Bayesian Analysis

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

Gareth W. Peters, Balakrishnan Kannan, Ben Lasscock, Chris Mellen, and Simon Godsill

Full-text: Open access


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.

Article information

Bayesian Anal., Volume 6, Number 4 (2011), 755-792.

First available in Project Euclid: 13 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Cointegrated Vector Autoregression $\alpha$-stable Approximate Bayesian Computation


Peters, Gareth W.; Kannan, Balakrishnan; Lasscock, Ben; Mellen, Chris; Godsill, Simon. Bayesian Cointegrated Vector Autoregression Models Incorporating alpha-stable Noise for Inter-day Price Movements Via Approximate Bayesian Computation. Bayesian Anal. 6 (2011), no. 4, 755--792. doi:10.1214/11-BA628.

Export citation


  • Ackert, L. and Racine, M. (1999). "Stochastic trends and cointegration in the market for equities"." Journal of Economics and Business, 51(2): 133–143.
  • Alder, R., Feldman, R., and Taqqu, M. S. (1998). A practical guide to heavy-tails: Statistical techniques for analysing heavy-tailed distributions. Birkhäuser, Boston.
  • Andrieu, C. and Moulines, É. (2006). "On the ergodicity properties of some adaptive MCMC algorithms." The Annals of Applied Probability, 16(3): 1462–1505.
  • Atchadé, Y. and Rosenthal, J. (2005). "On adaptive Markov chain Monte Carlo algorithms." Bernoulli, 11(5): 815–828.
  • Bauwens, L. and Lubrano, M. (1994). "Identification restrictions and posterior densities in cointegrated Gaussian VAR systems"." Universite catholique de Louvain, Center for Operations Research and Econometrics Discussion Papers, Report - 1994018..
  • Beaumont, M., Cornuet, J., Marin, J., and Robert, C. (2009). "Adaptivity for ABC algorithms: the ABC-PMC scheme"." Biometrika, 96(4): 983–990.
  • Bock, M. and Mestel, R. (2009). "A Regime-Switching Relative Value Arbitrage Rule"." Operations Research Proceedings 2008, 9–14.
  • Chambers, J., Mallows, C., and Stuck, B. (1976). "A method for simulating stable random variables." Journal of the American Statistical Association, 71: 340–334. Correction (1987), 82, 704.
  • Chen, P. and Hsiao, C. (2010). "Subsampling the Johansen test with stable innovations"." Australian & New Zealand Journal of Statistics, 52(1): 61–73.
  • De Lathauwer, L. and Vandewalle, J. (2004). "Dimensionality reduction in higher-order signal processing and rank-(R1, R2,..., RN) reduction in multilinear algebra"." Linear Algebra and its Applications, 391: 31–55.
  • Del Moral, P., Doucet, A., and Jasra, A. (2011). "An adaptive sequential Monte Carlo method for approximate Bayesian computation"." Statistics and Computing - to appear.
  • Engle, R. and Granger, C. (1987). "Co-integration and error correction: representation, estimation, and testing"." Econometrica: Journal of the Econometric Society, 55(2): 251–276.
  • Fama, E. and Roll, R. (1968). "Some properties of symmetric stable distributions." Journal of the American Statistical Association, 63: 817–83.
  • Fearnhead, P. and Prangle, D. (2010). "Semi-automatic Approximate Bayesian Computation"." Arxiv preprint arXiv:1004.1112v2 [stat.ME].
  • Friedlandera, M. and Hatzb, K. (2008). "Computing non-negative tensor factorizations"." Optimization Methods and Software, 23(4): 631–647.
  • Gatev, E., Goetzmann, W., and Rouwenhorst, K. (2006). "Pairs trading: Performance of a relative-value arbitrage rule." Review of Financial Studies, 19(3): 797.
  • Geweke, J. (1996). "Bayesian reduced rank regression in econometrics* 1"." Journal of Econometrics, 75(1): 121–146.
  • Godsill, S. (2000). "Inference in symmetric alpha-stable noise using MCMC" and the slice sampler. In Proceedings IEEE International Conference on Acoustics, Speech and Signal Processing, volume VI, 3806–3809.
  • Granger, C. and Hyung, N. (2004). "Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns"." Journal of Empirical Finance, 11(3): 399–421.
  • Granger, C. and Weiss, A. (2001). "Time series analysis of error correction models"." Spectral analysis, seasonality, nonlinearity, methodology and forecasting: collected papers of Clive WJ Granger, 129.
  • Grelaud, A., Robert, C., and Marin, J. (2009). "ABC methods for model choice in Gibbs random fields"." Comptes Rendus Mathematique, 347(3-4): 205–210.
  • Gupta, A. and Nagar, D. (1999). Matrix variate distributions. Monograph and Surveys in Pure and Applied Mathematics. Chapman Hall CRC.
  • Haario, H., Saksman, E., and Tamminen, J. (2001). "An adaptive Metropolis algorithm." Bernoulli, 7(2): 223–242.
  • Harville, D. (2008). Matrix algebra from a statistician's perspective. Springer Verlag, New York.
  • Johansen, S. (1988). "Statistical analysis of cointegration vectors"." Journal of Economic Dynamics and Control, 12(2-3): 231–254.
  • Kleibergen, F. and Paap, R. (2002). "Priors, posteriors and Bayes factors for a Bayesian analysis of cointegration"." Journal of Econometrics, 111(2): 223–249.
