1. INTRODUCTION

The empirical analysis of durations between the occurrences of certain financial events is important in understanding the market behavior. To describe the evolution of such durations, Bauwens and Veredas [1] proposed a class of models called the stochastic conditional duration (SCD) models. One can refer Engle and Russell [2], Pacurar [3] for details on such duration models. The SCD model assumes that the conditional mean of durations between events is generated by a latent stochastic process. The likelihood based inference for such models requires evaluation of multiple integral with respect to latent variables. In view of these difficulties Bauwens and Veredas [1] estimated the parameters by quasi-maximum likelihood (ML) by expressing the model into a linear state space form and then applying Kalman filter method. Feng, Jiang and Song [4] proposed an extension to Bauwens and Veredas's [1] SCD model in order to capture the asymmetric behavior or leverage effect in the duration process. They adopted the Markov Chain Maximum Likelihood (MCML) approach proposed by Durbin and Koopman [5]. Strickland, Forbes and Martin [6] proposed a Bayesian Markov Chain Monte Carlo (MCMC) method in which the sampling scheme employed is a hybrid of the Gibbs and Metropolis-Hastings (MH) algorithm, with the latent vector sampled in blocks. Knight and Ning [7] discussed the empirical characteristic function (ECF) and the generalized method of moments estimation. Bauwens and Galli [8] developed a ML estimation based on the efficient importance sampling (EIS) method to estimate the parameters. Xu, Knight and Wirjanto [9] extended the SCD model of Bauwens and Veredas [1] by imposing mixtures of bivariate normal distributions on the innovations of the observation and latent equations of the duration process. Men, Kolkiewicz and Wirjanto [10] introduced a correlation between the error process and the innovations of the duration process and adopted Monte Carlo methods for estimation. Ramanathan, Mishra and Abraham [11] introduced a new procedure for estimation, filtering and smoothing in SCD models, based on estimating functions. Majority of the literature on SCD models assume that the errors follow either a Weibull, Gamma or exponential distribution. Balakrishna and Rahul [12] proposed a SCD model having inverse Gaussian (IG) distribution for innovations. In the present paper, we consider the Bayesian analysis of SCD model with IG error random variables. We propose Bayesian MCMC methods to estimate the parameters of the model. Here we follow the algorithm mentioned in Men, Kolkiewicz and Wirjanto [13], Edwards and Sokal [14] and Neal [15]. Since it is difficult to obtain the analytical conditional densities of observed data, the auxiliary particle filter (APF) in Pitt and Shephard [16] is employed to evaluate the likelihood function.

A brief review of IG SCD model is given in Section 2. The Bayesian estimation methodology and MCMC algorithm are discussed in Section 3. Simulation studies are carried out in Section 4. Section 5 presents the application of proposed method to real life data sets.

2. THE SCD MODEL

Let τi be the time of occurrence of an event (or transaction) of interest with τ0=0 and let Xi=τi−τi−1, i=1,2,…,n be the ith trade duration, which is defined as the waiting time between two consecutive transactions of an underlying asset from time i−1 to i.

The SCD model is defined by

where |ϕ|<1 to ensure the stationarity of the process. Further, ηi follows independent and identically distributed (iid) N(0,σ2), so that {ψi} follows a Gaussian AR(1) sequence and {ϵi} is an iid sequence on the positive support with common pdf f(ϵi) and ηj is independent of ϵi ∀ i,j. Note that the model depends on the unobservable ψi, called the latent variable. Most of the SCD models available in the literature assume that the innovations follow iid exponential, gamma or Weibull distributions. One can refer Bauwens and Veredas [1], Strickland, Forbes and Martin [6], Knight and Ning [7], Durbin and Koopman [5] and so on for the details on such models. Xu, Knight and Wirjanto [9] proposed a SCD model by assuming a bivariate mixture of normal distribution for (ϵi,ηi). Balakrishna and Rahul [12] assumed an IG distribution for ϵi and estimated the model parameters by EIS method. In this paper we propose a Bayesian method for this SCD model with IG innovations. We consider the unit mean IG distribution for ϵi in the model (1).

A random variable X is said to have an IG distribution with parameters μ and λ, and is denoted by IG(μ,λ) if its probability density function (pdf) is of the form

If we restrict ϵi to follow a unit mean IG distribution then its pdf is of the form

In order to develop an estimation procedure for the model let X=(x1,x2,x3,…,xn−1,xn)′ be a vector of observations from the model Eq. (1) and ψ=(ψ1,ψ2,…,ψn)′ be a vector of associated latent variables, where ′ denotes the transpose of a vector. If we denote the joint density function of (X,ψ) by f(X,ψ|θ), then the likelihood function of the parameter θ=(ϕ,σ,λ)′ based on the observations is

which is an n-fold integral with respect to ψ. We need to evaluate this integral to obtain the likelihood function. Toward that end some numerical methods such as MCMC are to be used. In the next section, we propose a Bayesian method for estimating the parameters.

