Markov processes, martingale estimating functions, misspecified models, statistical inference for stochastic processes, quasi likelihood. ∗MaPhySto Centre for Mathematical Physics and Stochastics, funded by a grant from The Danish
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 45 (9 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 43 (5 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
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 39 (3 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
Estimating Functions for Discretely Sampled DiffusionType Models
 IN FESTSCHRIFT FOR LUCIEN LE CAM: RESEARCH PAPERS IN PROBABILITY AND STATISTICS
, 2004
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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 28 (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