Nonparametric Estimation for Second-Order Jump-Diffusion Model in High Frequency Data

Song, Yisheng

doi:10.1142/s0217590817500102

Cited by 8 publications

(19 citation statements)

References 68 publications

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“…Remark Song () showed that under the assumptions as those in this article, h n n Δ n →0 and h n = O (( n Δ n ) −1/5 ), the following result holds with symmetric kernels

\sqrt{h_{n} n {normalΔ}_{n}} ({\hat{μ}}_{n}^{N W} (x) - μ (x) - h_{n}^{2} K_{1}^{2} (\frac{1}{2} μ^{'^{'}} (x) + μ^{'} (x) \frac{φ^{'} (x)}{φ (x)})) \overset{d}{\to} N (0, K_{2}^{0} \frac{M (x)}{p (x)}),

where

{\overset{μ}{true}}_{n}^{N W} false(x false) = \frac{\sum_{i = 1}^{n} K ((), \frac{{\overset{X}{true}}_{i - 1} - x}{h_{n}}) ((), \frac{{\overset{X}{true}}_{i + 1} - {\overset{X}{true}}_{i}}{Δ_{n}})}{\sum_{i = 1}^{n} K ((), \frac{{\overset{X}{true}}_{i - 1} - x}{h_{n}})} .

When K is symmetric, for local linear estimators we obtain that

\sqrt{h_{n} n {normalΔ}_{n}} ({\hat{μ}}_{n} (x) - μ (x) - \frac{1}{2} h_{<...>})

…”

Section: Local Linear Estimators and Large Sample Propertiesmentioning

confidence: 83%

“…c ( x , y ) and f ( y ) are usually estimated together such as

\int_{ℰ} c^{2} false(x, y false) f false(y false) normald y

in M ( x ) or

\int_{ℰ} c^{4} false(x, y false) f false(y false) normald y,

which reflects the conditional expectation of jumps. The consistent estimators for p ( x ) is

{\hat{p}}_{n} (x) = \frac{1}{n h_{n}} {falsefalse}_{\sum}^{i = 1} K (\frac{{\tilde{X}}_{(i - 1) {normalΔ}_{n}} - x}{h_{n}}),

(see Song, ). The consistent estimators for M ( x ), one can refer to the estimator constructed in this article.…”

Section: Local Linear Estimators and Large Sample Propertiesmentioning

confidence: 99%

“…A simple Monte Carlo simulation experiment is constructed aimed at the finite‐sampling performance between local linear estimators constructed as and denoted as LL and Nadaraya–Watson estimators constructed in Song () denoted as NW. Throughout this section, various lengths of time intervals T ( = 50,100,500) and sample sizes n ( = 500,1000,2500) with

Δ_{n} = \frac{T}{n}

will be considered.…”

Section: Monte Carlo Simulation Studymentioning

confidence: 99%

See 2 more Smart Citations

Bias Correction Estimation for a Continuous‐Time Asset Return Model with Jumps

Song

Chen

Wang

2018

Journal Time Series Analysis

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In this article, local linear estimators are adapted for the unknown infinitesimal coefficients associated with continuous‐time asset return models with jumps, which can correct the bias automatically due to their simple bias representation. The integrated diffusion models with jumps, especially infinite activity jumps, are mainly investigated. In addition, under mild conditions, the weak consistency and asymptotic normality are provided through the conditional Lindeberg theorem as the time span T→∞ and the sample interval Δ n →0. Furthermore, our method presents advantages in bias correction through simulation whether jumps belong to the finite activity case or infinite activity case. Finally, the estimators are illustrated empirically through the returns of stock index under 5‐minute high sampling frequency for real application.

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“…Remark Song () showed that under the assumptions as those in this article, h n n Δ n →0 and h n = O (( n Δ n ) −1/5 ), the following result holds with symmetric kernels

\sqrt{h_{n} n {normalΔ}_{n}} ({\hat{μ}}_{n}^{N W} (x) - μ (x) - h_{n}^{2} K_{1}^{2} (\frac{1}{2} μ^{'^{'}} (x) + μ^{'} (x) \frac{φ^{'} (x)}{φ (x)})) \overset{d}{\to} N (0, K_{2}^{0} \frac{M (x)}{p (x)}),

where

{\overset{μ}{true}}_{n}^{N W} false(x false) = \frac{\sum_{i = 1}^{n} K ((), \frac{{\overset{X}{true}}_{i - 1} - x}{h_{n}}) ((), \frac{{\overset{X}{true}}_{i + 1} - {\overset{X}{true}}_{i}}{Δ_{n}})}{\sum_{i = 1}^{n} K ((), \frac{{\overset{X}{true}}_{i - 1} - x}{h_{n}})} .

When K is symmetric, for local linear estimators we obtain that

\sqrt{h_{n} n {normalΔ}_{n}} ({\hat{μ}}_{n} (x) - μ (x) - \frac{1}{2} h_{<...>})

…”

Section: Local Linear Estimators and Large Sample Propertiesmentioning

confidence: 83%

“…c ( x , y ) and f ( y ) are usually estimated together such as

\int_{ℰ} c^{2} false(x, y false) f false(y false) normald y

in M ( x ) or

\int_{ℰ} c^{4} false(x, y false) f false(y false) normald y,

which reflects the conditional expectation of jumps. The consistent estimators for p ( x ) is

{\hat{p}}_{n} (x) = \frac{1}{n h_{n}} {falsefalse}_{\sum}^{i = 1} K (\frac{{\tilde{X}}_{(i - 1) {normalΔ}_{n}} - x}{h_{n}}),

(see Song, ). The consistent estimators for M ( x ), one can refer to the estimator constructed in this article.…”

Section: Local Linear Estimators and Large Sample Propertiesmentioning

confidence: 99%

Δ_{n} = \frac{T}{n}

will be considered.…”

Section: Monte Carlo Simulation Studymentioning

confidence: 99%

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Bias Correction Estimation for a Continuous‐Time Asset Return Model with Jumps

Song

Chen

Wang

2018

Journal Time Series Analysis

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“…In a comparable model, Song (2017) investigated the nonparametric pointwise estimation of the unknown drift b as well as of the function σ 2 + ξ 2 in an integrated jump diffusion model using a kernel based approach. The resulting estimator is consistent and asymptotically normal distributed possessing a rate of convergence of √ n∆h.…”

Section: Estimation Of the Drift Functionmentioning

confidence: 99%

“…In contrast, empirical likelihood inference for this model has been conducted in . Moreover, a re-weighted kernel estimation procedure has been used by for estimating the function σ 2 + ξ 2 and a kernel based approach for estimating b pointwisely has been used in Song (2017).…”

Section: Introductionmentioning

confidence: 99%

Adaptive nonparametric drift estimation of an integrated jump diffusion process

Funke

Schmisser²

2018

ESAIM: PS

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In the present article, we investigate nonparametric estimation of the unknown drift function b in an integrated Lévy driven jump diffusion model. Our aim will be to estimate the drift on a compact set based on a high-frequency data sample. Instead of observing the jump diffusion process V itself, we observe a discrete and high-frequent sample of the integrated process Vsds.Based on the available observations of Xt, we will construct an adaptive penalized least-squares estimate in order to compute an adaptive estimator of the corresponding drift function b. Under appropriate assumptions, we will bound the L 2 -risk of our proposed estimator. Moreover, we study the behavior of the proposed estimator in various Monte Carlo simulation setups.MSC 2010 Classification: 62M09, 62G08

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