Bayesian inference for the log-symmetric autoregressive conditional duration model

Abstract

This paper adapts Hamiltonian Monte Carlo methods for application in log-symmetric autoregressive conditional duration models. These recent models are based on a class of log-symmetric distributions. In this class, it is possible to model both median and skewness of the duration time distribution. We use the Bayesian approach to estimate the model parameters of some log-symmetric autoregressive conditional duration models and evaluate their performance using a Monte Carlo simulation study. The usefulness of the estimation methodology is demonstrated by analyzing a high frequency financial data set from the German DAX of 2016.

Key words
ACD models; Bayesian inference; high frequency financial data; log-symmetric distributions

INTRODUCTION

Bibliographical review and preliminaries

The concept of log-symmetry appears when a random variable presents the same distribution as its reciprocal, or in terms of ordinary symmetry regarding the distribution of the logged random variable; see Jones 2008. The class of distributions having this property is called log-symmetric and it has been used to describe the behavior of strictly positive data. The log-symmetric class encompasses bimodal distributions as special cases, and distributions that possess either lighter or heavier tails than the log-normal distribution, which is a particular case of this class; see e.g., Vanegas & Paula ( 2016b). Some examples of log-symmetric distributions are: log-normal, log-Student-$t$, log-logistic, log-Laplace, log-Cauchy, log-power-exponential, log-slash, harmonic law, Birnbaum-Saunders, and Birnbaum-Saunders-$t$; see e.g., Crow & Shimizu (1988), Birnbaum & Saunders (1969), Rieck & Nedelman (1991), Johnson et al. (1994), 1995, Díaz-García & Leiva (2005), Marshall & Olkin (2007), Jones (2008), and Vanegas & Paula (2016b).

The class of log-symmetric distributions has been primarily used in the regression context. Vanegas & Paula ( 2016a) proposed log-symmetric regression models which allow both the median and skewness (or the relative dispersion) be described using an arbitrary number of non-parametric additive components. Vanegas & Paula ( 2016b) studied some interesting properties of the log-symmetric class of distributions. Vanegas & Paula ( 2016c) proposed an extension to allow the presence of non-informative left or right-censored data in log-symmetric regression models. Medeiros & Ferrari 2017 discussed the issue of testing hypothesis in symmetric and log-symmetric regression models. In special, the authors considered the Wald, likelihood ratio, score and gradient tests for this purpose. Finally, Ventura et al. 2019 analyzed movie business data using log-symmetric regression models.

High frequency financial data on transactions have been modeled primarily by autoregressive conditional duration (ACD) models, which were proposed by Engle & Russell 1998. These models are commonly used to capture the clustering structure and they can be seen as the counterpart of GARCH models for modeling trade duration (TD) data; see Liu & Heyde 2008. We strongly recommend reading the works by Pacurar 2008 and Bhogal & Variyam 2019, which are literature reviews on ACD models. Some characteristics concerning TD data are: (C1) the irregular nature with respect to the way they are collected; (C2) the diurnal intra-day pattern; (C3) the large number of observations; (C4) the probability density function (PDF) with asymmetric shape; and (C5) the hazard rate (HR) with inverse bathtub shape (unimodal); see e.g., Leiva et al. 2014. Extensions of the original ACD model proposed by Engle & Russell 1998 are basically based on the following aspects: (A1) the distributional assumption in order to yield an asymmetric PDF and an unimodal HR; (A2) the linear form for the conditional mean or median dynamics; (A3) and the time series properties; see, for example, Bauwens & Giot 2000, Grammig & Maurer 2000, Meitz & Terasvirta 2006, Chiang 2007, Pacurar 2008, Bhatti 2010, Leiva et al. 2014, Diana 2015, Dionne et al. 2015, Zheng et al. 2016, Saulo et al. 2019, and Mishra & Ramanathan 2017.

