Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T17:18:57.344Z Has data issue: false hasContentIssue false
  • English
  • Français

MIXED CAUSAL-NONCAUSAL AR PROCESSES AND THE MODELLING OF EXPLOSIVE BUBBLES

Published online by Cambridge University Press:  28 January 2019

Sébastien Fries  and
Jean-Michel Zakoian
Sébastien Fries
Affiliation:
CREST and Paris-Saclay University
Jean-Michel Zakoian*
Affiliation:
CREST and University of Lille
*
*Address correspondence to Jean-Michel Zakoian, CREST and University of Lille, CREST, 5 Avenue Henri Le Chatelier, 91120 Palaiseau, France; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 6 , December 2019 , pp. 1234 - 1270
Copyright
Copyright © Cambridge University Press 2019 

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

The authors are grateful to three anonymous referees, the Co-editor, and the Editor for insightful comments, suggestions, and useful criticisms. The second author acknowledges financial support from the Agence Nationale de la Recherche (ANR), which supported this work via the Project MultiRisk (ANR CE26 2016 - CR), the Labex ECODEC and the Chaire ACPR “Régulation et risques systémiques”

References

REFERENCES

An, H.Z. & Chen, Z.G. (1982) On convergence of LAD estimates in autoregression with infinite variance. Journal of Multivariate Analysis 12, 335–345.CrossRefGoogle Scholar
Andrews, B., Calder, M., & Davis, R. (2009) Maximum likelihood estimation for α-stable autoregressive process. Annals of Statistics 37, 1946–1982.CrossRefGoogle Scholar
Andrews, B. & Davis, R.A. (2013) Model identification for infinite variance autoregressive processes. Journal of Econometrics 172(2), 222–234.CrossRefGoogle Scholar
Breidt, F.J., Davis, R.A., & Trindade, A.A. (2001) Least absolute deviation estimation for all-pass time series models. Annals of Statistics 29, 919–946.Google Scholar
Brock, W.A., Dechert, W.D., LeBaron, B., & Scheinkman, J.A. (1996) A test for independence based on the correlation dimension. Econometric Reviews 15(3), 197–235.CrossRefGoogle Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed. Springer-Verlag.CrossRefGoogle Scholar
Cavaliere, G., Nielsen, H.B., & Rahbek, A. (2018) Bootstrapping non-causal autoregressions: With applications to explosive bubble modelling. Journal of Business and Economic Statistics, forthcoming.CrossRefGoogle Scholar
Chavez-Demoulin, V. & Davison, A.C. (2012) Modelling time series extremes. REVSTAT-Statistical Journal 10, 109–133.Google Scholar
Chen, Y., Phillips, P.C.B., & Yu, J. (2017) Inference in continuous systems with mildly explosive regressors. Journal of Econometrics 201, 400–416.CrossRefGoogle Scholar
Cline, D. (1983) Estimation and Linear Prediction for Regression, Autoregression and ARMA with Infinite Variance Data. Ph.D. thesis, Department of Statistics, Colorado State University.Google Scholar
Daley, D.J. & Vere-Jones, D. (2007) An Introduction to the Theory of Point Processes, vol. 1. Springer.Google Scholar
Davis, R., Knight, K., & Liu, J. (1992) M-estimation for autoregressions with infinite variance. Stochastic Processes and their Applications 40, 145–180.CrossRefGoogle Scholar
Davis, R. & Liu, H. (2012) Theory and inference for a class of observation driven models with application to time series of counts. ArXiv e-prints 1204.3915.Google Scholar
Davis, R. & Resnick, S. (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Annals of Probability 13, 179–195.CrossRefGoogle Scholar
Davis, R. & Resnick, S. (1986) Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14, 533–558.CrossRefGoogle Scholar
Davis, R. & Song, L. (2012) Noncausal Vector AR Process with Application to Economic Time Series. Discussion paper, Columbia University.Google Scholar
De Haan, L., Mercadier, C., & Zhou, C. (2016) Adapting extreme value statistics to financial time series: Dealing with bias and serial dependence. Finance and Stochastics 20, 321–354.CrossRefGoogle Scholar
Donoho, D. (1981) On minimum entropy deconvolution. In Findley, D.F. (ed.), Applied Time Series Analysis II, pp. 565–608. Academic Press.CrossRefGoogle Scholar
Duchesne, P., Ghoudi, K., & Rémillard, B. (2012) On testing for independence between the innovations of several time series. Canadian Journal of Statistics 40(3), 447–479.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997) Modelling Extremal Events. Applications of Mathematics. Springer-Verlag.CrossRefGoogle Scholar
Ferro, C.A. & Segers, J. (2003) Inference for clusters of extreme values. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65(2), 545–556.CrossRefGoogle Scholar
Francq, C. & Zakoian, J-M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley.CrossRefGoogle Scholar
