# Parametric Estimation of Diffusion Processes: A Review and Comparative Study

^{*}

## Abstract

**:**

## 1. Introduction

`estsde`R package [24], implemented in C and C++ for the sake of efficiency.

## 2. Estimation Methods

**Assumption**

**1**

**θ**such that

**Assumption**

**2**

**θ**such that

**Assumption**

**3**

#### 2.1. Exact Maximum Likelihood

#### 2.2. Discrete Maximum Likelihood

#### 2.2.1. Euler Method

#### 2.2.2. Local Linearization

#### 2.3. Hermite Polynomial Expansion

#### 2.4. Kalman Filter

#### 2.5. Markov Chain Monte Carlo

#### 2.6. Generalized Method of Moments

## 3. Simulation Study

#### 3.1. Experimental Design

- (i)
- Exact maximum likelihood (EML);
- (ii)
- Euler method (DML);
- (iii)
- Local linearization (LL);
- (iv)
- Hermite polynomial expansion (HP);
- (v)
- Generalized Method of Moments (GMM);
- (vi)
- Kalman Filter (KF);
- (vii)
- Markov Chain Monte Carlo (MCMC).

#### 3.2. Implementation Details

#### 3.3. Discussion

- (i)
- The dynamic of the process is governed by the drift parameter $\kappa $, which determines the persistence of the process by controlling the reversion towards the unconditional mean. As $\kappa \to 0$, mean reversion goes to zero and correlation between observations approaches one. This increases persistence, which introduces sample bias in parametric estimation [45]. Simulations show that increasing $\kappa $ lowered persistence and, therefore, estimation bias was also diminished (see Figure 2). High persistence scenarios, near unit-root cases, revealed significant estimation bias in the drift parameter $\kappa $, but almost negligible in the diffusion parameter $\sigma $.
- (ii)
- Increasing the volatility parameters had minor effect on the estimators performance. In the Vasicek model, higher values in the volatility parameter slightly increased RMSE in the estimation of $\sigma $. On the other hand, the estimation of the parameters in the CKLS diffusion function benefit from richer volatility dynamics, reducing RMSE.
- (iii)
- In discrete time series, the bias and variance of estimators is controlled by the sample size n, so that they reduce as $n\to \infty $. In continuous-time models sampled at discrete time points, bias and variance in the estimation of the drift parameter $\kappa $ is dominated by the total observation time $T=n\Delta $. Under quite general conditions, the estimators of the drift parameters are of order $\mathcal{O}\left({T}^{-1}\right)$, while the diffusion parameter $\sigma $ is of order ${n}^{-1}$[46]. The simulated scenarios corroborate this, as estimation bias with $T=50$ and 43 years were close despite the different frequency (weekly and monthly, respectively) and sample size n (2600 and 520, respectively).
- (iv)
- Discretization bias arises in DML and KF methods in scenarios with low sampling frequency and low persistence (see Figure 3), and correcting the DML estimates with local linearization does not always correct the bias and both $\kappa $ and $\sigma $ are underestimated.
- (v)
- There appears to be similar estimation bias in the drift parameters for both Vasicek and CKLS models. However, the more flexible parametric form of the CKLS volatility function makes estimation more challenging, and bias and RMSE for those parameters are higher than for the Vasicek model.
- (vi)
- Regarding efficiency, as exact ML is available for the Vasicek model, it can be regarded as a benchmark. Overall, the estimations of the parameters are close to the EML performance, being the GMM the less efficient. The estimations differ when discretization bias arises.

## 4. Application to Euribor Series

## 5. Conclusions

`estsde`R package.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**a**) Marginal density for Vasicek model, scenario 1 (3, 5 and 7) and 2 (4, 6 and 8) and (

**b**) Marginal density for CKLS model, scenario 1 and 2.

**Table A1.**Monte Carlo simulation for Vasicek model with ${(\mu ,\kappa ,\sigma )}^{\prime}={(0.09,0.2,0.02)}^{\prime}$. Boldfaces denote the best results in terms of bias, standard deviation and RMSE.

Scenario 2 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

EML | $\widehat{\mu}$ | $\mathbf{0.0938}$ | $\mathbf{0.0363}$ | $\mathbf{0.0365}$ | $\mathbf{0.0900}$ | $\mathbf{0.0145}$ | $\mathbf{0.0145}$ | $\mathbf{0.0906}$ | $\mathbf{0.0158}$ | $\mathbf{0.0158}$ |

$\widehat{\kappa}$ | $0.6900$ | $0.4817$ | $0.6871$ | $0.2862$ | $0.1283$ | $0.1546$ | $0.2972$ | $0.1388$ | $0.1695$ | |

$\widehat{\sigma}$ | $0.0200$ | $6.27\times {10}^{-4}$ | $6.28\times {10}^{-4}$ | $\mathbf{0.0200}$ | $2.63\times {10}^{-4}$ | $\mathbf{2.63}\times {\mathbf{10}}^{-\mathbf{4}}$ | $\mathbf{0.0200}$ | $6.28\times {10}^{-4}$ | $6.29\times {10}^{-4}$ | |

DML | $\widehat{\mu}$ | $0.0938$ | $0.0366$ | $0.0368$ | $0.0899$ | $0.0146$ | $0.0146$ | $0.0906$ | $0.0158$ | $0.0158$ |

$\widehat{\kappa}$ | $0.6812$ | $\mathbf{0.4715}$ | $0.6737$ | $0.2838$ | $0.1262$ | $0.1515$ | $0.2925$ | $\mathbf{0.1345}$ | $\mathbf{0.1632}$ | |

$\widehat{\sigma}$ | $0.0199$ | $6.22\times {10}^{-4}$ | $6.29\times {10}^{-4}$ | $0.0200$ | $2.62\times {10}^{-4}$ | $2.65\times {10}^{-4}$ | $0.0198$ | $6.17\times {10}^{-4}$ | $6.51\times {10}^{-4}$ | |

LL | $\widehat{\mu}$ | $0.0943$ | $0.0412$ | $0.0415$ | $0.0897$ | $0.0206$ | $0.0206$ | $0.0892$ | $0.0235$ | $0.0235$ |

$\widehat{\kappa}$ | $\mathbf{0.6781}$ | $0.4758$ | $0.6745$ | $\mathbf{0.2796}$ | $0.1302$ | $0.1526$ | $\mathbf{0.2856}$ | $0.1436$ | $0.1672$ | |

$\widehat{\sigma}$ | $0.0199$ | $6.18\times {10}^{-4}$ | $6.23\times {10}^{-4}$ | $0.0200$ | $2.62\times {10}^{-4}$ | $2.64\times {10}^{-4}$ | $0.0198$ | $6.21\times {10}^{-4}$ | $6.56\times {10}^{-4}$ | |

HP | $\widehat{\mu}$ | $0.0938$ | $0.0363$ | $0.0365$ | $0.0900$ | $0.0145$ | $0.0145$ | $0.0906$ | $0.0158$ | $0.0158$ |

$\widehat{\kappa}$ | $0.6875$ | $0.4784$ | $0.6830$ | $0.2858$ | $0.1277$ | $0.1538$ | $0.2972$ | $0.139$ | $0.1696$ | |

$\widehat{\sigma}$ | $\mathbf{0.0200}$ | $6.27\times {10}^{-4}$ | $6.28\times {10}^{-4}$ | $0.0200$ | $2.63\times {10}^{-4}$ | $2.63\times {10}^{-4}$ | $0.0200$ | $6.28\times {10}^{-4}$ | $6.29\times {10}^{-4}$ | |

KF | $\widehat{\mu}$ | $0.0939$ | $0.0370$ | $0.0372$ | $0.0900$ | $0.0146$ | $0.0146$ | $0.0906$ | $0.0158$ | $0.0158$ |

$\widehat{\kappa}$ | $0.6804$ | $0.4719$ | $\mathbf{0.6733}$ | $0.2836$ | $\mathbf{0.1261}$ | $\mathbf{0.1513}$ | $0.2924$ | $0.1346$ | $0.1633$ | |

$\widehat{\sigma}$ | $0.0199$ | $\mathbf{6.18}\times {\mathbf{10}}^{-\mathbf{4}}$ | $\mathbf{6.21}\times {\mathbf{10}}^{-\mathbf{4}}$ | $0.0200$ | $\mathbf{2.62}\times {\mathbf{10}}^{-\mathbf{4}}$ | $2.64\times {10}^{-4}$ | $0.0198$ | $\mathbf{6.15}\times {\mathbf{10}}^{-\mathbf{4}}$ | $6.38\times {10}^{-4}$ | |

MCMC | $\widehat{\mu}$ | $0.0941$ | $0.0424$ | $0.0426$ | $0.0899$ | $0.0146$ | $0.0146$ | $0.0906$ | $0.0158$ | $0.0158$ |

$\widehat{\kappa}$ | $0.6899$ | $0.482$ | $0.6873$ | $0.2848$ | $0.1271$ | $0.1528$ | $0.2977$ | $0.1389$ | $0.1699$ | |

$\widehat{\sigma}$ | $0.0201$ | $6.28\times {10}^{-4}$ | $6.33\times {10}^{-4}$ | $0.0200$ | $2.64\times {10}^{-4}$ | $2.65\times {10}^{-4}$ | $0.0201$ | $6.24\times {10}^{-4}$ | $\mathbf{6.26}\times {\mathbf{10}}^{-\mathbf{4}}$ | |

GMM | $\widehat{\mu}$ | $0.0960$ | $0.0578$ | $0.0581$ | $0.0900$ | $0.0146$ | $0.0146$ | $0.0906$ | $0.0172$ | $0.0172$ |

$\widehat{\kappa}$ | $0.7161$ | $0.5101$ | $0.7257$ | $0.2856$ | $0.1275$ | $0.1536$ | $0.3081$ | $0.1462$ | $0.1818$ | |

$\widehat{\sigma}$ | $0.0199$ | $7.14\times {10}^{-4}$ | $7.22\times {10}^{-4}$ | $0.0200$ | $2.67\times {10}^{-4}$ | $2.68\times {10}^{-4}$ | $0.0199$ | $7.09\times {10}^{-4}$ | $7.20\times {10}^{-4}$ |

**Table A2.**Monte Carlo simulation for Vasicek model with ${(\mu ,\kappa ,\sigma )}^{\prime}={(0.09,0.9,0.0424)}^{\prime}$.

