Indirect estimation of stochastic differential equation models: some computational experiments

Bianchi, Carlo; Cleur, Eugene M.

doi:10.1007/bf00121638

Cited by 13 publications

(8 citation statements)

References 18 publications

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“…For our purposes, a value δ ≤ 0.1 proved to be sufficiently accurate (we have used δ = 0.05; see on this problem also the empirical applications in Bianchi et al (1995), Broze et al (1995, and Bianchi and Cleur (1996)). A time unit corresponds to the frequency of the actually observed data.…”

Section: Square-root Processmentioning

confidence: 98%

“…Control variates for variance reduction in indirect inference C107 Taking (3.1) as a valid characterization of the process generating interest rates, regularity conditions ensure that the discretized model, with a conveniently small discretization step δ, exhibits negligible differences from the corresponding continuous-time model. For our purposes, a value δ ≤ 0.1 proved to be sufficiently accurate (we have used δ = 0.05; see on this problem also the empirical applications in Bianchi et al (1995), Broze et al (1995), and Bianchi and Cleur (1996)). A time unit corresponds to the frequency of the actually observed data.…”

Section: Square-root Processmentioning

confidence: 99%

“…Indirect estimation of diffusion processes is widely exemplified in the recent econometric literature (e.g. Bianchi and Cleur 1996, Broze et al 1994, Broze et al 1995, Pastorello et al 1994. So our discussion will be focussed on the implementation of the control variates.…”

Section: Introductionmentioning

confidence: 99%

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Control variates for variance reduction in indirect inference: Interest rate models in continuous time

Calzolari

Iorio

Fiorentini

1998

The Econometrics Journal

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Simulation estimators, such as indirect inference or simulated maximum likelihood, are successfully employed for estimating stochastic differential equations. They adjust for the bias (inconsistency) caused by discretization of the underlying stochastic process, which is in continuous time. The price to be paid is an increased variance of the estimated parameters. The variance suffers from an additional component, which depends on the stochastic simulation involved in the estimation procedure. To reduce this undesirable effect, one could increase the number of simulations (or the length of each simulation) and thus the computational cost. Alternatively, this paper shows how variance reduction can be achieved, at virtually no additional computational cost, by use of control variates. The Ornstein–Uhlenbeck process, used by Vasicek to model the short‐term interest rate in continuous time, and the square‐root process, used by Cox, Ingersoll and Ross, are explicitly considered and experimented with. Monte Carlo experiments show a global efficiency gain of almost 50% over the simple indirect estimator.

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Section: Square-root Processmentioning

confidence: 98%

Section: Square-root Processmentioning

confidence: 99%

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Control variates for variance reduction in indirect inference: Interest rate models in continuous time

Calzolari

Iorio

Fiorentini

1998

The Econometrics Journal

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“…Moreover, we show in Section 4 that Nowman's method can be useful in practice and why it makes sense to analyse its asymptotic and finite sample properties. Tang and Chen () propose the use of the bootstrap method and Bianchi and Cleur () use indirect inference. However, both the indirect inference and bootstrap methods are computationally expensive versus Nowman's method.…”

Section: Model Estimators and Asymptotic Theorymentioning

confidence: 99%

Further Results on Pseudo‐Maximum Likelihood Estimation and Testing in the Constant Elasticity of Variance Continuous Time Model

Iglesias

Phillips

2019

Journal Time Series Analysis

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Constant elasticity volatility processes have been shown to be useful, for example, to encompass a number of existing models that have closed‐form likelihood functions. In this article, we extend the existing literature in two directions: first we find explicit closed form solutions of the pseudo maximum likelihood estimators (MLEs) by discretizing the diffusion function and we provide their asymptotic theory in the context of the constant elasticity of variance (CEV) model characterized by a general CEV parameter ρ ≥ 0. Second we obtain bias expansions for those pseudo MLEs also in terms of ρ ≥ 0. We provide a general framework since only the cases with ρ = 0 and ρ = 0.5 have been considered in the literature so far. When the time series is not positive almost surely, we need to impose the restriction that ρ is a non‐negative integer.

