Parameter estimation and model testing for Markov processes via conditional characteristic functions

Parameter estimation and model testing for Markov processes via conditional characteristic functions

[ [    [ [    [ [ Department of Statistics, Iowa State University, Ames, IA 50011-1210, USA.
Guanghua School of Management and Center for Statistical Science, Peking University, Beijing 100871, China School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
\smonth5 \syear2011\smonth8 \syear2011
\smonth5 \syear2011\smonth8 \syear2011
\smonth5 \syear2011\smonth8 \syear2011

Markov processes are used in a wide range of disciplines, including finance. The transition densities of these processes are often unknown. However, the conditional characteristic functions are more likely to be available, especially for Lévy-driven processes. We propose an empirical likelihood approach, for both parameter estimation and model specification testing, based on the conditional characteristic function for processes with either continuous or discontinuous sample paths. Theoretical properties of the empirical likelihood estimator for parameters and a smoothed empirical likelihood ratio test for a parametric specification of the process are provided. Simulations and empirical case studies are carried out to confirm the effectiveness of the proposed estimator and test.


0 \volume19 \issue1 2013 \firstpage228 \lastpage251 \doi10.3150/11-BEJ400


CCF-based estimation and testing


1,2]\fnmsSong X. \snmChen\thanksref1,2,e1label=e1,mark,], 3]\fnmsLiang \snmPeng\corref\thanksref3label=e2,] and 1]\fnmsCindy L. \snmYu\thanksref1,e3label=e3,mark]

conditional characteristic function \kwddiffusion processes \kwdempirical likelihood \kwdkernel smoothing \kwdLévy-driven processes

1 Introduction

Let be a parametric -dimensional Markov process defined by


where is a -dimensional drift function, is a matrix-valued function of , is a Lévy process in and . When is a standard Brownian motion, (1) is a diffusion process having a continuous sample path. When contains the Brownian motion and a compound Poisson process, (1) becomes the jump diffusion process. A stochastic process of form (1) has long been used to model stochastic systems arising in physics, biology and other natural sciences. It has also been the fundamental tool in financial modeling. We refer to Sundaresan Sun00 () and Fan Fan05 () for overviews, Barndorff-Nielsen, Mikosch and Resnick Ba01 () for recent developments on Lévy-driven processes and Sørensen Sre91 () for statistical inference. Important subclasses of (1) include (i) the multivariate diffusion process defined by


where is the standard Brownian motion in (Stroock and Varadhan StrVar79 () and Øksendal kse00 ()); (ii) the Vasicek with Merton Jump model (VSK-MJ) defined by


where , and are unknown parameters and represent the mean reverting rate, long-run mean and volatility of the process, respectively, is a Poisson process with intensity and is the random jump size independent of the filtration up to time and has a normal density (Merton Mer76 ()); (iii) Lévy driven Ornstein–Uhlenbeck process defined by


where is a Lévy process with no Brownian part, a non-negative drift and a Lévy measure which is zero on the negative half line, and the parameter is positive (see Barndorff-Nielsen and Shephard BarShe01 ()).

Often a closed form expression for the transition density of process (1) is not available except for some special processes, even if the transition density exists and is unique. This fact prevents the use of the maximum likelihood estimation (MLE) and the specification tests based on the exact transition density. Recently Aït-Sahalia AtS02 (), AtS08 () established expansions for the transition densities so that parameter estimation could be based on the approximate likelihood functions. Testing may be also formulated via the approximate density; see Chen, Gao and Tang CheGaoTan08 () and Aït-Sahalia, Fan and Peng AtSFanPen09 () for such tests. The conditional characteristic functions (CCF) are more likely available than the transition densities for the continuous-time models, especially for the Lévy-driven processes through the celebrated Lévy–Khintchine representation. For instance, Duffie, Pan and Singleton DufPanSin00 () derived the explicit form of the CCF for multivariate affine jump processes, which include the Vasicek with Merton jump process given in (3). The CCF for the Lévy-driven Ornstein–Uhlenbeck process (4) is established in Barndorff-Nielsen and Shephard BarShe01 ().