Scenario 4 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

EML | $\widehat{\mu}$ | $0.0907$ | $0.0155$ | $0.0155$ | $\mathbf{0.0900}$ | $\mathbf{0.0068}$ | $\mathbf{0.0068}$ | $\mathbf{0.0903}$ | $\mathbf{0.0074}$ | $\mathbf{0.0074}$ |

$\widehat{\kappa}$ | $1.3214$ | $0.6079$ | $0.7397$ | $0.9757$ | $0.2141$ | $0.2271$ | $0.9877$ | $0.2383$ | $0.2539$ | |

$\widehat{\sigma}$ | $0.0425$ | $1.33\times {10}^{-3}$ | $1.34\times {10}^{-3}$ | $0.0425$ | $5.59\times {10}^{-4}$ | $5.61\times {10}^{-4}$ | $0.0425$ | $1.37\times {10}^{-3}$ | $1.37\times {10}^{-3}$ | |

DML | $\widehat{\mu}$ | $0.0906$ | $0.0155$ | $0.0155$ | $0.0900$ | $0.0068$ | $0.0068$ | $0.0903$ | $0.0074$ | $0.0074$ |

$\widehat{\kappa}$ | $1.2997$ | $\mathbf{0.5882}$ | $0.7112$ | $0.9631$ | $\mathbf{0.2085}$ | $\mathbf{0.2178}$ | $0.9465$ | $0.2178$ | $0.2227$ | |

$\widehat{\sigma}$ | $0.0420$ | $1.31\times {10}^{-3}$ | $1.39\times {10}^{-3}$ | $0.0421$ | $5.52\times {10}^{-4}$ | $6.57\times {10}^{-4}$ | $0.0408$ | $1.27\times {10}^{-3}$ | $2.06\times {10}^{-3}$ | |

LL | $\widehat{\mu}$ | $0.0888$ | $0.0190$ | $0.0190$ | $0.0876$ | $0.0106$ | $0.0109$ | $0.0884$ | $0.0107$ | $0.0109$ |

$\widehat{\kappa}$ | $\mathbf{1.2717}$ | $0.6329$ | $0.7340$ | $\mathbf{0.8910}$ | $0.2752$ | $0.2754$ | $\mathbf{0.9157}$ | $0.2766$ | $0.2770$ | |

$\widehat{\sigma}$ | $0.0419$ | $\mathbf{1.31}\times {\mathbf{10}}^{-\mathbf{3}}$ | $1.40\times {10}^{-3}$ | $0.0420$ | $5.56\times {10}^{-4}$ | $6.84\times {10}^{-4}$ | $0.0406$ | $1.28\times {10}^{-3}$ | $2.21\times {10}^{-3}$ | |

HP | $\widehat{\mu}$ | $0.0907$ | $0.0155$ | $0.0155$ | $0.0900$ | $0.0068$ | $0.0068$ | $0.0903$ | $0.0074$ | $0.0074$ |

$\widehat{\kappa}$ | $1.3207$ | $0.6077$ | $0.7391$ | $0.9739$ | $0.2131$ | $0.2256$ | $0.9886$ | $0.2382$ | $0.2542$ | |

$\widehat{\sigma}$ | $0.0425$ | $1.33\times {10}^{-3}$ | $1.33\times {10}^{-3}$ | $0.0425$ | $5.60\times {10}^{-4}$ | $5.61\times {10}^{-4}$ | $0.0425$ | $1.37\times {10}^{-3}$ | $1.37\times {10}^{-3}$ | |

KF | $\widehat{\mu}$ | $0.0907$ | $0.0155$ | $0.0155$ | $0.0900$ | $0.0068$ | $0.0068$ | $0.0903$ | $0.0074$ | $0.0074$ |

$\widehat{\kappa}$ | $1.2992$ | $0.5886$ | $\mathbf{0.7112}$ | $0.9635$ | $0.2085$ | $0.2180$ | $0.9465$ | $\mathbf{0.2178}$ | $\mathbf{0.2227}$ | |

$\widehat{\sigma}$ | $0.0420$ | $1.31\times {10}^{-3}$ | $1.39\times {10}^{-3}$ | $0.0421$ | $\mathbf{5.52}\times {\mathbf{10}}^{-\mathbf{4}}$ | $6.49\times {10}^{-4}$ | $0.0408$ | $\mathbf{1.27}\times {\mathbf{10}}^{-\mathbf{3}}$ | $2.06\times {10}^{-3}$ | |

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

MCMC | $\widehat{\mu}$ | $\mathbf{0.0906}$ | $\mathbf{0.0154}$ | $\mathbf{0.0155}$ | $0.0900$ | $0.0068$ | $0.0068$ | $0.0903$ | $0.0074$ | $0.0074$ |

$\widehat{\kappa}$ | $1.3239$ | $0.6069$ | $0.7403$ | $0.9722$ | $0.2120$ | $0.2240$ | $0.9867$ | $0.2358$ | $0.2513$ | |

$\widehat{\sigma}$ | $\mathbf{0.0425}$ | $1.32\times {10}^{-3}$ | $\mathbf{1.33}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{0.0424}$ | $5.57\times {10}^{-4}$ | $\mathbf{5.57}\times {\mathbf{10}}^{-\mathbf{4}}$ | $0.0423$ | $1.34\times {10}^{-3}$ | $\mathbf{1.35}\times {\mathbf{10}}^{-\mathbf{3}}$ | |

GMM | $\widehat{\mu}$ | $0.0907$ | $0.0179$ | $0.0179$ | $0.0900$ | $0.0069$ | $0.0069$ | $0.0904$ | $0.0090$ | $0.0091$ |

$\widehat{\kappa}$ | $1.4695$ | $0.7586$ | $0.9486$ | $0.9716$ | $0.2144$ | $0.2260$ | $1.0457$ | $0.3062$ | $0.3390$ | |

$\widehat{\sigma}$ | $0.0423$ | $1.64\times {10}^{-3}$ | $1.65\times {10}^{-3}$ | $0.0423$ | $5.72\times {10}^{-4}$ | $5.97\times {10}^{-4}$ | $\mathbf{0.0424}$ | $1.77\times {10}^{-3}$ | $1.77\times {10}^{-3}$ |

**Table A3.**Monte Carlo simulation for Vasicek model with ${(\mu ,\kappa ,\sigma )}^{\prime}={(0.09,5,0.1)}^{\prime}$.

Scenario 6 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

EML | $\widehat{\mu}$ | $\mathbf{0.0902}$ | $\mathbf{0.0065}$ | $\mathbf{0.0065}$ | $\mathbf{0.0900}$ | $\mathbf{0.0029}$ | $\mathbf{0.0029}$ | $\mathbf{0.0901}$ | $\mathbf{0.0031}$ | $\mathbf{0.0031}$ |

$\widehat{\kappa}$ | $5.3788$ | $1.1529$ | $1.2136$ | $5.0732$ | $0.4992$ | $0.5045$ | $5.0939$ | $0.6115$ | $0.6186$ | |

$\widehat{\sigma}$ | $0.1002$ | $0.0033$ | $0.0033$ | $0.1001$ | $0.0014$ | $0.0014$ | $\mathbf{0.1001}$ | $0.0038$ | $\mathbf{0.0038}$ | |

DML | $\widehat{\mu}$ | $0.0902$ | $0.0065$ | $0.0065$ | $0.0900$ | $0.0029$ | $0.0029$ | $0.0901$ | $0.0031$ | $0.0031$ |

$\widehat{\kappa}$ | $5.0997$ | $1.0319$ | $\mathbf{1.0367}$ | $4.8317$ | $0.4518$ | $0.4822$ | $4.1421$ | $0.3959$ | $0.9448$ | |

$\widehat{\sigma}$ | $0.0952$ | $0.0030$ | $0.0056$ | $0.0954$ | $0.0013$ | $0.0048$ | $0.0822$ | $\mathbf{0.0026}$ | $0.0180$ | |

LL | $\widehat{\mu}$ | $0.0892$ | $0.0074$ | $0.0074$ | $0.0889$ | $0.0034$ | $0.0036$ | $0.0891$ | $0.0033$ | $0.0035$ |

$\widehat{\kappa}$ | $\mathbf{5.0206}$ | $1.2585$ | $1.2587$ | $4.6468$ | $0.6269$ | $0.7195$ | $4.6304$ | $0.6236$ | $0.7249$ | |

$\widehat{\sigma}$ | $0.0946$ | $0.0030$ | $0.0061$ | $0.0948$ | $0.0013$ | $0.0053$ | $0.0810$ | $0.0025$ | $0.0192$ | |

HP | $\widehat{\mu}$ | $0.0902$ | $0.0065$ | $0.0065$ | $0.0900$ | $0.0029$ | $0.0029$ | $0.0899$ | $0.0033$ | $0.0033$ |

$\widehat{\kappa}$ | $5.3814$ | $1.1493$ | $1.2109$ | $5.0743$ | $0.4977$ | $0.5032$ | $5.1696$ | $0.7052$ | $0.7253$ | |

$\widehat{\sigma}$ | $0.1002$ | $0.0033$ | $0.0033$ | $0.1001$ | $0.0014$ | $0.0014$ | $0.0996$ | $0.0041$ | $0.0041$ | |

KF | $\widehat{\mu}$ | $0.0902$ | $0.0065$ | $0.0065$ | $0.0900$ | $0.0029$ | $0.0029$ | $0.0901$ | $0.0031$ | $0.0031$ |

