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LIMITED TIME SERIES WITH A UNIT ROOT

Published online by Cambridge University Press:  22 August 2005

Giuseppe Cavaliere
Giuseppe Cavaliere
Affiliation:
University of Bologna
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Abstract

This paper develops an asymptotic theory for integrated and near-integrated time series whose range is constrained in some ways. Such a framework arises when integration and cointegration analyses are applied to time series that are bounded either by construction or because they are subject to control. The asymptotic properties of some commonly used integration tests are discussed; the bounded unit root distribution is introduced to describe the limiting distribution of the sample first-order autoregressive coefficient of a random walk under range constraints. The theoretical results show that the presence of such constraints can lead to drastically different asymptotics. Because deviations from the standard unit root theory are measured through two noncentrality parameters that can be consistently estimated, simple measures of the impact of range constraints on the asymptotic distributions are obtained. Generalizations of standard unit root tests that are robust to the presence of range constraints are also provided. Finally, it is shown that the proposed asymptotic framework provides an adequate approximation to the finite-sample properties of the unit root statistics under range constraints.Partial financial support from Italian MIUR grants is gratefully acknowledged. I thank, without implicating, Pentti Saikkonen (the co-editor), an anonymous referee, Attilio Gardini, Martin Jacobsen, Robert de Jong, Paolo Paruolo, Anders Rahbek, and participants at the 58th European Meeting of the Econometric Society, Stockholm, August 21–24, 2003, for helpful comments. I also thank the Bank of International Settlements for providing the European monetary system exchange rate data.

Type
Research Article
Information
Econometric Theory , Volume 21 , Issue 5 , October 2005 , pp. 907 - 945
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Abadir, K.M. & A.M.R. Taylor (1999) On the definition of (co-)integration. Journal of Time Series Analysis 20, 129–137.Google Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.Google Scholar
Andrews, D.W.K. & C.J. McDermott (1995) Nonlinear econometric models with deterministically trending variables. Review of Economic Studies 62, 343–360.Google Scholar
Anthony, M. & R. MacDonald (1998) On the mean-reverting properties of target zone exchange rates: Some evidence from the ERM. European Economic Review 42, 1493–1523.Google Scholar
Asmussen, S., P. Glynn, & J. Pitman (1995) Discretization error in simulation of one-dimensional reflecting Brownian motion. Annals of Applied Probability 5, 875–896.Google Scholar
Barr, D.G. & K. Cuthbertson (1991) Neo-classical consumer demand theory and the demand for money. Economic Journal 101, 855–876.Google Scholar
Bec, F. & A. Rahbek (2004) Vector equilibrium correction models with non-linear discontinuous adjustments. Econometrics Journal 7, 628–651.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Blanchard, O.J. & L.H. Summers (1986) Hysteresis and European unemployment. In S. Fischer (ed.), NBER Macroeconomics Annual, pp. 15–77. MIT Press.
Caner, M. & B.E. Hansen (2001) Threshold autoregression with a unit root. Econometrica 69, 1555–1596.Google Scholar
Cavaliere, G. (2001) Testing the unit root hypothesis using generalized rescaled range statistics. Econometrics Journal 4, 70–88.Google Scholar
Cavaliere, G. (2003) The asymptotic distribution of the Dickey-Fuller statistic under a non-negativity constraint. Econometric Theory, Problem, 19, 691–692; Solution, 20, 808–810.Google Scholar
Cox, D.R. & H.D. Miller (1965) The Theory of Stochastic Processes. Chapman and Hall.
Davidson, J. (2002) Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. Journal of Econometrics 106, 243–269.Google Scholar
De Jong, R.J. & J. Davidson (2000) Consistency of kernel estimators of heteroskedastic and autocorrelated covariance matrices. Econometrica 68, 407–424.Google Scholar
Delgado, F. & B. Dumas (1992) Target zones, broad and narrow. In P. Krugman & M. Miller (eds.), Exchange Rate Targets and Currency Bands, pp. 35–56. Cambridge University Press.
Dixit, A. (1993) The Art of Smooth Pasting. Harwood Academic Publishers.
Elliott, G., T.J. Rothenberg, & J.H. Stock (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Hansen, B.E. (1992) Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60, 967–972.Google Scholar
Harrison, M.J. (1985) Brownian Motion and Stochastic Flow Systems. Wiley.
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459.Google Scholar
Karatzas, I. & S.E. Shreve (1988) Brownian Motion and Stochastic Calculus. Springer-Verlag.
Lindbeck, A. & D.J. Snower (1989) Macroeconomic policy and insider power. American Economic Review 79, 370–376.Google Scholar
Nelson, C.R. & C.I. Plosser (1982) Trends and random walks in macroeconomic time series. Journal of Monetary Economics 10, 139–162.Google Scholar
Ng, S. & P. Perron (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.Google Scholar
Nicolau, J. (2002) Stationary processes that look like random walks—the bounded random walk process in discrete and continuous time. Econometric Theory 18, 99–118.Google Scholar
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Phillips, P.C.B. (1987b) Toward a unified asymptotic theory for autoregression. Biometrika 74, 535–547.Google Scholar
Phillips, P.C.B. (2001) Descriptive econometrics for non-stationary time series with empirical illustrations. Journal of Applied Econometrics 16, 389–413.Google Scholar
Phillips, P.C.B. & S. Ouliaris (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165–193.Google Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Pollard, D. (1990) Empirical Processes: Theory and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics, vol. 2. Hayward: Institute of Mathematical Statistics; Alexandria: American Statistical Association.
Saikkonen, P. & I. Choi (2004) Cointegrating smooth transition regressions. Econometric Theory 20, 301–340.Google Scholar
Sargan, J.D. & A. Bhargava (1983) Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica 51, 153–174.Google Scholar
Svensson, L.E.O. (1993) Assessing target zone credibility: Mean reversion and devaluation expectations in the ERM, 1979–1992. European Economic Review 37, 763–802.Google Scholar