Autoregressive Conditional Duration Model Research Articles

Hierarchical modeling of irregularly spaced financial returns

Volatility modeling is crucial in finance, especially when dealing with intraday transaction‐level asset returns. The irregular and high‐frequency nature of the data presents unique challenges. While stochastic volatility (SV) models are widely used for understanding patterns in volatility of daily stock returns which constitute regularly spaced time series, new classes of models must be introduced for analyzing volatility in irregularly spaced intraday data. Specifically these models must accommodate the random gaps between successive transactional events. By modeling the gaps using autoregressive conditional duration (ACD) models, we describe a hierarchical irregular SV autoregressive conditional duration (IR‐SV‐ACD) model for estimating and forecasting intertransaction gaps and the volatility of log‐returns. We carry out the analysis in the Bayesian framework via the Hamiltonian Monte Carlo (HMC) algorithm with No‐U‐turn sampler (NUTS) in R using the cmdstanr package. The fits and forecasts are obtained using Monte Carlo averages based on the posterior samples. We illustrate this approach using simulation studies and real data analysis for intraday prices available at microseconds level of health stocks traded on the New York Stock Exchange (NYSE). The log‐returns and gaps are calculated for the stocks and are used for modeling.

Recursive generalized gamma kernel density estimation for nonnegative dependent data

In this article, we consider the recursive generalized gamma (GG) kernel density estimator (KDE) for nonnegative dependent data from stationary α-mixing process. Asymptotic bias, variance and mean integrated squared error are provided. Simulation experiments from an autoregressive conditional duration model and a stochastic volatility model are conducted to compare the recursive GG KDE with non-recursive GG KDE. Two applications are provided for nonnegative time series data.

Modeling high-frequency financial data using R and Stan: A bayesian autoregressive conditional duration approach

In econometrics, Autoregressive Conditional Duration (ACD) models use high-frequency economic or financial duration data, which mostly exhibit irregular time intervals. The ACD model is widely used to examine the duration of transaction volume and duration of price variations in stock markets. In this work, our goal is to devise testing that will aid in the identification of the best potential duration model among a set of four models using Bayesian approach. We test three models that rely on conditional mean duration (Weibull ACD, Log Weibull ACD, Generalized Gamma ACD) and one conditional median duration model (Birnbaum-Saunders ACD), and are being compared each other. The study was done in Rstan, a programming language for statistical inference, and the simulation uses the Hamiltonian Monte Carlo (HMC) algorithm of Markov Chain Monte Carlo (MCMC) to sample from the posterior distribution. Our findings show that Log Weibull ACD (second-generation model) as best among the four models followed by Birnbaum-Saunders ACD (third-generation model). The result offers methodological implications for algorithmic trading (algo-trading), high-frequency trading and risk management.

Tail behavior of ACD models and consequences for likelihood-based estimation

We establish new results for estimation and inference in financial durations models, where events are observed over a given time span, such as a trading day, or a week. For the classical autoregressive conditional duration (ACD) models by Engle and Russell (1998), we show that the large sample behavior of likelihood estimators is highly sensitive to the tail behavior of the financial durations. In particular, even under stationarity, asymptotic normality breaks down for tail indices smaller than one or, equivalently, when the clustering behavior of the observed events is such that the unconditional distribution of the durations has no finite mean. Instead, we find that estimators are mixed Gaussian and have non-standard rates of convergence. The results are based on exploiting the crucial fact that for duration data the number of observations within any given time span is random. Our results apply to general econometric models where the number of observed events is random.

Zero-Inflated Autoregressive Conditional Duration Model for Discrete Trade Durations with Excessive Zeros

Abstract In finance, durations between successive transactions are usually modeled by the autoregressive conditional duration model based on a continuous distribution omitting zero values. Zero or close-to-zero durations can be caused by either split transactions or independent transactions. We propose a discrete model allowing for excessive zero values based on the zero-inflated negative binomial distribution with score dynamics. This model allows to distinguish between the processes generating split and standard transactions. We use the existing theory on score models to establish the invertibility of the score filter and verify that sufficient conditions hold for the consistency and asymptotic normality of the maximum likelihood of the model parameters. In an empirical study, we find that split transactions cause between 92 % and 98 % of zero and close-to-zero values. Furthermore, the loss of decimal places in the proposed approach is less severe than the incorrect treatment of zero values in continuous models.