$\widehat{\kappa}$ | $5.1001$ | $\mathbf{1.0318}$ | $1.0367$ | $4.8322$ | $\mathbf{0.4516}$ | $\mathbf{0.4818}$ | $4.1418$ | $\mathbf{0.3957}$ | $0.9450$ | |

$\widehat{\sigma}$ | $0.0952$ | $\mathbf{0.0030}$ | $0.0056$ | $0.0954$ | $\mathbf{0.0013}$ | $0.0048$ | $0.0822$ | $0.0026$ | $0.0180$ | |

MCMC | $\widehat{\mu}$ | $0.0902$ | $0.0065$ | $0.0065$ | $0.0900$ | $0.0029$ | $0.0029$ | $0.0901$ | $0.0031$ | $0.0031$ |

$\widehat{\kappa}$ | $5.3614$ | $1.1402$ | $1.1961$ | $\mathbf{5.0311}$ | $0.4889$ | $0.4899$ | $\mathbf{4.9286}$ | $0.5693$ | $\mathbf{0.5737}$ | |

$\widehat{\sigma}$ | $0.0995$ | $0.0032$ | $\mathbf{0.0032}$ | $0.0992$ | $0.0014$ | $0.0016$ | $0.0965$ | $0.0034$ | $0.0049$ | |

GMM | $\widehat{\mu}$ | $0.0904$ | $0.0084$ | $0.0084$ | $0.0900$ | $0.0029$ | $0.0029$ | $0.0901$ | $0.0041$ | $0.0041$ |

$\widehat{\kappa}$ | $5.6980$ | $1.5548$ | $1.7043$ | $4.9583$ | $0.4878$ | $0.4895$ | $5.0970$ | $0.7941$ | $0.8000$ | |

$\widehat{\sigma}$ | $\mathbf{0.1000}$ | $0.0043$ | $0.0043$ | $\mathbf{0.1000}$ | $0.0014$ | $\mathbf{0.0014}$ | $0.0997$ | $0.0047$ | $0.0047$ |

**Table A4.**Monte Carlo simulation for Vasicek model with ${(\mu ,\kappa ,\sigma )}^{\prime}={(0.09,9.8,0.14)}^{\prime}$.

Scenario 8 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

EML | $\widehat{\mu}$ | $0.0902$ | $\mathbf{0.0046}$ | $\mathbf{0.0046}$ | $\mathbf{0.0900}$ | $\mathbf{0.0021}$ | $\mathbf{0.0021}$ | $\mathbf{0.0901}$ | $\mathbf{0.0023}$ | $\mathbf{0.0023}$ |

$\widehat{\kappa}$ | $10.1822$ | $1.6165$ | $1.6610$ | $9.8776$ | $0.7359$ | $0.7400$ | $\mathbf{9.9341}$ | $1.0861$ | $1.0943$ | |

$\widehat{\sigma}$ | $0.1402$ | $0.0048$ | $\mathbf{0.0048}$ | $0.1401$ | $0.0021$ | $\mathbf{0.0021}$ | $\mathbf{0.1403}$ | $0.0064$ | $\mathbf{0.0064}$ | |

DML | $\widehat{\mu}$ | $0.0902$ | $0.0046$ | $0.0046$ | $0.0900$ | $0.0021$ | $0.0021$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $9.2274$ | $\mathbf{1.3166}$ | $\mathbf{1.4357}$ | $8.9929$ | $\mathbf{0.6049}$ | $1.0086$ | $6.7332$ | $0.4699$ | $3.1026$ | |

$\widehat{\sigma}$ | $0.1275$ | $\mathbf{0.0040}$ | $0.0131$ | $0.1278$ | $\mathbf{0.0017}$ | $0.0123$ | $0.0981$ | $0.0030$ | $0.0421$ | |

LL | $\widehat{\mu}$ | $0.0892$ | $0.0050$ | $0.0051$ | $0.0890$ | $0.0023$ | $0.0025$ | $0.0890$ | $0.0024$ | $0.0026$ |

$\widehat{\kappa}$ | $9.4201$ | $1.7549$ | $1.7955$ | $9.0671$ | $0.8327$ | $1.1093$ | $8.7862$ | $1.0441$ | $1.4553$ | |

$\widehat{\sigma}$ | $0.1263$ | $0.0039$ | $0.0143$ | $0.1265$ | $0.0017$ | $0.0136$ | $0.0966$ | $0.0031$ | $0.0435$ | |

HP | $\widehat{\mu}$ | $\mathbf{0.0901}$ | $0.0047$ | $0.0047$ | $0.0900$ | $0.0021$ | $0.0021$ | $0.0898$ | $0.0027$ | $0.0027$ |

$\widehat{\kappa}$ | $10.1842$ | $1.6113$ | $1.6564$ | $\mathbf{9.8722}$ | $0.7334$ | $0.7369$ | $9.5056$ | $1.2285$ | $1.2632$ | |

$\widehat{\sigma}$ | $\mathbf{0.1401}$ | $0.0049$ | $0.0049$ | $\mathbf{0.1401}$ | $0.0021$ | $0.0021$ | $0.1339$ | $0.0075$ | $0.0097$ | |

KF | $\widehat{\mu}$ | $0.0902$ | $0.0046$ | $0.0046$ | $0.0900$ | $0.0021$ | $0.0021$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $9.2281$ | $1.3174$ | $1.4362$ | $8.9941$ | $0.6060$ | $1.0083$ | $6.7348$ | $\mathbf{0.4690}$ | $3.1009$ | |

$\widehat{\sigma}$ | $0.1275$ | $0.0040$ | $0.0131$ | $0.1278$ | $0.0017$ | $0.0123$ | $0.0981$ | $\mathbf{0.0030}$ | $0.0421$ | |

MCMC | $\widehat{\mu}$ | $0.0902$ | $0.0046$ | $0.0046$ | $0.0900$ | $0.0021$ | $0.0021$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $\mathbf{10.0552}$ | $1.5661$ | $1.5867$ | $9.7084$ | $0.7076$ | $\mathbf{0.7135}$ | $9.2813$ | $0.9545$ | $\mathbf{1.0864}$ | |

$\widehat{\sigma}$ | $0.1381$ | $0.0046$ | $0.0050$ | $0.1376$ | $0.0020$ | $0.0031$ | $0.1303$ | $0.0053$ | $0.0110$ | |

GMM | $\widehat{\mu}$ | $0.0903$ | $0.0064$ | $0.0064$ | $0.0900$ | $0.0021$ | $0.0021$ | $0.0901$ | $0.0030$ | $0.0030$ |

$\widehat{\kappa}$ | $10.4527$ | $2.1184$ | $2.2167$ | $9.4325$ | $0.7327$ | $0.8197$ | $9.6680$ | $1.4521$ | $1.4581$ | |

$\widehat{\sigma}$ | $0.1399$ | $0.0062$ | $0.0062$ | $0.1395$ | $0.0021$ | $0.0021$ | $0.1384$ | $0.0079$ | $0.0081$ |

**Table A5.**Monte Carlo simulation for CKLS model with ${(\mu ,\kappa ,\sigma ,\gamma )}^{\prime}={(0.09,0.2,1,1.5)}^{\prime}$.

Scenario 2 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

DML | $\widehat{\mu}$ | $0.0984$ | $0.0933$ | $0.0937$ | $0.1002$ | $0.0687$ | $0.0695$ | $0.1086$ | $0.0974$ | $0.0991$ |

$\widehat{\kappa}$ | $\mathbf{0.6577}$ | $\mathbf{0.4887}$ | $\mathbf{0.6696}$ | $0.2807$ | $0.1567$ | $0.1762$ | $\mathbf{0.2860}$ | $0.1679$ | $\mathbf{0.1887}$ | |

$\widehat{\sigma}$ | $1.0244$ | $0.3720$ | $0.3728$ | $\mathbf{0.9944}$ | $0.1044$ | $0.1045$ | $0.9690$ | $0.2392$ | $0.2412$ | |

$\widehat{\gamma}$ | $1.4873$ | $0.1372$ | $0.1378$ | $1.4956$ | $0.0407$ | $0.0409$ | $1.4771$ | $0.0958$ | $0.0985$ | |

LL | $\widehat{\mu}$ | $\mathbf{0.0977}$ | $0.0926$ | $0.0929$ | $0.0994$ | $0.0605$ | $0.0612$ | $0.1102$ | $0.1213$ | $0.1229$ |

$\widehat{\kappa}$ | $0.6650$ | $0.4984$ | $0.6816$ | $0.2819$ | $0.1580$ | $0.1780$ | $0.2916$ | $0.1734$ | $0.1961$ | |

$\widehat{\sigma}$ | $1.0638$ | $0.3859$ | $0.3911$ | $1.0074$ | $\mathbf{0.1034}$ | $\mathbf{0.1037}$ | $1.0247$ | $\mathbf{0.2361}$ | $\mathbf{0.2374}$ | |

$\widehat{\gamma}$ | $1.5038$ | $0.1350$ | $0.1351$ | $1.5019$ | $\mathbf{0.0395}$ | $\mathbf{0.0396}$ | $1.5057$ | $\mathbf{0.0871}$ | $\mathbf{0.0873}$ | |

HP | $\widehat{\mu}$ | $0.0988$ | $0.1091$ | $0.1095$ | $0.0997$ | $0.0616$ | $0.0623$ | $0.1133$ | $0.1425$ | $0.1444$ |

$\widehat{\kappa}$ | $0.6608$ | $0.4932$ | $0.6750$ | $0.2812$ | $0.1575$ | $0.1772$ | $0.2890$ | $0.1724$ | $0.1941$ | |

$\widehat{\sigma}$ | $1.0706$ | $0.4013$ | $0.4075$ | $1.0061$ | $0.1040$ | $0.1042$ | $\mathbf{1.0223}$ | $0.2437$ | $0.2447$ | |

$\widehat{\gamma}$ | $1.5032$ | $\mathbf{0.1327}$ | $\mathbf{0.1328}$ | $\mathbf{1.4999}$ | $0.0399$ | $0.0399$ | $1.4977$ | $0.0899$ | $0.0899$ | |