Statistical inference based on the characteristic functions was proposed by Feuerverger and Mureika FeuMur77 (), Feuerverger and McDunnough FeuMcD81 () for independent observations and Feuerverger Feu90 () for discrete time series. Singleton Sin01 () introduced the approach to inference for parametric continuous-time Markov processes and showed that estimation can be carried out based on the CCF without having to carry out the the Fourier inversion. Chacko and Viceira ChaVic03 () proposed a generalized method of moment estimator (GMM) for parameters at a finite number of frequencies of the CCF. Carrasco et al. Caretal07 () carried out GMM estimation on a slowly diverging number of frequencies of the CCF to achieve the optimal estimation efficiency offered by the MLE. Jiang and Knight JiaKni02 () proposed GMM estimators based on the joint characteristic function of the observed state variables. Chen and Hong CheHon10 () proposed a test for multivariate processes based on the CCF via a generalized spectral density approach.

In this paper, we first propose an empirical likelihood (Owen Owe88 ()) approach for parameter estimation and model specification testing of a parametric Markov process via the CCF. An empirical likelihood ratio is formulated for the unknown parameters assuming specification (1), which leads to a non-parametric maximum likelihood estimator. The proposed estimator may be viewed as a compromise between Chacko and Viceira’s ChaVic03 () GMM, based on a finite number of frequencies, and that of Carrasco et al. Caretal07 (), of a high-dimensional GMM. The high-dimensional GMM approach requires ridging a high-dimensional weighting matrix in order to avoid its singularity, and the selecting the ridging parameter can be computationally expensive. The proposed estimation utilizes a wide range of frequency information in the parametric CCF, while having the computation easily managed.

We then formulate an empirical likelihood CCF-based model specification test for the parametric process (1) via kernel smoothing. The proposed test extends the transition density based tests of Qin and Lawless Qin94 (), Chen, Gao and Tang CheGaoTan08 () and Aït-Sahalia, Fan and Peng AtSFanPen09 () to the CCF based. This largely increases the range of the continuous-time Markov processes which can be tested directly without replying on the transition density approximation. The proposed test provides an alternative formulation of the CCF-based test of Chen and Hong CheHon10 (), which is based on an explicit measure between an kernel estimator of the CCF and its parametric counter-part. It is largely distinct from the above mentioned tests, except Chen and Hong CheHon10 (), by targeting directly on CCF, which is more readily available for continuous-time models than the transition density functions. Another advantage of the proposed test is the empirical likelihood (EL) formulation, which can produce an integrated likelihood ratio test in a nonparametric setting. The proposed test utilizes some of the attractive properties of the EL, like internal studentizing without an explicit variance estimation and good power performance. How to extend the proposed methods to the case of latent variables is quite challenging and will be a part of our future research.

The paper is organized as follows. In Section 2, we introduce and evaluate the CCF-based empirical likelihood estimator. The model specification test is given in Section 3. Section 4 reports results from simulation studies. An empirical study for a set of 3-month treasury bill rate data is analyzed in Section 5. All technical details are reported in the Appendix.

2 Parameter estimation

Let be discretely sampled observations of (1). For notation simplification, we denote as , where the sampling interval is any fixed quantity. Let , for , be the conditional characteristic function. We use and to denote the conjugate of a complex number and the conjugate transpose of the complex matrix , respectively.

Let for , where is a weight factor. Here can be regarded as “residuals” between and the parametric CCF . The complex weight factor satisfies and for any , whose use is aimed to utilize more model information. Let be the true parameter and the unique solution of


From the Markov property and (5), for any ,


Let and be the real and imaginary parts of respectively, and be the real bivariate vector corresponding to .

We now formulate an empirical likelihood for based on the CCF . The empirical likelihood (EL) introduced in Owen Owe88 () is a technique that allows construction of a non-parametric likelihood for parameters of interest. Despite that the EL method is intrinsically non-parametric, it possesses two important properties of a parametric likelihood, the Wilks theorem and the Bartlett correction; see Chen and Van Keilegom CheVan09 () for a latest overview and Kitamura, Tripathi and Ahn KitTriAhn04 () for a formulation with conditional moments.

Let be probability weights allocated to the “residuals” . A local EL for at is


subject to and . Here the second constraint reflects (5). The maximum empirical likelihood is attained at for all such that the maximum likelihood . Let be the local log-EL ratio of at .

Employing the EL algorithm (Owen Owe88 ()), the optimal of the above optimization problem (7) is

where is a Lagrange multiplier in that satisfies


Hence, the local EL ratio becomes


Integrating against a probability weight , which is supported on a compact set in , an integrated empirical likelihood ratio for is


The maximum EL estimator (MELE) for is defined as

by noting that has been multiplied in the EL ratio .

