An Inflated Multivariate Integer Count Hurdle Model: An Application to Bid and Ask Quote Dynamics

Abstract

In this paper we develop a model for the conditional inflated multivariate density of integer count variables with domain Z n . Our modelling framework is based on a copula approach and can be used for a broad set of applications where the primary characteristics of the data are: (i) discrete domain, (ii) the tendency to cluster at certain outcome values and (iii) contemporaneous dependence. These kind of properties can be found for high or ultra-high frequent data describing the trading process on financial markets. We present a straightforward method of sampling from such an inflated multivariate density through the application of an Independence Metropolis-Hastings sampling algorithm. We demonstrate the power of our approach by modelling the conditional bivariate density of bid and ask quote changes in a high frequency setup. We show how to derive the implied conditional discrete density of the bid-ask spread, taking quote clusterings (at multiples of 5 ticks) into account.

A Note on Identification of Bivariate Copulas for Discrete Count Data

“…The copula approach for constructing joint distributions has gained popularity in recent years in applied econometric studies, including models with discrete outcomes (Van Ophen (1999) [1]; Cameron et al (2004) [2]; Zimmer and Trivedi (2006) [3]; Bien et al (2011) [4]; Winkelmann (2012) [5]). While copula researchers have long understood that a multivariate discrete distribution does not possess a unique copula representation (Marshall (1996) [6]), recent research also indicates that any copula applied to discrete data is not identified.…”

The predictive distributions of thinning‐based count processes

“…Nonnegative integer‐valued count process models are widely used in domains such as marketing (Böckenholt, 1998), economics (Blundell, Griffith, & Van Reenen, 1999; Brännäs & Hellström, 2001; Harris & McCabe, 2019), finance (Bien, Nolte, & Pohlmeier, 2011; Heinen & Rengifo, 2007; Kirchner, 2017), and insurance (Gouriéroux & Jasiak, 2004). The benchmark model, introduced by McKenzie (1985) and Al‐Osh and Alzaid (1987) in the first‐order case (called INAR(1)) postulates that: where the thinning operator is defined as follows: conditionally on X t −1 , variable α∘ X t −1 has the distribution , that is the binomial distribution with probability parameter α and size parameter X t −1 .…”