KF | $\widehat{\mu}$ | $0.0980$ | $\mathbf{0.0904}$ | $\mathbf{0.0908}$ | $0.1003$ | $0.0692$ | $0.0699$ | $0.1088$ | $0.0992$ | $0.1010$ |

$\widehat{\kappa}$ | $0.6579$ | $0.4888$ | $0.6698$ | $\mathbf{0.2805}$ | $\mathbf{0.1566}$ | $\mathbf{0.1761}$ | $0.2861$ | $0.1680$ | $0.1888$ | |

$\widehat{\sigma}$ | $\mathbf{1.0241}$ | $\mathbf{0.3719}$ | $\mathbf{0.3726}$ | $0.9943$ | $0.1042$ | $0.1044$ | $0.9688$ | $0.2394$ | $0.2414$ | |

$\widehat{\gamma}$ | $1.4872$ | $0.1372$ | $0.1378$ | $1.4956$ | $0.0406$ | $0.0408$ | $1.4770$ | $0.0959$ | $0.0986$ | |

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

MCMC | $\widehat{\mu}$ | $0.0988$ | $0.1054$ | $0.1057$ | $0.1003$ | $0.0689$ | $0.0697$ | $0.1109$ | $0.1198$ | $0.1216$ |

$\widehat{\kappa}$ | $0.6651$ | $0.4972$ | $0.6808$ | $0.2813$ | $0.1576$ | $0.1773$ | $0.2888$ | $0.1721$ | $0.1936$ | |

$\widehat{\sigma}$ | $1.1357$ | $0.5257$ | $0.5429$ | $1.0122$ | $0.1140$ | $0.1146$ | $1.0606$ | $0.2895$ | $0.2957$ | |

$\widehat{\gamma}$ | $\mathbf{1.4998}$ | $0.1595$ | $0.1595$ | $1.5002$ | $0.0430$ | $0.0430$ | $\mathbf{1.4999}$ | $0.0999$ | $0.0999$ | |

GMM | $\widehat{\mu}$ | $0.1003$ | $0.1872$ | $0.1875$ | $\mathbf{0.0882}$ | $\mathbf{0.0218}$ | $\mathbf{0.0219}$ | $\mathbf{0.0890}$ | $\mathbf{0.0232}$ | $\mathbf{0.0233}$ |

$\widehat{\kappa}$ | $0.8836$ | $0.4953$ | $0.8442$ | $0.4766$ | $0.1604$ | $0.3198$ | $0.4813$ | $\mathbf{0.1579}$ | $0.3226$ | |

$\widehat{\sigma}$ | $0.9625$ | $0.4548$ | $0.4563$ | $0.9631$ | $0.1923$ | $0.1958$ | $0.8381$ | $0.3223$ | $0.3606$ | |

$\widehat{\gamma}$ | $1.4440$ | $0.1787$ | $0.1873$ | $1.4690$ | $0.0889$ | $0.0942$ | $1.3820$ | $0.1609$ | $0.1995$ |

**Table A6.**Monte Carlo simulation for CKLS model with ${(\mu ,\kappa ,\sigma ,\gamma )}^{\prime}={(0.09,0.9,1.414,1.5)}^{\prime}$.

Scenario 4 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

DML | $\widehat{\mu}$ | $0.0965$ | $0.0625$ | $0.0628$ | $0.0907$ | $0.0078$ | $0.0079$ | $0.0909$ | $0.0080$ | $0.0081$ |

$\widehat{\kappa}$ | $1.2752$ | $0.6674$ | $0.7657$ | $\mathbf{0.9594}$ | $\mathbf{0.2535}$ | $\mathbf{0.2604}$ | $\mathbf{0.9493}$ | $\mathbf{0.2692}$ | $\mathbf{0.2737}$ | |

$\widehat{\sigma}$ | $1.3874$ | $\mathbf{0.4298}$ | $\mathbf{0.4307}$ | $1.3765$ | $0.1661$ | $0.1703$ | $1.2486$ | $0.3849$ | $0.4190$ | |

$\widehat{\gamma}$ | $1.4762$ | $0.1241$ | $0.1263$ | $1.4876$ | $0.0488$ | $0.0503$ | $1.4382$ | $0.1245$ | $0.1390$ | |

LL | $\widehat{\mu}$ | $0.0953$ | $\mathbf{0.0421}$ | $\mathbf{0.0424}$ | $0.0904$ | $0.0078$ | $0.0078$ | $\mathbf{0.0902}$ | $\mathbf{0.0078}$ | $\mathbf{0.0078}$ |

$\widehat{\kappa}$ | $1.2993$ | $0.6905$ | $0.7976$ | $0.9759$ | $0.2601$ | $0.2709$ | $1.0009$ | $0.2939$ | $0.3107$ | |

$\widehat{\sigma}$ | $1.4853$ | $0.4496$ | $0.4552$ | $1.4397$ | $0.1668$ | $\mathbf{0.1688}$ | $1.4915$ | $0.3856$ | $\mathbf{0.3933}$ | |

$\widehat{\gamma}$ | $1.5076$ | $\mathbf{0.1184}$ | $\mathbf{0.1187}$ | $1.5085$ | $\mathbf{0.0468}$ | $\mathbf{0.0476}$ | $1.5265$ | $\mathbf{0.1048}$ | $\mathbf{0.1081}$ | |

HP | $\widehat{\mu}$ | $0.0955$ | $0.0441$ | $0.0444$ | $0.0912$ | $0.0150$ | $0.0150$ | $0.0925$ | $0.0452$ | $0.0453$ |

$\widehat{\kappa}$ | $1.2912$ | $0.6911$ | $0.7941$ | $0.9664$ | $0.2584$ | $0.2668$ | $0.9838$ | $0.2968$ | $0.3084$ | |

$\widehat{\sigma}$ | $1.4823$ | $0.4685$ | $0.4734$ | $\mathbf{1.4308}$ | $0.1770$ | $0.1778$ | $\mathbf{1.4720}$ | $0.4144$ | $0.4185$ | |

$\widehat{\gamma}$ | $\mathbf{1.4998}$ | $0.1209$ | $0.1209$ | $\mathbf{1.5010}$ | $0.0494$ | $0.0494$ | $\mathbf{1.4989}$ | $0.1126$ | $0.1126$ | |

KF | $\widehat{\mu}$ | $0.0963$ | $0.0576$ | $0.0580$ | $0.0907$ | $0.0078$ | $0.0079$ | $0.0909$ | $0.0080$ | $0.0081$ |

$\widehat{\kappa}$ | $\mathbf{1.2743}$ | $0.6663$ | $\mathbf{0.7643}$ | $0.9600$ | $0.2537$ | $0.2607$ | $0.9493$ | $0.2694$ | $0.2738$ | |

$\widehat{\sigma}$ | $\mathbf{1.3874}$ | $0.4299$ | $0.4308$ | $1.3757$ | $\mathbf{0.1659}$ | $0.1704$ | $1.2482$ | $\mathbf{0.3846}$ | $0.4189$ | |

$\widehat{\gamma}$ | $1.4762$ | $0.1240$ | $0.1263$ | $1.4874$ | $0.0488$ | $0.0504$ | $1.4381$ | $0.1245$ | $0.1390$ | |

MCMC | $\widehat{\mu}$ | $0.0961$ | $0.0494$ | $0.0498$ | $0.0907$ | $0.0078$ | $0.0079$ | $0.0911$ | $0.0081$ | $0.0082$ |

$\widehat{\kappa}$ | $1.2958$ | $0.6871$ | $0.7930$ | $0.9665$ | $0.2570$ | $0.2654$ | $0.9799$ | $0.2891$ | $0.3000$ | |

$\widehat{\sigma}$ | $1.5353$ | $0.5573$ | $0.5703$ | $1.4465$ | $0.1839$ | $0.1867$ | $1.5296$ | $0.4573$ | $0.4717$ | |

$\widehat{\gamma}$ | $1.4944$ | $0.1345$ | $0.1347$ | $1.5036$ | $0.0512$ | $0.0513$ | $1.5040$ | $0.1198$ | $0.1199$ | |

GMM | $\widehat{\mu}$ | $\mathbf{0.0952}$ | $0.1351$ | $0.1352$ | $\mathbf{0.0899}$ | $\mathbf{0.0076}$ | $\mathbf{0.0076}$ | $0.0903$ | $0.0078$ | $0.0078$ |

$\widehat{\kappa}$ | $1.6327$ | $\mathbf{0.6541}$ | $0.9822$ | $1.1897$ | $0.2754$ | $0.3998$ | $1.1551$ | $0.2834$ | $0.3812$ | |

$\widehat{\sigma}$ | $1.2845$ | $0.5483$ | $0.5635$ | $1.3012$ | $0.2754$ | $0.2977$ | $0.9994$ | $0.3995$ | $0.5759$ | |

$\widehat{\gamma}$ | $1.4259$ | $0.1689$ | $0.1845$ | $1.4541$ | $0.0905$ | $0.1015$ | $1.3218$ | $0.1594$ | $0.2391$ |

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**Figure 1.**Augmented data in the discretization scheme: the observed data, ${X}_{{t}_{i}}$ and ${X}_{{t}_{i+1}}$, is augmented by introducing $(m-1)$ unobserved data points.

**Figure 2.**Relative bias (in percentage) for the drift parameter $\kappa $ with weekly data ($\Delta =1/52$) and $n=2600$.

**Figure 3.**Vasicek model log-likelihood (ℓ) for the drift parameter $\kappa $ with known (${\ell}_{\mathit{\theta}}$) and estimated (${\ell}_{\widehat{\mathit{\theta}}}$) parameters, where ${\kappa}_{0}$ is the true value. The true density and its discretized version ($\Delta =1/52$) is illustrated, along with the estimate obtained (${\widehat{\kappa}}_{\mathrm{EML}}$ and ${\widehat{\kappa}}_{\mathrm{DML}}$, respectively). Note that scenarios with same ${\kappa}_{0}$ but different volatility parameter (scenarios 2, 4, 5 and 7, respectively) are not included, as the estimates were very close and figures very similar to the ones displayed.

