Conditional duration model and the unobserved market heterogeneity of traders: an infinite mixture of non-exponentials

Revista Colombiana de Estadística July 2016, Volume 39, Issue 2, pp. 307 to 325 DOI: http://dx.doi.org/10.15446/rce.v39n2.51584 Conditional Duration Model and the Unobserved Market Heterogeneity of Traders: An Infinite Mixture of Non-Exponentials Modelo de duración condicionada y heterogeneidad inobservada de los agentes. Una mezcla infinita de distribuciones no exponenciales Emilio Gómez-Déniz1,a, Jorge V. Pérez-Rodríguez2,b 1Department of Quantitative Methods, University of Las Palmas de Gran Canaria and TIDES Institute, Las Palmas de Gran Canaria, Spain 2Department of Quantitative Methods, University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain Abstract This paper extends the conditional duration model proposed by Luca & Zuccolotto (2003) proposing an infinite mixture of distributions based on non-exponentials that account for the unobserved market heterogeneity of traders. The model we propose takes into account the fact that reaction times follow a gamma distribution and that the intensity parameter follows the reciprocal of an inverse Gaussian distribution. This extension allows us to capture, not only various density shapes of durations, but also nonmonotonic shapes of hazard functions. The model also allows us to test the unobserved heterogeneity of traders. This mixture model is easy to fit and characterises the behaviour of the conditional durations reasonably well. Key words:Autoregressive conditional duration model, Exponential distribution, Gamma distribution, Heterogeneity, Reciprocal inverse gaussian distribution. Resumen Este trabajo extiende el modelo de duración condicionada propuesto por Luca & Zuccolotto (2003) introduciendo una mezcla infinita de distribuciones no exponenciales que permite incorporar la heterogeneidad inobservada en el mercado por los agentes. El modelo propuesto tiene en cuenta el hecho de que el tiempo de respuesta sigue una distribución gamma y que el parámetro que mide la intensidad sigue una distribución recíproca inversa aProfessor. E-mail: [email protected] bProfessor. E-mail: jv.perez-ro[email protected] 307 308 Emilio Gómez-Déniz & Jorge V. Pérez-Rodríguez Gaussiana. Esta modelización permite no sólo capturar distintas formas de la distribución de la duración sino que también captura funciones de azar no monótonas. El modelo propuesto es fácil de ajustar a datos de duración proporcionando resultados razonables y competitivos con otros modelos utilizados en la literatura. Palabras clave:modelo de duración autorregresivo condicional, distribución exponencial, distribución Gamma, heterogeneidad, distribución recíproca inversa gaussiana. 1. Introduction The most popular econometric model for the intraday trading process has traditionally been the autoregressive conditional duration (ACD) model (Engle & Russell 1998) and its extensions: logarithmic ACD (Bauwens & Giot 2000), stochastic conditional duration (Bauwens & Veredas 2004) and stochastic volatility duration. To cope with higher-order dynamics in the duration process (Ghysels, Gouriéroux & Jasiak 2004), a nonlinear version based on self-exciting threshold autoregressive processes (Zhang, Russell & Tsay 2001), or a family of ACD models that encompasses most specifications proposed in the literature (Fernandes & Grammig 2006), among others. However, the recent focus of several studies of the financial duration of transactions has taken into account the heterogeneity of agents. This heterogeneity plays an important role in determining the size of price movements, the amount being exchanged, and the frequency at which orders are presented and executed, among other factors (Luca & Gallo 2009). For example, financial market microstructure theories divide traders into informed and non-informed traders. The assumption of interaction among agents, i.e, the coexistence of informed traders who possess