Fries, Sébastien and Zakoian, JeanMichel (2017): Mixed CausalNoncausal AR Processes and the Modelling of Explosive Bubbles.
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Abstract
Noncausal autoregressive models with heavytailed errors generate locally explosive processes and therefore provide a natural framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causalnoncausal autoregressive processes, assuming the errors follow a stable nonGaussian 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 lightertailed 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 noni.i.d. errors can be consistently estimated by classical leastsquares. 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.
Item Type:  MPRA Paper 

Original Title:  Mixed CausalNoncausal AR Processes and the Modelling of Explosive Bubbles 
Language:  English 
Keywords:  Noncausal process, Stable process, Extreme clustering, Explosive bubble, Portmanteau test. 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes 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 
Item ID:  86926 
Depositing User:  Sébastien Fries 
Date Deposited:  28 May 2018 18:56 
Last Modified:  03 Oct 2019 15:13 
References:  An, H.Z., and Z.G. Chen (1982): On Convergence of LAD Estimates in Autoregression with Infinite Variance. Journal of Multivariate Analysis 12, 335345. Andrews, B., Calder, M., and R., Davis (2009): Maximum Likelihood Estimation for � Stable Autoregressive Process. Annals of Statistics, 37, 19461982. Andrews, B. and R. A., Davis (2013): Model Identification for Infinite Variance Autoregressive Processes. Journal of Econometrics, 172(2), 222234. Bierens, H. (1982): Consistent Model Specification Tests. Journal of Econometrics, 20, 105134. Breidt, F.J., Davis, R.A. and A.A. Trindade (2001): Least Absolute Deviation Estimation for AllPass Time Series Models. Annals of Statistics, 29, 919946. Brock, W. A., Dechert, W. D., LeBaron, B., and J. A. Scheinkman (1996): A test for independence based on the correlation dimension. Econometric reviews, 15(3), 197235. Brockwell, P.J. and R.A., Davis (1991): Time Series: Theory and Methods, SpringerVerlag, New York, 2nd edition. Cavaliere, G., Nielsen, H.B, and A. Rahbek (2018): Bootstrapping NonCausal Autoregressions: with Applications to Explosive Bubble Modelling. Forthcoming in Journal of Business and Economic Statistics. ChavezDemoulin, V., and A. C., Davison (2012): Modelling Time Series Extremes. REVSTATStatistical Journal, 10, 109133. Chen, Y., Phillips, P.C.B. and J. Yu (2017): Inference in Continuous Systems with Mildly Explosive Regressors. Unpublished document, Singapore Management University. 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. Daley, D. J. and D., VereJones (2007): An introduction to the Theory of Point Processes, Springer, New York, vol. 1. Davis, R., Knight, K. and J., Liu (1992). Mestimation for Autoregressions with Infinite Variance. Stochastic Processes and their Applications, 40, 145–180. Davis, R., and H, Liu (2012): Theory and Inference for a Class of Observation Driven Models with Application to Time Series of Counts. ArXiv preprint. arXiv:1204.3915. Davis, R. and S., Resnick (1985): Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities. Annals of Probability, 13, 179195. Davis, R. and S., Resnick (1986): Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages. Annals of Statistics, 14, 533558. Davis, R. and L., Song (2012): Noncausal Vector AR Process with Application to Economic Time Series. Discussion Paper, Columbia University. De Haan, L., Mercadier, C., and C. Zhou (2016): Adapting Extreme Value Statistics to Financial Time Series: Dealing with Bias and Serial Dependence. Finance and Stochastics, 20, 321354. Donoho, D. (1981): On Minimum Entropy Deconvolution. In Applied Time Series Analysis II (D. F. Findley, ed.) 565608. Academic Press, New York. Duchesne, P., Ghoudi, K. and B. Rémillard (2012): On Testing for Independence between the Innovations of Several Time Series. Canadian Journal of Statistics, 40(3), 447479. Embrechts, P., Klüppelberg, C. and T. Mikosch (1997): Modelling Extremal Events. Applications of Mathematics, SpringerVerlag, Berlin. Ferro, C. A., and J., Segers (2003): Inference for Clusters of Extreme Values. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65(2), 545556. Francq, C. and JM. Zakoian (2010): GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley. Gourieroux, C., and J., Jasiak (2016): Filtering, Prediction and Simulation Methods for Noncausal Processes. Journal of Time Series Analysis, 37(3), 405430. Gourieroux, C., Jasiak, J. and A., Monfort (2016): Stationary Bubble Equilibria in Rational Expectation Models. Working Papers CREST, No 201631. Gouriéroux, C., and J.M., Zakoian (2015): On Uniqueness of Moving Average Representations of Heavytailed Stationary Processes. Journal of Time Series Analysis, 36, 876887. Gouriéroux, C. and J.M., Zakoian (2017): Local Explosion Modelling by NonCausal Process. Journal of the Royal Statistical Society: Series B (Statistical Methodology). 79, 737756. Hecq, A., Issler, J. V., and S. M., Telg (2017): Mixed CausalNoncausal Autoregressions with Strictly Exogenous Regressors. MPRA Paper 80767, Germany. Hecq, A., Lieb, L., and S. M., Telg (2016): Identification of Mixed CausalNoncausal Models in Finite Samples. Annals of Economics and Statistics, 123/124, 307331. Hecq, A., Telg, S., and L., Lieb (2017a): Do Seasonal Adjustments Induce Noncausal Dynamics in Inflation Rates? Econometrics, 5, 1–22. Hecq, A., Telg, S., and L, Lieb (2017b): Simulation, Estimation and Selection of Mixed CausalNoncausal Autoregressive Models: the MARX Package. Available at SSRN: https://ssrn.com/abstract=3015797. Hencic, A., and C., Gouriéroux (2015): Noncausal Autoregressive Model in Application to Bitcoin/USD Exchange Rates. Econometrics of Risk, Springer International Publishing, 1740. Hill, B.M. (1975): A Simple General Approach to Inference about the Tail of a Distribution. Annals of Mathematical Statistics 3, 11631174. Hill, J.B. (2010): On Tail Index Estimation For Dependent, Heterogeneous Data. Econometric Theory, 26, 13981436. Hsing, T., Hüsler, J., and M. R., Leadbetter (1988): On the Exceedance Point Process for a Stationary Sequence. Probability Theory and Related Fields, 78, 97112. Hsueh, A.C., and J. M. Mendel (1985): MinimumVariance and MaximumLikelihood Deconvolution for Noncausal Channel Models. IEEE Trans. Geoscience and Remote Sensing, 23, 797808. Lanne, M., Nyberg, H., and E., Saarinen (2012): Does Noncausality Help in Forecasting Economic Time Series? Economics Bulletin, 32, 28492859. Lanne, M., and P., Saikkonen (2011): NonCausal Autogressions for Economic Time Series. Journal of Time Series Econometrics, 3. Leadbetter, M. R., and S., Nandagopalan (1989): On Exceedance Point Processes for Stationary Sequences Under Mild Oscillation Restrictions. Lecture Notes in Statistics, 51, 6980. Lin, J. W., and A. I., McLeod (2008): Portmanteau Tests for ARMA Models with Infinite Variance. Journal of Time Series Analysis, 29, 600617. Ling, S. (2007): A Double AR(p) Model: Structure and Estimation. Statistica Sinica, 17, 161 175. Markovich, N. M. (2014): Modelling Clusters of Extreme Values. Extremes, 17, 97125. Mason, D. (1982): Laws of Large Numbers for Sums of Extreme Values. Annals of Probability 10, 754764. McCulloch, J. H. (1986): Simple Consistent Estimators of Stable Distribution Parameters. Communication in Statistics–Computation and Simulation 15, 1109–36. Nielsen, H. B., and A., Rahbek (2014): Unit Root Vector Autoregression with Volatility Induced Stationarity. Journal of Empirical Finance, 29, 144167. Phillips, P.C.B., Shi, S. and J. Yu (2015): Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P500. International Economic Review, 56, 10431077. Phillips, P.C.B., Wu, Y. and J. Yu (2011): Explosive Behaviour in the 1990’s Nasdaq: When Did Exuberance Escalate Asset Values? International Economic Review, 52, 201226. Robert, C. Y., Segers, J., and Ferro, C. A. (2009): A Sliding Blocks Estimator for the Extremal Index. Electronic Journal of Statistics, 3, 9931020. Rosenblatt, M. (2000): Gaussian and NonGaussian Linear, Time Series and Random Fields, Springer Verlag, NewYork. Samorodnitsky G. and M. S., and Taqqu (1994): Stable NonGaussian Random Processes, Chapman & Hall, London. Scargle, J. D. (1981): PhaseSensitive Deconvolution to Model Random Processes, with Special Reference to Astronomical Data. In Applied Time Series Analysis II (D. F. Findley, ed.) 549564. Academic Press, New York. Wiggins, R. A. (1978): Minimum Entropy Deconvolution. Geoexploration, 16, 2135. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/86926 
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Mixed CausalNoncausal AR Processes and the Modelling of Explosive Bubbles. (deposited 16 Sep 2017 09:01)
 Mixed CausalNoncausal AR Processes and the Modelling of Explosive Bubbles. (deposited 28 May 2018 18:56) [Currently Displayed]