**Figure 4.**Euribor series. Daily evolution for the time period between 15th October 2001 and 30th December 2005. Sample size for each data set is $n=1077$. From left to right, Euribor 3, 6, 9 and 12 months, respectively.

**Figure 5.**Bootstrap histogram of ${\widehat{\mathit{\theta}}}_{i}$, with $i\in \{1,2,3,4\}$, and asymptotic Gaussian distribution for the CKLS model, $\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{X}_{t}=({\mathit{\theta}}_{1}-{\mathit{\theta}}_{2}{X}_{t})\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}t+{\mathit{\theta}}_{3}{X}_{t}^{{\mathit{\theta}}_{4}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{W}_{t}$, fitted to Euribor 3 months series.

**Table 1.**Scenarios for the Monte Carlo study for the Vasicek, $\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{X}_{t}=\kappa (\mu -{X}_{t})\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}t+\sigma \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{W}_{t}$, and CKLS, $\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{X}_{t}=\kappa (\mu -{X}_{t})\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}t+\sigma {X}_{t}^{\gamma}\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{W}_{t}$, models with $\Delta =1/52$.

Vasicek | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 |
---|---|---|---|---|

$(\mu ,\kappa ,{\sigma}^{2})$ | $(0.09,0.2,4\times {10}^{-5})$ | $(0.09,0.2,4\times {10}^{-4})$ | $(0.09,0.9,1.8\times {10}^{-4})$ | $(0.09,0.9,1.8\times {10}^{-3})$ |

$\mathbb{E}\left\{{X}_{t}\right\}$ | $0.090$ | $0.090$ | $0.090$ | $0.090$ |

$\mathbb{V}\mathrm{ar}\left\{{X}_{t}\right\}$ | ${10}^{-4}$ | ${10}^{-3}$ | ${10}^{-4}$ | ${10}^{-3}$ |

$\mathbb{E}\left(\right)open="\{"\; close="\}">{X}_{(i+1)\Delta}\mid {X}_{i\Delta}$ | $3.4\times {10}^{-4}+0.996{X}_{t}$ | $3.4\times {10}^{-4}+0.996{X}_{t}$ | $0.002+0.98{X}_{t}$ | $0.002+0.98{X}_{t}$ |

$\mathbb{V}\mathrm{ar}\left(\right)open="\{"\; close="\}">{X}_{(i+1)\Delta}\mid {X}_{i\Delta}$ | $7.6\times {10}^{-7}$ | $7.6\times {10}^{-6}$ | $3.4\times {10}^{-6}$ | $3.4\times {10}^{-5}$ |

Vasicek | Scenario 5 | Scenario 6 | Scenario 7 | Scenario 8 |

$(\mu ,\kappa ,{\sigma}^{2})$ | $(0.09,5,{10}^{-3})$ | $(0.09,5,{10}^{-2})$ | $(0.09,9.8,1.96\times {10}^{-3})$ | $(0.09,9.8,1.96\times {10}^{-2})$ |

$\mathbb{E}\left\{{X}_{t}\right\}$ | $0.090$ | $0.090$ | $0.090$ | $0.090$ |

$\mathbb{V}\mathrm{ar}\left\{{X}_{t}\right\}$ | ${10}^{-4}$ | ${10}^{-3}$ | ${10}^{-4}$ | ${10}^{-3}$ |

$\mathbb{E}\left(\right)open="\{"\; close="\}">{X}_{(i+1)\Delta}\mid {X}_{i\Delta}$ | $8.3\times {10}^{-3}+0.908{X}_{t}$ | $8.3\times {10}^{-3}+0.908{X}_{t}$ | $0.015+0.83{X}_{t}$ | $0.015+0.83{X}_{t}$ |

$\mathbb{V}\mathrm{ar}\left(\right)open="\{"\; close="\}">{X}_{(i+1)\Delta}\mid {X}_{i\Delta}$ | $1.7\times {10}^{-5}$ | $1.7\times {10}^{-4}$ | $3.1\times {10}^{-5}$ | $3.1\times {10}^{-4}$ |

CKLS | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 |

$(\mu ,\kappa ,{\sigma}^{2},\gamma )$ | $(0.09,0.2,0.25,1.5)$ | $(0.09,0.2,1,1.5)$ | $(0.09,0.9,0.5,1.5)$ | $(0.09,0.9,2,1.5)$ |

**Table 2.**Monte Carlo simulation for Vasicek model, $(\mu ,\kappa ,\sigma )={(0.09,0.2,0.00632)}^{\prime}$, scenario 1: low mean reversion and low volatility. Boldfaces denote the best results in terms of bias, standard deviation and RMSE.

Scenario 1 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

EML | $\widehat{\mu}$ | $0.0915$ | $0.0177$ | $0.0178$ | $\mathbf{0.0900}$ | $\mathbf{0.0046}$ | $\mathbf{0.0046}$ | $\mathbf{0.0902}$ | $\mathbf{0.0050}$ | $\mathbf{0.0050}$ |

$\widehat{\kappa}$ | $0.6762$ | $0.4579$ | $0.6606$ | $0.2865$ | $0.1274$ | $0.1540$ | $0.2973$ | $0.1377$ | $0.1686$ | |

$\widehat{\sigma}$ | $0.0063$ | $1.98\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $\mathbf{0.0063}$ | $8.30\times {10}^{-5}$ | $\mathbf{8.33}\times {\mathbf{10}}^{-\mathbf{5}}$ | $\mathbf{0.0063}$ | $1.99\times {10}^{-4}$ | $\mathbf{1.99}\times {\mathbf{10}}^{-\mathbf{4}}$ | |

DML | $\widehat{\mu}$ | $0.0915$ | $0.0180$ | $0.0180$ | $0.0900$ | $0.0046$ | $0.0046$ | $0.0902$ | $0.0050$ | $0.0050$ |

$\widehat{\kappa}$ | $0.6703$ | $\mathbf{0.4511}$ | $0.6517$ | $0.2837$ | $0.1262$ | $0.1514$ | $\mathbf{0.2923}$ | $\mathbf{0.1345}$ | $\mathbf{0.1632}$ | |

$\widehat{\sigma}$ | $0.0063$ | $1.97\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $0.0063$ | $8.29\times {10}^{-5}$ | $8.37\times {10}^{-5}$ | $0.0063$ | $1.95\times {10}^{-4}$ | $2.06\times {10}^{-4}$ | |

LL | $\widehat{\mu}$ | $0.0915$ | $0.0179$ | $0.0180$ | $0.0900$ | $0.0046$ | $0.0046$ | $0.0902$ | $0.0050$ | $0.0050$ |

$\widehat{\kappa}$ | $0.6696$ | $0.4518$ | $0.6517$ | $0.2836$ | $0.1261$ | $0.1513$ | $0.2924$ | $0.1346$ | $0.1633$ | |

$\widehat{\sigma}$ | $0.0063$ | $1.97\times {10}^{-4}$ | $1.98\times {10}^{-4}$ | $0.0063$ | $\mathbf{8.29}\times {\mathbf{10}}^{-\mathbf{5}}$ | $8.34\times {10}^{-5}$ | $0.0063$ | $\mathbf{1.95}\times {\mathbf{10}}^{-\mathbf{4}}$ | $2.03\times {10}^{-4}$ | |

HP | $\widehat{\mu}$ | $0.0915$ | $0.0177$ | $0.0178$ | $0.0900$ | $0.0046$ | $0.0046$ | $0.0902$ | $0.0050$ | $0.0050$ |

$\widehat{\kappa}$ | $0.6720$ | $0.4548$ | $0.6554$ | $0.2864$ | $0.1271$ | $0.1537$ | $0.2939$ | $0.1349$ | $0.1644$ | |

$\widehat{\sigma}$ | $0.0063$ | $1.98\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $0.0063$ | $8.31\times {10}^{-5}$ | $8.33\times {10}^{-5}$ | $0.0063$ | $1.99\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | |

KF | $\widehat{\mu}$ | $0.0915$ | $0.0179$ | $0.0180$ | $0.0900$ | $0.0046$ | $0.0046$ | $0.0902$ | $0.0050$ | $0.0050$ |

$\widehat{\kappa}$ | $\mathbf{0.6696}$ | $0.4518$ | $\mathbf{0.6517}$ | $\mathbf{0.2836}$ | $\mathbf{0.1261}$ | $\mathbf{0.1513}$ | $0.2924$ | $0.1346$ | $0.1633$ | |

$\widehat{\sigma}$ | $\mathbf{0.0063}$ | $\mathbf{1.97}\times {\mathbf{10}}^{-\mathbf{4}}$ | $\mathbf{1.98}\times {\mathbf{10}}^{-\mathbf{4}}$ | $0.0063$ | $\mathbf{8.29}\times {\mathbf{10}}^{-\mathbf{5}}$ | $8.34\times {10}^{-5}$ | $0.0063$ | $1.95\times {10}^{-4}$ | $2.03\times {10}^{-4}$ | |

MCMC | $\widehat{\mu}$ | $\mathbf{0.0913}$ | $\mathbf{0.0148}$ | $\mathbf{0.0148}$ | $0.0900$ | $0.0046$ | $0.0046$ | $0.0902$ | $0.0050$ | $0.0050$ |

$\widehat{\kappa}$ | $0.6789$ | $0.4609$ | $0.6647$ | $0.2847$ | $0.1270$ | $0.1527$ | $0.2978$ | $0.1389$ | $0.1698$ | |

$\widehat{\sigma}$ | $0.0063$ | $1.99\times {10}^{-4}$ | $2.01\times {10}^{-4}$ | $0.0063$ | $8.33\times {10}^{-5}$ | $8.36\times {10}^{-5}$ | $0.0063$ | $1.98\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | |

GMM | $\widehat{\mu}$ | $0.0917$ | $0.0178$ | $0.0178$ | $0.0900$ | $0.0046$ | $0.0046$ | $0.0902$ | $0.0051$ | $0.0051$ |

$\widehat{\kappa}$ | $0.6701$ | $0.4518$ | $0.6520$ | $0.2837$ | $0.1261$ | $0.1513$ | $0.2932$ | $0.1347$ | $0.1638$ | |

$\widehat{\sigma}$ | $0.0063$ | $2.04\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $0.0063$ | $8.31\times {10}^{-5}$ | $8.41\times {10}^{-5}$ | $0.0063$ | $2.05\times {10}^{-4}$ | $2.15\times {10}^{-4}$ |

**Table 3.**Monte Carlo simulation for Vasicek model, $(\mu ,\kappa ,\sigma )={(0.09,0.9,0.0134)}^{\prime}$, scenario 3: high mean reversion and low volatility. Boldfaces denote the best results in terms of bias, standard deviation and RMSE.

