Linking Frequentist and Bayesian Change-Point Methods

Ardia, David and Dufays, Arnaud and Ordás Criado, Carlos (2023): Linking Frequentist and Bayesian Change-Point Methods.

Abstract

We show that the two-stage minimum description length (MDL) criterion widely used to estimate linear change-point (CP) models corresponds to the marginal likelihood of a Bayesian model with a specific class of prior distributions. This allows results from the frequentist and Bayesian paradigms to be bridged together. Thanks to this link, one can rely on the consistency of the number and locations of the estimated CPs and the computational efficiency of frequentist methods, and obtain a probability of observing a CP at a given time, compute model posterior probabilities, and select or combine CP methods via Bayesian posteriors. Furthermore, we adapt several CP methods to take advantage of the MDL probabilistic representation. Based on simulated data, we show that the adapted CP methods can improve structural break detection compared to state-of-the-art approaches. Finally, we empirically illustrate the usefulness of combining CP detection methods when dealing with long time series and forecasting.

• Item Type: MPRA Paper
• Language: English
• Keywords: Change-point; Minimum description length; Model selection/combination; Structural change.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods

References

Bai, J., 1997. Estimating multiple breaks one at a time. Econometric theory 13, 315–352.

Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structure changes. Econometrica 66, 47–78.

Bai, J., Perron, P., 2003. Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, 1–22.

Bauwens, L., Dufays, A., De Backer, B., 2014a. Estimating and forecasting structural breaks in financial time series. Journal of Empirical Finance 29, 207–229.

Bauwens, L., Dufays, A., Rombouts, J., 2014b. Marginal likelihood for Markov-switching and change-point GARCH models. Journal of Econometrics 178, 508–522.

Bauwens, L., Koop, G., Korobilis, D., Rombouts, J., 2015. The contribution of structural break models to forecasting macroecomic series. Journal of Applied Econometrics 30, 596–620.

Bernardo, J., Bayarri, M., Berger, J., Dawid, A., Heckerman, D., Smith, A., West, M., 2007. Objective Bayesian analysis of multiple changepoints for linear models, in: Bayesian statistics 8: proceedings of the eighth Valencia International Meeting, June 2-6, 2006, Oxford University Press, USA. pp. 1–27.

Carlin, B.P., Gelfand, A.E., Smith, A.F.M., 1992. Hierarchical Bayesian analysis of changepoint problems. Journal of the Royal Statistical Society C 41, 389–405.

Chan, N., Yip, C., Zhang, R., 2014. Group LASSO for structural break time series. Journal of the American Statistical Association 109, 590–599.

Chib, S., 1998. Estimation and comparison of multiple change-point models. Journal of Econometrics 86, 221–241.

Ciuperca, G., 2011. A general criterion to determine the number of change-points. Statistics & Probability Letters 81, 1267–1275.

Davis, R.A., Lee, T.C.M., Rodriguez-Yam, G.A., 2006. Structural break estimation for nonstationary time series models. Journal of the American Statistical Association 101, 223–239.

Davis, R.A., Yau, C.Y., et al., 2013. Consistency of minimum description length model selection for piecewise stationary time series models. Electronic Journal of Statistics 7, 381–411.

Du, C., Kao, C.M., Kou, S.C., 2016. Stepwise signal extraction via marginal likelihood. Journal of the American Statistical Association 111, 314–330.

Dufays, A., Houndetoungan, E.A., Coën, A., 2022. Selective linear segmentation for detecting relevant parameter changes. Journal of Financial Econometrics 116, 762–805.

Eckley, I.A., Fearnhead, P., Killick, R., 2011. Analysis of changepoint models. Cambridge University Press, Cambridge. chapter 10. pp. 205–224.

Fama, E., French, K., 2015. A five-factor asset pricing model. Journal of Financial Economics 116, 1–22.

Fernandez, C., Ley, E., Steel, M.F., 2001. Benchmark priors for Bayesian model averaging. Journal of Econometrics 100, 381–427.

Fryzlewicz, P., 2014. Wild binary segmentation for multiple change-point detection. Annals of Statistics 42, 2243–2281.

Geweke, J., Jiang, Y., 2011. Inference and prediction in a multiple-structural-break model. Journal of Econometrics 163, 172–185.

Ghalanos, A., 2022. rugarch: Univariate GARCH models. R package version 1.4-9.

Giordani, P., Kohn, R., 2008. Efficient Bayesian inference for multiple change-point and mixture innovation models. Journal of Business & Economic Statistics 26, 66–77.

Gupta, A., Chen, J., 1996. Detecting changes of mean in multidimensional normal sequences with applications to literature and geology. Computational Statistics 11, 211–221.