3. BAYESIAN ESTIMATION

We consider the problem of estimation for the SCD model when the innovations of the durations follow an IG distribution in a Bayesian framework. By Bayes theorem, the joint conditional distribution of ψ and θ given the observations is

where f(X|ψ,θ) is the density of X given (θ,ψ), f(ψ|θ) is the density of ψ and f(θ) is the prior density of θ. Bayesian inference on (θ,ψ) is based upon the posterior distribution f(θ,ψ|X). From this joint posterior, the marginal f(θ|X) can be used to make inferences about the parameters of SCD model with IG innovations and the marginal f(ψ|X) provides the inference about the latent variables. We assume that the prior distributions of ϕ,σ,λ are mutually independent. We take prior distribution of ϕ as a normal truncated in the interval (−1,1) which results in flat density over the supporting region. For σ2 we take an inverse gamma prior distribution as in Pitt and Shephard [16], which is a conjugate prior. For the parameter λ, we use a truncated Cauchy prior distribution with density which was also considered by Bauwens and Lubrano [17].

To construct an appropriate Markov Chain sampler, the joint posterior should be expressed as proportional to various conditional distributions. So in Eq. (5), the joint posterior, f(ψ|θ) is further decomposed into a set of conditionals f(ψi|ψ−i,θ,X), where ψ−i means all terms of ψ except ψi. Here, since ψ is a vector of latent variables, it is difficult to sample from these univariate conditional distributions. So we follow the MCMC algorithm proposed by Men, Kolkiewicz and Wirjanto [13] for estimation.

While developing MCMC algorithm, the latent variables are augmented with the vector of parameters and then the estimation is carried out. We start drawing samples from the conditional posterior distribution of the latent variable ψ and the parameters θ. The sampling algorithm required to draw the samples from each conditional posterior and the associated steps are detailed below.

The parameters θ=(ϕ,σ,λ)′ are initialized with values (0.5,0.5,1)′ and then the initial value of ψ is generated from latent AR(1) model by using these values of θ.

Step 1. Sample ψ=(ψ1,ψ2,…,ψn) by drawing ψi from f(ψi|ψ−i,X,θ) for i=1,2,3,…,n. By employing the markovian structure of the model in (1), f(ψi|ψ−i,X,θ) can be expressed as f(ψi|ψi−1,ψi+1,X,θ). A single-move MH algorithm is used to sample ψi. The conditional distribution of latent random variables ψi, given the other parameters in the model have been previously obtained is given by

where f(xi|ψi), i=1,2,…,n are the conditional density functions of the durations, f(ψ1|θ) is the density of the latent state ψ1, f(ψi|ψi−1,xi−1,θ) is the conditional density of ψi given ψi−1 and f(ψi+1|ψi,xi,θ) is the conditional density of ψi+1 given ψi. Since xn is the last observation, the posterior distribution of ψn depends only on xn, xn−1 and ψn−1. The conditional distribution of ψ1 and ψn are given in Appendix A. If the conditional distribution does not possess a simple form as in the present case then it is not possible to draw the samples directly. In such cases one of the options is to consider an accept/reject method. Following Men, Kolkiewicz and Wirjanto [13] we use a single-move Metropolis Hastings algorithm to sample the latent states, where the proposal distribution is simulated by slice sampler method. The slice sampler method proposed by Edwards and Sokal [14] and Neal [15] is a method to sample random variables which do not have simple pdfs or their pdfs are known up to a normalizing constant. As the slice sampler adapts to the analytical structure of the underlying density, it is more efficient. Also it ensures faster convergence to the underlying distribution. So, to generate random variates from the conditional distribution we employ the method of slice sampler. While analyzing a data, we estimate these unobservable latent variables ψ, usingAPF method proposed by Pitt and Shephard [16]. In the following discussion, we obtain specific forms of the required samplers.

For i=2,3,…,n−1,

Substituting for the corresponding normal densities and on simplifying the square terms we get

where ω1=ϕ1+ϕ2ψi−1+ψi+1 and σ12=σ21+ϕ2.

Hence the conditional posterior distribution of ψi can be represented as

where ai=ϕ(ψi+ψi+1)1+ϕ2 and b=σ21+ϕ2.

The posterior distribution in (8) is proportional to a product of three positive functions and data cannot be simulated directly from them. So we propose a method of slice sampler whose algorithm is given below.

Slice sampler algorithm for ψi. Let us rewrite the conditional distribution in (8) as

where a1i=ai+b2 and follow the algorithm given below.

To start the slice sampling procedure, the sampled value of ψi from the last MCMC step is set as the initial value.

1. Initialize ψi(0). Set t = 0.

2. Draw a random observation u1 uniformly from the interval 0,exp−λ(xi−eψi(t))22eψi(t)xi. Then we define an interval for ψi through the inequality u1≤exp−λ(xi−eψi)22eψixi, which is equivalent to

   (10)ψi≥logxi(1+1ϵi2)21−(log(u1))λ.

