Regime-switching Pareto distributions for ACD models

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Abstract

Refinements have been proposed for the autoregressive conditional duration model within the framework of financial durations. It is argued that a Pareto distribution is a meaningful representation for durations. The model is analyzed under the hypothesis of regime-switching parameters with different transition functions governed both by an observable and a latent variable.

Introduction

A financial duration is the time between two consecutive events occurring in a financial market (a trade, a price change, etc.). The analysis of financial durations involves an irregularly spaced time series, requiring new specific statistical techniques. The autoregressive conditional duration (ACD) model, proposed by Engle and Russell (1998), has been widely used. Experiments were performed mainly using data from the New York Stock Exchange. The choice of a random variable to describe the durations probability law is crucial for a satisfactory goodness-of-fit of the model. The simplest assumptions are the exponential or the Weibull random variable. Both, however, have a poor performance, so that many authors proposed more flexible distributions or pursued a semiparametric approach in order to improve the model (Drost and Werker, 2004). In this paper the use of the Pareto distribution is analyzed assuming both constant and time-varying parameters. Particularly, a model with regime-switching parameters in the innovation component is proposed.

The paper is organized as follows: Section 2 reviews the Pareto autoregressive conditional duration model (PACD). In Sections 3 and 4 the regime-switching PACD models are introduced and analyzed. An application to Italian data is presented in Section 5. In Section 6 some conclusions are drawn.

Section snippets

The Pareto ACD model

Let Xi be the ith duration, that is Xi=ti-ti-1, where ti is the time of the ith market event. After removing the daily seasonal effect, we get the deseasonalized durations, denoted as xi.

The basic ACD(p,q) model is given by xi=Ψiεi,εii.i.d.with E(εi)=1 and Ψi=f(xi-1,,xi-p,Ψi-1,,Ψi-q).Because of the unit expected value of εi, Ψi is the expected value of the (seasonality-adjusted) durations, conditional on past information. The attention is then focused both on the functional form of Ψi and on

Regime-switching Pareto ACD models

In this paper we argue that the parameter θ characterizing the Pareto distribution could itself be variable, according to a set Si of predetermined variables. The basic hypothesis is that the dynamics of the durations xi is not completely summarized by their conditional mean Ψi but past events somehow affect the distribution of the innovations. Therefore we assume that, conditionally on Si, the εi's are independently and identically distributed as Pareto II type random variables with parameters

Transforming innovations with regime-switching Pareto distribution

With the assumed regime-switching Pareto distribution, the innovations εi are i.i.d. conditionally on the regime, but not unconditionally, so that the usual diagnostics based upon the estimated residuals ε^i=xi/Ψ^i are not correct.

The problem could be solved by properly transforming the innovations, in order to obtain a standardized distribution not depending on the parameter θi. In other words, given the general ACD model with regime-switching Pareto distribution xi=Ψiεi,εi|Sii.i.d.Pareto(θi),

Case study

As a case study, (seasonality-adjusted) durations between consecutive price changes for the stock Comit (Milan Stock Exchange) are analyzed. The observation period is February 2000 (21 trading days—sample size 8222). All the computations are made in Gauss using the Newton–Raphson algorithm implemented in the constrained maximum likelihood (CML) library.

A preliminary analysis is carried out on data to check the performance of the two basic ACD models, respectively, with exponential (EACD) and

Concluding remarks

One of the main topics in the context of ACD models is the formulation of a correct distributional assumption for the innovation component. In this respect, a microstructure argument could be the fundamental heterogeneity of traders in the financial market. Following this line of reasoning, a proposal to use finite or infinite mixtures of distributions has been made by De Luca and Zuccolotto (2003). An infinite mixture of exponentials with inverse gamma mixing distribution leads to the

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