Recently, Saulo & Leao 2017 proposed a family of ACD models based on the class of log-symmetric models. The log-symmetric ACD models encompass all the log-symmetric distributions cited at the beginning of this introduction as special cases, that is, they encompass highly competitive performance models in the literature. For example, the log-normal-ACD, Birnbaum-Saunders-ACD and Birnbaum-Saunders-$t$-ACD models; see Xu 2013 and Leiva et al. 2014. The log-symmetric ACD models are written in terms of a conditional median duration rather than a conditional mean duration - the typical parameter used in the literature is the mean. The use of the median is more interesting because it is a measure of central tendency better than the mean, besides the median-based approach provides additional protection against outliers; see Saulo et al. 2019. On the other hand, the log-symmetric family provides a wide range of asymmetric distributions with HR with inverse bathtub shape. Therefore, characteristics (C4) and (C5) and aspects (A1) and (A2) are addressed by the log-symmetric ACD models.

The flexibility provided by the log-symmetric family makes its corresponding ACD models an important area to be explored in the literature. In this context, this paper deals with the problem of Bayesian inference for log-symmetric-ACD models. The estimation methodology is based on the Hamiltonian Monte Carlo (HMC) method, which generates chains both with little dependence and high probability of acceptance (Neal 2011). The main advantage of log-symmetric ACD models is the robustness property of the median, namely, it is not affected by extremes or outliers. In terms of predictions, it implies that they will not be significantly affected by freak events; see Saulo et al. 2019.

Log-symmetric distributions

A continuous and positive random variable $X$ follows a log-symmetric distribution if its PDF is given by

$$f_X(x;\theta ,\varphi ,g(\cdot ))=\{\begin{array}{cc}\hfill \frac{1}{\sqrt{\varphi }x}g\left(a^2\left(x\right)\right),& \hfill x>0,\\ \hfill 0,& \hfill \mathrm{\text{otherwise}},\end{array}$$(1)

where $a\left(x\right)=log\left({[x/\theta ]}^{1/\sqrt{\varphi }}\right)$, $\theta >0$ is a scale parameter, $\varphi$ is a power parameter and $g$ is a density generator with $g\left(u\right)>0$ for $u>0$ and ${\int }_0^{\infty }{u}^{-1/2}g\left(u\right)\mathrm{\text{d}}u=1$. Note that $g$ may involve an extra parameter $\xi$ or an extra parameter vector $𝛏$. In this case, the notation $X\sim \mathrm{\text{LS}}(\theta ,\varphi ,g)$ is used. For example, in order to obtain a random variable $X$ following a log-normal, log-Student-$t$ (having $\xi$ degrees of freedom), log-Laplace or log-slash distribution, we use, respectively, $g\left(u\right)\propto exp(-\frac{1}{2}u)$, $g\left(u\right)\propto {(1+\frac{u}{\xi })}^{-\frac{\xi +1}{2}}$, $g\left(u\right)\propto exp(-\frac{1}{2}{u}^{\frac{1}{2}})$, $g\left(u\right)\propto \mathrm{\text{IGF}}(\xi +\frac{1}{2},\frac{u}{2})$, where $\mathrm{\text{IGF}}(a,x)=\frac{1}{x^a}{\int }_0^{x}exp(-t)t^{a-1}\mathrm{\text{d}}t$ is the incomplete gamma function for $a>0$ and $x\ge 0$. The quantile function of the log-symmetric distribution is given by

$$t_X(q;\theta ,\varphi ,g(\cdot ))=\theta exp\left[\sqrt{\varphi }{v}_{\xi }\left(q\right)\right],$$(2)

where $v_{\xi }\left(q\right)$ is the $q\times 100$th quantile of $V=\frac{(Y-\mu )}{\sqrt{\varphi }}\sim \mathrm{\text{S}}(\mu =0,\varphi =1,g(\cdot ))$, with the notation $\mathrm{\text{S}}$ referring to symmetrical distributions.

Some statistical properties related to a random variable $X$ following a log-symmetric distribution, namely $X\sim \mathrm{\text{LS}}(\theta ,\varphi ,g)$, are: (P1) $X^{\star }={\left(\frac{X}{\theta }\right)}^{1/\sqrt{\varphi }}\sim \mathrm{\text{LS}}(\theta =1,\varphi =1,g)$ is standard log-symmetric distributed; (P2) $cX\sim \mathrm{\text{LS}}(c\theta ,\varphi ,g)$, with $c>0$; (P3) $X^c\sim \mathrm{\text{LS}}({\theta }^c,c^2\varphi ,g)$, with $c\ne 0$; (P4) $\theta$ is the the median of the distribution of $X$; and (P5) setting $Y=log\left(X\right)$; (Vanegas & Paula 2015). With this, we obtain a symmetric random variable whose distribution belongs to the symmetric class with PDF given by

