Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-12T08:24:03.010Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIT ROOT INFERENCE FOR NON-STATIONARY LINEAR PROCESSES DRIVEN BY INFINITE VARIANCE INNOVATIONS

Published online by Cambridge University Press:  03 May 2016

Giuseppe Cavaliere,
Iliyan Georgiev and
A.M.Robert Taylor
Giuseppe Cavaliere
Affiliation:
University of Bologna
Iliyan Georgiev
Affiliation:
University of Bologna
A.M.Robert Taylor*
Affiliation:
University of Essex
*
*Address correspondence to Robert Taylor, Essex Business School, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK; e-mail: rtaylor@essex.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The contribution of this paper is two-fold. First, we derive the asymptotic null distribution of the familiar augmented Dickey-Fuller [ADF] statistics in the case where the shocks follow a linear process driven by infinite variance innovations. We show that these distributions are free of serial correlation nuisance parameters but depend on the tail index of the infinite variance process. These distributions are shown to coincide with the corresponding results for the case where the shocks follow a finite autoregression, provided the lag length in the ADF regression satisfies the same o(T1/3) rate condition as is required in the finite variance case. In addition, we establish the rates of consistency and (where they exist) the asymptotic distributions of the ordinary least squares sieve estimates from the ADF regression. Given the dependence of their null distributions on the unknown tail index, our second contribution is to explore sieve wild bootstrap implementations of the ADF tests. Under the assumption of symmetry, we demonstrate the asymptotic validity (bootstrap consistency) of the wild bootstrap ADF tests. This is done by establishing that (conditional on the data) the wild bootstrap ADF statistics attain the same limiting distribution as that of the original ADF statistics taken conditional on the magnitude of the innovations.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 2 , April 2018, pp. 302 - 348
Copyright
Copyright © Cambridge University Press 2016 

Footnotes

We thank two anonymous referees, the Editor, Peter Phillips, and the Co-Editor, Michael Jansson, for their helpful and constructive comments on previous versions of the paper. Cavaliere and Georgiev gratefully acknowledge financial support provided by the Fundação para a Ciência e a Tecnologia, Portugal, through grant PTDC/EGE-ECO/108620/2008. Cavaliere thanks the Italian Ministry of Education, University and Research (MIUR) for financial support (PRIN project: “Multivariate statistical models for risk assessment”). Cavaliere and Taylor would also like to thank the Danish Council for Independent Research, Sapere Aude | DFF Advanced Grant (Grant nr: 12-124980) for financial support.