Scenario 3 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

EML | $\widehat{\mu}$ | $\mathbf{0.0902}$ | $\mathbf{0.0049}$ | $\mathbf{0.0049}$ | $\mathbf{0.0900}$ | $\mathbf{0.0022}$ | $\mathbf{0.0022}$ | $\mathbf{0.0901}$ | $\mathbf{0.0023}$ | $\mathbf{0.0023}$ |

$\widehat{\kappa}$ | $1.3193$ | $0.6031$ | $0.7346$ | $0.9782$ | $0.2116$ | $0.2256$ | $0.9851$ | $0.2362$ | $0.2511$ | |

$\widehat{\sigma}$ | $0.0134$ | $4.21\times {10}^{-4}$ | $4.22\times {10}^{-4}$ | $\mathbf{0.0134}$ | $1.77\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $0.0134$ | $4.34\times {10}^{-4}$ | $4.35\times {10}^{-4}$ | |

DML | $\widehat{\mu}$ | $0.0902$ | $0.0049$ | $0.0049$ | $0.0900$ | $0.0022$ | $0.0022$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $\mathbf{1.2982}$ | $\mathbf{0.5884}$ | $\mathbf{0.7105}$ | $0.9637$ | $0.2088$ | $0.2183$ | $\mathbf{0.9462}$ | $0.2182$ | $0.2230$ | |

$\widehat{\sigma}$ | $0.0133$ | $4.14\times {10}^{-4}$ | $4.38\times {10}^{-4}$ | $0.0133$ | $1.75\times {10}^{-4}$ | $2.08\times {10}^{-4}$ | $0.0129$ | $\mathbf{4.02}\times {\mathbf{10}}^{-\mathbf{4}}$ | $6.52\times {10}^{-4}$ | |

LL | $\widehat{\mu}$ | $0.0902$ | $0.0049$ | $0.0049$ | $0.0900$ | $0.0022$ | $0.0022$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $1.2984$ | $0.5888$ | $0.7110$ | $\mathbf{0.9635}$ | $\mathbf{0.2085}$ | $\mathbf{0.2180}$ | $0.9465$ | $\mathbf{0.2178}$ | $\mathbf{0.2227}$ | |

$\widehat{\sigma}$ | $0.0133$ | $\mathbf{4.14}\times {\mathbf{10}}^{-\mathbf{4}}$ | $4.31\times {10}^{-4}$ | $0.0133$ | $\mathbf{1.75}\times {\mathbf{10}}^{-\mathbf{4}}$ | $2.05\times {10}^{-4}$ | $0.0129$ | $4.03\times {10}^{-4}$ | $6.33\times {10}^{-4}$ | |

HP | $\widehat{\mu}$ | $0.0902$ | $0.0049$ | $0.0049$ | $0.0900$ | $0.0022$ | $0.0022$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $1.3063$ | $0.5893$ | $0.7158$ | $0.9737$ | $0.2088$ | $0.2215$ | $0.9664$ | $0.237$ | $0.2461$ | |

$\widehat{\sigma}$ | $\mathbf{0.0134}$ | $4.21\times {10}^{-4}$ | $\mathbf{4.22}\times {\mathbf{10}}^{-\mathbf{4}}$ | $\mathbf{0.0134}$ | $1.77\times {10}^{-4}$ | $\mathbf{1.77}\times {\mathbf{10}}^{-\mathbf{4}}$ | $\mathbf{0.0134}$ | $4.32\times {10}^{-4}$ | $\mathbf{4.32}\times {\mathbf{10}}^{-\mathbf{4}}$ | |

KF | $\widehat{\mu}$ | $0.0902$ | $0.0049$ | $0.0049$ | $0.0900$ | $0.0022$ | $0.0022$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $1.2984$ | $0.5888$ | $0.7110$ | $0.9635$ | $0.2085$ | $0.2180$ | $0.9465$ | $0.2178$ | $0.2227$ | |

$\widehat{\sigma}$ | $0.0133$ | $4.14\times {10}^{-4}$ | $4.31\times {10}^{-4}$ | $0.0133$ | $1.75\times {10}^{-4}$ | $2.05\times {10}^{-4}$ | $0.0129$ | $4.03\times {10}^{-4}$ | $6.33\times {10}^{-4}$ | |

MCMC | $\widehat{\mu}$ | $0.0902$ | $0.0049$ | $0.0049$ | $0.0900$ | $0.0022$ | $0.0022$ | $0.0901$ | $0.0023$ | $0.0023$ |

$\widehat{\kappa}$ | $1.3224$ | $0.6077$ | $0.7401$ | $0.9724$ | $0.2125$ | $0.2245$ | $0.9868$ | $0.2358$ | $0.2513$ | |

$\widehat{\sigma}$ | $0.0135$ | $4.21\times {10}^{-4}$ | $4.22\times {10}^{-4}$ | $0.0134$ | $1.77\times {10}^{-4}$ | $1.77\times {10}^{-4}$ | $0.0134$ | $4.26\times {10}^{-4}$ | $4.28\times {10}^{-4}$ | |

GMM | $\widehat{\mu}$ | $0.0902$ | $0.0051$ | $0.0051$ | $0.0900$ | $0.0022$ | $0.0022$ | $0.0901$ | $0.0025$ | $0.0025$ |

$\widehat{\kappa}$ | $1.3127$ | $0.5937$ | $0.7231$ | $0.9646$ | $0.2098$ | $0.2195$ | $0.9704$ | $0.2365$ | $0.2467$ | |

$\widehat{\sigma}$ | $0.0133$ | $4.47\times {10}^{-4}$ | $4.60\times {10}^{-4}$ | $0.0133$ | $1.78\times {10}^{-4}$ | $1.95\times {10}^{-4}$ | $0.0132$ | $4.95\times {10}^{-4}$ | $5.53\times {10}^{-4}$ |

Time (Seconds) | EML | DML | LL | HP | KF | MCMC | GMM |
---|---|---|---|---|---|---|---|

Vasicek | 0.0534 | 0.0469 | 0.0424 | 0.4921 | 0.1097 | 50.8442 | 1.0108 |

CKLS | - | 0.0818 | 0.2011 | 1.9543 | 0.2017 | 140.5385 | 2.0237 |

Implementation: | R | R | R | R | C | C++ | R/C++ |

**Table 5.**Monte Carlo simulation for CKLS model, $(\mu ,\kappa ,\sigma ,\gamma )={(0.09,0.2,0.5,1.5)}^{\prime}$, scenario 1: low mean reversion and low volatility. Boldfaces denote the best results in terms of bias, standard deviation and RMSE.

Scenario 1 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

DML | $\widehat{\mu}$ | $\mathbf{0.0950}$ | $\mathbf{0.0420}$ | $\mathbf{0.0423}$ | $0.0912$ | $0.0131$ | $0.0131$ | $0.0930$ | $0.0145$ | $0.0148$ |

$\widehat{\kappa}$ | $0.6596$ | $0.4630$ | $0.6524$ | $\mathbf{0.2816}$ | $0.1356$ | $\mathbf{0.1583}$ | $\mathbf{0.2844}$ | $0.1446$ | $\mathbf{0.1674}$ | |

$\widehat{\sigma}$ | $0.5776$ | $0.3783$ | $0.3862$ | $0.5015$ | $0.0865$ | $0.0865$ | $0.4722$ | $0.1690$ | $0.1713$ | |

$\widehat{\gamma}$ | $1.4904$ | $0.2453$ | $0.2455$ | $1.4956$ | $0.0701$ | $0.0702$ | $1.4543$ | $0.1466$ | $0.1535$ | |

LL | $\widehat{\mu}$ | $0.0953$ | $0.0483$ | $0.0486$ | $0.0911$ | $0.0131$ | $0.0131$ | $0.0929$ | $\mathbf{0.0143}$ | $\mathbf{0.0146}$ |

$\widehat{\kappa}$ | $0.6683$ | $0.4766$ | $0.6681$ | $0.2829$ | $0.1368$ | $0.1599$ | $0.2892$ | $0.1497$ | $0.1743$ | |

$\widehat{\sigma}$ | $0.5985$ | $0.3938$ | $0.4060$ | $0.5095$ | $0.0875$ | $0.0880$ | $0.5055$ | $0.1785$ | $0.1786$ | |

$\widehat{\gamma}$ | $1.5052$ | $0.2459$ | $0.2460$ | $1.5025$ | $\mathbf{0.0694}$ | $\mathbf{0.0695}$ | $\mathbf{1.4846}$ | $\mathbf{0.1434}$ | $\mathbf{0.1442}$ | |

HP | $\widehat{\mu}$ | $0.0967$ | $0.0547$ | $0.0551$ | $0.0912$ | $0.0131$ | $0.0131$ | $0.0936$ | $0.0236$ | $0.0239$ |

$\widehat{\kappa}$ | $\mathbf{0.6438}$ | $0.4649$ | $\mathbf{0.6427}$ | $0.2828$ | $0.1378$ | $0.1608$ | $0.2891$ | $0.1516$ | $0.1758$ | |

