Duration time‐series models with proportional hazard

Abstract.  The analysis of liquidity in financial markets is generally performed by means of the dynamics of the observed intertrade durations (possibly weighted by price or volume). Various dynamic models for duration data have been considered in the literature, such as the Autoregressive Conditional Duration (ACD) model. These models are often excessively constrained, introducing, for example, a deterministic link between conditional expectation and variance in the case of the ACD model. Moreover, the stationarity properties and the patterns of the stationary distributions are often unknown. The aim of this article is to solve these difficulties by considering a duration time series satisfying the proportional hazard property. We describe in detail this class of dynamic models, discuss its various representations and provide the ergodicity conditions. The proportional hazard copula can be specified either parametrically, or nonparametrically. We discuss estimation methods in both contexts, and explain why they are efficient, that is, why they reach the parametric (respectively, nonparametric) efficiency bound.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  D. Cox.  Some Statistical Methods Connected with Series of Events , 1955 .

[3]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[4]  Takemi Yanagimoto,et al.  Partial orderings of permutations and monotonicity of a rank correlation statistic , 1969 .

[5]  P. Clark.  A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices , 1973 .

[6]  Y. Davydov.  Mixing Conditions for Markov Chains , 1974 .

[7]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[8]  M. Shaked.  A Family of Concepts of Dependence for Bivariate Distributions , 1977 .

[9]  R. John,et al.  Boundary modification for kernel regression , 1984 .

[10]  Paul D. Feigin,et al.  RANDOM COEFFICIENT AUTOREGRESSIVE PROCESSES:A MARKOV CHAIN ANALYSIS OF STATIONARITY AND FINITENESS OF MOMENTS , 1985 .

[11]  P. Newbold,et al.  Estimation and Prediction , 1985 .

[12]  G. Kimeldorf,et al.  Positive dependence orderings , 1987 .

[13]  J. Stock.  Estimating Continuous-Time Processes Subject to Time Deformation: An Application to Postwar U.S. GNP , 1988 .

[14]  G. Kimeldorf,et al.  A framework for positive dependence , 1989, Annals of the Institute of Statistical Mathematics.

[15]  Christian Genest,et al.  Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles , 1990 .

[16]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .

[17]  Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles@@@Concepts de dependance et ordres stochastiques pour des lois bidimensionnelles , 1990 .

[18]  E. T. Olsen,et al.  Copulas and Markov processes , 1992 .

[19]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[20]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[21]  Harry Joe,et al.  Parametric families of multivariate distributions with given margins , 1993 .

[22]  M. C. Jones,et al.  Simple boundary correction for kernel density estimation , 1993 .

[23]  P. Doukhan.  Mixing: Properties and Examples , 1994 .

[24]  E. Ghysels,et al.  Stochastic Volatility and time Deformation: an Application of trading Volume and Leverage Effects , 1994 .

[25]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[26]  ON THE DECREASE IN DEPENDENCE WITH LAG FOR STATIONARY MARKOV CHAINS , 1994 .

[27]  C. Gouriéroux,et al.  Série Scientifique Scientific Series N o 95 s-32 MARKET TIME AND ASSET PRICE MOVEMENTS THEORY AND ESTIMATION , 1995 .

[28]  Robert F. Engle,et al.  The Econometrics of Ultra-High Frequency Data , 1996 .

[29]  Denis Bosq,et al.  Nonparametric statistics for stochastic processes , 1996 .

[30]  R. Nelsen.  An Introduction to Copulas , 1998 .

[31]  Jeffrey R. Russell,et al.  Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data , 1998 .

[32]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[33]  Nikolaus Hautsch,et al.  Analyzing the Time between Trades with a Gamma Compounded Hazard Model. An Application to LIFFE Bund Future Transactions , 1999 .

[34]  Joann Jasiak,et al.  Persistence in Intertrade Durations , 1999 .

[35]  H. Geman,et al.  Order Flow, Transaction Clock, and Normality of Asset Returns , 2000 .

[36]  L. Bauwens,et al.  The Logarithmic Acd Model: An Application to the Bid-Ask Quote Process of Three NYSE Stocks , 2000 .

[37]  Christian Genest,et al.  Conditions for the Asymptotic Semiparametric Efficiency of an Omnibus Estimator of Dependence Parame , 2000 .

[38]  Pierre Giot,et al.  Time transformations, intraday data and volatility models , 2000 .

[39]  John Odenckantz,et al.  Nonparametric Statistics for Stochastic Processes: Estimation and Prediction , 2000, Technometrics.

[40]  Thomas A. Severini,et al.  A simpli"ed approach to computing e$ciency bounds in semiparametric models , 2001 .

[41]  Ruey S. Tsay,et al.  A nonlinear autoregressive conditional duration model with applications to financial transaction data , 2001 .

[42]  T. Severini,et al.  A simplified approach to computing efficiency bounds in semiparametric models , 2001 .

[43]  C. Gouriéroux,et al.  Nonlinear Autocorrelograms: an Application to Inter‐Trade Durations , 2002 .

[44]  Xiaohong Chen,et al.  MIXING AND MOMENT PROPERTIES OF VARIOUS GARCH AND STOCHASTIC VOLATILITY MODELS , 2002, Econometric Theory.

[45]  Eric Bouyé,et al.  Investigating Dynamic Dependence Using Copulae , 2002 .

[46]  Xiaohong Chen,et al.  Evaluating Density Forecasts via the Copula Approach , 2003 .

[47]  Eric Ghysels,et al.  Stochastic volatility duration models , 2004 .

[48]  Xiaohong Chen,et al.  Estimation of Copula-Based Semiparametric Time Series Models , 2006 .

[49]  C. Robert,et al.  STOCHASTIC UNIT ROOT MODELS , 2006, Econometric Theory.

[50]  M. Rockinger,et al.  The Copula-GARCH model of conditional dependencies: An international stock market application , 2006 .

[51]  Christian Gourieroux,et al.  Structural Laplace Transform and Compound Autoregressive Models , 2006 .

[52]  Christian Gourieroux,et al.  An efficient nonparametric estimator for models with nonlinear dependence , 2007 .

[53]  Brendan K. Beare.  COPULAS AND TEMPORAL DEPENDENCE , 2008 .