References

REFERENCES

Adler, R.J., Feldman, R.E., & Taqqu, M.S. (1998) A Practical Guide to Heavy Tails. Birkhauser.Google Scholar
Andrews, D.W.K. & Buchinsky, M. (2000) A three-step method for choosing the number of bootstrap repetitions. Econometrica 68, 23–51.CrossRefGoogle Scholar
Arcones, M. & Giné, E. (1989) The bootstrap of the mean with arbitrary bootstrap sample size. Annales de l’I.H.P. Probabilités et statistiques, Section B 25, 457–481.Google Scholar
Arcones, M. & Giné, E. (1991) Additions and correction to “The bootstrap of the mean with arbitrary bootstrap sample size”. Annales de l’I.H.P. Probabilités et statistiques, Section B 27, 583–595.Google Scholar
Athreya, K.B. (1987) Bootstrap of the mean in the infinite variance case. The Annals of Statistics 15, 724–731.CrossRefGoogle Scholar
Aue, A., Berkes, I., & Horváth, L. (2008) Selection from a stable box. Bernoulli 14, 125–139.CrossRefGoogle Scholar
Berk, K. (1974) Consistent autoregressive spectral estimates. The Annals of Statistics 2, 489–502.CrossRefGoogle Scholar
Brillinger, D. (2001) Time Series. Data Analysis and Theory. SIAM.CrossRefGoogle Scholar
Burridge, P. & Hristova, D. (2008) Consistent estimation and order selection for nonstationary autoregressive processes with stable innovations. Journal of Time Series Analysis 29, 695–718.CrossRefGoogle Scholar
Caner, M. (1998) Tests for cointegration with infinite variance errors. Journal of Econometrics 86, 155–175.CrossRefGoogle Scholar
Cavaliere, G. & Georgiev, I. (2013) Exploiting infinite variance through dummy variables in nonstationary autoregressions. Econometric Theory 29, 1162–1195.CrossRefGoogle Scholar
Cavaliere, G., Georgiev, I., & Taylor, A.M.R. (2013) Wild bootstrap of the sample mean in the infinite variance case. Econometric Reviews 32, 204–219.CrossRefGoogle Scholar
Cavaliere, G., Georgiev, I., & Taylor, A.M.R. (2016a) Sieve-based inference for infinite-variance linear processes. The Annals of Statistics, forthcoming, with on-line supplement.CrossRefGoogle Scholar
Cavaliere, G., Georgiev, I., & Taylor, A.M.R. (2016b) Unit root inference for nonstationary linear processes driven by infinite variance innovations. Quaderni di Dipartimento 1, Department of Statistics, University of Bologna. https://ideas.repec.org/p/bot/quadip/wpaper130.htmlGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2008) Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory 24, 43–71.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2009) Bootstrap M unit root tests. Econometric Reviews 28, 393–421.CrossRefGoogle Scholar
Chan, N.H. & Tran, L.T. (1989) On the first order autoregressive process with infinite variance. Econometric Theory 5, 354–362.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2003) A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis 24, 379–400.CrossRefGoogle Scholar
Charemza, W.W., Hristova, D., & Burridge, P. (2005) Is inflation stationary? Applied Economics 37, 901–903.CrossRefGoogle Scholar
Cornea-Madeira, A. & Davidson, R. (2015) A parametric bootstrap for heavy-tailed distributions. Econometric Theory 31, 449–470.CrossRefGoogle Scholar
Daley, D. & Vere-Jones, D. (2008) An Introduction to the Theory of Point Processes. General Theory and Structure, vol. II. Springer-Verlag.CrossRefGoogle Scholar
Davidson, R. & MacKinnon, J. (2000) Bootstrap tests: How many bootstraps? Econometric Reviews 19, 55–68.CrossRefGoogle Scholar
Davis, R.A. (2010) Heavy tails in financial time series. In Cont, R. (ed.), Encyclopedia Quantitative Finance. Wiley.Google Scholar
Davis, R. & Resnick, S. (1985a) Limit theory for moving averages of random variables with regularly varying tail probabilities. The Annals of Probability 13, 179–195.CrossRefGoogle Scholar
Davis, R. & Resnick, S. (1985b) More limit theory for the sample correlation function of moving averages. Stochastic Processes and their Applications 20, 257–279.CrossRefGoogle Scholar
Davis, R. & Resnick, S. (1986) Limit theory for the sample covariance and correlation functions of moving averages. The Annals of Statistics 14, 533–558.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer-Verlag.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, vol. 2. Wiley.Google Scholar
Finkenstädt, B. & Rootzén, H. (2003) Extreme Values in Finance, Telecommunication and the Environment. Chapman & Hall.CrossRefGoogle Scholar
Fuller, W.A. (1996) Introduction to Statistical Time Series, 2nd ed. Wiley.Google Scholar
Gabaix, X. (2009) Power laws in economics and finance. Annual Review of Economics 1, 255–293.CrossRefGoogle Scholar