$$f_Y(y;\mu ,\varphi ,g(\cdot ))=\frac{1}{\sqrt{\varphi }}g\left(\frac{{(y-\mu )}^2}{\varphi }\right),y\in ℝ,$$(3)

where $\mu =log(\theta )\in ℝ$ is a location parameter, $\varphi >0$ is a dispersion parameter and $g$ is as in (2); see Fang et al. 1990 and we write $Y\sim \mathrm{\text{S}}(\mu ,\varphi ,g)$. The properties (P2) and (P3) say that the log-symmetric distribution is closed under scale and reciprocal transformations, respectively.

Organization of the paper

The rest of this paper proceeds as follows. The log-symmetric ACD models are formulated in Section LOG-SYMMETRIC ACD MODELS, with parameters estimated by the Bayesian approach using HMC algorithm. In Section NUMERICAL EVALUATION we present a numerical evaluation of the proposed model considering the (i) evaluation of this model via Monte Carlo (MC) simulations and; (ii) application of a real-world high-frequency financial data. Some concluding remarks and possible future research are mentioned in Section CONCLUDING REMARKS.

LOG-SYMMETRIC ACD MODELS

Consider a sequence of successive times $T_1,\dots ,T_n$ at which market events, or trades, occur. Then, the duration or time elapsed between $X_i$ and $X_{i-1}$, for $i=1,\dots ,n$, is given by $X_i=T_i-T_{i-1}$. The family of log-symmetric ACD models introduced by Saulo & Leao 2017 considers a dynamic point process in terms of a conditional median duration

$${\theta }_i=t_X(0.5;{\theta }_i,{\varphi }_i,g),$$

where $t_X(\cdot )$ is the quantile function (QF) of the log-symmetric distribution presented in (2) and ${\Omega }_{i-1}$ is an information set including all information available until time $T_{i-1}$. The log-symmetric ACD model is defined by

$$X_i={\theta }_i{ϵ}_i^{\sqrt{{\varphi }_i}},i=1,\dots ,n,$$(4)

where $\{{ϵ}_i\}$ are independent identically distributed (IID) random variables following a log-symmetric distribution with median and power equal to one, denoted by $${ϵ}_i\stackrel{\text{IID}}{\sim }\text{LS}(1,1,g)$$, and ${\theta }_i$ and ${\varphi }_i$ are the median and skewness of the $X_i$ distribution, respectively, as $$X_i\stackrel{\text{IND}}{\sim }\text{LS}({\theta }_i,{\varphi }_i,g)$$, then the $X_{i}s$ are independent (IND) not identically distributed. The linear form of (4) is given by

$$\underset{Y_i}{\underbrace{log\left(X_i\right)}}=\underset{{\mu }_i}{\underbrace{log\left({\theta }_i\right)}}+\sqrt{{\varphi }_i}\underset{{\epsilon }_i}{\underbrace{log\left({ϵ}_i\right)}},i=1,\dots ,n,$$(5)

where $${\epsilon }_i\stackrel{\text{IND}}{\sim }\text{S}(0,1,g(\cdot ))$$, namely $\{{\epsilon }_i\}$, are IID random variables following a standard symmetric distribution with PDF given (3). Then, we write $$Y_i\stackrel{\text{IND}}{\sim }\text{S}({\mu }_i,{\varphi }_i,g)$$.

The component ${\theta }_i$ in (4) is defined in terms of autoregressive (AR) and moving average (MA) processes, of order $p$ and $q$ respectively, as

$$\begin{array}{cc}\hfill & log\left({\theta }_i\right)=\varpi +\sum _{j=1}^p{\alpha }_{j}log\left({\theta }_{i-j}\right)+\sum _{j=1}^q{\beta }_j\left(\frac{X_{i-j}}{{\theta }_{i-j}}\right),\hfill \end{array}$$(6)

where $\varpi >0$, ${\alpha }_j\ge 0$ and ${\beta }_j\ge 0$. Then, the notation LSACD($p,q$) is used. The order of lags for LSACD($p,q$) models, in general, are set as $p=1$ and $q=1$, because a higher order does not improve the model fit; see Bhatti 2010. Thereby, in the following, any LSACD($p=1,q=1$) model is simply denoted as LSACD. Moreover, for simplicity’s sake, it is assumed that ${\varphi }_i=\varphi$, for $i=1,\dots ,n$.