private information and liquidity traders whose information set is publicly available (O’Hara (1995) and Ghysels (2000)) suggests that financial durations may obey different probability laws. In addition, there are many reasons to believe that arrival rates for informed and uninformed agents exhibit temporal dependence: each having its own distinct pattern. Since durations reflect the heterogeneity of traders, modelling their baseline distribution requires us to take into account random effects that account for unobserved heterogeneity, rather than fixed effects, which can suffer from the incidental parameter problem. The distribution of the duration is assumed to be derived from a mixture of distributions given that heterogeneity could reflect differing rates of information arrival (i.e., Clark 1973, Luca & Gallo 2009 and references therein). The basic idea of enabling the inclusion of the unobservable heterogeneity of individuals in models, by means of diverse probability laws, is not new in microeconometric literature (Lancaster 1990). For example, this approach is taken in shared-frailty models, which are the survival-data analogue to regression models with random effects.1 1A frailty is a latent random effect that enters multiplicatively on the hazard function. Revista Colombiana de Estadística 39 (2016) 307–325 Conditional Duration Model and Unobserved Heterogeneity 309 However, in a financial context, durations could also reflect the heterogeneity of traders. Therefore, modelling the baseline distribution requires a mixture of distributions. For example, Luca & Zuccolotto (2003), Luca & Gallo (2004) and Luca & Gallo (2009) have proposed a link between statistical and financial aspects within a set of distributional assumptions based on financial market microstructure theories. These are related to a wide variety, or heterogeneity, of agents and also of trading conditions (for example, different degrees of information possessed by various traders and differing attitudes toward risk or budget constraints, among others). These authors have proposed a mixture of distributions based on exponentials in the context of ACD models, such as finite mixtures (i.e., the traders are assumed to be divided into a finite number of groups),2or infinite mixtures (i.e., when every trader is considered to have an individual behaviour). However, unlike Luca & Zuccolotto (2003), our paper contributes to the financial duration literature by considering another baseline distribution for ACD models. More specifically, we propose an infinite mixture of non-exponential distributions for reaction times, thus we draw attentran to a more complex unobserved heterogeneity based on the intensity of duration we do this by using a reciprocal inverse Gaussian distribution, which is related to the inverse Gaussian distribution. The reciprocal inverse Gaussian distribution, just like the connected inverse Gaussian distribution, has a flexible shape and location in the positive support. This shape is allowed to vary according to the position of the data. There are several motivations for doing so. First, the inverse Gaussian distribution (also known as the Wald distribution) has many applications in studying lifetime and number of event occurrences (Lancaster 2003, Jørgensen 1982, Zhang, Russell & Tsay 1983, Seshadri 1993, Chhikara & Folks 1989, Abraham & Balakrishnan 1998, Balakrishnan & Nevzorov 2003, among others).3When modelling the unobserved heterogeneity of individuals, the inverse Gaussian distribution is also employed in frailty duration models (Hougard 1984, Hanagal & Dabade 2013). However, the reciprocal of the inverse Gaussian distribution (which is not commonly used by statisticians and has been little explored in the literature) allows us to introduce more flexibility and tractability into the modelling of scale parameters and to obtain closed form expressions for the infinite mixture. Second, following Grammig & Maurer (2000), it is well known that a good candidate distribution should allow for both greater flexibility in modelling and non-monotonic hazard shapes (i.e., QML estimation may provide very inaccurate estimates if the baseline hazard function is non-monotonic). This argument sug2For example, Luca & Gallo (2009) suggest using a mixture of two distributions with timevarying weights (Mixture ACD model). These authors show that the limitations of the standard base model, and its inadequate modelling of the behaviour in the tail of the distribution, are resolved by their model. 