  • Kleibergen, F. and Van Dijk, H. (2009). "On the shape of the likelihood/posterior in cointegration models"." Econometric Theory, 10(3-4): 514–551.
  • Koop, G., Strachan, R., Van Dijk, H., and Villani, M. (2006). "Bayesian approaches to cointegration"." Palgrave Handbook on Econometrics, 1: 871–898.
  • Krolzig, H. (1997). "Statistical analysis of cointegrated VAR processes with Markovian regime shifts"." unpublished, Nuffield College, Oxford.
  • Luetkepohl, H. (2005). New introduction to multiple time series analysis. Springer-Verlag, Berlin.
  • Mills, T. and Markellos, R. (2008). The econometric modelling of financial time series. Cambridge University Press.
  • Neslehova, J., Embrechts, P., and Chavez-Demoulin, V. (2006). "Infinite mean models and the LDA for operational risk"." Journal of Operational Risk, 1(1): 3–25.
  • Nolan, J. P. (1997). "Numerical computation of stable densities and distributions." Communications in Statistics, Stochastic Models, 13: 759–774.
  • –- (2012). "Stable Distributions - Models for Heavy Tailed Data." In progress, Chapter 1 online at
  • Peters, G., Kannan, B., Lasscock, B., and Mellen, C. (2010). "Model Selection and Adaptive Markov chain Monte Carlo for Bayesian Cointegrated VAR model"." Bayesian Analysis, 5(3): 465–492.
  • Peters, G., Nevat, I., Sisson, S., Fan, Y., and Yuan, J. (2010). "Bayesian symbol detection in wireless relay networks via likelihood-free inference." Signal Processing, IEEE Transactions on, 58(10): 5206–5218.
  • Peters, G. and Sisson, S. (2006). "Bayesian inference, Monte Carlo sampling and operational risk"." Journal of Operational Risk, 1(3): 27–50.
  • Peters, G., Wuethrich, M., and Shevchenko, P. (2010). "Chain ladder method: Bayesian bootstrap versus classical bootstrap." Insurance: Mathematics and Economics, 47(1): 36–51.
  • Peters, G. W., Fan, Y., and Sisson, S. A. (2008). "On sequential Monte Carlo", partial rejection control and approximate Bayesian computation. Technical report, Tech. rep. UNSW.
  • Qiou, Z. and Ravishanker, N. (1998). "Bayesian inference for time series with infinite variance stable innovations." Journal of Time Series Analysis, 19: 235–249.
  • Ratmann, O., Andrieu, C., Hinkley, T., Wiuf, C., and Richardson, S. (2009). "Model criticism based on likelihood-free inference, with an application to protein network evolution." Proceedings of the National Academy of Science USA, 106: 10576–10581.
  • Reeves, R. and Pettitt, A. (2005). "A theoretical framework for approximate Bayesian computation"." In Statistical Solutions to Modern Problems: Proceedings of the 20th International Workshop on Statistical Modelling, Sydney, Australia, July 10–15, 2005, 393–396.
  • Roberts, G. and Rosenthal, J. (2009). "Examples of adaptive MCMC." Journal of Computational and Graphical Statistics, 18(2): 349–367.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable non-Gaussian random processes: Stochastic models with infinite variance. Chapman and Hall/CRC.
  • Sisson, S. and Fan, Y. (2010). "Likelihood-free Markov chain Monte Carlo"." Arxiv preprint arXiv:1001.2058v1 [stat.ME].
  • Strachan, R. (2003). "Valid Bayesian estimation of the cointegrating error correction model"." Journal of Business and Economic Statistics, 21(1): 185–195.
  • Strachan, R. and Inder, B. (2004). "Bayesian analysis of the error correction model"." Journal of Econometrics, 123(2): 307–325.
  • Sugita, K. (2002). "Testing for cointegration rank using Bayes factors"." In Royal Economic Society Annual Conference. Royal Economic Society.
  • –- (2008). "Bayesian analysis of a Markov switching temporal cointegration model"." Japan and the World Economy, 20(2): 257–274.
  • –- (2009). "A Monte Carlo comparison of Bayesian testing for cointegration rank"." Economics Bulletin, 29(3): 2145–2151.
  • Tavaré, S., Balding, D. J., Griffiths, R. C., and Donnelly, P. (1997). "Inferring coalescence times from DNA" sequence data. Genetics, 145: 505–518.
  • Toni, T., Welch, D., Strelkowa, N., Ipsen, A., and Stumpf, M. P. H. (2009). "Approximate Bayesian" computation scheme for parameter inference and model selection in dynamical systems. Journal of the Royal Society Interface, 6: 187–202.
  • Villani, M. (2005). "Bayesian reference analysis of cointegration"." Econometric Theory, 21(02): 326–357.
  • Wang, J. and Zivot, E. (2000). "A Bayesian time series model of multiple structural changes in level, trend, and variance"." Journal of Business & Economic Statistics, 18(3): 374–386.
  • Wilkinson, R. (2008). "Approximate Bayesian computation (ABC) gives exact results under the assumption of model error." Arxiv preprint arXiv:0811.3355v1 [stat.CO].
  • Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Translations of Mathematical Monographs. American Mathematical Society, Providence, Rhode Island.