Bayesian inference

The Bayesian inference is based on the Bayes theorem

$$\pi (\theta ,{\varphi }_i\mid X)\propto f_{X|\theta ,\varphi }(x_i|{\theta }_i,{\varphi }_i;g(\cdot ))\pi (\theta ,{\varphi }_i),$$

where $\pi (\theta ,{\varphi }_i\mid X)$ is the posterior distribution and $\pi (\theta ,{\varphi }_i)$ is the prior distribution. We will use the following estimator

$$\hat{\theta }={𝔼}_{\theta \mid y}\left[\theta \right],$$(7)

which minimizes the expected squared error of the estimate. The Equation (7) is analytically intractable. Therefore, we adopt HMC sampling strategies for obtaining samples from the joint posterior distributions and adopt the sample mean of the HMC simulation as an estimator of ${𝔼}_{\theta \mid y}\left[\theta \right]$. In the next subsection, we present the HMC methodology.

The HMC is a method which combines alternately Gibbs updates and Metropolis ones and avoids the random walk behavior. The main advantage of using HMC as opposed to other methods is to generate chains both with little dependence and high probability of acceptance (Neal 2011), especially when compared to the Metropolis-Hastings algorithm. The HMC method is implemented in the R package rstan; see www.R-project.org and R-Team 2019. Consider a random vector $\theta \in {ℝ}^k$ as position variables (parameters) and $$rr\in {\mathbb{R}}^k$$ an independent auxiliary random vector, with $$rr\sim N_k(0,M)$$. The joint PDF of $$(\theta \theta ,rr)$$ is given by

$$\pi (\theta \theta ,rr)\propto \exp (-H(\theta \theta ,rr)),$$(8)

where $$H(\theta \theta ,rr)=U(\theta \theta )+K(rr)$$ is a Hamiltonian function, $U\left(\theta \right)=-log\left[\pi (\theta \mid y)\pi \left(\theta \right)\right]$ is called the potential energy and $$K(rr)=rr{M}^{-1}rr$$ is called the kinetic energy.

This method considers a candidate to $$(\theta \theta ,rr)$$ which is generated in two stages before being subjected to a Metropolis acceptance step. These stages are presented in Algorithm 1

NUMERICAL EVALUATION

In this section, we carry out a simulation study to evaluate the performance of the Bayesian estimators of some log-symmetric ACD models. Then, we illustrate the proposed methodology by applying it to a real-world high-frequency financial data set. This data set refers to price durations of BASF-SE stock on 19th April 2016 downloaded from the Dukascopy site (www.dukascopy.com). We consider the ACD models based on the following log-symmetric distributions: log-normal (LNACD), log-Laplace (LLACD), log-Student-$t$ (L$t$ACD) and log-slash (LSACD). To estimate the parameters of the model, we used the HMC algorithm using rstan in the R software. In Stan’s programming language the convergence of a Markov chain to a stationary distribution by no-U-turn sampler (NUTS) which is even more efficient at exploring the posterior; see Hoffman & Gelman 2014. The interested reader in Stan programming language and HMC algorithm is referred to Hoffman & Gelman 2014 and Carpenter et al. 2017.

A simulation study

The scenario considers: sample size $n\in \{500,2000,5000\}$, vector of true parameters ${(\omega ,\alpha ,\beta ,\varphi )}^{\top }=\{0.10$, $0.90$, $0.10,0.50\}$ for LNACD and LLACD models; ${(\omega ,\alpha ,\beta ,\varphi ,\xi )}^{\top }=\{0.10$, $0.90$, $0.10,4.00\}$ for L$t$ACD and LSACD models, and the following independent prior distributions: $\omega \sim \mathrm{\text{N}}(0,100)𝕀(\omega >0)$; $\alpha \sim \mathrm{\text{N}}(0,100)𝕀(0\le \alpha \le 1)$; $\beta \sim \mathrm{\text{N}}(0,100)𝕀(0\le \beta \le 1)$; $\mu \sim \mathrm{\text{N}}(0,100)$; $\xi \sim \mathrm{\text{N}}(0,100)𝕀(\xi >2)$ if ${ϵ}_t\sim {\mathrm{\text{log-}}t}_{\xi }$; and $\xi \sim \mathrm{\text{N}}(0,100)𝕀(\xi >1)$ if ${ϵ}_t\sim {\mathrm{\text{log-slash}}}_{\xi }$.

