Bayesian Analysis

Variable Selection in Seemingly Unrelated Regressions with Random Predictors

David Puelz, P. Richard Hahn, and Carlos M. Carvalho

Full-text: Open access


This paper considers linear model selection when the response is vector-valued and the predictors, either all or some, are randomly observed. We propose a new approach that decouples statistical inference from the selection step in a “post-inference model summarization” strategy. We study the impact of predictor uncertainty on the model selection procedure. The method is demonstrated through an application to asset pricing.

Article information

Bayesian Anal. Volume 12, Number 4 (2017), 969-989.

First available in Project Euclid: 7 March 2017

Permanent link to this document

Digital Object Identifier

decoupling shrinkage and selection seemingly unrelated regressions penalized utility selection

Creative Commons Attribution 4.0 International License.


Puelz, David; Hahn, P. Richard; Carvalho, Carlos M. Variable Selection in Seemingly Unrelated Regressions with Random Predictors. Bayesian Anal. 12 (2017), no. 4, 969--989. doi:10.1214/17-BA1053.

Export citation


  • Barbieri, M. M. and Berger, J. O. (2004). “Optimal predictive model selection.”Annals of Statistics, 870–897.
  • Bayarri, M., Berger, J., Forte, A., and Garcia-Donato, G. (2012). “Criteria for Bayesian model choice with application to variable selection.”The Annals of Statistics, 40(3): 1550–1577.
  • Brown, P. J., Vannucci, M., and Fearn, T. (1998). “Multivariate Bayesian variable selection and prediction.”Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(3): 627–641.
  • Clyde, M. and George, E. (2004). “Model uncertainty.”Statistical Science, 19: 81–94.
  • Cochrane, J. H. (2011). “Presidential address: Discount rates.”The Journal of Finance, 66(4): 1047–1108.
  • Dobra, A., Hans, C., Jones, B., Nevins, J. R., Yao, G., and West, M. (2004). “Sparse graphical models for exploring gene expression data.”Journal of Multivariate Analysis, 90(1): 196–212.
  • Efron, B., Hastie, T., Johnstone, I., Tibshirani, R., et al. (2004). “Least angle regression.”The Annals of Statistics, 32(2): 407–499.
  • Ericsson, J. and Karlsson, S. (2004). “Choosing Factors in a Multifactor Asset Pricing Model: A Bayesian Approach.” Technical report, Stockholm School of Economics.
  • Fama, E. F. and French, K. R. (1992). “The cross-section of expected stock returns.”The Journal of Finance, 47(2): 427–465.
  • Fama, E. F. and French, K. R. (2015). “A five-factor asset pricing model.”Journal of Financial Economics, 116(1): 1–22.
  • Garcia-Donato, G. and Martinez-Beneito, M. (2013). “On sampling strategies in Bayesian variable selection problems with large model spaces.”Journal of the American Statistical Association, 108(501): 340–352.
  • George, E. I. and McCulloch, R. E. (1993). “Variable selection via Gibbs sampling.”Journal of the American Statistical Association, 88(423): 881–889.
  • Hahn, P. R. and Carvalho, C. M. (2015). “Decoupling shrinkage and selection in Bayesian linear models: a posterior summary perspective.”Journal of the American Statistical Association, 110(509): 435–448.
  • Hans, C., Dobra, A., and West, M. (2007). “Shotgun stochastic search for “large p” regression.”Journal of the American Statistical Association, 102(478): 507–516.
  • Harvey, C. R. and Liu, Y. (2015). “Lucky factors.”Available at SSRN 2528780.
  • Jeffreys, H. (1961). “Theory of Probability (3rd edt.) Oxford University Press.”
  • Jones, B., Carvalho, C., Dobra, A., Hans, C., Carter, C., and West, M. (2005). “Experiments in stochastic computation for high-dimensional graphical models.”Statistical Science, 388–400.
  • Liang, F., Paulo, R., Molina, G., Clyde, M., and Berger, J. (2008a). “Mixtures of g Priors for Bayesian Variable Selection.”Journal of the American Statistical Association, 103: 410–423.
  • Liang, F., Paulo, R., Molina, G., Clyde, M. A., and Berger, J. O. (2008b). “Mixtures of g priors for Bayesian variable selection.”Journal of the American Statistical Association, 103(481).
  • Murray, J. S., Dunson, D. B., Carin, L., and Lucas, J. E. (2013). “Bayesian Gaussian copula factor models for mixed data.”Journal of the American Statistical Association, 108(502): 656–665.
  • Puelz, D., Hahn, P. R., and Carvalho, C. M. (2017). “Supplement for Variable selection in seemingly unrelated regressions with random predictors.”Bayesian Analysis.
  • Ross, S. A. (1976). “The arbitrage theory of capital asset pricing.”Journal of Economic Theory, 13(3): 341–360.
  • Scott, J. and Berger, J. (2006). “An exploration of aspects of Bayesian multiple testing.”Journal of Statistical Planning and Inference, 136: 2144–2162.
  • Wang, H. (2010). “Sparse seemingly unrelated regression modelling: Applications in finance and econometrics.”Computational Statistics & Data Analysis, 54(11): 2866–2877.
  • Wang, H. and West, M. (2009). “Bayesian analysis of matrix normal graphical models.”Biometrika, 96(4): 821–834.
  • Zellner, A. (1962). “An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias.”Journal of the American Statistical Association, 57(298): 348–368.
  • Zellner, A. (1986). “On assessing prior distributions and Bayesian regression analysis with g-prior distributions.”Bayesian inference and decision techniques: Essays in Honor of Bruno De Finetti, 6: 233–243.
  • Zellner, A. and Siow, A. (1980). “Posterior odds ratios for selected regression hypotheses.”Trabajos de estadística y de investigación operativa, 31(1): 585–603.
  • Zellner, A. and Siow, A. (1984).Basic issues in econometrics. University of Chicago Press Chicago.

Supplemental materials