$\widehat{\sigma}$ | $0.6331$ | $0.4799$ | $0.4980$ | $0.5078$ | $0.0885$ | $0.0888$ | $\mathbf{0.5043}$ | $0.1814$ | $0.1815$ | |

$\widehat{\gamma}$ | $1.5130$ | $0.2588$ | $0.2591$ | $\mathbf{1.4997}$ | $0.0711$ | $0.0711$ | $1.4783$ | $0.1466$ | $0.1482$ | |

KF | $\widehat{\mu}$ | $0.0956$ | $0.0518$ | $0.0522$ | $0.0912$ | $0.0131$ | $0.0131$ | $0.0930$ | $0.0145$ | $0.0148$ |

$\widehat{\kappa}$ | $0.6590$ | $\mathbf{0.4615}$ | $0.6509$ | $0.2820$ | $0.1357$ | $0.1586$ | $0.2845$ | $0.1448$ | $0.1677$ | |

$\widehat{\sigma}$ | $0.5774$ | $\mathbf{0.3778}$ | $\mathbf{0.3857}$ | $\mathbf{0.5012}$ | $\mathbf{0.0863}$ | $\mathbf{0.0863}$ | $0.4722$ | $\mathbf{0.1688}$ | $\mathbf{0.1711}$ | |

$\widehat{\gamma}$ | $1.4905$ | $\mathbf{0.2446}$ | $\mathbf{0.2447}$ | $1.4954$ | $0.0700$ | $0.0702$ | $1.4543$ | $0.1464$ | $0.1534$ | |

MCMC | $\widehat{\mu}$ | $0.0963$ | $0.0730$ | $0.0733$ | $0.0912$ | $0.0131$ | $0.0131$ | $0.0930$ | $0.0144$ | $0.0147$ |

$\widehat{\kappa}$ | $0.6688$ | $0.4729$ | $0.6659$ | $0.2828$ | $0.1365$ | $0.1597$ | $0.2888$ | $0.1492$ | $0.1736$ | |

$\widehat{\sigma}$ | $0.6860$ | $0.5906$ | $0.6192$ | $0.5169$ | $0.1078$ | $0.1091$ | $0.5395$ | $0.2521$ | $0.2552$ | |

$\widehat{\gamma}$ | $\mathbf{1.4961}$ | $0.2913$ | $0.2913$ | $1.5004$ | $0.0832$ | $0.0832$ | $1.4739$ | $0.1702$ | $0.1721$ | |

GMM | $\widehat{\mu}$ | $0.1001$ | $0.2047$ | $0.2050$ | $\mathbf{0.0904}$ | $\mathbf{0.0114}$ | $\mathbf{0.0114}$ | $\mathbf{0.0919}$ | $0.0153$ | $0.0154$ |

$\widehat{\kappa}$ | $0.7208$ | $0.4675$ | $0.6998$ | $0.3216$ | $\mathbf{0.1353}$ | $0.1819$ | $0.3248$ | $\mathbf{0.1437}$ | $0.1904$ | |

$\widehat{\sigma}$ | $\mathbf{0.5534}$ | $0.3965$ | $0.4001$ | $0.4972$ | $0.1166$ | $0.1167$ | $0.4498$ | $0.2068$ | $0.2128$ | |

$\widehat{\gamma}$ | $1.4605$ | $0.2644$ | $0.2673$ | $1.4868$ | $0.0954$ | $0.0963$ | $1.4196$ | $0.1803$ | $0.1974$ |

**Table 6.**Monte Carlo simulation for CKLS model, ${(\mu ,\kappa ,\sigma ,\gamma )}^{\prime}={(0.09,0.9,0.707,1.5)}^{\prime}$, scenario 3: high mean reversion and low volatility. Boldfaces denote the best results in terms of bias, standard deviation and RMSE.

Scenario 3 | $\mathbf{\Delta}=1/52$, $\mathit{n}=520$ | $\mathbf{\Delta}=1/52$, $\mathit{n}=2600$ | $\mathbf{\Delta}=1/12$, $\mathit{n}=520$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Method | $\widehat{\mathit{\theta}}$ | Mean | SD | RMSE | Mean | SD | RMSE | Mean | SD | RMSE |

DML | $\widehat{\mu}$ | $0.0908$ | $0.0086$ | $0.0087$ | $0.0901$ | $0.0032$ | $0.0032$ | $0.0902$ | $0.0034$ | $0.0034$ |

$\widehat{\kappa}$ | $1.2893$ | $0.6072$ | $0.7213$ | $\mathbf{0.9623}$ | $\mathbf{0.2200}$ | $\mathbf{0.2287}$ | $\mathbf{0.9481}$ | $\mathbf{0.2304}$ | $\mathbf{0.2353}$ | |

$\widehat{\sigma}$ | $0.7701$ | $\mathbf{0.4582}$ | $\mathbf{0.4625}$ | $\mathbf{0.6958}$ | $0.1523$ | $0.1527$ | $\mathbf{0.6411}$ | $0.3094$ | $0.3163$ | |

$\widehat{\gamma}$ | $1.4769$ | $0.2277$ | $0.2289$ | $1.4863$ | $0.0896$ | $0.0907$ | $1.4251$ | $0.2005$ | $0.2140$ | |

LL | $\widehat{\mu}$ | $0.0908$ | $0.0087$ | $0.0087$ | $\mathbf{0.0900}$ | $\mathbf{0.0032}$ | $\mathbf{0.0032}$ | $\mathbf{0.0901}$ | $0.0035$ | $0.0035$ |

$\widehat{\kappa}$ | $1.3112$ | $0.6277$ | $0.7504$ | $0.9777$ | $0.2244$ | $0.2375$ | $0.9917$ | $0.2514$ | $0.2676$ | |

$\widehat{\sigma}$ | $0.8260$ | $0.4938$ | $0.5080$ | $0.7421$ | $0.1564$ | $0.1603$ | $0.7908$ | $0.3763$ | $0.3855$ | |

$\widehat{\gamma}$ | $\mathbf{1.5073}$ | $\mathbf{0.2268}$ | $\mathbf{0.2270}$ | $1.5142$ | $\mathbf{0.0868}$ | $\mathbf{0.0880}$ | $1.5156$ | $\mathbf{0.1999}$ | $\mathbf{0.2005}$ | |

HP | $\widehat{\mu}$ | $0.0909$ | $0.0089$ | $0.0089$ | $0.0902$ | $0.0043$ | $0.0043$ | $0.0904$ | $0.0043$ | $0.0043$ |

$\widehat{\kappa}$ | $1.2927$ | $0.6182$ | $0.7324$ | $0.9688$ | $0.2268$ | $0.2370$ | $0.9738$ | $0.2565$ | $0.2669$ | |

$\widehat{\sigma}$ | $0.9005$ | $0.6338$ | $0.6627$ | $0.7396$ | $0.2030$ | $0.2056$ | $0.8134$ | $0.4269$ | $0.4400$ | |

$\widehat{\gamma}$ | $1.5281$ | $0.2332$ | $0.2349$ | $1.5050$ | $0.1007$ | $0.1008$ | $\mathbf{1.5052}$ | $0.2055$ | $0.2055$ | |

KF | $\widehat{\mu}$ | $0.0908$ | $0.0086$ | $0.0086$ | $0.0901$ | $0.0032$ | $0.0032$ | $0.0902$ | $\mathbf{0.0034}$ | $\mathbf{0.0034}$ |

$\widehat{\kappa}$ | $\mathbf{1.2882}$ | $\mathbf{0.6059}$ | $\mathbf{0.7197}$ | $0.9629$ | $0.2209$ | $0.2297$ | $0.9481$ | $0.2304$ | $0.2354$ | |

$\widehat{\sigma}$ | $0.7711$ | $0.4605$ | $0.4650$ | $0.6948$ | $\mathbf{0.1516}$ | $\mathbf{0.1521}$ | $0.6407$ | $\mathbf{0.3089}$ | $\mathbf{0.3159}$ | |

$\widehat{\gamma}$ | $1.4773$ | $0.2278$ | $0.2289$ | $1.4857$ | $0.0895$ | $0.0906$ | $1.4249$ | $0.2001$ | $0.2137$ | |

MCMC | $\widehat{\mu}$ | $0.0909$ | $0.0089$ | $0.0090$ | $0.0901$ | $0.0032$ | $0.0032$ | $0.0903$ | $0.0035$ | $0.0035$ |

$\widehat{\kappa}$ | $1.3121$ | $0.6274$ | $0.7506$ | $0.9717$ | $0.2249$ | $0.2361$ | $0.9852$ | $0.2498$ | $0.2640$ | |

$\widehat{\sigma}$ | $0.9479$ | $0.8601$ | $0.8932$ | $0.7402$ | $0.2014$ | $0.2041$ | $0.8705$ | $0.5436$ | $0.5676$ | |

$\widehat{\gamma}$ | $1.4900$ | $0.2805$ | $0.2807$ | $\mathbf{1.4985}$ | $0.1099$ | $0.1099$ | $1.4894$ | $0.2406$ | $0.2409$ | |

GMM | $\widehat{\mu}$ | $\mathbf{0.0905}$ | $\mathbf{0.0075}$ | $\mathbf{0.0075}$ | $0.0900$ | $0.0032$ | $0.0032$ | $0.0902$ | $0.0034$ | $0.0034$ |

$\widehat{\kappa}$ | $1.3760$ | $0.6121$ | $0.7755$ | $1.0023$ | $0.2400$ | $0.2608$ | $0.9759$ | $0.2450$ | $0.2565$ | |

$\widehat{\sigma}$ | $\mathbf{0.7452}$ | $0.5017$ | $0.5031$ | $0.6837$ | $0.1913$ | $0.1928$ | $0.5984$ | $0.3535$ | $0.3698$ | |

$\widehat{\gamma}$ | $1.4486$ | $0.2514$ | $0.2566$ | $1.4732$ | $0.1128$ | $0.1160$ | $1.3820$ | $0.2253$ | $0.2544$ |

Method | Authors | Asymptotic Properties | Finite Sample Performance |
---|---|---|---|

DML | Florens–Zmirou [29] | Asymptotically normal and consistent | Biased when $\Delta $ is large |

LL | Ozaki [17] Shoji and Ozaki [18] | As ML estimators | Outperforms discrete ML and KF |

HP | Aït–Sahalia [19,47] | Asymptotically normal and consistent [19] | Outperforms LL |

KF | Kalman [20,48] | Asymptotically normal and consistent [30,34] | Similar to discrete ML |

MCMC | Elerian et al. [21] Eraker [22] | Simulation based | Efficient but the most computationally intensive |

GMM | Hansen [23] Chan et al. [6] | Asymptotically normal and consistent [23,41] | The least efficient and efficiency depends on moments |

**Table 8.**Estimated parameters and standard errors (in parentheses) for the CKLS model, $\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{X}_{t}=({\mathit{\theta}}_{1}-{\mathit{\theta}}_{2}{X}_{t})\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}t+{\mathit{\theta}}_{3}{X}_{t}^{{\mathit{\theta}}_{4}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{d}{W}_{t}$, fitted to Euribor series using six estimation methods.