3This is a member of the natural exponential family of distributions and can be considered an alternative to the exponential, log-normal, Frechet and Weibull distributions, among others. However, it also provides flexibility in modelling when early occurrences of failure are dominant in a lifetime distribution and the failure rate is expected to be non-monotonic. Revista Colombiana de Estadística 39 (2016) 307–325 310 Emilio Gómez-Déniz & Jorge V. Pérez-Rodríguez gests that another way to increase the flexibility of ACD models is to use a mixture of distributions. Finally, parameter estimates are very sensitive to the choice of mixing distribution and hence it is important to consider one that takes this non-monotonicity in the baseline hazard into consideration. The proposed approach facilitates the natural parameterisation of a point process in terms of a conditional mean duration for the ACD model4and provides estimates with a high degree of fit in terms of the log likelihood function for the family of exponentials. The rest of this paper is organised as follows: in Section 2, we describe two mixture models based on the exponential and gamma-reciprocal inverse Gaussian ACD model. An empirical example is developed in Section 3, and some conclusions are drawn in Section 4. 2. The New Mixture ACD Model Let xi=ti−ti−1be the duration between two consecutive periods, where tiis the time for period i, and the expected conditional duration for the ith trade is expressed as ψi=E(xi|xi−1, . . . , x1;θ1), where xi=ψiεi. Therefore, standardised or excess durations are described by xi/ψi≡εi∼iid D(θ2), where Dis a general distribution defined within the interval (0,∞)with E(εi)=1, and where θ1and θ2are vectors of unknown parameters. ψiis called the conditional duration and can be expressed as a linear function of past durations and lagged conditional durations. Hence, the ACD (p, q)model can be written as: ψi=ω+ q X j=1 αjxi−j+ p X j=1 βjψi−j, i = 1,2, . . . , N, where ω > 0,αj≥0and βj≥0for all j. Although not necessary, these sign restrictions are convenient to ensure the positivity of ψiin the estimation. In this paper, we focus particularly on D(θ2). Any distribution defined as a positive support can be specified for Dto estimate ACD models. Simple distributional assumptions for the conditional excess durations have been employed, such as exponential and Weibull distributions (Engle & Russell 1998). However, these pdfs are far from capturing the most salient features of the errors, namely their variability. Therefore, alternative hypotheses have been considered, such as the Generalised gamma distribution (Lunde 1999), or the Burr distribution (Grammig & Maurer 2000) (both nest Weibull and exponential as special cases), and other standardised financial duration distributions such as BirnbaumSaunders (Bhatti 2010). There are distributions which accommodate certain stylised facts such as over-dispersion (standard deviation greater than the mean), slowly-decreasing autocorrelations (Bauwens, Giot, Gramming & Veredas 2004) or 4In the basic formulation, ACD considers that all heterogeneity is captured by the conditional expectation term, which is linear in lagged durations and exhibits persistence decay at an exponential rate. Revista Colombiana de Estadística 39 (2016) 307–325 Conditional Duration Model and Unobserved Heterogeneity 311 a finite and infinite mixture of