The Annals of Applied Statistics

Power-weighted densities for time series data

Daniel McCarthy and Shane T. Jensen

Full-text: Open access


While time series prediction is an important, actively studied problem, the predictive accuracy of time series models is complicated by nonstationarity. We develop a fast and effective approach to allow for nonstationarity in the parameters of a chosen time series model. In our power-weighted density (PWD) approach, observations in the distant past are down-weighted in the likelihood function relative to more recent observations, while still giving the practitioner control over the choice of data model. One of the most popular nonstationary techniques in the academic finance community, rolling window estimation, is a special case of our PWD approach. Our PWD framework is a simpler alternative compared to popular state–space methods that explicitly model the evolution of an underlying state vector. We demonstrate the benefits of our PWD approach in terms of predictive performance compared to both stationary models and alternative nonstationary methods. In a financial application to thirty industry portfolios, our PWD method has a significantly favorable predictive performance and draws a number of substantive conclusions about the evolution of the coefficients and the importance of market factors over time.

Article information

Ann. Appl. Stat., Volume 10, Number 1 (2016), 305-334.

Received: November 2014
Revised: September 2015
First available in Project Euclid: 25 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Time series analysis power prior forecasting finance


McCarthy, Daniel; Jensen, Shane T. Power-weighted densities for time series data. Ann. Appl. Stat. 10 (2016), no. 1, 305--334. doi:10.1214/15-AOAS893.

Export citation


  • Aiolfi, M. and Timmermann, A. (2006). Persistence in forecasting performance and conditional combination strategies. J. Econometrics 135 31–53.
  • Avramov, D. (2002). Stock return predictability and model uncertainty. Journal of Financial Economics 64 423–458.
  • Berry, D. A. and Stangl, D. K. (1996). Bayesian methods in health-related research. In Bayesian Biostatistics. Statistics: A Series of Textbooks and Monographs 15 3–66. Marcel Dekker, New York, NY.
  • Berry, S. M., Carlin, B. P., Lee, J. J. and Muller, P. (2010). Bayesian Adaptive Methods for Clinical Trials 38. CRC press, Boca Raton.
  • Brian, N. (2010). Bayesian analysis using power priors with application to pediatric quality of care. Journal of Biometrics & Biostatistics 1:103. DOI:10.4172/2155-6180.1000103.
  • Carhart, M. M. (1997). On persistence in mutual fund performance. J. Finance 52 57–82.
  • Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika 81 541–553.
  • Chen, Y. and Singpurwalla, N. D. (1994). A non-Gaussian Kalman filter model for tracking software reliability. Statist. Sinica 4 535–548.
  • Dangl, T. and Halling, M. (2012). Predictive regressions with time-varying coefficients. Journal of Financial Economics 106 157–181.
  • Dawid, A. P. (1992). Prequential data analysis. In Current Issues in Statistical Inference: Essays in Honor of D. Basu. Institute of Mathematical Statistics Lecture Notes—Monograph Series 17 113–126. IMS, Hayward, CA.
  • Fama, E. F. and French, K. R. (1989). Business conditions and expected returns on stocks and bonds. Journal of Financial Economics 25 23–49.
  • Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33 3–56.
  • Figlewski, S. (1984). Hedging performance and basis risk in stock index futures. J. Finance 39 657–669.
  • Gelfand, A. E. and Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 56 501–514.
  • Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2003). Bayesian Data Analysis. CRC press, Boca Raton.
  • Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transaction on Pattern Analysis and Machine Intelligence 6 721–741.
  • Grunwald, G. K., Raftery, A. E. and Guttorp, P. (1993). Time series of continuous proportions. J. R. Stat. Soc. Ser. B. Stat. Methodol. 55 103–116.
  • Hobbs, B. P., Carlin, B. P., Mandrekar, S. J. and Sargent, D. J. (2011). Hierarchical commensurate and power prior models for adaptive incorporation of historical information in clinical trials. Biometrics 67 1047–1056.
  • Hoeting, J. A., Raftery, A. E. and Madigan, D. (2002). Bayesian variable and transformation selection in linear regression. J. Comput. Graph. Statist. 11 485–507.
  • Ibrahim, J. G. and Chen, M.-H. (2000). Power prior distributions for regression models. Statist. Sci. 15 46–60.
  • Kass, R. E. and Raftery, A. E. (1995). Bayes factors. J. Amer. Statist. Assoc. 90 773–795.
  • Lewellen, J. and Nagel, S. (2006). The conditional CAPM does not explain asset-pricing anomalies. Journal of Financial Economics 82 289–314.
  • Madigan, D. and Raftery, A. E. (1994). Model selection and accounting for model uncertainty in graphical models using Occam’s window. J. Amer. Statist. Assoc. 89 1535–1546.
  • McCarthy, D. and Jensen, S. T. (2015). Supplement to “Power-weighted densities for time series data.” DOI:10.1214/15-AOAS893SUPP.
  • Paez, M. S. and Gamerman, D. (2013). Hierarchical dynamic models. In The SAGE Handbook of Multilevel Modeling 335–357. SAGE Publications Ltd, London.
  • Petkova, R. and Zhang, L. (2005). Is value riskier than growth? Journal of Financial Economics 78 187–202.
  • Petris, G., Petrone, S. and Campagnoli, P. (2009). Dynamic Linear Models with R. Springer, New York.
  • Raftery, A. E. and Zheng, Y. (2003). Discussion: Performance of Bayesian model averaging. J. Amer. Statist. Assoc. 98 931–938.
  • Rapach, D. E., Strauss, J. K. and Zhou, G. (2010). Out-of-sample equity premium prediction: Combination forecasts and links to the real economy. Rev. Financ. Stud. 23 821-862.
  • Shephard, N. (1994). Local scale models: State space alternative to integrated GARCH processes. J. Econometrics 60 181–202.
  • Shiller, R. (2014). Online data. Available at
  • Smith, J. Q. (1979). A generalization of the Bayesian steady forecasting model. J. R. Stat. Soc. Ser. B. Stat. Methodol. 41 375–387.
  • Smith, J. Q. (1981). The multiparameter steady model. J. R. Stat. Soc. Ser. B. Stat. Methodol. 43 256–260.
  • Stock, J. H. and Watson, M. W. (2004). Combination forecasts of output growth in a seven-country data set. J. Forecast. 23 405–430.
  • Tan, S.-B., Machin, D., Tai, B.-C., Foo, K.-F. and Tan, E.-H. (2002). A Bayesian re-assessment of two Phase II trials of gemcitabine in metastatic nasopharyngeal cancer. Br. J. Cancer 86 843–850.
  • Welch, I. and Goyal, A. (2008). A comprehensive look at the empirical performance of equity premium prediction. Rev. Financ. Stud. 21 1455–1508.
  • West, M. and Harrison, J. (1998). Bayesian forecasting and dynamic models (2nd edn). Journal of the Operational Research Society 49 179–179.
  • Wilmott, P., Howison, S. and Dewynne, J. (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge Univ. Press, Cambridge.

Supplemental materials

  • Discussion of “Power-weighted densities for time series data”. In McCarthy and Jensen (2015), we show the conjugacy for exponential families under our PWD approach and the Kullback–Leibler optimality of the general PWD setup. We provide additional results for computational cost and simulations comparing additional PWD variants to competing models. An adaptive PWD variant which switches between linear and exponentially decaying weights is also explored.