Bayesian analysis of the stochastic conditional duration model

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Abstract

A Bayesian Markov chain Monte Carlo methodology is developed for estimating the stochastic conditional duration model. The conditional mean of durations between trades is modelled as a latent stochastic process, with the conditional distribution of durations having positive support. Regressors are included in the model for the latent process in order to allow additional variables to impact on durations. The sampling scheme employed is a hybrid of the Gibbs and Metropolis-Hastings algorithms, with the latent vector sampled in blocks. Candidate draws for the latent process are generated by applying a Kalman filtering and smoothing algorithm to a linear Gaussian approximation of the non-Gaussian state space representation of the model. Monte Carlo sampling experiments demonstrate that the Bayesian method performs better overall than an alternative quasi-maximum likelihood approach. The methodology is illustrated using Australian intraday stock market data, with Bayes factors used to discriminate between different distributional assumptions for durations.

Introduction

The increased availability of data at the transaction level for financial commodities has allowed researchers to model the microstructure of financial markets. New models and inferential methods have been developed to enable the analysis of intraday patterns and the testing of certain microstructure hypotheses to occur.

The present paper contributes to this growing literature by presenting a methodology for estimating a particular dynamic model for durations between stock market trades: the stochastic conditional duration (SCD) model. In contrast to the autoregressive conditional duration (ACD) model of Engle and Russell (1998), in which the conditional mean of the durations is modelled as a deterministic function of past information, the SCD model treats the conditional mean of durations as a function of a stochastic latent process, with the conditional distribution of durations defined on a positive support. As such, the contrast between the two specifications is similar to the contrast between the generalized autoregressive conditional heteroscedasticity (GARCH) and stochastic volatility (SV) frameworks for capturing the conditional volatility of financial returns.

Whilst several modifications of the original ACD specification have been put forward (see Bauwens et al., 2004, for a recent summary), the literature that focuses on the SCD model is less advanced. Although Durbin and Koopman (2001) suggest the use of a latent variable model for durations, they do not develop the idea further, with the first published analysis of the model occurring in Bauwens and Veredas (2004). The latter authors present a quasi-maximum likelihood (QML) technique for estimating the SCD model. They also compare the empirical performance of the SCD model and a particular specification of the ACD model, concluding that the SCD model is preferable according to a number of different criteria. Variants of the SCD model include a two-factor version of the model developed in Ghysels et al. (2004) (first appearing in draft form in 1998) and the SCD model with ‘leverage effect’ outlined in Feng et al. (2004), with these models estimated using simulated method of moments and Monte Carlo maximum likelihood, respectively. A comparison of the forecasting performance of a range of duration models, including the SCD and ACD specifications, is presented in Bauwens et al. (2004).

As is the case with the SV model, the ‘parameter-driven’ SCD model presents a potentially more complex estimation problem than its ‘observation-driven’ alternative, by augmenting the set of unknowns with a set of unobservable latent variables. However, as is also argued in the case of the SV/GARCH dichotomy, with the advent of more sophisticated computing resources, the extra computational burden associated with that complexity is no longer such an issue. Moreover, as is highlighted in Ghysels et al. (2004), the ability of the latent variable framework, in which the SCD model is nested, to be extended to the multi-factor case is a crucial advantage over the ACD framework, in which the dynamic behavior in all higher order moments is necessarily tied to the dynamic behavior in the conditional mean. This is a particularly important point as it relates to the first two moments of durations, given their association with the separate features of liquidity and liquidity risk, respectively. Consequently, the SCD model should be viewed both as a serious competitor to the ACD model and as the starting point for more sophisticated modelling of durations data.

In this paper, a Bayesian methodology for estimating the SCD model is presented. The unobservable latent variables are integrated out of the joint posterior distribution via a hybrid Gibbs/Metropolis–Hastings (MH) Markov chain Monte Carlo (MCMC) sampling scheme. Along the lines suggested in Durbin and Koopman, 2000, Durbin and Koopman, 2001, the non-Gaussian state-space representation of the model is approximated by a linear Gaussian model in the neighborhood of the posterior mode of the latent process. This approximating model defines the candidate distribution from which blocks of the latent process are drawn, via the application of the Kalman filter and simulation smoother; see, for example, Carter and Kohn (1994), Frühwirth-Schnatter (1994), De Jong and Shephard (1995) and Durbin and Koopman (2002). The latent variable draws are then accepted with a particular probability, according to the MH algorithm. The MH subchains associated with the latent variable blocks are embedded into an outer Gibbs chain in the usual way, with estimates of all posterior quantities of interest produced from the draws after convergence of the hybrid algorithm.

The methodology presented here is very general. In particular, with relatively minor alterations, the MCMC algorithm can accommodate a range of conditional distributions. This means that we can readily allow for alternative specifications for the durations data and use Bayes factors to choose between the different models. In addition, this flexibility means that the methodology is applicable beyond the durations context. For example, in the case of financial returns, the use of an appropriate conditional distribution in combination with a latent variable model for the variance, constitutes a valid SV specification. Hence, the methodology that we present is an alternative to existing methods for estimating SV models; see, for example, Jacquier et al., 1994, Jacquier et al., 2004, Kim et al. (1998), Meyer and Yu (2000) and Liesenfeld and Richard (2003). Furthermore, the algorithm is equally applicable to different data types, including discrete and binary data. As such, in the context of transaction data, it could be used to estimate, for example, the dynamic behavior in tick count or binary price change data; see Rydberg and Shephard (2003). Whilst this level of generality is also a feature, in principle, of the Durbin and Koopman, 2000, Durbin and Koopman, 2001 importance sampling-based algorithm, as is noted below, that algorithm is likely to be computationally inefficient compared to MCMC when applied to the large data sets associated with transaction data.