distributions to model the behaviour in the tail of the distribution (Luca & Zuccolotto 2003, Luca & Gallo 2004, Luca & Gallo 2009). Regarding the consistency of the estimation, (Engle & Russell 1998) show that consistent and asymptotically Normal estimates of vector parameters are obtained by maximising by QML. This is the case even if the distribution of the standardised duration, D(θ2), is not exponential. Drost & Werker (2004) show that consistent estimates are obtained when the QML estimation method is based on the standard gamma family (including the exponential). In this paper, we examine exponential and gamma distributions with reciprocal inverse Gaussian heterogeneity. Unlike Luca & Zuccolotto (2003), who studied an exponential inverse gamma distribution, we take into account the non-monotonic hazard function (i.e., that the intensity function conditional on past durations could be constant, increasing or decreasing with respect to duration (like Grammig & Maurer 2000). 2.1. The mixture model A mixture of distributions is usually employed for modelling situations with characteristics that differ from those that would be anticipated under a simple component distribution. This is what occurs with the exponential distribution, where the variance is determined by the mean. For this reason, general families of distributions, such as mixtures, are often taken as alternative models that offer greater flexibility. Apart from this flexibility, a mixture model can be thought of as a market which is heterogeneous in which the mixing distribution represents a measure of this heterogeneity. Let the pdf of the gamma distribution be f(x) = 1 θσΓ(σ)xσ−1e−x/θ, with a scale parameter θ > 0and shape parameter σ > 0. A new class of probability distributions with a domain in IR+is now introduced by mixing the θparameter with the reciprocal of the inverse Gaussian distribution. This family can be considered as an alternative to the exponential-inverse Gaussian distribution described in Bhattacharya & Kumar (1986) and Frangos & Karlis (2004), and also to the gamma-generalised inverse Gaussian distribution proposed in GómezDéniz, Calderín & Sarabia (2013). Distribution mixtures have often been used in statistical research, especially in the construction of duration models (see Luca & Zuccolotto 2003, Luca & Gallo 2004, Luca & Gallo 2009) because they make it possible to model heterogeneity. It is straightforward to show that g(z) = γ √2πz exp −(γz +δ)2 2z, z > 0, γ > 0, δ > 0(1) is the probability density function of the reciprocal of a variable distributed according to the inverse Gaussian distribution with parameters γ > 0and δ > 0. That is, if Y,Y > 0, follows an inverse Gaussian distribution with parameters γ > 0and δ > 0, then the random variable Z= 1/Y follows the distribution Revista Colombiana de Estadística 39 (2016) 307–325 312 Emilio Gómez-Déniz & Jorge V. Pérez-Rodríguez given in (1). See, for instance, Jørgensen, Seshadri & Whitmore (1991).5Henceforth, when a positive random variable Zfollows the pdf given in (1) we write Z∼ RIG(γ, δ)and X∼ G(σ, θ)to denote that the random variable Xfollows a gamma distribution. Additionally, the reciprocal inverse Gaussian distribution can be derived, as the inverse Gaussian distribution, as a particular case of the generalised inverse Gaussian distribution (see Mohtashami & Mohtashami 2011, for details). We begin with the definition of the gamma-reciprocal inverse Gaussian distribution. Definition 1. A random variable Xfollows a gamma-reciprocal inverse Gaussian distribution if it admits the following stochastic representation: X|σ, θ ∼ G(σ, θ)(2) θ∼ RIG(δ, γ), θ ∈Θ = (0,∞),(3) where σ, δ, γ > 0. Henceforth, this distribution is denoted by X∼ GRIG(σ, δ, γ). The next result gives us closed expressions for the pdf of the new distribution. Theorem 1. Let εi∼ GRIG(σ, δ, γ), i.e. εifollow the representation given by expressions (2)-(3). Then its pdf is given by: f(εi) = r2γ π γσeγ δ Γ(σ)xσ−1[φ(εi, δ)]1/2−σKσ−1/2(γφ(εi, δ)) ,(4) where φ(εi, δ) = √δ2+ 2εiand Kν(z)represents the modified Bessel function of the second kind (see Jørgensen 1982). Proof .The pdf can be obtained directly by using the well-known compounding formula f(εi) = Z∞ 0 f(εi|θ)g(θ)dθ, (5) and by arranging the parameters. The mean and variance of the new distribution can also be obtained by compounding and are given by E(εi) = λ=σ(1 + γδ) γ2,(6) var(εi) = σ(σ+ 1) γ2δ2+ 3+ 3γδ−σ(γδ + 1)2 γ4. The next result gives the closed expression for the cumulative distribution function (cdf) of this new distribution, in which the parameter σis assumed to be integer and known. 5Note that there are several different parameterizations of the inverse Gaussian distribution (see, for example Zhang et al. 1983). Revista Colombiana de Estadística 39 (2016) 307–325 Conditional Duration Model and Unobserved Heterogeneity 313 Theorem 2. Let σ > 0be an integer and known and εi∼ GRIG(δ, γ), i.e. εi follows the representation given by expressions (2)-(3). Then its cdf is given by: F(εi) = 1 −r2 πγ3/2eγ δ σ−1 X j=0 (γx)j j!(φ(εi, δ))1/2−jKj−1/2(γφ(εi, δ)) .(7) Proof .The cdf can be computed by using: F(εi) = Zεi 0Z∞ 0 f(ti|θ)f(θ)dt dθ. Now, by applying Fubini’s theorem and taking into account the expression, for integer values of σ, 1 Γ(σ)Zεi 0 1 θσtσ−1e−t/θdt = 1 − σ−1 X j=0 (εi/θ)je−εi/θ j!, is satisfied (see for instance (Castillo, Hadi, Balakrishnan & Sarabia 2005), p. 82), then the desired result is obtained after some algebra. 2.2. Two Simple Sub-Models Two simple sub-models obtained from the model above are now studied in some detail. 2.2.1. The Exponential-Reciprocal Inverse Gaussian It is known that the exponential distribution is a particular case of the gamma distribution and obtained by taking σ= 1 in (2). In this case, after the compounding process with the reciprocal inverse Gaussian distribution we obtain the exponential-inverse Gaussian distribution mixture, the pdf of which is obtained from (4). It should be taken into account that K1/2(z) = pπ 2zexp(−z)and thus, after some simple algebra, we obtain f(εi) = γeγ δ φ(εi, δ)exp [−γφ(εi, δ)] ,(8) for εi>0, δ > 0and γ > 0, which are the heterogeneity parameters. It is straightforward to see that d2 dx2(log f(εi)) = 2 + γφ(εi, δ) [φ(εi, δ)]4>0, and so the distribution is log-convex. Therefore, the cumulative distribution function, F(εi), and the survival function, 1−F(εi)are also log-convex and the hazard rate function is nonincreasing. Revista Colombiana de Estadística 39 (2016) 307–325 314 Emilio Gómez-Déniz & Jorge V. Pérez-Rodríguez To further investigate the properties of this mixture model, we examine the behaviour of some parameters of interest. The mode is always at the origin of the support of the Distribution. Some pdf curves are shown in Figure 1. Figure 1: Some examples of the probability density function of the gamma-reciprocal inverse Gaussian mixture distribution for selected parameter values, δ= 0.5 (thick), δ= 1 (dashed) and δ= 2 (dotdashed). The mean and the variance are given by E(εi) = κ=1 + γ δ γ2,(9) var(εi) = γ2δ2+ 4γδ + 5 γ4. Revista Colombiana de Estadística 39 (2016) 307–325 Conditional Duration Model and Unobserved Heterogeneity 315 The survival function of the exponential-reciprocal inverse Gaussian mixture distribution is obtained from (7). This gives: ¯ F(εi) = exp {γ[δ−φ(εi, δ)]}.(10) By applying (8) in conjunction with (10) we obtain the hazard rate function given by: h(εi) = γ φ(εi, δ), which is clearly decreasing, as expected from a mixture of the exponential distribution. For many insurance and financial risks, the right-tail risk, representing lowfrequency and large-loss events, is usually measured in terms of the right-tail index (Frangos & Karlis 2004), given by: ζ(εi) = 1 E(εi)Z∞ 0q¯ F(t)dt −1. In some computations, for the exponential-reciprocal inverse Gaussian mixture distribution, this value is given by ζ(εi) = 1 + 2 1 + γδ , and, therefore a value larger than 