Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-27T04:23:27.670Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZED LAPLACE INFERENCE IN MULTIPLE CHANGE-POINTS MODELS

Published online by Cambridge University Press:  23 February 2021

Alessandro Casini  and
Pierre Perron
Alessandro Casini*
Affiliation:
University of Rome Tor Vergata
Pierre Perron
Affiliation:
Boston University
*
Address Correspondence to Alessandro Casini, Department of Economics and Finance, University of Rome Tor Vergata, Via Columbia 2, Rome00133, Italy; e-mail: alessandro.casini@uniroma2.it.
Get access Rights & Permissions [Opens in a new window]

Abstract

Under the classical long-span asymptotic framework, we develop a class of generalized laplace (GL) inference methods for the change-point dates in a linear time series regression model with multiple structural changes analyzed in, e.g., Bai and Perron (1998, Econometrica 66, 47–78). The GL estimator is defined by an integration rather than optimization-based method and relies on the LS criterion function. It is interpreted as a classical (non-Bayesian) estimator, and the inference methods proposed retain a frequentist interpretation. This approach provides a better approximation about the uncertainty in the data of the change-points relative to existing methods. On the theoretical side, depending on some input (smoothing) parameter, the class of GL estimators exhibits a dual limiting distribution, namely the classical shrinkage asymptotic distribution or a Bayes-type asymptotic distribution. We propose an inference method based on highest density regions using the latter distribution. We show that it has attractive theoretical properties not shared by the other popular alternatives, i.e., it is bet-proof. Simulations confirm that these theoretical properties translate to good finite-sample performance.

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 1 , February 2022 , pp. 35 - 65
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This paper is based on the fourth chapter of the first author’s doctoral dissertation at Boston University. We thank the Editor and a Co-Editor for guiding the review, and three anonymous referees for constructive comments. We also thank Zhongjun Qu for useful comments.