Method | $\widehat{\mathit{\theta}}$ | 3 Months | 6 Months | 9 Months | 12 Months | ||||
---|---|---|---|---|---|---|---|---|---|

DML | ${\widehat{\mathit{\theta}}}_{1}$ | $1.5063$ | $\left(0.4510\right)$ | $1.3956$ | $\left(0.6671\right)$ | $1.5620$ | $\left(0.9484\right)$ | $1.7607$ | $\left(1.1623\right)$ |

${\widehat{\mathit{\theta}}}_{2}$ | $0.7097$ | $\left(0.1974\right)$ | $0.6323$ | $\left(0.2828\right)$ | $0.6713$ | $\left(0.3920\right)$ | $0.7232$ | $\left(0.4668\right)$ | |

${\widehat{\mathit{\theta}}}_{3}$ | $0.0297$ | $\left(0.0031\right)$ | $0.0667$ | $\left(0.0068\right)$ | $0.1141$ | $\left(0.0117\right)$ | $0.1613$ | $\left(0.0169\right)$ | |

${\widehat{\mathit{\theta}}}_{4}$ | $1.8143$ | $\left(0.1118\right)$ | $1.3900$ | $\left(0.1084\right)$ | $1.1987$ | $\left(0.1081\right)$ | $1.0521$ | $\left(0.1078\right)$ | |

LL | ${\widehat{\mathit{\theta}}}_{1}$ | $1.5083$ | $\left(0.4537\right)$ | $1.4027$ | $\left(0.6696\right)$ | $1.5723$ | $\left(0.9517\right)$ | $1.7803$ | $\left(1.1667\right)$ |

${\widehat{\mathit{\theta}}}_{2}$ | $0.7109$ | $\left(0.1989\right)$ | $0.6352$ | $\left(0.2842\right)$ | $0.6763$ | $\left(0.3940\right)$ | $0.7311$ | $\left(0.4695\right)$ | |

${\widehat{\mathit{\theta}}}_{3}$ | $0.0286$ | $\left(0.0030\right)$ | $0.0653$ | $\left(0.0067\right)$ | $0.1109$ | $\left(0.0114\right)$ | $0.1560$ | $\left(0.0164\right)$ | |

${\widehat{\mathit{\theta}}}_{4}$ | $1.8520$ | $\left(0.1140\right)$ | $1.4114$ | $\left(0.1098\right)$ | $1.2264$ | $\left(0.1086\right)$ | $1.0843$ | $\left(0.1084\right)$ | |

HP | ${\widehat{\mathit{\theta}}}_{1}$ | $1.5255$ | $\left(0.4553\right)$ | $1.4092$ | $\left(0.6701\right)$ | $1.5650$ | $\left(0.9521\right)$ | $1.7844$ | $\left(1.1673\right)$ |

${\widehat{\mathit{\theta}}}_{2}$ | $0.7182$ | $\left(0.1999\right)$ | $0.6380$ | $\left(0.2844\right)$ | $0.6732$ | $\left(0.3942\right)$ | $0.7332$ | $\left(0.4698\right)$ | |

${\widehat{\mathit{\theta}}}_{3}$ | $0.0280$ | $\left(0.0029\right)$ | $0.0652$ | $\left(0.0067\right)$ | $0.1110$ | $\left(0.0115\right)$ | $0.1563$ | $\left(0.0164\right)$ | |

${\widehat{\mathit{\theta}}}_{4}$ | $1.8764$ | $\left(0.1129\right)$ | $1.4156$ | $\left(0.1097\right)$ | $1.2267$ | $\left(0.1086\right)$ | $1.0841$ | $\left(0.1084\right)$ | |

KF | ${\widehat{\mathit{\theta}}}_{1}$ | $1.4964$ | $\left(0.4492\right)$ | $1.3954$ | $\left(0.6669\right)$ | $1.5603$ | $\left(0.9483\right)$ | $1.9177$ | $\left(1.1629\right)$ |

${\widehat{\mathit{\theta}}}_{2}$ | $0.7049$ | $\left(0.1964\right)$ | $0.6321$ | $\left(0.2826\right)$ | $0.6711$ | $\left(0.3920\right)$ | $0.7827$ | $\left(0.4672\right)$ | |

${\widehat{\mathit{\theta}}}_{3}$ | $0.0303$ | $\left(0.0031\right)$ | $0.0670$ | $\left(0.0068\right)$ | $0.1141$ | $\left(0.0117\right)$ | $0.1609$ | $\left(0.0168\right)$ | |

${\widehat{\mathit{\theta}}}_{4}$ | $1.7899$ | $\left(0.1121\right)$ | $1.3861$ | $\left(0.1088\right)$ | $1.1985$ | $\left(0.1079\right)$ | $1.0547$ | $\left(0.1078\right)$ | |

MCMC | ${\widehat{\mathit{\theta}}}_{1}$ | $1.5256$ | $\left(0.4536\right)$ | $1.4058$ | $\left(0.6657\right)$ | $1.5398$ | $\left(0.9508\right)$ | $1.7899$ | $\left(1.1700\right)$ |

${\widehat{\mathit{\theta}}}_{2}$ | $0.7183$ | $\left(0.2016\right)$ | $0.6366$ | $\left(0.2821\right)$ | $0.6657$ | $\left(0.3949\right)$ | $0.7325$ | $\left(0.4683\right)$ | |

${\widehat{\mathit{\theta}}}_{3}$ | $0.0290$ | $\left(0.0032\right)$ | $0.0673$ | $\left(0.0062\right)$ | $0.1173$ | $\left(0.0108\right)$ | $0.1548$ | $\left(0.0159\right)$ | |

${\widehat{\mathit{\theta}}}_{4}$ | $1.8444$ | $\left(0.1059\right)$ | $1.3869$ | $\left(0.1029\right)$ | $1.1756$ | $\left(0.1012\right)$ | $1.0993$ | $\left(0.1095\right)$ | |

GMM | ${\widehat{\mathit{\theta}}}_{1}$ | $1.7081$ | $\left(0.6782\right)$ | $1.1439$ | $\left(0.6100\right)$ | $1.2429$ | $\left(0.6646\right)$ | $1.3074$ | $\left(0.6633\right)$ |

${\widehat{\mathit{\theta}}}_{2}$ | $0.7431$ | $\left(0.2389\right)$ | $0.5216$ | $\left(0.2214\right)$ | $0.5400$ | $\left(0.2503\right)$ | $0.5465$ | $\left(0.2590\right)$ | |

${\widehat{\mathit{\theta}}}_{3}$ | $0.0242$ | $\left(0.0167\right)$ | $0.0676$ | $\left(0.0213\right)$ | $0.1126$ | $\left(0.0306\right)$ | $0.1661$ | $\left(0.0438\right)$ | |

${\widehat{\mathit{\theta}}}_{4}$ | $1.9562$ | $\left(0.6262\right)$ | $1.3704$ | $\left(0.2735\right)$ | $1.2068$ | $\left(0.2230\right)$ | $1.0227$ | $\left(0.2118\right)$ |

**Table 9.**p-values for the Monsalve-Cobis et al. [51] goodness-of-fit test for the CKLS parametric form of the drift and diffusion functions.

Maturity: | 3 Months | 6 Months | 9 Months | 12 Months |
---|---|---|---|---|

p-value drift function | $0.267$ | $0.723$ | $0.911$ | $0.950$ |

p-value volatility function | <0.001 | <0.001 | $0.001$ | $0.004$ |

**Table 10.**Kalman filter estimates, asymptotic standard error and bootstrap standard error of the CKLS model fitted to Euribor 3 months serie.

Parameters: | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{4}$ |
---|---|---|---|---|

Estimation | 1.4964 | 0.7049 | 0.0303 | 1.7899 |

Asymptotic standard error | 0.4492 | 0.1964 | 0.0031 | 0.1121 |

Bootstrap standard error | 0.4794 | 0.2105 | 0.0653 | 0.4110 |

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López-Pérez, A.; Febrero-Bande, M.; González-Manteiga, W.
Parametric Estimation of Diffusion Processes: A Review and Comparative Study. *Mathematics* **2021**, *9*, 859.
https://doi.org/10.3390/math9080859

**AMA Style**

López-Pérez A, Febrero-Bande M, González-Manteiga W.
Parametric Estimation of Diffusion Processes: A Review and Comparative Study. *Mathematics*. 2021; 9(8):859.
https://doi.org/10.3390/math9080859

**Chicago/Turabian Style**

López-Pérez, Alejandra, Manuel Febrero-Bande, and Wencesalo González-Manteiga.
2021. "Parametric Estimation of Diffusion Processes: A Review and Comparative Study" *Mathematics* 9, no. 8: 859.
https://doi.org/10.3390/math9080859