1, which is that of the exponential distribution. Moreover, in recent years there has been increasing interest (especially in actuarial and financial settings) in computing quantiles of probability of the distribution of a particular risk. Two measures of risk that are of particular importance in this framework are the value of risk (VaR) and the tail of the value of risk (TVaR); see Furman & Zitikis (2008) for details. For the distribution we are considering, these measures are obtained in simple forms, and are given by: VaR(q) = 1 2"δ−1 γlog q2 −δ2#, TVaR(q) = µ+qlog q γ21 + γδ −1 2log q, for 0< q < 1. Finally, the mean residual life, which is defined as m(t) = E(X−t|x > t) gives m(t) = 1 γ2[1 + γφ(t)] . Because d dt m(t)>0for all t, the resulting mean residual life is an increasing function on t. Revista Colombiana de Estadística 39 (2016) 307–325 322 Emilio Gómez-Déniz & Jorge V. Pérez-Rodríguez 4. Conclusions In this paper, we have proposed a baseline ACD model based on a mixture of gamma and reciprocal inverse Gaussian distributions to take into account the more complex unobserved heterogeneity arising from the variety of agents and trading conditions in financial markets. The closed form solution obtained for the mixture distribution means that our proposed model is easy to fit. In this respect, the statistical measures used, the autocorrelation tests performed on standardised residuals and the Vuong closeness test conducted for non-nested models show that the gamma-reciprocal inverse Gaussian ACD model performs better than those without heterogeneity, such as the exponential, Weibull, and Burr models, but also with regard to other infinite exponential mixtures. Therefore, we conclude that our model characterises the behaviour of conditional durations reasonably well. Acknowledgements The authors are indebted to the anonymous referee for helpful comments which, without doubt, helped to improve an earlier version of the paper. The authors thank to the Ministerio de Economía y Competitividad (EGD project ECO201347092 and JVPR project ECO2011-23189) for partial support of this work. Received: June 2015 — Accepted: March 2016 References Abraham, B. & Balakrishnan, N. (1998), ‘Inverse gaussian autoregressive models’, Working Paper. University of Waterloo. Balakrishnan, N. & Nevzorov, V. (2003), A Primer on Statistical Distributions, John Wiley and Sons, New York. Bauwens, L. & Giot, P. (2000), ‘The logarithmic acd model: an application to the bid-ask quote process of three nyse stocks’, Annales d’Economie et de Statistique 60, 117–150. Bauwens, L., Giot, P., Gramming, J. & Veredas, D. (2004), ‘Comparison of financial duration models via density forecasts’, International Journal of Forecasting 20, 598–609. Bauwens, L. & Veredas, D. (2004), ‘The stochastic conditional duration model: a latent factor model for the analysis of financial durations’, Journal of Econometrics 119, 381–412. Bhattacharya, S. & Kumar, S. (1986), ‘E-ig model in life testing’, Calcutta Statistical Association Bulletin 35, 85–90. Revista Colombiana de Estadística 39 (2016) 307–325 Conditional Duration Model and Unobserved Heterogeneity 323 Bhatti, C. R. (2010), ‘The birnbaum-saunders autoregressive conditional duration model’, Mathematics and Computers in Simulation 79, 2250–2257. Castillo, E., Hadi, A., Balakrishnan, N. & Sarabia, J. (2005), Extreme Value and Related Models with Applications in Engineering and Science, Wiley. Chhikara, R. & Folks, T. (1989), The inverse Gaussian Distribution, Marcel Dekker, New York. Clark, P. (1973), ‘A subordinated stochastic process model with finite variance for speculative prices’, Econometrica 41, 135–156. Drost, F. C. & Werker, B. J. M. (2004), ‘Semiparametric duration models’, Journal of Business and Economic Statistics 22, 40–50. Engle, R. & Russell, J. (1998), ‘Autoregressive conditional duration: a new model for irregularly-spaced transaction data’, Econometrica 66, 1127–1162. Fernandes, M. & Grammig, J. (2006), ‘A