HIGH-FREQUENCY TRANSACTION DATA: A COMPARISON BETWEEN TWO ASYMMETRIC MODELS

This paper compares two asymmetric market microstructure models, namely, the three-state asymmetric autoregressive conditional duration (AACD) model and the activity-direction-size (ADS) model. It is shown that the two models measure different aspects of the same underlying asymmetric nature of market microstructure. It is also shown that by extending the AACD model to include two size variables and adjusting for partial durations, each model’s parameter estimates can be used to estimate the other model’s parameters exactly. Thus, the two asymmetric market microstructure models are equivalent.

Autoregressive conditional durations: An application to the Surinamese dollar versus the US dollar exchange rate

AbstractThe parallel market nominal exchange rate of the United States dollar vis‐à‐vis the Surinamese dollar (USD/SRD) exhibited periods of severe volatility which were often followed by episodes of stability, usually at a cost of sharp depreciations. This study seeks to model this exchange rate using autoregressive conditional duration models. These models are suitable for modelling events occurring with irregular intervals. Exchange rates in developing countries have distinct features compared to exchange rates in countries with well‐established and accessible financial markets. A key feature is that for these developing countries, exchange rates only occasionally experience jumps. Our findings suggest that past exchange rate changes appear to be a significant driver of future exchange rate jumps. Furthermore, our results show that money, international reserves, and commodity prices can explain jumps in the market USD/SRD exchange rate.

Autoregressive conditional proportion: A multiplicative‐error model for (0,1)‐valued time series

We propose a multiplicative autoregressive conditional proportion (ARCP) model for (0,1)‐valued time series, in the spirit of GARCH (generalized autoregressive conditional heteroscedastic) and ACD (autoregressive conditional duration) models. In particular, our underlying process is defined as the product of a (0,1)‐valued independent and identically distributed (i.i.d.) sequence and the inverted conditional mean, which, in turn, depends on past reciprocal observations in such a way that is larger than unity. The probability structure of the model is studied in the context of the stochastic recurrence equation theory, while estimation of the model parameters is performed with the exponential quasi‐maximum likelihood estimator (EQMLE). The consistency and asymptotic normality of the EQMLE are both established under general regularity assumptions. Finally, the usefulness of our proposed model is illustrated with two real datasets.

On a quantile autoregressive conditional duration model

Autoregressive conditional duration (ACD) models are primarily used to deal with data arising from times between two successive events. These models are usually specified in terms of a time-varying conditional mean or median duration. In this work, we relax this assumption and consider a conditional quantile approach to facilitate the modeling of different percentiles. The proposed ACD quantile model is based on a skewed version of Birnbaum–Saunders distribution, which yields better fit of the tails than the traditional Birnbaum–Saunders distribution, in addition to facilitating the implementation of an expectation conditional maximization (ECM) algorithm. A Monte Carlo simulation study is performed to assess the behavior of the model as well as the parameter estimation method and the evaluation of a form of residuals. Two real financial transaction data sets are finally analyzed to illustrate the proposed approach.

Frequency and severity estimation of cyber attacks using spatial clustering analysis

In this paper, a cluster-based method is developed to investigate the risk of cyber attacks in the continental United States. The proposed analysis considers geographical information of cyber incidents for clustering. By clustering state-based observations, the frequency and severity of cyber losses demonstrate a simplified structure: independent structure between inter-arrival time and size of cyber breaches. The independence between frequency and severity is significant in the state level instead of national level. Within clustered subcategories, the inter-arrival time is modelled by the family of Autoregressive Conditional Duration models (ACD) and log-transformed size of breach is described by an ARMA-GARCH model. Under multiple statistical tests, it is shown that the cluster-based models have better fitting and are more robust than the aggregate model, where all incidents are considered together. Finally, a numerical analysis is presented to illustrate the performance of the approach. Accordingly, the prediction of total losses are compared with other dependent models. The differences of key cyber risk features among clusters are illustrated.

Bootstrap inference for Hawkes and general point processes

Inference and testing in general point process models such as the Hawkes model is predominantly based on asymptotic approximations for likelihood-based estimators and tests. As an alternative, and to improve finite sample performance, this paper considers bootstrap-based inference for interval estimation and testing. Specifically, for a wide class of point process models we consider a novel bootstrap scheme labeled ‘fixed intensity bootstrap’ (FIB), where the conditional intensity is kept fixed across bootstrap repetitions. The FIB, which is very simple to implement and fast in practice, extends previous ideas from the bootstrap literature on time series in discrete time, where the so-called ‘fixed design’ and ‘fixed volatility’ bootstrap schemes have shown to be particularly useful and effective. We compare the FIB with the classic recursive bootstrap, which is here labeled ‘recursive intensity bootstrap’ (RIB). In RIB algorithms, the intensity is stochastic in the bootstrap world and implementation of the bootstrap is more involved, due to its sequential structure. For both bootstrap schemes, we provide new bootstrap (asymptotic) theory which allows to assess bootstrap validity, and propose a ‘non-parametric’ approach based on resampling time-changed transformations of the original waiting times. We also establish the link between the proposed bootstraps for point process models and the related autoregressive conditional duration (ACD) models. Lastly, we show effectiveness of the different bootstrap schemes in finite samples through a set of detailed Monte Carlo experiments, and provide applications to both financial data and social media data to illustrate the proposed methodology.