For each value of the parameter and sample size, we report the empirical values for the mean, median, standard deviation (SD) and the percentage of data sets where the true parameter value was contained inside the Bayesian 90% and 95% credible intervals: this is the MC estimate of frequentist coverage for an interval estimator. The estimates computed by the Bayesian approach are presented by Tables I and II. As the sample size increases, the Bayesian estimators become more efficient. Therefore, in general, all of these results show the good performance of the Bayesian estimators of the corresponding parameters.

Table I
Summary statistics from simulated LNACD and LLACD data for the indicated estimators and $𝐧$.

Statistic	LNACD			LLACD
	$n$			$n$
	500	2000	5000	500	2000	5000
	$\hat{\omega }$			$\hat{\omega }$
True value	0.1000	0.1000	0.1000	0.1000	0.1000	0.1000
Mean	0.1607	0.1176	0.0973	0.1928	0.1071	0.0998
Median	0.1467	0.1104	0.0963	0.1848	0.0956	0.0992
SD	0.1196	0.0571	0.0275	0.0892	0.0614	0.0165
CR90%	0.8838	0.8866	0.9022	0.7606	0.8930	0.9004
CR95%	0.9194	0.9318	0.9486	0.8430	0.9404	0.9474
	$\hat{\alpha }$			$\hat{\alpha }$
True value	0.9000	0.9000	0.9000	0.9000	0.9000	0.9000
Mean	0.8195	0.8831	0.9004	0.8346	0.9135	0.8988
Median	0.8268	0.8872	0.9011	0.8386	0.9197	0.8986
SD	0.0608	0.0310	0.0143	0.0437	0.0322	0.0073
CR90%	0.6774	0.8552	0.9022	0.6056	0.9050	0.8932
CR95%	0.7760	0.9072	0.9502	0.7220	0.9400	0.9506
	$\hat{\beta }$			$\hat{\beta }$
True value	0.1000	0.1000	0.1000	0.1000	0.1000	0.1000
Mean	0.1686	0.1146	0.1026	0.1344	0.0684	0.1052
Median	0.1682	0.1138	0.1025	0.1340	0.0667	0.1052
SD	0.0289	0.0157	0.0083	0.0203	0.0126	0.0038
CR90%	0.2262	0.7710	0.8830	0.4852	0.6102	0.8932
CR95%	0.3326	0.8552	0.9346	0.6202	0.7276	0.9455
	$\hat{\varphi }$			$\hat{\varphi }$
True value	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000
Mean	0.5212	0.5022	0.4945	0.4391	0.4479	0.4985
Median	0.5209	0.5021	0.4944	0.4383	0.4477	0.4987
SD	0.0161	0.0078	0.0049	0.0195	0.0129	0.0086
CR90%	0.6374	0.8878	0.8956	0.3196	0.8684	0.8974
CR95%	0.7506	0.9392	0.9492	0.4300	0.9264	0.9448

Table II
Summary statistics from simulated L$𝐭$ACD and LSACD data for the indicated estimators and $𝐧$.