Like Qin and Lawless Qin94 (), we first show that there exists a consistent estimator with a certain rate of convergence as follows.

Lemma 1

Under Conditions C1–C4 given in the Appendix, with probability one, attains its minimum at in the interior of the ball , and and satisfy


where is defined in (8) and


Before deriving the asymptotic normality of the , we define




Theorem 1

Under Conditions C1–C4 given in the Appendix, for the estimator in Lemma 1, we have where .

The proposed estimator attains the -rate of convergence. It is computationally stable because computing for one at a time is essentially one-dimensional problem. Note that Carrasco et al. Caretal07 () considered CCF-based generalized method of moment estimation by considering a continuum of s in a functional space via covariance operator, but the covariance operator may not be invertible due to zero eigenvalues. Hence, Carrasco et al. Caretal07 () needed ridging to avoid the invertible issue, which makes the computation quite involved.

3 Test for model specification

In this section we consider testing for the validity of (1) via testing for the parametric specification of the CCF . Tests for model specification of a continuous-time Markov process have been proposed by Chen, Gao and Tang CheGaoTan08 () and Aït-Sahalia, Fan and Peng AtSFanPen09 (). Despite the fact that parameter estimation based on the transition density is asymptotically efficient, it is unclear if a test based on the transition density is more powerful than one based on the CCF. The choice is clearer when the transition density does not admit a closed form while the CCF does, since the latter is a test valid at any level of the sampling interval .

Let the underlying process that generates the observed sample path be


whose CCF is . The process (1) is a parametric specification of (14). To emphasize the dependence of the CCF on , we write in this section as , as and other quantities in a similar fashion. We consider testing

against a sequence of local alternative hypotheses

where is a sequence of non-random real constants converging to zero at a certain rate, and is a sequence of bounded complex functions which are continuous at and ; see Condition C6 in the Appendix for extra restrictions.

Since the target of inference is a conditional quantity, we need to work with a kernel smoothed version of . Let be a kernel function which is a symmetric probability density in , and  be a smoothing bandwidth that tends to as . A smoothed version of is


subject to and .

Let be the log-EL ratio. Then the integrated log-EL ratio for is

where and are probability weight functions on the frequency space and the state space, respectively. We can choose to be the same as the in Section 3.

The test statistic is , where is the empirical likelihood estimator proposed in Section 3. As a matter of fact, we can employ any estimator with -rate of convergence. To appreciate the meaning of the test statistic, let be the Nadaraya–Watson kernel weight, be the kernel smooth of the residuals, and . It can be shown by a similar derivation in Chen, Härdle and Li CheHarLi03 () that

where , and is the density of . So, the test statistic is asymptoticly equivalent to a -measure of the averaged “residuals” , inversely weighted by the covariance matrix function . Hence the proposed test is similar in tune to Fan and Zhang Fa03 () for testing diffusion processes, and of Härdle and Mammen Ha93 () and Wang and Van Keilegom Wa07 () for testing regression functions.

We need the following notations to describe the power property. Let , then , defined earlier. Express the matrices

Furthermore, we choose and define



where is the th convolution of the kernel function .

The asymptotic normality of is given in the following theorem.

Theorem 2

Under Conditions C1–C6 given in the Appendix,


We note that under . Under , since is non-vanishing with respect to , is non-vanishing with respect to for all in the support of , which leads to a positive quantity , due to being a Hermitian matrix. Since no restriction has been imposed on the functional form of , it means that the test is powerful for a wide range of local alternatives. Indeed, if is a consistent estimator of , the asymptotic normality-based test for with -level of significance rejects if

where is the quantile of the standard normal distribution. Theorem 2 implies that the power of the test under is

where is the standard normal distribution function.

It is known that the choice of bandwidth is important in any test based on the kernel smoothing technique. To make the test less sensitive to the choice of smoothing bandwidth, we propose carrying out the test based on a set of bandwidths, say , for a fixed integer such that for some constants . Here is a reference bandwidth which may be obtained via the cross-validation method.

This means that we have a set of the EL ratios corresponding to the bandwidth set, and the overall test statistic is


To describe the asymptotic distribution of , let be a generalization to the convolution of , and

Theorem 3

Under Conditions C1–C6, as , where

Let be the quantile of , where is the nominal size of the test. The following parametric bootstrap procedure is employed to approximate :

Step 1: Simulate a sample path at the same frequency according to the model under  with the CCF based estimate .