A higher-order correct fast moving-average bootstrap for dependent data

We develop the theory of a novel fast bootstrap for dependent data. Our scheme deploys i.i.d. resampling of smoothed moment indicators. We characterize the class of parametric and semiparametric estimation problems for which the method is valid. We show the asymptotic refinements of the new procedure, proving that it is higher-order correct under mild assumptions on the time series, the estimating functions, and the smoothing kernel. We illustrate the applicability and the advantages of our procedure for M-estimation, generalized method of moments, and generalized empirical likelihood estimation. In a Monte Carlo study, we consider an autoregressive conditional duration model and we compare our method with other extant, routinely-applied first- and higher-order correct methods. The results provide numerical evidence that the novel bootstrap yields higher-order accurate confidence intervals, while remaining computationally lighter than its higher-order correct competitors. A real-data example on dynamics of trading volume of US stocks illustrates the empirical relevance of our method.

Autoregressive conditional duration models for high frequency financial data: an empirical study on mid cap exchange traded funds

PurposeThe purpose of this study is to identify appropriate autoregressive conditional duration (ACD) models that can capture the dynamics of tick-by-tick mid-cap exchange traded funds (ETFs) for the period July 2017 to December 2017 and accurately predict future trade duration values. The forecasted durations are then used to demonstrate the practical usefulness of the ACD models in quantifying an intraday time-based risk measure.Design/methodology/approachThrough six functional forms and six error distributions, 36 ACD models are estimated for eight mid-cap ETFs. The Akaike information criterion and Bayesian information criterion and the Ljung-Box test are used to evaluate goodness-of-fit while root mean square error and the Superior predictive ability test are applied to assess forecast accuracy.FindingsThe Box-Cox ACD (BACD), augmented Box-Cox ACD (ABACD) and additive and multiplicative ACD (AMACD) extensions are among the best fits. The results obtained prove that higher degrees of flexibility do not necessarily enhance goodness of fit and forecast accuracy does not always depend on model adequacy. BACD and AMACD models based on the generalised-F distribution generate the best forecasts, irrespective of the trading frequencies of the ETFs.Originality/valueTo the best of the authors’ knowledge, this is the first study that analyses the empirical performance of ACD models for high-frequency ETF data. Additionally, in comparison to previous works, a wider range of ACD models is considered on a reasonably longer sample period. The paper will be of interest to researchers in the area of market microstructure and to practitioners engaged in high-frequency trading.

Stationarity and ergodicity of Markov switching positive conditional mean models

A general Markov‐Switching autoregressive conditional mean model, valued in the set of non‐negative numbers, is considered. The conditional distribution of this model is a finite mixture of non‐negative distributions whose conditional mean follows a GARCH‐like dynamics with parameters depending on the state of a Markov chain. Three different variants of the model are examined depending on how the lagged‐values of the mixing variable are integrated into the conditional mean equation. The model includes, in particular, Markov mixture versions of various well‐known non‐negative time series models such as the autoregressive conditional duration model, the integer‐valued GARCH (INGARCH) model, and the Beta observation driven model. For the three variants of the model, conditions are given for the existence of a stationary and ergodic solution. The proposed conditions match those already known for Markov‐switching GARCH models. We also give conditions for finite marginal moments. Applications to various mixture and Markov mixture count, duration and proportion models are provided.

Periodic autoregressive conditional duration

We propose an autoregressive conditional duration (ACD) model with periodic time‐varying parameters and multiplicative error form. We name this model periodic autoregressive conditional duration (PACD). First, we study the stability properties and the moment structures of it. Second, we estimate the model parameters, using (profile and two‐stage) Gamma quasi‐maximum likelihood estimates (QMLEs), the asymptotic properties of which are examined under general regularity conditions. Our estimation method encompasses the exponential QMLE, as a particular case. The proposed methodology is illustrated with simulated data and two empirical applications on forecasting Bitcoin trading volume and realized volatility. We found that the PACD produces better in‐sample and out‐of‐sample forecasts than the standard ACD.