References

REFERENCES

Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change-point. Econometrica 61(4), 821–856.CrossRefGoogle Scholar
Andrews, D.W.K. & Monahan, J.C. (1992) An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60(4), 953–966.CrossRefGoogle Scholar
Andrews, D.W.K. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62(6), 1383–1414.CrossRefGoogle Scholar
Bai, J. (1995) Least absolute deviation estimation of a shift. Econometric Theory 11(3), 403–436.CrossRefGoogle Scholar
Bai, J. (1997) Estimation of a change-point in multiple regression models. The Review of Economics and Statistics 79(4), 551–563.CrossRefGoogle Scholar
Bai, J. & Perron, P. (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66(1), 47–78.CrossRefGoogle Scholar
Bai, J. & Perron, P. (2003) Computation and analysis of multiple structural changes. Journal of Applied Econometrics 18, 1–22.CrossRefGoogle Scholar
Bhattacharya, P.K. (1987) Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case. Journal of Multivariate Analysis 23(2), 183–208.CrossRefGoogle Scholar
Bickel, P.J. & Yahav, J.A. (1969) Some contributions to the asymptotic theory of Bayes solutions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 11(4), 257–276.CrossRefGoogle Scholar
Box, G.E.P. & Tiao, G.C. (1973) Bayesian Inference in Bayesian Analysis. Addison-Wesley.Google Scholar
Buehler, R.J. (1959) Some validity criteria for statistical inferences. Annals of Mathematical Statistics 30(4), 845–863.CrossRefGoogle Scholar
Casini, A. (2018) Tests for Forecast Instability and Forecast Failure Under a Continuous Record Asymptotic Framework. arXiv preprint arXiv:1803.10883.Google Scholar
Casini, A. (2020) Theory of Evolutionary Spectra for Heteroskedasticity and Autocorrelation Robust Inference in Possibly Misspecified and Nonstationary Models. Unpublished manuscript, Department of Economics and Finance, University of Rome Tor Vergata.Google Scholar
Casini, A., Deng, T., & Perron, P. (2020) Theory of Low-Frequency Contamination from Unaccounted Nonstationarity: Consequences for HAR Inference. Unpublished manuscript, Department of Economics and Finance, University of Rome Tor Vergata.Google Scholar
Casini, A. & Perron, P. (2019) Structural Breaks in Time Series. Oxford Research Encyclopedia of Economics and Finance, Oxford University Press.CrossRefGoogle Scholar
Casini, A. & Perron, P. (2020a) Continuous record asymptotics for structural change models. arXiv preprint arXiv:1803.10881.Google Scholar
Casini, A. & Perron, P. (2020b) Continuous record Laplace-based inference about the break date in structural change models. Journal of Econometrics, forthcoming.Google Scholar
Chang, S.Y. & Perron, P. (2018) A comparison of alternative methods to construct confidence intervals for the estimate of a break date in linear regression models. Econometric Reviews 37(6), 577–601.CrossRefGoogle Scholar
Chernozhukov, V. & Hong, H. (2003) An MCMC approach to classical estimation. Journal of Econometrics 115(2), 293–346.CrossRefGoogle Scholar
Cornfield, J. (1969) The Bayesian outlook and its application. Biometrics 25(4), 617–657.CrossRefGoogle ScholarPubMed
Cox, D.R. (1958) Some problems connected with statistical inference. Annals of Mathematical Statistics 29(2), 357–372.CrossRefGoogle Scholar
Crainiceanu, C.M. & Vogelsang, T.J. (2007) Nonmonotonic power for tests of a mean shift in a time series. Journal of Statistical Computation and Simulation 77(6), 457–476.CrossRefGoogle Scholar
Csörgő, M. & Horváth, L. (1997) Limit Theorems in Change-Point Analysis. John Wiley and Sons.Google Scholar
Deng, A. & Perron, P. (2006) A comparison of alternative asymptotic frameworks to analyse a structural change in a linear time trend. Econometrics Journal 9(3), 423–447.CrossRefGoogle Scholar
Elliott, G. & Müller, U.K. (2007) Confidence sets for the date of a single break in linear time series regressions. Journal of Econometrics 141(2), 1196–1218.CrossRefGoogle Scholar
Eo, Y. & Morley, J. (2015) Likelihood-ratio-based confidence sets for the timing of structural breaks. Quantitative Economics 6(2), 463–497.CrossRefGoogle Scholar
Forneron, J.J. & Ng, S. (2018) The ABC of simulation estimation with auxiliary statistics. Journal of Econometrics 205(1), 112–139.CrossRefGoogle Scholar
Fossati, S. (2018) Testing for State-Dependent Predictive Ability. Unpublished manuscript, Department of Economics, University of Alberta.Google Scholar
Ghosal, S., Ghosh, J.K., & Samanta, T. (1995) On convergence of posterior distributions. Annals of Statistics 23(6), 2145–2152.CrossRefGoogle Scholar
Hall, A., Han, S., & Boldea, O. (2010) Inference regarding multiple structural changes in linear models with endogenous regressors. Journal of Econometrics 170(2), 281–302.CrossRefGoogle Scholar
Hawkins, D.M. (1976) Point estimation of the parameters of piecewise regression models. Journal of Applied Statistics 25(1), 51–57.CrossRefGoogle Scholar
Hawkins, D.M. (1977) Testing a sequence of observations for a shift in location. Journal of the American Statistical Association 72(357), 180–186.CrossRefGoogle Scholar
Hinkley, D.V. (1971) Inference about the change-point from cumulative sum tests. Biometrika 58(3), 509–523.CrossRefGoogle Scholar
Hirano, K. & Porter, J.R. (2003) Asymptotic efficiency in parametric structural models with parameter-dependent support. Econometrica 71(5), 1307–1338.CrossRefGoogle Scholar
Horváth, L. (1993) The maximum likelihood method for testing changes in the parameters of normal observations. Annals of Statistics 21(2), 671–680.CrossRefGoogle Scholar