Step 2: Let be the estimate of under using the resample path obtained in Step 1, and be the version of for the resampled path.

Step 3: For a large positive integer , repeat Steps 1 and 2 times and obtain, after ranking, .

Then, the Monte Carlo approximation of is . The proposed test rejects if . The justification of the above bootstrap procedure can be made based on Theorem 3 via the standard techniques for instance those given in Chen, Gao and Tang CheGaoTan08 ().

4 Simulation study

We report in this section the results from our simulation studies which are designed to verify the proposed parameter estimator and model testing procedure. To evaluate the quality of the proposed EL estimator, we first chose two univariate diffusion processes with known transition densities, so that the MLEs can be compared with the proposed EL estimates. The two processes are the Vasicek model (Vasicek Vas77 ()) (VSK),


and the Cox–Ingersoll–Ross model (Cox, Ingersoll and Ross CoxIngRos85 ()) (CIR),


where , and are unknown parameters which represent the mean reverting rate, long-run mean and volatility of the process, respectively. Both processes are widely used in interest rate modeling and various option price formulation. For the Vasicek model, the transition distribution of is a normal distribution . For the CIR model, when , is a multiple of a non-central Chi-square random variable with degrees of freedom and non-centrality parameter , where the multiplier is with . The CCFs of these two models can easily be derived from their known transitional densities.

We then considered estimation for the jump diffusion model VSK-MJ as given in (3) based on its CCF function


where . For comparison, we approximated its transition density by a mixture of normal distributions, which is a first order approximation proposed in Aït-Sahalia, Fan and Peng AtSFanPen09 (). Here, and The approximate MLEs were obtained based on the mixture approximation given above.

We also consider the Inverse Gaussian OU process (IG-OU) in (4), that is, the process follows the inverse Gaussian law , for every when is generated from . The CCF of this process is


Since neither the exact transition density nor its approximation is available, we were content with carrying out estimation with the proposed methods.

The last simulation model considered for the estimation is a bivariate extension of the univariate Ornstein–Uhlenbeck process (BI-OU),


where , Under the condition that the eigenvalues of the matrix have positive real parts, the process is stationary with transition distribution being a bivariate normal , where , and

The CCF of the process is known to be for .

We then carried out simulations to evaluate the ability of the proposed tests in detecting model deviations. When we chose the simulation models, we had in mind two issues in finance that have drawn considerable research attention recently. The first issue is whether the process is subject to jumps, and the second is whether we could differentiate two processes with different jump rates. Our simulation study formulated two settings of hypotheses to address these two issues. In the first setting, we tested

In the second setting, we tested

For computing the powers, in the first setting we used the data simulated from the jump diffusion model VSK-MJ to test the null model which does not have jumps; in the second setting, we used the data simulated from the inverse Gaussian OU model which has infinite-activity jumps to test the null hypothesis that prescribes a finite-activity jump process.

For each model, we simulated 500 sample paths which were observed at monthly observations () for , respectively. The choices of parameter values were motivated by Chen, Gao and Tang CheGaoTan08 () and Ait-Sahalia, Fan and Peng AtSFanPen09 ().

In parameter estimation, we discovered that for both real and imaginary parts of the CCF, their nonparametric smoothing estimators are wave-like functions and roughly diminish to zero at the same points, which creates a region denoted as (here the subscript indicates that the region depends on ). In practice, we searched on a couple of grid points in the data range of  and picked the union of as the support region for the frequency domain of in the estimation. We then chose the uniform density as the weight function over the support region.

In model testing, a similar effort was initially made to obtain the support region of the nonparametric CCF estimate, denoted as , and the support region of the theoretical CCF under , denoted as . Here the theoretical CCF under used from our EL method. Then the support region of the frequency domain in testing was taken as the union of and . We chose the uniform density as the weight function over this support region for testing. There is little contribution to the integrated empirical likelihood ratio from outside the support region. The biweight Kernel was used for smoothing in testing. The bandwidth selection is described in Section 3. The bandwidth sets were specified in Tables 3 and 4 for the two test settings. It is observed that the values of the bandwidths were quite small, which was due to the rapid oscillation of the CCF curves which favored smaller bandwidth in the curve fitting.

We chose throughout our simulation study as it is the optimal instrument suggested in Carrasco et al. Caretal07 (). Some numerical exploration (not reported) indicated the choice of the function is not crucial in the context of the paper. For testing, we picked the unit instrument to reduce computing burden.