Statistic	L$t$ACD			LSACD
	$n$			$n$
	500	2000	5000	500	2000	5000
	$\hat{\omega }$			$\hat{\omega }$
True value	0.1000	0.1000	0.1000	0.1000	0.1000	0.1000
Mean	0.0557	0.0985	0.1035	0.2890	0.1224	0.0960
Median	0.0530	0.0985	0.1034	0.2639	0.1221	0.0926
SD	0.0355	0.0021	0.0046	0.1769	0.0425	0.0289
CR90%	0.6292	0.7988	0.8116	0.7702	0.8104	0.9066
CR95%	0.7622	0.8804	0.9540	0.8506	0.8826	0.9536
	$\hat{\alpha }$			$\hat{\alpha }$
True value	0.9000	0.9000	0.9000	0.9000	0.9000	0.9000
Mean	0.9162	0.9001	0.8999	0.7501	0.8857	0.8991
Median	0.9180	0.9001	0.8998	0.7636	0.8864	0.9007
SD	0.0180	0.0007	0.0022	0.0872	0.0218	0.0148
CR90%	0.7542	0.8998	0.8982	0.5314	0.7608	0.9080
CR95%	0.8588	0.9470	0.9476	0.6543	0.8438	0.9528
	$\hat{\beta }$			$\hat{\beta }$
True value	0.1000	0.1000	0.1000	0.1000	0.1000	0.1000
Mean	0.1020	0.0993	0.0991	0.1975	0.1061	0.1042
Median	0.1019	0.0992	0.0991	0.1974	0.1059	0.1038
SD	0.0121	0.0013	0.0016	0.0320	0.0119	0.0077
CR90%	0.8396	0.8520	0.8926	0.7920	0.7970	0.8814
CR95%	0.9094	0.9192	0.9430	0.8065	0.8750	0.9348
	$\hat{\varphi }$			$\hat{\varphi }$
True value	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000
Mean	0.4912	0.5070	0.5137	0.5987	0.4986	0.4995
Median	0.4909	0.5069	0.5138	0.6006	0.4943	0.4987
SD	0.0296	0.0140	0.0087	0.0269	0.0303	0.0123
CR90%	0.5236	0.8556	0.8854	0.3420	0.8960	0.9036
CR95%	0.6430	0.9210	0.9416	0.5260	0.9476	0.9496
	$\hat{\xi }$			$\hat{\xi }$
True value	4.0000	4.0000	4.0000	4.0000	4.0000	4.0000
Mean	3.8302	3.5650	3.9848	5.5243	4.1830	3.9987
Median	3.7381	3.5486	3.9756	5.6100	4.9140	3.9790
SD	0.7377	0.2866	0.2217	0.8520	0.4132	1.4082
CR90%	0.5270	0.9092	0.9024	0.5809	0.9096	0.9023
CR95%	0.6512	0.9584	0.9536	0.6460	0.9160	0.9429

Analysis of high-frequency financial transaction data

We now illustrate the log-symmetric ACD models by analyzing the BASF-SE data set. A data adjustment is necessary due to the fact that these data often exhibit certain diurnal patterns; see Tsay 2002. We used the R package ACDm see Belfrage 2015 to perform the diurnal adjustment of the BASF-SE data. We simulated two chains with 1500 iterations and discarded the first 500 as burn-in. Table III presents some descriptive statistics for the BASF-SE data set, including central tendency statistics, standard deviation (SD), coefficient of variation (CV), skewness (CS) and kurtosis (CK). From Table III, we observe the right skewed nature and high kurtosis level of the data distribution. The skewness is ratified by the histogram showed by Figure 1(a).

Figure 1
Histogram (left), TTT plot (center) and boxplots (right) for the BASF-SE data.

Table III
Summary statistics for the BASF-SE data.

$n$	Minimum	Median	Mean	Maximum	SD	CV	CS	CK
2194	0.061	0.682	1.067	9.776	1.167	109.35%	2.521	8.902

A tool to characterize the shape of a hazard rate is the scaled total time on test (TTT) function; see Aarset 1987. The hazard rate of a random variable $X$ is defined by $h\left(x\right)=f\left(x\right)/[1-F\left(x\right)]$, where $f$ and $F$ are the PDF and cumulative distribution function (CDF) of $X$. The scaled TTT function is defined by $W\left(u\right)=H^{-1}\left(u\right)/H^{-1}\left(1\right)$, for $0\le u\le 1$, where $H^{-1}\left(u\right)={\int }_0^{F^{-1}\left(u\right)}[1-F\left(y\right)]\mathrm{\text{d}}y$, with $F^{-1}$ being the inverse function of the CDF of $X$. An approximation for $W$ is obtained by plotting the points $[k/n,W_n\left(k/n\right)]$, with

$$W_n\left(k/n\right)=\frac{\sum _{i=1}^k{x}_{\left(i\right)}+[n-k]x_k}{\sum _{i=1}^n{x}_{\left(i\right)}},k=1,\dots ,n,$$

and $x_{\left(i\right)}$ being the $i$th observed order statistic. Figure 1(b) suggests that the hazard rate for the BASF-SE data set is inverse-bathtub-shaped, as expected; see Leiva et al. 2014.