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“…Indirect Inference was originally introduced for complex econometric models to overcome the estimation problem of an intractable likelihood function, as for continuous time models with stochastic volatility (see Bianchi & Cleur, ; Jiang, ; Laurini & Hotta, ; Raknerud & Skare, ; Wahlberg, Welsh, & Ljung, ). Indirect Inference can also be used as a vehicle to produce estimators that are robust, when there are outliers in the observations (see de Luna & Genton, , for robust estimation of a discrete time ARMA, and see Fasen‐Hartmann & Kimmig, , for robust estimation of a continuous time ARMA).…”

Section: Introductionmentioning

confidence: 99%

Indirect Inference for Lévy‐driven continuous‐time GARCH models

Sousa

Haug

Klüppelberg

2019

Scandinavian J Statistics

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We advocate the use of an Indirect Inference method to estimate the parameter of a COGARCH(1,1) process for equally spaced observations. This requires that the true model can be simulated and a reasonable estimation method for an approximate auxiliary model. We follow previous approaches and use linear projections leading to an auxiliary autoregressive model for the squared COGARCH returns. The asymptotic theory of the Indirect Inference estimator relies on a uniform strong law of large numbers and asymptotic normality of the parameter estimates of the auxiliary model, which require continuity and differentiability of the COGARCH process with respect to its parameter and which we prove via Kolmogorov's continuity criterion. This leads to consistent and asymptotically normal Indirect Inference estimates under moment conditions on the driving Lévy process. A simulation study shows that the method yields a substantial finite sample bias reduction compared with previous estimators. DO RÊGO SOUSA ET AL. with parameter (to be specified in Section 2), L is a Lévy process with Lévy measure L ≢ 0 and having càdlàg sample paths. The volatility process ( s ( )) s ≥ 0 is predictable and its stochasticity depends only on L. The COGARCH process satisfies many stylized features of financial time series and is suited for modeling high-frequency data (see In many practical problems, one observes the log-price process (P iΔ ( 0 )) n i=1 on a fixed grid of size Δ > 0 and the question of interest is how to estimate the true parameter 0 . The data used for estimation are returnsSeveral methods have been proposed to estimate the parameter of a COGARCH process. A method of moments was proposed in Haug et al. (2007), Bibbona and Negri (2015) used prediction based estimation as developed in Sørensen (2000), and Maller et al. (2008) proposed a pseudo maximum likelihood (PML) method, which also works for nonequally spaced observations. Both momentand prediction-based estimators are consistent and asymptotically normal under certain regularity conditions. The asymptotic properties of the PML estimator were studied in Iannace (2014) and in Kim and Lee (2013), which require that Δ ↓ 0 as n → ∞. For the COGARCH process, Bayracı and Ünal (2014) used Indirect Inference with an auxiliary discrete-time GARCH model with Gaussian residuals. No theoretical results were proved, but their simulation study suggests that Indirect Inference estimators (IIEs) achieve a similar performance as the PML estimator of Maller et al. (2008) for fixed Δ > 0. Furthermore, Müller (2010) proposed a Markov chain Monte Carlo method, when L is a compound Poisson process.In this paper, we advocate an Indirect Inference method, different to the one suggested in Bayracı and Ünal (2014), to estimate the COGARCH parameter and derive the asymptotic properties of the estimator. Such methods were introduced in Smith (1993) and generalized in Gouriéroux, Monfort, and Renault (1993), and they offer a way to overcome many estimation problems by a clever simulation method. In sho...

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Indirect estimation of stochastic differential equation models: some computational experiments

Cited by 13 publications

References 18 publications

Control variates for variance reduction in indirect inference: Interest rate models in continuous time

Control variates for variance reduction in indirect inference: Interest rate models in continuous time

Further Results on Pseudo‐Maximum Likelihood Estimation and Testing in the Constant Elasticity of Variance Continuous Time Model

Indirect Inference for Lévy‐driven continuous‐time GARCH models

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