3. Similarly draw a random observation u2 uniformly from the interval 0,exp−(ψi−a1i(t))22b, where a1i(t) is calculated from Eq. (9). We define an interval for ψi through the inequality u2≤exp−(ψi−a1i)22b, which is equivalent to

   (11)a1i(t)−−2blog(u2)≤ψi≤a1i(t)+−2blog(u2).

4. Draw ψi(t+1) uniformly from the interval maxlogxi(1+1ϵi2)21−(log(u1))λ,a1i(t)−−2blog(u2),a1i(t)+−2blog(u2) determined by the inequalities (10) and (11).

5. Stop, if a stopping criterion is met; otherwise, set t=t+1 and repeat from ii.

Step 2. Sample ϕ. The prior distribution of ϕ is assumed to follow a univariate normal distribution truncated in the interval (-1,1). Given a truncated normal prior distribution, ϕ∼N(αϕ,βϕ2) and the other parameters in the model have been previously sampled, the conditional distribution of ϕ is

where c=∑i=2nψi2σ2+1βϕ2, d=∑i=2nψiψi−1σ2+αϕβϕ2. The conditional distribution of ϕ is given in Appendix A. It is proportional to the product of a normal distribution and a positive function. Hence we can use the slice sampling method to sample the parameter ϕ.

Step 3. Sample σ2. We sample σ2 by taking an inverse gamma prior distribution, i.e., σ2∝ inverse Gamma (ασ2,βσ2), where ασ2 and βσ2 are hyperparameters. As the prior for σ2 is a conjugate prior the sampling is carried out by simulating from the inverse Gamma distribution with the corresponding parameters obtained. The conditional distribution of σ2 is presented in Appendix A.

Step 4. Sample λ. The conditional distribution of λ is

where f(λ) is a prior density of λ, given by (6). Here the samples cannot be simulated directly from (13). So we use a random walk MH algorithm with standard normal distribution as the proposal distribution to sample λ. The acceptance probability is computed using (13).

The following is a summary of the procedure used to sample (θ′,ψ′)′.

• Sample ψ′ using the single-move MH algorithm with proposal distribution simulated by the method of slice sampling which is briefly explained in Step 1.

• Sample ϕ from (12) following the explanations in Step 2.

• Sample σ2 directly from the inverse Gamma density using Step 3.

• Sample λ using a random walk MH algorithm with the acceptance probability computed through (13) given in Step 4.

In the next section we demonstrate the applications of the above methods through a simulation study.

4. SIMULATION STUDY

A simulation study is carried out to assess the performance of the Bayes estimators, described in the previous section. We generate 5000 observations from the model (1). Then the MCMC algorithm discussed in Section 3 is run and first 25000 iterations were discarded as burn-in from 100000 iterations. The parameters are estimated and the simulation results are tabulated in Table 1. The plots of histograms of posterior samples of ϕ, σ2, λ are shown in Figure 1 (a), 1 (b) and 1 (c) respectively. The trace plots are shown in Figure 2 (a), 2 (b) and 2 (c) respectively.

Figure 1

Histogram of the posterior samples using simulated data.

Figure 2

Trace plots of the posterior samples using simulated data.

To illustrate the application of Bayesian estimation method we analyze two sets of data. We perform model diagnosis based on the residuals to explain model adequacy. The residuals of SCD model with IG innovations are defined as ϵî=xi∕eψî, where ψî is the estimator of ψi. The estimators of the parameters ϕ, σ and λ can be obtained by the MCMC algorithm discussed in Section 3. To obtain the estimates of the unobservable component ψi, we employ an APF proposed by Pitt and Shephard [16]. This method is described in the following subsection:

5. DATA ANALYSIS

We demonstrate the applications of the model, by analysing two sets of data. Data sets are on intraday trades of IBM OHLC bar data downloaded from Algoseek Website and intraday trades of US Brent Crude Oil downloaded from the Website of a Swiss Forex bank. Only trades between 9:30:00 am and 4:00:00 pm are recorded as this is the normal trading hour.

6. CONCLUSION

In this article we considered the Bayesian estimation of SCD model with IG error random variables. The slice sampling within an MCMC estimation methodology and the APF method are utilized to estimate the parameters and the latent states of the model. Simulation study shows that the proposed estimation method is reliable. Two sets of financial data are considered to demonstrate the utility of the model. The non-parametric tests applied to residuals support the assumptions on the errors. However, more rigorous statistical tests are to be developed to perform model diagnosis in the presence of latent variables.

CONFLICT OF INTEREST

The authors declare no conflict of interest.

ACKNOWLEDGMENTS

The authors thank the editor and the reviewers for their careful reading of the article. Sri Ranganath acknowledge Kerala State Council for Science Technology & Environment, for the financial assistance for carrying out this research work. The research of N. Balakrishna is partially supported by the DST SERB under the project No. SR/S4/MS:837/13.