family of autoregressive conditional duration models’, Journal of Econometrics 130, 1–23. Frangos, N. & Karlis, D. (2004), ‘Modelling losses using an exponential-inverse gaussian distribution’, Insurance: Mathematics and Economics 35, 53–67. Furman, E. & Zitikis, R. (2008), ‘Weighted risk capital allocations’, Insurance: Mathematics and Economics 43(2), 263–269. Ghysels, E. (2000), ‘Some econometric recipes for high-frequency data cooking’, Journal of Business & Economic Statistics 18, 154–163. Ghysels, E., Gouriéroux, C. & Jasiak, J. (2004), ‘Stochastic volatility duration’, Journal of Econometrics 119, 413–433. Gómez-Déniz, E., Calderín, E. & Sarabia, J. (2013), ‘Gamma-generalized inverse gaussian class of distributions with applications’, Communications in Statistics–Theory and Methods 42, 919–933. Grammig, J. & Maurer, K. (2000), ‘Non-monotonic hazard functions and the autoregressive conditional duration model’, Econometrics Journal 3, 16–38. Hanagal, D. & Dabade, A. (2013), ‘Modeling inverse Gaussian frailty model for bivariate survival data’, Communication in Statistics-Theory and Methods 42, 3744–3769. Hougard, P. (1984), ‘Life table methods for heterogeneous populations: distributions describing the heterogeneity’, Biometrika 71, 75–83. Jørgensen, B. (1982), Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Springer-Verlag, New York. Jørgensen, B., Seshadri, V. & Whitmore, G. (1991), ‘On the mixture of the inverse gaussian distribution with its complementary reciprocal’, Scandinavian Journal of Statistics 18, 77–89. Revista Colombiana de Estadística 39 (2016) 307–325 324 Emilio Gómez-Déniz & Jorge V. Pérez-Rodríguez Lancaster, T. (1990), The Econometric Analysis of Transition Data, Cambridge University Press. Lancaster, T. (2003), ‘A stochastic model for the duration of a strike’, Journal of the Royal Statistical Society, Series A 135(2), 257–271. Luca, G. D. & Gallo, G. (2004), ‘Mixture processes for intradaily financial durations’, Studies in Nonlinear Dynamics and Econometrics 8(2), 1–18. Luca, G. D. & Gallo, G. (2009), ‘Time-varying mixing weights in mixture autoregressive conditional duration models’, Econometric Reviews 28(1), 102–120. Luca, G. D. & Zuccolotto, P. (2003), ‘Finite and infinite mixtures for financial durations’, Metron 61, 431–455. Lunde, A. (1999), ‘A generalized Gamma autoregressive conditional duration model’, Discussion Paper. Aalborg Universiteit. Mohtashami, G. R. & Mohtashami, H. A. (2011), ‘Log-concavity property for some well-known distributions’, Surveys in Mathematics and its Applications 6, 203–219. O’Hara, M. (1995), Market Microstruture Theory, Basil Blackwell Inc., Oxford. Seshadri, V. (1993), The Inverse Gaussian Distribution: A Case Study in Exponential Families, Oxford Science Publications. Tsay, R. (2002), Analysis of financial time series, John Wiley & Sons. Zhang, M., Russell, J. & Tsay, R. (1983), ‘The inverse gaussian distribution: Some properties and characterizations’, The Canadian Journal of Statistics 11, 131–136. Zhang, M., Russell, J. & Tsay, R. (2001), ‘A nonlinear autoregressive conditional duration model with applications to financial transaction data’, Journal of Econometrics 104(7), 179–207. Appendix The integral representation of the Bessel function of the second kind is given by: Kν(z) = zν√π 2νΓ (ν+ 1/2) Z∞ 1 e−zt(t2−1)ν−1/2dt, which, after changing the variable t= 1/y, can be rewritten as Kν(z) = zν√π 2νΓ (ν+ 1/2) Z1 0 (1 −y2)ν−1/2 y2ν+1 exp −z ydy. Revista Colombiana de Estadística 39 (2016) 307–325 Conditional Duration Model and Unobserved Heterogeneity 325 The latter integral can now be approximated by using the composite trapezoidal rule: Z1 0 (1 −y2)ν−1/2 y2ν+1 exp −z ydy ≈y−y 2M χ2(y) + χ2(y)+2 M X j=0 χ2(y+jh) , where yand yare appropriate values that are near to 0 and 1, respectively, χ2(y) = (1 −y2)ν−1/2 y2ν+1 exp −z y Mis the number of grid points considered and h= 1/M. Revista Colombiana de Estadística 39 (2016) 307–325