Gourieroux, C. & Jasiak, J. (2016) Filtering, prediction and simulation methods for noncausal processes. Journal of Time Series Analysis 37(3), 405–430.CrossRefGoogle Scholar
Gourieroux, C., Jasiak, J., & Monfort, A. (2016) Stationary Bubble Equilibria in Rational Expectation Models. CREST Working paper 2016-31.Google Scholar
Gouriéroux, C. & Zakoian, J.-M. (2017) Local explosion modelling by non-causal process. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79, 737–756.CrossRefGoogle Scholar
Hecq, A., Issler, J.V., & Telg, S.M. (2017) Mixed Causal-Noncausal Autoregressions with Strictly Exogenous Regressors. MPRA paper 80767.Google Scholar
Hecq, A., Lieb, L., & Telg, S.M. (2016) Identification of mixed causal-noncausal models in finite samples. Annals of Economics and Statistics 123/124, 307–331.CrossRefGoogle Scholar
Hecq, A., Telg, S., & Lieb, L. (2017a) Do seasonal adjustments induce noncausal dynamics in inflation rates? Econometrics 5, 1–22.CrossRefGoogle Scholar
Hecq, A., Telg, S., & Lieb, L. (2017b) Simulation, Estimation and Selection of Mixed Causal-Noncausal Autoregressive Models: The MARX Package. Available at SSRN: https://ssrn.com/abstract=3015797.CrossRefGoogle Scholar
Hencic, A. & Gouriéroux, C. (2015) Noncausal autoregressive model in application to bitcoin/USD exchange rates. In Huynh, V.-N., Kreinovich, V., Sriboonchitta, S., Suriya, K. (eds.), Econometrics of Risk, pp. 17–40. Springer International Publishing.CrossRefGoogle Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Annals of Mathematical Statistics 3, 1163–1174.CrossRefGoogle Scholar
Hill, J.B. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Theory 26, 1398–1436.CrossRefGoogle Scholar
Hsing, T., Hüsler, J., & Leadbetter, M.R. (1988) On the exceedance point process for a stationary sequence. Probability Theory and Related Fields 78, 97–112.CrossRefGoogle Scholar
Hsueh, A.C. & Mendel, J.M. (1985) Minimum-variance and maximum-likelihood deconvolution for noncausal channel models. IEEE Transactions on Geoscience and Remote Sensing 23, 797–808.CrossRefGoogle Scholar
Lanne, M., Nyberg, H., & Saarinen, E. (2012) Does noncausality help in forecasting economic time series? Economics Bulletin 32, 2849–2859.Google Scholar
Lanne, M. & Saikkonen, P. (2011) Noncausal autogressions for economic time series. Journal of Time Series Econometrics 3.CrossRefGoogle Scholar
Leadbetter, M.R. & Nandagopalan, S. (1989) On exceedance point processes for stationary sequences under mild oscillation restrictions. Lecture Notes in Statistics 51, 69–80.CrossRefGoogle Scholar
Lin, J.W. & McLeod, A.I. (2008) Portmanteau tests for ARMA models with infinite variance. Journal of Time Series Analysis 29, 600–617.CrossRefGoogle Scholar
Ling, S. (2007) A double AR(𝑝) model: Structure and estimation. Statistica Sinica 17, 161–175.Google Scholar
Markovich, N.M. (2014) Modelling clusters of extreme values. Extremes 17, 97–125.CrossRefGoogle Scholar
Mason, D. (1982) Laws of large numbers for sums of extreme values. Annals of Probability 10, 754–764.CrossRefGoogle Scholar
McCulloch, J.H. (1986) Simple consistent estimators of stable distribution parameters. Communication in Statistics–Computation and Simulation 15, 1109–1136.CrossRefGoogle Scholar
Nielsen, H.B. & Rahbek, A. (2014) Unit root vector autoregression with volatility induced stationarity. Journal of Empirical Finance 29, 144–167.CrossRefGoogle Scholar
Phillips, P.C.B., Shi, S., & Yu, J. (2015) Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P500. International Economic Review 56, 1043–1077.CrossRefGoogle Scholar
Phillips, P.C.B., Wu, Y., & Yu, J. (2011) Explosive behavior in the 1990’s Nasdaq: When did exuberance escalate asset values? International Economic Review 52, 201–226.CrossRefGoogle Scholar
Robert, C.Y., Segers, J., & Ferro, C.A. (2009) A sliding blocks estimator for the extremal index. Electronic Journal of Statistics 3, 993–1020.CrossRefGoogle Scholar
Rosenblatt, M. (2000) Gaussian and Non-Gaussian Linear, Time Series and Random Fields. Springer Verlag.CrossRefGoogle Scholar
Samorodnitsky, G. & Taqqu, M.S. (1994) Stable Non-Gaussian Random Processes. Chapman & Hall.Google Scholar
Scargle, J.D. (1981) Phase-sensitive deconvolution to model random processes, with special reference to astronomical data. In Findley, D.F. (ed.), Applied Time Series Analysis II, pp. 549–564. Academic Press.CrossRefGoogle Scholar
Wiggins, R.A. (1978) Minimum entropy deconvolution. Geoexploration 16, 21–35.CrossRefGoogle Scholar
Supplementary material: PDF

Fries and Zakoian supplementary material

Fries and Zakoian supplementary material
Download Fries and Zakoian supplementary material(PDF)
PDF 618.2 KB