(a) Vasicek model
125 MLE 1.383 (0.603) 0.090 (0.015) 0.047 (0.003)
EL 1.305 (0.643) 0.090 (0.017) 0.046 (0.004)
250 MLE 1.118 (0.397) 0.090 (0.011) 0.047 (0.002)
EL 1.052 (0.410) 0.089 (0.013) 0.046 (0.002)
500 MLE 0.966 (0.240) 0.089 (0.008) 0.047 (0.002)
EL 0.951 (0.273) 0.089 (0.009) 0.047 (0.002)
(b) CIR model
125 MLE 1.372 (0.644) 0.091 (0.019) 0.183 (0.012)
EL 1.290 (0.719) 0.093 (0.023) 0.178 (0.014)
250 MLE 1.127 (0.374) 0.090 (0.013) 0.182 (0.008)
EL 1.089 (0.435) 0.091 (0.015) 0.179 (0.009)
500 MLE 1.000 (0.245) 0.091 (0.010) 0.182 (0.006)
EL 0.977 (0.290) 0.092 (0.011) 0.180 (0.007)
(c) Jump diffusion VSK-MJ model
125 AMLE 1.056 (0.381) 0.093 (0.020) 0.046 (0.005) 1.770 (0.723) 0.060 (0.016)
EL 1.090 (0.261) 0.084 (0.031) 0.048 (0.009) 1.851 (0.323) 0.066 (0.020)
250 AMLE 0.977 (0.226) 0.093 (0.013) 0.047 (0.003) 1.659 (0.466) 0.059 (0.010)
EL 1.043 (0.201) 0.090 (0.023) 0.048 (0.007) 1.825 (0.236) 0.068 (0.015)
500 AMLE 0.939 (0.145) 0.092 (0.009) 0.047 (0.002) 1.620 (0.311) 0.060 (0.007)
EL 1.018 (0.115) 0.089 (0.018) 0.049 (0.005) 1.801 (0.163) 0.068 (0.012)
(d) Inverse Gaussian OU model
125 EL 10.328 (3.665) 1.048 (0.106) 20.722 (2.146)
250 EL 11.154 (1.976) 1.059 (0.043) 21.380 (0.878)
500 EL 11.489 (1.652) 1.031 (0.024) 20.846 (0.461)
Table 1: Empirical averages and their standard errors (in parentheses) of the maximum (MLE) or approximate maximum (AMLE) likelihood estimates and the proposed empirical likelihood estimates (EL) under the four univariate models

Table 1 reports the empirical averages of the parameter estimates and their standard errors as well as the true parameter values used for simulation. When the sample size increases, standard errors of all the proposed estimates decrease, indicating the consistency of the estimators. We observe from Table 1(a)–(b) for the VSK and CIR models where the MLEs are available, the proposed EL estimates are quite close to the MLEs. Although the EL estimates tend to have larger standard errors than the MLEs, we do note that under the VSK model in Table 1(a), the bias of EL estimates for the mean reverting parameter are smaller than the corresponding MLEs for all , and . For the jump diffusion model VSK-MJ (Table 1(c)), we see the EL estimates are consistently more efficient than the approximate MLEs in the estimation of and the Poisson intensity . For the Inverse Gaussian OU model, which does not have the MLE to compare with, the proposed estimates as reported in Table 1(d) are close to the true values, and the standard errors converge as the sample size increases.

125 MLE 0.441 (0.197) 0.395 (0.270) 0.607 (0.176)
EL 0.381 (0.208) 0.525 (0.238) 0.594 (0.192)
250 MLE 0.353 (0.165) 0.307 (0.148) 0.563 (0.110)
EL 0.354 (0.178) 0.449 (0.184) 0.564 (0.153)
500 MLE 0.280 (0.118) 0.241 (0.104) 0.526 (0.068)
EL 0.261 (0.168) 0.383 (0.154) 0.487 (0.112)
125 MLE 0.145 (0.166) 0.099 (0.056) 0.167 (0.067) 0.080 (0.079)
EL 0.141 (0.141) 0.117 (0.085) 0.129 (0.044) 0.071 (0.034)
250 MLE 0.141 (0.151) 0.096 (0.036) 0.140 (0.065) 0.116 (0.074)
EL 0.142 (0.129) 0.094 (0.073) 0.095 (0.033) 0.094 (0.028)
500 MLE 0.102 (0.120) 0.092 (0.023) 0.115 (0.051) 0.146 (0.055)
EL 0.099 (0.108) 0.104 (0.064) 0.077 (0.024) 0.105 (0.028)
Table 2: Empirical averages and their standard errors (in parentheses) of the maximum (MLE) likelihood estimates and the proposed empirical likelihood estimates (EL) under the Bivariate OU model

