Bayesian Cointegrated Vector Autoregression Models Incorporating alpha-stable Noise for\n Inter-day Price Movements Via Approximate Bayesian Computation

Abstract

We consider a statistical model for pairs of traded assets, based on a Cointegrated Vector\nAuto Regression (CVAR) Model. We extend standard CVAR models to incorporate estimation of\nmodel parameters in the presence of price series level shifts which are not accurately modeled\nin the standard Gaussian error correction model (ECM) framework. This involves developing a\nnovel matrix-variate Bayesian CVAR mixture model, comprised of Gaussian errors intra-day and\n$\\alpha$-stable errors inter-day in the ECM framework. To achieve this we derive conjugate\nposterior models for the Scale Mixtures of Normals (SMiN CVAR) representation of\n$\\alpha$-stable inter-day innovations. These results are generalized to asymmetric intractable\nmodels for the innovation noise at inter-day boundaries allowing for skewed $\\alpha$-stable\nmodels via Approximate Bayesian computation.\n¶ Our proposed model and sampling methodology is general, incorporating the current CVAR\nliterature on Gaussian models, whilst allowing for price series level shifts to occur either\nat random estimated time points or known \\textit{a priori} time points. We focus analysis on\nregularly observed non-Gaussian level shifts that can have significant effect on estimation\nperformance in statistical models failing to account for such level shifts, such as at the\nclose and open times of markets. We illustrate our model and the corresponding estimation\nprocedures we develop on both synthetic and real data. The real data analysis investigates\nAustralian dollar, Canadian dollar, five and ten year notes (bonds) and NASDAQ price series.\nIn two studies we demonstrate the suitability of statistically modeling the heavy tailed noise\nprocesses for inter-day price shifts via an $\\alpha$-stable model. Then we fit the novel\nBayesian matrix variate CVAR model developed, which incorporates a composite noise model for\n$\\alpha$-stable and matrix variate Gaussian errors, under both symmetric and non-symmetric\n$\\alpha$-stable assumptions.