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

Abstract

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