Latent Factor-based Duration Models

Some literature treats the conditional duration as a latent variable instead of a deterministic variable. In the GARCH literature, it is well known that compared with GARCH frameworks, using unobserved latent variables in stochastic volatil- ity (SV) modelling yields favourable results, and the dynamics of the …nancial time series are better captured. The advantages of SV models over GARCH models were discussed in Danielson (1994), Kim, Shephard, and Chib, (1998), and Ghysels, Harvey, and Renault (1996).

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Based on the similarities to GARCH models, Bauwens and Veredas (1999) introduced the Stochastic Conditional Duration (SCD) model. The SCD model

follows: ₈ > > < > > : xi = i"i ln _i =!+ ln _{i 1}+ui; j j<1 (2.6)

where the two error terms "i i:i:dand ui i:i:d in equation (2.6) are the two

sources of unobservables in the observed and conditional durations. The unob- served latent variables in the SCD model can be treated as information ‡ows driving the duration process that cannot be observed directly. The second line of Equation (2.6) can be treated as a stationary autoregressive process. The pa- rameter of the lagged log conditional duration, ;is forced to be less than unity. Compared with traditional ACD models, the latent variables in SCD models can yield more complex shapes for the hazard functions. Also the distributional as- sumptions for "i and ui can be di¤erent, which makes the SCD model a mixture

model. The above conditions give greater ‡exibility in modelling the dynamics of the duration process and also make it possible for the SCD model to capture the unobservable information in the market. Bauwens and Veredas (1999) also …nd the SCD model yields favourable results in a comparison study with the log- ACD model. However, the more complex assumptions make the exact likelihood function very di¢ cult to locate and time consuming to estimate. The multidi- mensional integral requires heavy simulations which are especially extensive when the data set is large. Bauwens and Veredas (2004) use QML methods with the Kalman …lter1_{, but again this has the problem of estimate e¢ ciency since it is}

not using the true likelihood of durations.

1_{An alternative is the Monte Carlo Markov Chain (MCMC) technique in Strickland, Forbes,}

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Ghysels, Gourieroux, and Jasiak (2004) propose a more ‡exible Stochastic Volatility Duration (SVD) model, which allows one to estimate the dynamics for the conditional variance. Note that the traditional ACD model does not allow for independent dynamic parameterization for the conditional mean and conditional variance. The reason for this restriction is that the traditional ACD model as- sumes that the higher order conditional moments are directly linked to the …rst moment of conditional mean. Ghysels, Gourieroux, and Jasiak (2004) indicate that this assumption is too restrictive, and information from variance such as market liquidity and risk could be lost. The SVD model however, combines the dynamics of the conditional mean and conditional variance with two time vary- ing factors in the model. The SVD model starts from two independent Gaussian random factors, and analyses the conditional mean and conditional variance dy- namic patterns using a VAR representation. The initial SVD model is build on standard exponential duration model with gamma heterogeneity from cross- sectional and panel data literature. It assumes durationxi = _aVU , whereU follows

an exponential distribution with intensity one, V follows a gamma distribution and is independent of U. In terms of Gaussian factors, the SVD model can be expressed as:

xi =

H(1; F1)

aH(b; F2)

; (2.7)

where a and b are positive parameters, F1 and F2 are i.i.d. standard normal variables, andH(b; F) =G(b; (F))where is the c.d.f. of the standard normal distribution and G(b; ) is a quantile function of Gamma (b, b) distribution.

Since the SVD model belongs to the family of nonlinear ACD models, its like- lihood function is di¢ cult to evaluate. In fact there have been few applied studies

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of SVD models in the literature. Ghysels, Gourieroux, and Jasiak (2004) use a two-step procedure, which …rst assumes the marginal distribution of durationxi

follows a Pareto distribution depends only on a and b, and then uses simulated moments method to obtain the autoregressive parameters. Strong dynamics in both conditional mean and variance factors are found using Paris Stock Exchange data in Ghysels et al. (2004). Note that Bauwens, Giot, Grammig, and Veredas (2004) argue that the assumption of Pareto distribution in the …rst step of the above procedure might not be appropriate. The forecasting performance of the SVD is also found to be very poor compared with traditional ACD and log-ACD models in Bauwens et al. (2004).

Another latent variable based models is the discrete mixture ACD (MACD) model of Hujer and Vuletic (2004). Instead of treating the duration process as linear and following a particular form of distribution, they combine the idea of mixture models and ACD models. In common with other latent variable ACD models, the introduction of a discrete-valued latent regime variable increases the ‡exibility of the model, in order to capture the speci…c characteristics of intraday duration data. The discrete mixture ACD model proposed by Hujer and Vuletic (2004) can also be viewed as a compromise of the two extreme models of Markov switching ACD in Hujer, Vuletic,and Kokot (2003) and the discrete mixture exponential ACD model in De Luca and Gallo (2004).

Overall, the ACD literature is moving forwards in favour of the nonlinear models. The more comprehensive models give greater ‡exibility and better un- derstanding of the market information. However, the problem of evaluating the more complicated likelihood function is yet to be solved. Further research on ACD models is still needed.

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