The structure of the paper is as follows. Section 2 describes the SCD model, including the way in which microstructure effects can be readily incorporated by augmenting the state equation with observed regressors. Section 3 then outlines the MCMC scheme, with details of the approximation used in the production of a candidate distribution for the vector of latent variables provided in Appendix A. In Section 4, the sampling behavior of the Bayesian estimation method is compared with that of the QML approach adopted by Bauwens and Veredas (2004), via Monte Carlo (MC) experiments. The experiments are based on a sample size of N=10000, to be representative of the typically large sample sizes that are associated with transaction data. The findings indicate that the Bayesian method is, in general, superior to the QML approach, in terms of both bias and efficiency. Two empirical illustrations of the Bayesian method are then described in Section 5, based on durations data for two Australian firms, Broken Hill Proprietary (BHP) Limited and News Corporation (NCP), for the month of August 2001. In both cases three alternative conditional distributions are specified for durations, with Bayes factors, calculated using the Savage-Dickey density ratio, finding in favour of the gamma distribution for the BHP data and the Weibull distribution in the case of the NCP data. Trade volume is found to have no impact on durations for BHP yet does have an impact in the case of NCP. Some conclusions and proposed extensions are given in Section 6.

Section snippets

A stochastic conditional duration model

Denoting by τi the time of the ith transaction, the ith duration between successive trades at times τi-1 and τi is defined as xi=τi-τi-1. Defining x=x1,x2,,xN as the N–dimensional vector of durations, the SCD model for x is defined asp(x|ψ,λ)=i=1Npxi|ψi,λ,ψi=Wiδ+φψi-1+σηηi,ηii.i.d.N(0,1)for i=2,3,,N, where xi, conditional on ψi, is assumed to be independent of ηi for all i. The latent variable process in (2) is assumed to have finite variance, with |φ|<1. The (1×k) vector Wi contains the

Discussion

The similarity between the SCD model and the SV model provides some motivation for the use of an MCMC approach to estimation. Jacquier et al. (1994) show that an MCMC algorithm applied to a particular SV specification is superior, in terms of both bias and efficiency, to both maximum likelihood and generalized method of moment approaches, both of which have asymptotic justification only. An advantage of a Gibbs-based MCMC algorithm over other finite sample simulation methods, such as the

Sampling experiments

MC experiments are conducted to assess the sampling properties of the Bayesian simulation method and to compare these properties with those of the QML approach adopted by Bauwens and Veredas (2004) in their analysis of the SCD model. In these experiments we only consider the case where the conditional distribution is exponential. We also omit regressors from the state equation so that the scalar parameter δ represents the intercept in (2). Earlier research by Jacquier et al. (1994) in an SV

Data description

The Bayesian methodology for estimating the SCD model is illustrated using transaction data for two Australian listed companies: BHP and NCP. Trade durations are initially calculated for the month of August 2001, amounting to N=48190 and 17 691 observations for BHP and NCP, respectively. Only trades between 10:20 a.m. and 4:00 p.m. are recorded. Zero trade durations are not included, following Engle and Russell (1998). This filtering reduces the length of the time series to N=27746 observations

Conclusions

In this paper an MCMC estimation methodology for the SCD model has been introduced. The methodology exploits the state space representation of the latent variable model for durations and has been shown to be readily adaptable to different choices of distributional assumption for the measurement equation. The MCMC approach has been compared with the QML procedure using MC experiments. The results indicate that the MCMC approach tends to outperform the QML approach in terms of both bias and

Acknowledgements

This research has been supported by a Monash University Research Grant and an Australian Research Council Discovery Grant. The authors would like to thank an Associate Editor, three referees and Ralph Snyder for some very helpful suggestions and comments on an earlier version of the paper, as well as participants in seminars at Monash University and the Australasian Meetings of the Econometric Society in Sydney, July 2003. Note that most of the numerical results in the paper are produced using

References (34)

  • J. Daniellson

    Stochastic volatility in asset prices estimation with simulated maximum likelihood

    J. Econometrics

    (1994)
  • P. De Jong et al.

    The simulation smoother for time series models

    Biometrica

    (1995)
  • J. Durbin et al.

    Time series analysis of non-Gaussian observations based on state space models from both classical and Bayesian perspectives

    J. Roy. Statist. Soc. Ser. B

    (2000)
  • J. Durbin et al.

    Time Series Analysis by State Space Methods

    (2001)
  • J. Durbin et al.

    A simple and efficient simulation smoother for time series models

    Biometrika

    (2002)
  • D. Easley et al.

    Time and the process of security price adjustment

    J. Finance

    (1992)
  • O. Elerian et al.

    Likelihood inference for discretely observed nonlinear diffusions

    Econometrica

    (2001)
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