Haldrup, N. & Jansson, M. (2006) Improving power and size in unit root testing. In Mills, T.C. & Patterson, K. (eds.), Palgrave Handbooks of Econometrics, vol. 1, chap. 7, pp. 252–277. Palgrave MacMillan.Google Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis Econometrica 64, 413–430.CrossRefGoogle Scholar
Horváth, L. & Kokoszka, P. (2003) A bootstrap approximation to a unit root test statistic for heavy-tailed observations. Statistics & Probability Letters 62, 163–173.CrossRefGoogle Scholar
Jach, A. & Kokozska, P. (2004) Subsampling unit root tests for heavy-tailed observations. Methodology & Computing in Applied Probability 6, 73–97.CrossRefGoogle Scholar
Knight, K. (1989) Limit theory for autoregressive parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics 17, 261–278.CrossRefGoogle Scholar
Knight, K. (1991) Limit theory for M -estimates in an integrated infinite variance process. Econometric Theory 7, 200–212.CrossRefGoogle Scholar
LePage, R. & Podgórsky, K. (1996) Resampling permutations in regression without second moments. Journal of Multivariate Analysis 57, 119–141.CrossRefGoogle Scholar
LePage, R., Podgórski, K., & Ryznar, M. (1997) Strong and conditional invariance principles for samples attracted to stable laws. Probability Theory and Related Fields 108, 281–298.CrossRefGoogle Scholar
Lewis, R. & Reinsel, G. (1985) Prediction of multivariate time series by autoreressive model fitting. Journal of Multivariate Analysis 16, 393–411.CrossRefGoogle Scholar
Lütkepohl, H. (2005) New Introduction to Multiple Time Series Analysis. Springer.CrossRefGoogle Scholar
Maronna, R., Martin, D., & Yohai, V. (2006) Robust Statistics: Theory and Methods. Wiley.CrossRefGoogle Scholar
McCulloch, J.C. (1986) Simple consistent estimators of stable distribution parameters. Communications in Statistics, Simulation and Computation 15, 1109–1136.CrossRefGoogle Scholar
McCulloch, J. (1997) Measuring tail thickness to estimate the stable index α: A critique. Journal of Business and Economic Statistics 15, 74–81.Google Scholar
Moreno, M. & Romo, J. (2012) Unit root bootstrap tests under infinite variance. Journal of Time Series Analysis 33, 32–47.CrossRefGoogle Scholar
Müller, U. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.CrossRefGoogle Scholar
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.CrossRefGoogle Scholar
Perron, P. & Ng, S. (1996) Useful modifications to some unit root tests with dependent errors and their local asymptotic properties. Review of Economic Studies 63, 435–463.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1990) Time series regression with a unit root and infinite-variance errors. Econometric Theory 6, 44–62.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.CrossRefGoogle Scholar
Rachev, S.T. & Mittnik, S. (2000) Stable Paretian Models in Finance. Wiley.Google Scholar
Rachev, S.T., Mittnik, S., & Kim, J.R. (1998) Time series with unit roots and infinite variance disturbances. Applied Mathematics Letters 11, 69–74.CrossRefGoogle Scholar
Resnick, S. (1986) Point processes, regular variation and weak convergence. Advances in Applied Probability 18, 66–138.CrossRefGoogle Scholar
Resnick, S. (1997) Heavy tailed modeling and teletraffic data. Annals of Statistics 25, 1805–1869.CrossRefGoogle Scholar
Resnick, S. & Greenwood, P. (1979) A bivariate stable characterization and domains of attraction. Journal of Multivariate Analysis 9, 206–221.CrossRefGoogle Scholar
Romano, J.P. & Wolf, M. (1999) Inference for the mean in the heavy-tailed case. Metrika 50, 55–69.CrossRefGoogle Scholar
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599–608.CrossRefGoogle Scholar
Samarakoon, M. & Knight, K. (2009) A note on unit root tests with infinite variance noise. Econometric Reviews 28, 314–334.CrossRefGoogle Scholar
Schwert, G.W. (1989) Tests for unit roots: A Monte Carlo investigation. Journal of Business and Economic Statistics 7, 147–159.Google Scholar
Silverberg, G. & Verspagen, B. (2007) The size distribution of innovations revisited: an application of extreme value statistics to citation and value measures of patent significance. Journal of Econometrics 139, 318–339.CrossRefGoogle Scholar
Stock, J.H. (1999) A class of tests for integration and cointegration. In Engle, R.F. & White, H. (eds.), Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger, pp. 137–167. Oxford University Press.Google Scholar
Zarepour, M. & Knight, K. (1999) Bootstrapping unstable first order autoregressive process with errors in the domain of attraction of stable law. Communications in Statistics. Stochastic Models 15, 11–27.CrossRefGoogle Scholar
Zhang, R.-M., Sin, C.Y., & Ling, S. (2015) On functional limits of short- and long-memory linear processes with GARCH(1,1) noises. Stochastic Processes and Their Applications 125, 482–512.CrossRefGoogle Scholar