Abstract
In the framework of integrated processes, the problem of testing the presence of unknown boundaries which constrain the process to move within a closed interval is considered. To analyze this problem, the concept of bounded integrated process is introduced, thus allowing to formally define boundary conditions for I(1) processes. A new class of tests, which are based on the rescaled range of the process, is introduced in order to test the null hypothesis of no boundary conditions. The limit distribution of the test statistics involved can be expressed in terms of the distribution of the range of Brownian functionals, while the power properties are obtained by deriving some asymptotic results for I(1) processes with boundary conditions. Both theoretical and simulation investigations show that range-based tests outperform standard unit root tests significantly when used to detect the presence of boundary conditions.
Similar content being viewed by others
References
Andrews DWK (1991), Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59: 817–858
Bertola G (1994), Continuous time models of exchange rates and intervention. In: van der Ploeg (eds.) Handbook of international economics. Amsterdam, North Holland
Billingsley P (1968), Convergence of probability measures. John Wiley and Sons, New York
Borodin AN, Salminen P (1996), Handbook of Brownian motion—facts and formulae. Birkhäuser Verlag, Basel
Cavaliere G (2000), A rescaled range statistics approach to unit root tests. 8th World Congress of the Econometric Society, Seattle, 11–16 August 2000
Cavaliere G (2001 a), Testing the unit root hypothesis using generalized rescaled range statistics. Econometrics Journal 4: 70–88
Cavaliere G (2001 b), Asymptotics for nonstationary time series under range constraints. Department of Statistics and Operations Research, University of Copenhagen, preprint 7/2001
Chan NH, Wei CZ (1987), Asymptotic inference for nearly stationary AR(1) processes. Annals of Statistics 15: 1050–1063
Clarida RH (1999). G3 exchange rate relationships: a recap of the record and a review of proposals for change. NBER Working Paper, 7434
Cox DR, Miller HD (1965), The theory of stochastic processes. Chapman and Hall, London
Davidson J (2002), Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. Journal of Econometrics, 106: 243–269
Davidson J, de Jong R (2000), Consistency of kernel estimators of heteroskedastic and autocorrelated covariance matrices. Econometrica 68: 407–424
de Jong R (2000), A strong consistency proof for heteroskedasticity and autocorrelation consistent covariance matrix estimators. Econometric Theory 16: 262–268
Dickey DA, Fuller WA (1979), Distribution of the estimator for autoregressive time series with a unit root. Journal of the American Statistical Association 74: 427–431
Dixit AK (1993), The art of smooth pasting. Harwood Academic
Elliott G, Rothemberg J, Stock JH (1996), Efficient tests for an autoregressive unit root. Econometrica 64: 813–836
Gallant AR, White H (1988), A unified theory of estimation and inference for nonlinear dynamic models. Basil Blackwell, Oxford
Gardini A, Cavaliere G, Costa M (1999), A new approach to stock prices forecasting. Journal of the Italian Statistical Society 8: 25–47
Hansen BE (1992), Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60: 967–972
Harrison MJ (1985), Brownian motion and stochastic flow systems. John Wiley and Sons, New York
Herrndorf N (1984), A Functional Central Limit Theorem for Weakly Dependent Sequences of Random variables. Annals of Probability 11: 141–153
Hurst H (1951), Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116: 770–799
Karatzas I, Shreve SE (1988), Brownian motion and stochastic calculus. Springer-Verlag, New York
Kwiatkowski D, Phillips PCB, Schmidt P, Shin Y (1992), Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54: 159–178
Lo AW (1991), Long-term memory in stock market prices. Econometrica 59: 1279–1313
MacKinnon JG, Haug AA, Michelis L (1999), Numerical distribution functions of likelihood ratio tests for cointegration. Journal of Applied Econometrics 14: 563–577
Mandelbrot B (1972), Statistic methodology for non-periodic cycles: from the covariance to R/S analysis. Annals of Economics and Social Measurement 1: 259–290
Newey WK, West KD (1987), A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55: 703–708
Pesaran MH, Potter SM (1997), A floor and ceiling model of US output. Journal of Economic Dynamics and Control 21: 661–695
Phillips PCB (1987), Time series regression with a unit root. Econometrica 55: 277–301
Phillips PCB, Perron P (1988), Testing for a unit root in time series regression. Biometrika 75: 335–346
Phillips PCB, Solo V (1992), Asymptotics for linear processes. Annals of Statistics 20: 971–1001
Phillips PCB, Xiao Z (1998), A primer on unit root testing. Journal of Economic Surveys 12: 423–469
Shorack GR, Wellner JA (1987), Empirical processes and their applications to statistics. John Wiley and Sons, New-York
Savin NE (1994), Multiple hypothesis testing. Handbook of Econometrics, vol. 4, ch. 14. North Holland, Amsterdam
Schmidt P, Phillips PCB (1991), LM tests for a unit root in the presence of deterministic trends. Oxford Bullettin of Economics and Statistics 54: 257–287
Author information
Authors and Affiliations
Additional information
A previous draft of the paper (Cavaliere, 2000) was presented at the 8th World Congress of the Econometric Society, Seattle, 11–16 August 2000. I wish sincerely to thank: Martin Jacobsen for his patience in discussing weak convergence to regulated Brownian motions and his valuable suggestions; the Department of Theoretical Statistics of the University of Copenhagen whose hospitality is gratefully acknowledged; Tommaso Proietti for important suggestions; Silvano Bordignon and partecipants at the CIdE seminar, University of Padua, June 2000; two anonymous referees. Partial financial support from 60% M.U.R.S.T. research grants is acknowledged.
Rights and permissions
About this article
Cite this article
Cavaliere, G. Bounded integrated processes and unit root tests. Statistical Methods & Applications 11, 41–69 (2002). https://doi.org/10.1007/BF02511445
Issue Date:
DOI: https://doi.org/10.1007/BF02511445