Exponential Multivariate Autoregressive Conditional High Frequency Data Model

The availability of high frequency databases makes possible to understand financial market dynamics and test some of hypothesis brought up by the microstructure theory. In that way, many formulations have been suggested. One of the first proposals to model event based high frequency data has been the ACD (Autoregressive Conditional Duration) model by Engle and Russel , in 1998, where the time between events, called duration, has been described as a sequence of independent random variables with a time varying mean given by a GARCH type equation. After that, in 2000, Engle incorporated duration into a volatility context (UHF-GARCH). Then, in 2001, Zhang, Russell and Tsay extended the original models (Exponencial and Weibull ACD), through the using of thresholds to capture nonlinear effects. In a similar way, in 2001, Fernandes and Gramming (CORE – A family of autoregressive conditional duration models) had added changes to the model initially proposed by Engle and Russell through the application of a Box-Cox transformation. Recently, in 2002, Manganelli (ECB Working Paper Series – Duration, Volume and Volatility impact of trades) proposed the joint modeling of different variables (duration, volume and volatility) involved in the financial transaction process via a Vector Autoregressive Moving Average (VARMA). That was the first time volume had been explicitly modeled. In this paper, we extend the work of Manganelli by including the bid-ask spread into VARMA formulation. We called it the Exponential Multivariate Autoregressive Conditional Model (EMACM). Different constraint sets on the structure of the coefficient matrices of the VARMA model are tested via likelihood ratio tests, allowing to answer some of the questions raised in the microstructure literature about causality and dependency among variables. The article is divided into seven sections. The first section presents a literature overview. Section 2 presents the model formulation and the joint likelihood function. Section 3 describes the seasonal adjustment, based on a natural cubic spline (off-line estimation). Section 4 brings details of the nonlinear optimization algorithms used (Sequential Quadratic Programming – SQP and Nelder-Mead Simplex Method). In the fifth section, a Monte-Carlo simulation is carried out, in order to test the estimation process capability, through the analysis of the system impulse-response function. An empirical example with the IBM tick-data is shown in section 6 and section 7 concludes.