## Bayesian Analysis

### Model selection and adaptive Markov chain Monte Carlo for Bayesian cointegrated {VAR} models

#### 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 $n \times n$ full conditional posterior using a spline over a high dimensional $n \times n$ 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.

#### Article information

Source
Bayesian Anal., Volume 5, Number 3 (2010), 465-491.

Dates
First available in Project Euclid: 22 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1340380537

Digital Object Identifier
doi:10.1214/10-BA518

Mathematical Reviews number (MathSciNet)
MR2719663

Zentralblatt MATH identifier
1330.65022

#### Citation

Peters, Gareth W.; Kannan, Balakrishnan; Lasscock, Ben; Mellen, Chris. Model selection and adaptive Markov chain Monte Carlo for Bayesian cointegrated {VAR} models. Bayesian Anal. 5 (2010), no. 3, 465--491. doi:10.1214/10-BA518. https://projecteuclid.org/euclid.ba/1340380537

#### References

• Andrieu, C. and Atchadé, Y. (2007). "On the efficiency of adaptive MCMC algorithms"." Electronic Communications in Probability, 336–349.
• Andrieu, C. and Moulines, É. (2006). "On the ergodicity properties of some adaptive MCMC algorithms"." The Annals of Applied Probability, 1462–1505.
• Atchadé, Y. and Rosenthal, J. (2005). "On adaptive markov chain monte carlo algorithms"." Bernoulli, 11(5): 815–828.
• Bauwens, L. and Giot, P. (1998). "A Gibbs sampling approach to cointegration"." Computational Statistics, 13(3): 339–368.
• Bauwens, L. and Lubrano, M. (1996). "Identification restrictions and posterior densities in cointegrated Gaussian VAR systems"." Advances in Econometrics, 11: 3–28.
• Chib, S. (1995). "Marginal Likelihood from the Gibbs Output."" Journal of the American Statistical Association, 90(432).
• 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.
• Fan, Y., Peters, G., and Sisson, S. (2009). "Automating and evaluating reversible jump MCMC proposal distributions"." Statistics and Computing, 19(4): 409–421.
• Geweke, J. (1996). "Bayesian reduced rank regression in econometrics"." Journal of Econometrics, 75(1): 121–146.
• Giordani, P. and Kohn, R. (2010). "Adaptive independent Metropolis–Hastings by fast estimation of mixtures of normals"." Journal of Computational and Graphical Statistics, (ahead of print. doi:10.1198/jcgs.2009.07174): 1–17.
• Granger, C. (1981). "Some properties of time series data and their use in econometric model specification"." Journal of Econometrics, 16(1): 121–130.
• 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 – 144.
• Haario, H., Saksman, E., and Tamminen, J. (2001). "An adaptive Metropolis algorithm"." Bernoulli, 7(2): 223–242.
• –- (2005). "Componentwise adaptation for high dimensional MCMC"." Computational Statistics, 20(2): 265–273.
• 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.
• Luetkepohl, H. (2005). New introduction to multiple time series analysis. Springer.
• Peters, G., Kannan, B., Lasscock, B., and Mellen, C. (2009). "Model Selection and Adaptive Markov chain Monte Carlo for Bayesian Cointegrated VAR model"."
• Reinsel, G. and Velu, R. (1998). Multivariate reduced-rank regression. Springer New York.
• Roberts, G., Gelman, A., and Gilks, W. (1997). "Weak convergence and optimal scaling of random walk Metropolis algorithms"." The Annals of Applied Probability, 7(1): 110–120.
• Roberts, G. and Rosenthal, J. (2001). "Optimal scaling for various Metropolis-Hastings algorithms"." Statistical Science, 16(4): 351–367.
• –- (2009). "Examples of adaptive MCMC"." Journal of Computational and Graphical Statistics, 18(2): 349–367.
• Rosenthal, J. (2008). "Optimal Proposal Distributions and Adaptive MCMC"." Preprint - Chapter for MCMC Handbook, S. Brooks, A. Gelman, G. Jones, and X.-L. Meng, eds..
• Silva, R., Giordani, P., Kohn, R., and Pitt, M. (2009). "Particle filtering within adaptive Metropolis Hastings sampling"." Arxiv preprint arXiv:0911.0230.
• Strachan, R. and Inder, B. (2004). "Bayesian analysis of the error correction model"." Journal of Econometrics, 123(2): 307–325.
• Strachan, R. and van Dijk, H. (2003). "Bayesian model selection with an uninformative prior"." Oxford Bulletin of Economics and Statistics, 65(1): 863–876.
• –- (2007). "Bayesian model averaging in vector autoregressive processes with an investigation of stability of the US great ratios and risk of a liquidity trap in the USA, UK and Japan"." Econometric Institute Report, Erasmus University Rotterdam,Rotterdam, The Netherlands, 9: 47.
• Sugita, K. (2002). "Testing for cointegration rank using Bayes factors"." University of Warwick, Department of Economics, Economic Research Papers..
• –- (2009). "A Monte Carlo comparison of Bayesian testing for cointegration rank”"." Economics Bulletin, 29(3): 2145–2151.
• Verdinelli, I. and Wasserman, L. (1995). "Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio."" Journal of the American Statistical Association, 90(430): 614–618.
• Vermaak, J., Andrieu, C., Doucet, A., and Godsill, S. (2004). "Reversible jump Markov chain Monte Carlo strategies for Bayesian model selection in autoregressive processes"." Journal of Time Series Analysis, 25(6): 785–809.