Hyndman, R.J. (1996) Computing and graphing highest density regions. The American Statistician 50(2), 120–126.Google Scholar
Ibragimov, A. & Has’minskiǐ, R.Z. (1981) Statistical Estimation: Asymptotic Theory. Springer-Verlag.CrossRefGoogle Scholar
Jiang, L., Wang, X., & Yu, J. (2018) New distribution theory for the estimation of structural break point in mean. Journal of Econometrics 205(1), 156–176.CrossRefGoogle Scholar
Jiang, L., Wang, X., & Yu, J. (2020) In-fill asymptotic theory for structural break point in autoregression: A unified theory. Econometric Reviews, forthcoming.Google Scholar
Juhl, T. & Xiao, Z. (2009) Testing for changing mean with monotonic power. Journal of Econometrics 148(1), 14–24.CrossRefGoogle Scholar
Jun, S.J., Pinkse, J., & Wan, Y. (2015) Classical Laplace estimation for $\sqrt[3]{n}$ -consistent estimators: Improved convergence rates and rate-adaptive inference. Journal of Econometrics 187(1), 201–216.CrossRefGoogle Scholar
Jurečová, J (1977) Asymptotic relations of M-estimates and R-estimates in linear regression model. Annals of Statistics 5(3), 464–472.Google Scholar
Kendall, M.G. & Stuart, A. (1961) The Advanced Theory of Statistics: Vol. 2–Inference and Relationship. Hafner Publishing Company.CrossRefGoogle Scholar
Kim, D. & Perron, P. (2009) Assessing the relative power of structural break tests using a framework based on the approximate Bahadur slope. Journal of Econometrics 149(1), 26–51.CrossRefGoogle Scholar
Kim, H.J. & Pollard, D. (1990) Cube root asymptotics. Annals of Statistics 18(1), 191–219.CrossRefGoogle Scholar
Kim, H.J. & Siegmund, D. (1989) The likelihood ratio test for a change point in simple linear regression. Biometrika 76(3), 409–423.CrossRefGoogle Scholar
Laplace, P.S. (1774) Memoir on the probability of causes of events. Mémoires de Mathématique et de Physique 6, 621–646 (English translation by S. M. Stigler 1986. Statist. Sci., 1(19):364–378).Google Scholar
Martins, L. & Perron, P. (2016) Improved tests for forecast comparisons in the presence of instabilities. Journal of Time Series Analysis 37(5), 650–659.CrossRefGoogle Scholar
Mason, D.M. & Polonik, W. (2008) Asymptotic Normality of Plug-in Level Set Estimates. Extended version.CrossRefGoogle Scholar
Mason, D.M. & Polonik, W. (2009) Asymptotic normality of plug-in level set estimates. Annals of Applied Probability 19(3), 1108–1142.CrossRefGoogle Scholar
Müller, U.K. & Norest, A. (2016) Credibility of confidence sets in nonstandard econometric problems. Econometrica 84(6), 2183–2213.CrossRefGoogle Scholar
Newey, W.K. & West, K.D. (1987) A simple positive semidefinite, heteroskedastic and autocorrelation consistent covariance matrix. Econometrica 55(3), 703–708.CrossRefGoogle Scholar
Nyblom, J. (1989) Testing for the constancy of parameters over time. Journal of the American Statistical Association 89(451), 223–230.CrossRefGoogle Scholar
Oka, T. & Qu, Z. (2010) Estimating structural changes in regression quantiles. Journal of Econometrics 162(2), 248–267.CrossRefGoogle Scholar
Perron, P. (2006) Dealing with structural breaks. In Patterson, K. & Mills, T. (eds.), Palgrave Handbook of Econometrics, Volume 1: Econometric Theory, pp. 278–352. Palgrave Macmillan.Google Scholar
Perron, P. & Qu, Z. (2006) Estimating restricted structural change model. Journal of Econometrics 134(2), 373–399.CrossRefGoogle Scholar
Perron, P. & Yamamoto, Y. (2014) A note on estimating and testing for multiple structural changes in models with endogenous regressors via 2SLS. Econometric Theory 30(2), 491–507.CrossRefGoogle Scholar
Perron, P. & Yamamoto, Y. (2015) Using OLS to estimate and test for structural changes in models with endogenous regressors. Journal of Applied Econometrics 30(1), 119–144.CrossRefGoogle Scholar
Perron, P. & Yamamoto, Y. (2021) Testing for changes in forecast performance. Journal of Business and Economic Statistics 39(1), 148–165.CrossRefGoogle Scholar
Perron, P. & Zhu, X. (2005) Structural breaks with deterministic and stochastic trends. Journal of Econometrics 129(1–2), 65–119.CrossRefGoogle Scholar
Picard, D. (1985) Testing and estimating change-points in time series. Advances in Applied Probability 17(4), 841–867.CrossRefGoogle Scholar
Pierce, D.A. (1973) On some difficulties in a frequency theory of inference. Annals of Statistics 1(2), 241–250.CrossRefGoogle Scholar
Qu, Z. & Perron, P. (2007) Estimating and testing structural changes in multivariate regressions. Econometrica 75(2), 459–502.CrossRefGoogle Scholar
Robert, C. & Casella, G. (2004) Monte Carlo Statistical Methods, 2nd Edition, Springer Texts in Statistics. Springer-Verlag.CrossRefGoogle Scholar
Robinson, G.K. (1977) Conservative statistical inference . Journal of the Royal Statistical Society Series B 39(3), 381–386.Google Scholar
Samworth, R.J. & Wand, M.P. (2010) Asymptotics and optimal bandwidth selection for highest density region estimation. Annals of Statistics 38(3), 1767–1792.CrossRefGoogle Scholar
Vogelsang, T.J. (1999) Sources of nonmonotonic power when testing for a shift in mean of a dynamic time series. Journal of Econometrics 88(2), 283–299.CrossRefGoogle Scholar
Wallace, D.L. (1959) Conditional confidence level properties. Annals of Mathematical Statistics 30(4), 864–876.CrossRefGoogle Scholar
Yao, Y. (1987) Approximating the distribution of the ML estimate of the change-point in a sequence of independent random variables. Annals of Statistics 15, 1321–1328.CrossRefGoogle Scholar
Supplementary material: PDF

Casini and Perron supplementary material

Casini and Perron supplementary material

Download Casini and Perron supplementary material(PDF)
PDF 590.7 KB