The Annals of Applied Statistics

Good, great, or lucky? Screening for firms with sustained superior performance using heavy-tailed priors

Nicholas G. Polson and James G. Scott

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This paper examines historical patterns of ROA (return on assets) for a cohort of 53,038 publicly traded firms across 93 countries, measured over the past 45 years. Our goal is to screen for firms whose ROA trajectories suggest that they have systematically outperformed their peer groups over time. Such a project faces at least three statistical difficulties: adjustment for relevant covariates, massive multiplicity, and longitudinal dependence. We conclude that, once these difficulties are taken into account, demonstrably superior performance appears to be quite rare. We compare our findings with other recent management studies on the same subject, and with the popular literature on corporate success.

Our methodological contribution is to propose a new class of priors for use in large-scale simultaneous testing. These priors are based on the hypergeometric inverted-beta family, and have two main attractive features: heavy tails and computational tractability. The family is a four-parameter generalization of the normal/inverted-beta prior, and is the natural conjugate prior for shrinkage coefficients in a hierarchical normal model. Our results emphasize the usefulness of these heavy-tailed priors in large multiple-testing problems, as they have a mild rate of tail decay in the marginal likelihood m(y)—a property long recognized to be important in testing.

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Ann. Appl. Stat., Volume 6, Number 1 (2012), 161-185.

First available in Project Euclid: 6 March 2012

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Corporate benchmarking type-II beta distribution multiple testing normal scale mixtures sparsity


Polson, Nicholas G.; Scott, James G. Good, great, or lucky? Screening for firms with sustained superior performance using heavy-tailed priors. Ann. Appl. Stat. 6 (2012), no. 1, 161--185. doi:10.1214/11-AOAS512.

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  • Abramovich, F., Benjamini, Y., Donoho, D. L. and Johnstone, I. M. (2006). Adapting to unknown sparsity by controlling the false discovery rate. Ann. Statist. 34 584–653.
  • Abramowitz, M. and Stegun, I. A., eds. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55. National Bureau of Standards, Washington, DC. Reprinted in paperback by Dover (1974).
  • Armero, C. and Bayarri, M. (1994). Prior assessments for predictions in queues. J. Roy. Statist. Soc. Ser. D 43 139–153.
  • Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300.
  • Berger, J. (1980). A robust generalized Bayes estimator and confidence region for a multivariate normal mean. Ann. Statist. 8 716–761.
  • Bogdan, M., Chakrabarti, A. and Ghosh, J. K. (2008). Optimal rules for multiple testing and sparse multiple regression. Technical Report I-18/08/P-003, Wrocław Univ. Technology.
  • Bogdan, M., Ghosh, J. K. and Tokdar, S. T. (2008). A comparison of the Benjamini–Hochberg procedure with some Bayesian rules for multiple testing. In Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen. Inst. Math. Stat. Collect. 1 211–230. IMS, Beachwood, OH.
  • Carvalho, C. M., Polson, N. G. and Scott, J. G. (2010). The horseshoe estimator for sparse signals. Biometrika 97 465–480.
  • Dahl, D. B. and Newton, M. A. (2007). Multiple hypothesis testing by clustering treatment effects. J. Amer. Statist. Assoc. 102 517–526.
  • Denrell, J. (2005). Selection bias and the perils of benchmarking. Harvard Business Review 83 114–119.
  • Do, K.-A., Müller, P. and Tang, F. (2005). A Bayesian mixture model for differential gene expression. J. Roy. Statist. Soc. Ser. C 54 627–644.
  • Efron, B. (2008). Microarrays, empirical Bayes and the two-groups model. Statist. Sci. 23 1–22.
  • Fourdrinier, D., Strawderman, W. E. and Wells, M. T. (1998). On the construction of Bayes minimax estimators. Ann. Statist. 26 660–671.
  • Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Anal. 1 515–533 (electronic).
  • Gordy, M. B. (1998). A generalization of generalized beta distributions. Finance and Economics Discussion Series 1998-18, Board of Governors of the Federal Reserve System (U.S.).
  • Gradshteyn, I. and Ryzhik, I. (1965). Table of Integrals, Series, and Products. Academic Press, New York.
  • Gramacy, R. B. and Lee, H. K. H. (2008). Bayesian treed Gaussian process models with an application to computer modeling. J. Amer. Statist. Assoc. 103 1119–1130.
  • Griffin, J. and Brown, P. (2005). Alternative prior distributions for variable selection with very many more variables than observations. Technical report, Univ. Warwick.
  • Henderson, A. D., Raynor, M. E. and Ahmed, M. (2009). How long must a firm be great to rule out luck? Benchmarking sustained superior performance without being fooled by randomness. Strategic Manag. J. To appear. DOI:10.1002/smj.1943.
  • James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Univ. California Press, Berkeley, CA.
  • Jeffreys, H. (1961). Theory of Probability, 3rd ed. Clarendon Press, Oxford.
  • Johnstone, I. M. and Silverman, B. W. (2004). Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences. Ann. Statist. 32 1594–1649.
  • Liang, F., Paulo, R., Molina, G., Clyde, M. A. and Berger, J. O. (2008). Mixtures of g priors for Bayesian variable selection. J. Amer. Statist. Assoc. 103 410–423.
  • Maruyama, Y. (1999). Improving on the James–Stein estimator. Statist. Decisions 17 137–140.
  • Masreliez, C. (1975). Approximate non-Gaussian filtering with linear state and observation relations. IEEE Trans. Automat. Control 20 107–110.
  • McDonald, J. B. and Xu, Y. J. (1995). A generalization of the beta distribution with applications. J. Econometrics 66 133–152.
  • McGahan, A. M. and Porter, M. E. (1999). The persistence of shocks to profitability. Rev. Econom. Statist. 81 143–153.
  • Müller, P., Parmigiani, G. and Rice, K. (2007). FDR and Bayesian multiple comparisons rules. In Bayesian Statistics 8 349–370. Oxford Univ. Press, Oxford.
  • Park, J. and Ghosh, J. K. (2010). A guided random walk through some high dimensional problems. Sankhyā 72 81–100.
  • Pericchi, L. R. and Smith, A. F. M. (1992). Exact and approximate posterior moments for a normal location parameter. J. Roy. Statist. Soc. Ser. B 54 793–804.
  • Polson, N. G. (1991). A representation of the posterior mean for a location model. Biometrika 78 426–430.
  • Polson, N. K. and Scott, J. G. (2011). On the half-Cauchy prior for a global scale parameter. Technical report, Univ. Texas at Austin. Available at arXiv:1104.4937v2.
  • Scott, J. G. and Berger, J. O. (2006). An exploration of aspects of Bayesian multiple testing. J. Statist. Plann. Inference 136 2144–2162.
  • Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135–1151.
  • Strawderman, W. E. (1971). Proper Bayes minimax estimators of the multivariate normal mean. Ann. Math. Statist. 42 385–388.
  • Wiggins, R. R. and Ruefli, T. W. (2005). Schumpeter’s ghost: Is hypercompetition making the best of times shorter? Strategic Management Journal 26 887–911.
  • Zellner, A. and Siow, A. (1980). Posterior odds ratios for selected regression hypotheses. In Bayesian Statistics: Proceedings of the First International Meeting Held in Valencia (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 585–603. Valencia Univ. Press, Valencia.