Local Explosion Modelling by Noncausal Process

Gouriéroux, Christian and Zakoian, Jean-Michel (2016): Local Explosion Modelling by Noncausal Process. Forthcoming in: Journal of the Royal Statistical Society: Series B (Statistical Methodology)

Abstract

The noncausal autoregressive process with heavy-tailed errors possesses a nonlinear causal dynamics, which allows for %unit root, local explosion or asymmetric cycles often observed in economic and financial time series. It provides a new model for multiple local explosions in a strictly stationary framework. The causal predictive distribution displays surprising features, such as the existence of higher moments than for the marginal distribution, or the presence of a unit root in the Cauchy case. Aggregating such models can yield complex dynamics with local and global explosion as well as variation in the rate of explosion. The asymptotic behavior of a vector of sample autocorrelations is studied in a semi-parametric noncausal AR(1) framework with Pareto-like tails, and diagnostic tests are proposed. Empirical results based on the Nasdaq composite price index are provided.

• Item Type: MPRA Paper
• Language: English
• Keywords: Causal innovation; Explosive bubble; Heavy-tailed errors; Noncausal process; Stable process
• 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 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection

References

Andrews, B., Calder, M., and R., Davis (2009): Maximum Likelihood Estimation for $\alpha$-Stable Autoregressive Process. Ann. Statist., 37, 1946-1982.

Bierens, H. (1982) Consistent Model Specification Tests. Journal of Econometrics, 20, 105-134.

Breidt, F.J. and R.A., Davis (1992) Time-Reversibility, Identifiability and Independence of Innovations for Stationary Time-Series. J. Time Series Anal. 13, 377-390.

Brockwell, P.J. and R.A. Davis (1991) Time Series: Theory and Methods, Springer-Verlag, New York, 2nd edition.

Busetti, F., and A.M. Taylor (2004): Tests of Stationarity Against a Change in Persistence. J. Econometrics, 123, 33-66.

Cambanis,S., and I., Fakhre-Zakeri (1994): On Predictions of Heavy Tailed Autoregressive Sequences: Forward versus Reversed Time. Theory Prob. Appl. 39, 217-233.

Charemza, W., and D. Deadman (1995): Speculative Bubble with Stochastic Explosive Roots: The Failure of Unit Root Testing. J. Empir. Finance, 2, 153-163.

Chen, B., Choi, J., and J. C. Escanciano (2012): Testing for Fundamental Vector Moving Average Representations. Discussion Paper, Indiana University.

Cheng, Q. (1992): On the Unique Representation of Non-Gaussian Linear Processes. Ann. Statist., 20, 1143-1145.

Cooper, G. (2008): The Origin of Financial Crises: Central Banks, Credit Bubbles and the Efficient Market Fallacy. Vintage Books, New-York.

Davis, R. and S. Resnick (1985): Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities. Ann. Probab., 13, 179-195.

Davis, R. and S. Resnick (1986): Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages. Ann. Statist., 14, 533-558.

Davis, R. and L. Song (2012): Noncausal Vector AR Process with Application to Economic Time Series. Discussion Paper, Columbia University.

Diba, B. and H. Grossman (1988): Explosive Rational Bubbles in Stock Prices? The American Economic Review, 78, 520-530.

Embrechts, P., Klüppelberg, C. and T. Mikosch (1997) Modelling extremal events. Springer.

Evans, G. (1991): Pitfalls in Testing for Explosive Bubbles in Asset Prices. Am. Econ. Rev., 81, 922-930.

Flood, R., and P., Garber (1980): Market Fundamentals Versus Price Level Bubbles. The First Tests. Journal of Political Economy, 88, 745-770.

Francq, C. and J-M. Zakoïan (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley.

Gouriéroux, C., and A., Hencic (2015): Noncausal Autoregressive Model in Application to Bitcoin/USD Exchange rate. In Econometrics of Risk, Edited by V-N Huynh, V. Kreinovich, S. Sriboonchitta, K. Suriya, Springer.

Gouriéroux, C., and J., Jasiak (2005): Nonlinear Impulse Response Function. Annales d'Economie et de Statistique, 78, 1-33.

Gouriéroux, C., and J., Jasiak (2015): Filtering, Prediction and Simulation Methods for Noncausal Processes. J. Time Series Anal., forthcoming.

Gouriéroux, C., and J-M., Zakoïan (2015): On Uniqueness of Moving Average Representations of Heavy-Tailed Stationary Processes. J. Time Series Anal., 36, 876-887.

Heathcote, C.R. (1977) The integrated squared error estimation of parameters. Biometrika, 64, 255-264.

Homm, U. and J. Breitung (2012): Testing for Speculative Bubbles in Stock Markets: A Comparison of Alternative Methods. J. Financ. Econometrics, 10, 198-231.

Krengel, U. (1985) Ergodic theorems. De Gruyter, Berlin-New York.

Lanne, M., Nyberg, H., and E., Saarinen (2012): Does Noncausality Help in Forecasting Economic Time Series? Economics Bulletin, 32, 2849-2859.

Lanne, M., and P., Saikkonen (2011): NonCausal Autogressions for Economic Time Series. J. Time Ser. Econom., 3, Article 2.

Lin, J-W. and A.I. McLeod (2008) Portmanteau Tests for ARMA Models with Infinite Variance. J. Time Series Anal., 29, 600-617.

Ling, S. and D. Li (2008): Asymptotic Inference For a Nonstationary Double AR(1) Model. Biometrika, 95, 257-263.

Paulson, A.S., Holcomb, E.W. and R.A. Leitch (1975): The Estimation of the Parameters of the Stable Laws. Biometrika, 62, 163-170.

Pham, D.T. (1986): The Mixing Property of Bilinear And Generalized Random Coefficient Autoregressive Models. Stochastic Process. Appl., 20, 295-306.

Phillips, P.C.B. (1987): Towards a Unified Asymptotic Theory for Autoregression. Biometrika, 74, 535-547.

Phillips, P.C.B., Shi, S. and J. Yu (2015): Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S\&P500. Int. Econ. Rev., 56, 1043-1077.

Phillips, P.C.B., Wu, Y. and J. Yu (2011): Explosive Behavior in the 1990's Nasdaq: When Did Exuberance Escalate Asset Values? Int. Econ. Rev., 52, 201-226.

Ramsey, J., and P., Rothman (1996): Time Irreversibility and Business Cycle Asymmetry. Journal of Money and Banking, 28, 1-21.

Rosenblatt, M. (2000): Gaussian and Non-Gaussian Linear Time Series and Random Fields, Springer Verlag, New-York.

Samorodnitsky G. and M. S. and Taqqu (1994): Stable Non-Gaussian Random Processes, Chapman \& Hall, London.

Taufer, E. and N. Leonenko (2009): Characteristic Function Estimation of Non-Gaussian Ornstein-Uhlenbeck Processes. J. Statist. Plann. Inference, 139, 3050-3063.

Sims, C. (1980): Macroeconomics and Reality. Econometrica, 1-48.

Yu, J. (2004): Empirical Characteristic Function Estimation and Its Applications. Economet. Rev., 23, 93-123.