A Nonparametric Approach to the Estimation of Diffusion Processes, With an Application to a Short-Term Interest Rate Model

Jiang, George J.; Knight, John L.

doi:10.1017/s0266466600006101

Cited by 164 publications

(127 citation statements)

References 39 publications

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“…Aït-Sahalia (1996) proposed a semiparametric procedure for estimating the diffusion function, under the parametric speci cation of the drift function. Jiang and Knight (1997) developed a nonparametric kernel estimator for the diffusion function, and then derived a consistent nonparametric drift estimator. Using an in nitesimal generator and Taylor series expansion, Stanton (1997) constructed the rst-, second-, and third-order approximation formulas for OE4¢5 and ' 4¢5 and further claimed the superiority of higher-order approximations.…”

Section: Introductionmentioning

confidence: 99%

A Reexamination of Diffusion Estimators With Applications to Financial Model Validation

Fan

Zhang

2003

Journal of the American Statistical Association

167

111

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Time-homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables. Recently, Stanton proposed drift and diffusion estimators based on a higher-order approximation scheme and kernel regression method. He claimed that "higher order approximations must outperform lower order approximations" and concluded nonlinearity in the instantaneous return function of short-term interest rates. To examine the impact of higher-order approximations, we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators. We show that these estimators will reduce the numerical approximation errors in asymptotic biases, but their asymptotic variances escalate nearly exponentially with the order of approximation. Simulation studies also con rm our asymptotic results. This variance in ation problem arises not only from nonparametric tting, but also from parametric tting. Stanton's work also postulates the interesting question of whether the short-term rate drift is nonlinear. Based on empirical simulation studies, Chapman and Pearson suggested that the nonlinearity might be spurious, due partially to the boundary effect of kernel regression. This prompts us to use the local linear t based on the rst-order approximation, proposed by Fan and Yao, to ameliorate the boundary effect and to construct formal tests of parametric nancial models against the nonparametric alternatives. Our simulation results show that the local linear method indeed outperforms the kernel approach. Furthermore, our nonparametric "generalized likelihood ratio tests" are indeed versatile and powerful in detecting nonparametric alternatives. Using this formal testing procedure, we show that the evidence against the linear drift of the short-term interest rates is weak, whereas evidence against a family of popular models for the volatility function is very strong. Application to Standard & Poor 500 data is also illustrated.

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Section: Introductionmentioning

confidence: 99%

A Reexamination of Diffusion Estimators With Applications to Financial Model Validation

Fan

Zhang

2003

Journal of the American Statistical Association

167

111

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“…As is well known, the first to consider nonparametric estimation for the diffusion coefficient in model (1) with discrete-time observation was [17], where a kernel type estimator was considered. Thereafter, [18] proposed a nonparametric identification and an estimation procedure for the diffusion function after [17], and derived a consistent nonparametric estimator for the drift function by combining their estimator of the diffusion function. Reference [19] constructed the first-, second-, and third-order approximation formulas for drift and diffusion functions by using an infinitesimal generator and Taylor expansion.…”

Section: Introductionmentioning

confidence: 99%

Variable bandwidth local maximum likelihood type estimation for diffusion processes

Tang

Wang

2018

Adv Differ Equ

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The method of robust approach is applied to estimate drift function and diffusion function of diffusion processes with discrete-time observations. The proposed method combines the ideas of local linear regression technique and maximum likelihood type estimation technique, so the advantages of local linear estimators persist and overcome the disadvantages of least-squares estimator. Moreover, a variable bandwidth instead of a constant bandwidth is considered in the local maximum likelihood type estimators. The consistency and asymptotic normality of the local maximum likelihood type estimators for drift and diffusion functions are developed under some given conditions. We perform a simulation study to evaluate the robust performances of the proposed estimators. MSC: 62G35; 62G20

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“…On the basis that the literature is divided about the appropriate drift speci…cation, we estimate both linear and nonlinear drift functions of the short rate. The estimation of a parametric drift speci…cation quali…es this approach as a semiparametric method (Jiang and Knight, 1997). To estimate the latent volatility process, we use the algorithm developed by Bühlman and McNeil (2002) and apply it to a generalised additive model of Hastie and Tibshirani (1990).…”

Section: Introductionmentioning

confidence: 99%

Modelling and Forecasting Short-Term Interest Rate Volatility: A Semiparametric Approach

Hou

Suardi

2009

SSRN Journal

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This paper employs a semiparametric procedure to estimate the di¤usion process of short-term interest rates. The Monte Carlo study shows that the semiparametric approach produces more accurate volatility estimates than models that accommodate asymmetry, levels e¤ect and serial dependence in the conditional variance. Moreover, the semiparametric approach yields robust volatility estimates even if the short rate drift function and the underlying innovation distribution are misspeci…ed. Empirical investigation with the U.S. three-month Treasury bill rates suggests that the semiparametric procedure produces superior in-sample and out-of-sample forecast of short rate changes volatility compared with the widely used single-factor di¤usion models. This forecast improvement has implications for pricing interest rate derivatives.

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A Nonparametric Approach to the Estimation of Diffusion Processes, With an Application to a Short-Term Interest Rate Model

Cited by 164 publications

References 39 publications

A Reexamination of Diffusion Estimators With Applications to Financial Model Validation

A Reexamination of Diffusion Estimators With Applications to Financial Model Validation

Variable bandwidth local maximum likelihood type estimation for diffusion processes

Modelling and Forecasting Short-Term Interest Rate Volatility: A Semiparametric Approach

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