Figure 1(c) shows the usual and adjusted box-plots, where the latter is useful in cases when the data follow a skew distribution, since a significant number of observations can be classified as atypical when they are not; see Hubert & Vanderveeken 2008. From Figure 1(c), we note that potential outliers considered by the usual box-plot are not influential when we observe the adjusted box-plot.

Table IV reports the Bayesian estimates of the LNACD, L$t$ACD, LLACD and LSACD model parameters (with estimated standard errors in parentheses). Furthermore, we report the Expected Bayesian Information Criterion (EBIC), Deviance Information Criterion (DIC), Watanabe-Akaike Information Criterion (WAIC) and Bayesian Information Criterion (BIC); see Akaike 1992, Schwarz et al. 1978, Watanabe 2010 and Spiegelhalter et al. 2002.

Table IV
Estimates (with SE in parentheses) for fit to the BASF-SE data.

	LNACD	L$t$ACD	LLACD	LSACD
$\varpi$	$-$0.3228 (0.0831)	$-$0.3211 (0.0853)	$-$0.2122 (0.0507)	$-$0.2203 (0.0821)
$\alpha$	0.4748 (0.1673)	0.4809 (0.1681)	0.6646 (0.1145)	0.5819 (0.1632)
$\beta$	0.0507 (0.0107)	0.0516 (0.0117)	0.0564 (0.0117)	0.0512 (0.0113)
$\varphi$	1.0694 (0.0164)	1.0597 (0.0187)	0.8720 (0.0183)	0.9641 (0.0163)
$\xi$		6.7067 (1.0554)		7.8025 (4.9732)
AIC	6543.0070	6546.5520	6787.1950	6604.3071
BIC	6562.6380	6571.0910	6806.8260	6643.2816
WAIC	6539.0000	6541.1000	6789.8000	6671.2800
DIC	6524.3930	6590.7080	6815.9870	6695.7233

From Table IV, note that the LNACD model provides the better adjustment compared to the other models based on the values of EBIC, DIC, WAIC and BIC. Table V reports the estimate of effective sample size and $R$ statistic of Gelman et al. 1992, we observed that the generated chains were efficient. Figure 2 shows the sample autocorrelation function (ACF) and trace plots for the parameters from LNACD model. From this figure, we can notice the absence of autocorrelation and that the chains converge to their stationary distributions also observed by the statistic $R$ of the Table V.

Trace plots and ACF for the LNACD model.

Table V
Estimate of effective sample size and $R$ statistic.

	LNACD		L$t$ACD		LLACD		LSACD
	n_eff	$\hat{R}$	n_eff	$\hat{R}$	n_eff	$\hat{R}$	n_eff	$\hat{R}$
$\varpi$	380	1.00	563	1.01	531	1.01	571	1.01
$\alpha$	313	1.00	624	1.01	579	1.01	631	1.01
$\beta$	384	1.01	748	1.00	515	1.00	683	1.00
$\varphi$	358	1.00	1065	1.00	1087	1.00	895	1.00
$\xi$			2698	1.00	947	1.00	607	1.01

CONCLUDING REMARKS

We have discussed Bayesian inference for log-symmetric autoregressive conditional duration models, which are based on the conditional median duration. We have employed Hamiltonian Monte Carlo sampling strategies to obtain the estimates of the model parameters. A Monte Carlo simulation study was carried out to evaluate the behavior of the Bayesian estimates of the corresponding parameters. We have applied the proposed models to a real-world data set of financial transactions from the German DAX stock exchange.

As part of future research, it would be of interest to propose an outlier detection procedure to detect and estimate outlier effects for these models; see Chiang & Wang 2012. Also, influence diagnostic tools can be extended to log-symmetric autoregressive conditional duration models; see Leiva et al. 2014. Work on these issues is currently in progress and we hope to report some findings in a future paper.

ACKNOWLEDGMENTS

The research was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) grants from the Brazilian government and by Fundação de Amparo `a Pesquisa do Estado do Amazonas (FAPEAM) grants from the government of the state of Amazonas, Brazil.

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Publication Dates

• Publication in this collection
  20 Oct 2021
• Date of issue
  2021

History

• Received
  19 Mar 2019
• Accepted
  18 July 2019