Hansen, M.H., Yu, B., 2001. Model selection and the principle of minimum description length. Journal of the American Statistical Association 96, 746–774.

Hansen, P.R., Lunde, A., Nason, J.M., 2011. The model confidence set. Econometrica 79, 453–497.

Horváth, L., Miller, C., Rice, G., 2020. A new class of change point test statistics of rényi type. Journal of Business & Economic Statistics 38, 570–579.

Jackson, B., Scargle, J.D., Barnes, D., Arabhi, S., Alt, A., Gioumousis, P., Gwin, E., Sangtrakulcharoen, P., Tan, L., Tsai, T.T., 2005. An algorithm for optimal partitioning of data on an interval. IEEE Signal Processing Letters 12, 105–108.

Kass, R., Raftery, A., 1995. Bayes factors. Journal of the American Statistical Association 90, 773–795.

Killick, R., Fearnhead, P., Eckley, I.A., 2012. Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association 107, 1590–1598.

Koop, G., Potter, S., 2007. Estimation and forecasting with multiple breaks. Review of Economic Studies 74, 763–789.

Korkas, K., Fryzlewicz, P., 2020. wbsts: Multiple change-point detection for nonstationary time series. R package version 2.1.

Korkas, K.K., Fryzlewiczv, P., 2017. Multiple change-point detection for non-stationary time series using wild binary segmentation. Statistica Sinica 27, 287–311.

Li, S., Lund, R., 2012. Multiple changepoint detection via genetic algorithms. Journal of Climate 25, 674–686.

Liu, J., Wu, S., Zidek, J.V., 1997. On segmented multivariate regressions. Statistica Sinica 7, 497–525.

Maheu, J., Song, Y., 2014. A new structural break model, with an application to Canadian inflation forecasting. International Journal of Forecasting 30, 144–160.

Maheu, J.M., Gordon, S., 2008. Learning, forecasting and structural breaks. Journal of Applied Econometrics 23, 553–583.

Maheu, J.M., Song, Y., 2018. An efficient Bayesian approach to multiple structural change in multivariate time series. Journal of Applied Econometrics 33, 251–270.

Meligkotsidou, L., Vrontos, I.D., 2008. Detecting structural breaks and identifying risk factors in hedge fund returns: A Bayesian approach. Journal of Banking & Finance 32, 2471–2481.

Nemes, G., 2015. Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 145, 571–596.

Ng, W.L., Pan, S., Yau, C.Y., 2022. Bootstrap inference for multiple change-points in time series. Econometric Theory 38, 752–792.

Pástor, L., Stambaugh, R.F., 2001. The equity premium and structural breaks. Journal of Finance 56, 1207–1239.

Pesaran, M.H., Pettenuzzo, D., Timmermann, A., 2006. Forecasting time series subject to multiple structural breaks. Review of Economic Studies 73, 1057–1084.

Rigaill, G., Lebarbier, E., Robin, S., 2012. Exact posterior distributions and model selection criteria for multiple change-point detection problems. Statistics and Computing 22, 917–929.

Safikhani, A., Shojaie, A., 2020. Joint structural break detection and parameter estimation in high-dimensional non-stationary VAR models. Journal of the American Statistical Association 117, 1–26.

Smith, A.F., Spiegelhalter, D.J., 1980. Bayes factors and choice criteria for linear models. Journal of the Royal Statistical Society B 42, 213–220.

Smith, S.C., 2022. Structural breaks in grouped heterogeneity. Forthcoming in the Journal of Business & Economic Statistics.

Smith, S.C., Bulkley, G., Leslie, D.S., 2020. Equity premium forecasts with an unknown number of structural breaks. Journal of Financial Econometrics 18, 59–94.

Stephens, D.A., 1994. Bayesian retrospective multiple-changepoint identification. Applied Statistics 1, 159–178.

Yao, Y.C., Au, S.T., 1989. Least-squares estimation of a step function. Sankhya: The Indian Journal of Statistics 51, 370–381.

Yau, C.Y., Zhao, Z., 2016. Inference for multiple change points in time series via likelihood ratio scan statistics. Journal of the Royal Statistical Society B 78, 895–916.

Young Yang, T., Kuo, L., 2001. Bayesian binary segmentation procedure for a Poisson process with multiple changepoints. Journal of Computational and Graphical Statistics 10, 772–785.

Zellner, A., 1986. On assessing prior distributions and Bayesian regression analysis with g-prior distributions, in: Goel, P., Zellner, A. (Eds.), Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. Elsevier Science, New York, pp. 233–243.