Model Selection and Adaptive Markov Chain Monte Carlo for Bayesian Cointegrated VAR Model

Abstract

This paper develops a matrix-variate adaptive Markov chain Monte Carlo (MCMC) methodology for Bayesian Cointegrated Vector Auto Regressions (CVAR). We replace the popular approach to sampling Bayesian CVAR models, involving griddy Gibbs, with an automated efficient alternative, based on the Adaptive Metropolis algorithm of Roberts and Rosenthal, (2009). Developing the adaptive MCMC framework for Bayesian CVAR models allows for efficient estimation of posterior parameters in significantly higher dimensional CVAR series than previously possible with existing griddy Gibbs samplers. For a n-dimensional CVAR series, the matrix-variate posterior is in dimension 3n 2 + n, with significant correlation present between the blocks of matrix random variables. Hence, utilizing a griddy Gibbs sampler for large n becomes computationally impractical as it involves approximating an n × n full conditional posterior using a spline over a high dimensional n × n grid. The adaptive MCMC approach is demonstrated to be ideally suited to learning on-line a proposal to reflect the posterior correlation structure, therefore improving the computational efficiency of the sampler.We also treat the rank of the CVAR model as a random variable and perform joint inference on the rank and model parameters. This is achieved with a Bayesian posterior distribution defined over both the rank and the CVAR model parameters, and inference is made via Bayes Factor analysis of rank.Practically the adaptive sampler also aids in the development of automated Bayesian cointegration models for algorithmic trading systems considering instruments made up of several assets, such as currency baskets. Previously the literature on financial applications of CVAR trading models typically only considers pairs trading (n=2) due to the computational cost of the griddy Gibbs. We are able to extend under our adaptive framework to n >> 2 and demonstrate an example with n = 10, resulting in a posterior distribution with parameters up to dimension 310. By also considering the rank as a random quantity we can ensure our resulting trading models are able to adjust to potentially time varying market conditions in a coherent statistical framework.

Bayesian Cointegrated Vector Autoregression Models Incorporating -Stable Noise for Inter-Day Price Movements Via Approximate Bayesian Computation

“…This provides us with statistical modeling of the inter-day left shifts via generalized α-stable models for each asset pair. We then take the parameter estimates for the α-stable model and study the impact of naively applying the standard Johansen procedure and the Bayesian model of Peters et al [2010a] to a price series with intra-day level shifts generated from one of the more extreme currency pair α-stable fits. This study is performed for one hundred independently generated data sets and the impact on the frequentist and Bayesian point estimators is studied.…”

“…Noise modeling via α-stable distributions has been suggested in several areas, such as wireless communications and in financial data analysis, see Fama and Roll [1968], Godsill [2000], Neslehova et al [2006] and Peters et al [2010a]. α-stable distributions possess several useful properties, including infinite mean and infinite variance, skewness and heavy tails Zolotarev [1986] and Samorodnitsky and Taqqu [1994].…”

“…We consider the S0 parameterization, see Peters et al [2010a] for details. Considered as generalizations of the Gaussian distribution, they are defined as the class of location-scale distributions which are closed under convolutions.…”

“…Bayesian analysis of CVAR models has been addressed in several papers, see Bauwens and Lubrano [1994], Geweke [1996], Kleibergen and Van Dijk [2009], Ackert and Racine [1999] and Sugita [2002]. In practice, Bayesian and non-Bayesian CVAR models are used extensively in pairs trading, see an example in Peters et al [2010a]. We demonstrate that when estimating even the basic CVAR models using data which is sampled at a frequency less than one day, on real price series pairs, the accuracy and robustness of the statistical model fit and estimation and therefore the stability of the selected portfolio weights, is strongly affected by level shifts or jumps in price series due to inter-day movements.…”

“…Their approach generalizes the Johannsen procedure Johansen and Juselius [1990] to the symmetric α-stable innovation setting for testing the rank of the cointegrated VAR model. We will demonstrate a more flexible model removing the symmetry assumption for the stable noise, introducing a more realistic mixture noise model and providing a novel Approximate Bayesian Computation (ABC) sampling methodology for estimation and rank selection, generalizing the approach of Peters et al [2010a].…”