Table 2 reports the estimates for the bivariate OU process and shows that the EL estimates are close to the corresponding MLEs, providing the further evidence of the effectiveness of our EL estimator for multivariate process estimation. We also found that the EL estimates for the long run mean and the volatility of the first process have smaller biases and standard errors than the MLEs for all , and .

(a) Size evaluation (in percentage)
Bandwidth Overall
Size 4.8
Bandwidth Overall
Size 5.4
Bandwidth Overall
Size 5.0
(b) Power evaluation (in percentage)
Bandwidth Overall
Power 72.2
Bandwidth Overall
Power 82.6
Bandwidth Overall
Power 94.8
Table 3: : VSK versus : the jump diffusion model VSK-MJ
(a) Size evaluation (in percentage)
Bandwidth Overall
Size 4.6
Bandwidth Overall
Size 4.8
Bandwidth Overall
Size 5.0
(b) Power evaluation (in percentage)
Bandwidth Overall
Power 74.4
Bandwidth Overall
Power 84.4
Bandwidth Overall
Power 90.2
Table 4: : the jump diffusion model VSK-MJ versus the inverse Gaussian OU model

Tables 3 and 4 report the empirical size and power of the proposed test based on bootstrap resampled paths for each simulation. They contain the sizes and powers for the overall test that is based on the five bandwidth set, and for the tests that only use one bandwidth. We observe that the tests gave satisfactory sizes under both testing settings. In the first test where we used the data from the jump diffusion model VSK-MJ to test the continuous diffusion model VSK, the powers range from to across the different sample sizes and bandwidths. In the second test where we used data simulated from the infinity-activity jump process (the inverse Gaussian OU) to test the finite-activity jump process (the jump diffusion VSK-MJ), the powers range from to across the different sample sizes and bandwidth choices.

We also compared our methods with Carrasco et al. Caretal07 () for estimation, and with Chen, Gao and Tang CheGaoTan08 () for testing. To save space, we reported the results in details in the supplemental article (Chen, Peng and Yu ChePenYu ()).

5 A case study

In this section, we examine empirically the capability of our testing procedure in detecting jumps using the secondary market quotes of the 3-month Treasury Bill (T-bill) between January 1, 1965 and February 2, 1999. This bill was sampled at monthly frequency, and in total we had 410 observations. The mean of these bills is , the volatility is , the mean of the differences is very close to zero () and the standard deviation of the differences is . The sample period contains some large movements that turn out to coincide with arrivals of macroeconomic news (Johannes Joh04 ()). The goal of this empirical study was to test whether the underlying process is subject to jumps or not.

(a) VSK model
(b) CIR model
(c) VSK-MJ model
(d) Inverse Gaussian OU model
Table 5: Empirical estimation for the 3-month T-bill Data

The proposed parameter estimates under each of the four univariate models considered in the simulation study are reported in Table 5. For comparison, the MLEs or the approximate MLEs are also reported except for the Inverse Gaussian OU model. For the univariate diffusion models VSK and CIR, and the jump diffusion model VSK-MJ, the proposed parameter estimates based on CCF are very similar to the MLEs or the approximate MLEs. The EL estimates of the long-run mean are for VSK and for CIR, both of which are close to the summary statistic of mean rates (). In VSK, the average volatility of 3-month T-bill monthly return (difference) is estimated to be , which is also close to the summary statistic of volatility for the change (). However the conditional volatility of monthly change in CIR model is , and has a long-run average which is less than . Therefore, the process needs to have higher () to bring up the average volatility of monthly change to the same level reflected by the real data. In the jump diffusion model VSK-MJ, our estimate of suggests on average about 2 jumps per year. Relative to VSK and CIR models, the estimate for parameter in the jump diffusion VSK-MJ model is much smaller (), indicating that allowing jumps in the process helps to capture large movements in the interest rate, and, as a result, the continuous part of the process does not have to be as volatile as the one in VSK or CIR models.

0.010 0.012 0.014 0.016 0.018 Overall
VSK Test Stats