Self-weighted quantile estimation of autoregressive conditional duration model

An efficient market is often related to the market liquidity in a certain sense. In this paper, the autoregressive conditional duration (ACD) model is used for modeling and analyzing the market liquidity based on high-frequency financial data, which takes the volume duration as its measure index. Considering the high peak and heavy tail of high-frequency financial data, the self-weighted quantile regression (SQR) estimators for the unknown parameters in ACD model are constructed. The consistency and asymptotic properties of the estimators are proved. Furthermore, Monte Carlo simulation shows that the SQR estimators with data-driven weights are more accurate than those by traditional quantile regression (QR). Moreover, the performance of SQR estimation performs better with the increase of the proportion of outliers. The mean deviation and mean square error are reduced up to 96.24% and 91.83%, respectively. Finally, we illustrate the SQR method by an empirical analysis of the volume duration for Industrial And Commercial Bank Of China (ICBC) and PingAn Bank stocks in China. Through the Akaike Information Criterion (AIC) and other evaluation criteria, the SQR estimators at different quantiles all possess better performance.

Ensemble binary segmentation for irregularly spaced data with change-points

We propose a new technique for consistent estimation of the number and locations of the change-points in the structure of an irregularly spaced time series. The core of the segmentation procedure is the ensemble binary segmentation method (EBS), a technique in which a large number of multiple change-point detection tasks using the binary segmentation method are applied on sub-samples of the data of differing lengths, and then the results are combined to create an overall answer. We do not restrict the total number of change-points a time series can have, therefore, our proposed method works well when the spacings between change-points are short. Our main change-point detection statistic is the time-varying autoregressive conditional duration model on which we apply a transformation process in order to decorrelate it. To examine the performance of EBS we provide a simulation study for various types of scenarios. A proof of consistency is also provided. Our methodology is implemented in the R package eNchange, available to download from CRAN.

Clustering of arrivals in queueing systems: autoregressive conditional duration approach

Arrivals in queueing systems are typically assumed to be independent and exponentially distributed. Our analysis of an online bookshop, however, shows that there is an autocorrelation structure present. First, we adjust the inter-arrival times for diurnal and seasonal patterns. Second, we model adjusted inter-arrival times by the generalized autoregressive score (GAS) model based on the generalized gamma distribution in the spirit of the autoregressive conditional duration (ACD) models. Third, in a simulation study, we investigate the effects of the dynamic arrival model on the number of customers, the busy period, and the response time in queueing systems with single and multiple servers. We find that ignoring the autocorrelation structure leads to significantly underestimated performance measures and consequently suboptimal decisions. The proposed approach serves as a general methodology for the treatment of arrivals clustering in practice.

Testing for a serial correlation in VaR failures through the exponential autoregressive conditional duration model

Abstract Although regulatory standards, currently developed by the Basel Committee on Banking Supervision, anticipate a shift from VaR to ES, the evaluation of risk models currently remains based on the VaR measure. Motivated by the Basel regulations, we address the issue of VaR backtesting and contribute to the debate by exploring statistical properties of the exponential autoregressive conditional duration (EACD) VaR test. We show that, under the null, the tested parameter lies at the boundary of the parameter space, which can profoundly affect the accuracy of this test. To compensate for this deficiency, a mixture of chi-square distributions is applied. The resulting accuracy improvement allows for the omission of the Monte Carlo simulations used to implement the EACD VaR test in earlier studies, which dramatically improves the computational efficiency of the procedure. We demonstrate that the EACD approach to testing VaR has the potential to enhance statistical inference in most problematic cases – for small samples and for those close to the null.

Improved ACD-Based Financial Trade Durations Prediction Leveraging LSTM Networks and Attention Mechanism

The liquidity risk factor of security market plays an important role in the formulation of trading strategies. A more liquid stock market means that the securities can be bought or sold more easily. As a sound indicator of market liquidity, the transaction duration is the focus of this study. We concentrate on estimating the probability density function p Δ t i + 1 | G i , where Δ t i + 1 represents the duration of the (i + 1)-th transaction and G i represents the historical information at the time when the (i + 1)-th transaction occurs. In this paper, we propose a new ultrahigh-frequency (UHF) duration modelling framework by utilizing long short-term memory (LSTM) networks to extend the conditional mean equation of classic autoregressive conditional duration (ACD) model while retaining the probabilistic inference ability. And then, the attention mechanism is leveraged to unveil the internal mechanism of the constructed model. In order to minimize the impact of manual parameter tuning, we adopt fixed hyperparameters during the training process. The experiments applied to a large-scale dataset prove the superiority of the proposed hybrid models. In the input sequence, the temporal positions which are more important for predicting the next duration can be efficiently highlighted via the added attention mechanism layer.