Skip to main content
×
Home
    • Aa
    • Aa

LIMITED TIME SERIES WITH A UNIT ROOT

  • Giuseppe Cavaliere (a1)
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.

Copyright
Corresponding author
Address correspondence to Giuseppe Cavaliere, Department of Statistical Sciences, University of Bologna, Via Belle Arti 41, 40126 Bologna, Italy; e-mail: cavaliere@stat.unibo.it.
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Abadir, K.M. & A.M.R.Taylor (1999) On the definition of (co-)integration. Journal of Time Series Analysis20, 129137.

Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica59, 817858.

Andrews, D.W.K. & C.J.McDermott (1995) Nonlinear econometric models with deterministically trending variables. Review of Economic Studies62, 343360.

Anthony, M. & R.MacDonald (1998) On the mean-reverting properties of target zone exchange rates: Some evidence from the ERM. European Economic Review42, 14931523.

Asmussen, S., P.Glynn, & J.Pitman (1995) Discretization error in simulation of one-dimensional reflecting Brownian motion. Annals of Applied Probability5, 875896.

Barr, D.G. & K.Cuthbertson (1991) Neo-classical consumer demand theory and the demand for money. Economic Journal101, 855876.

Bec, F. & A.Rahbek (2004) Vector equilibrium correction models with non-linear discontinuous adjustments. Econometrics Journal7, 628651.

Caner, M. & B.E.Hansen (2001) Threshold autoregression with a unit root. Econometrica69, 15551596.

Cavaliere, G. (2001) Testing the unit root hypothesis using generalized rescaled range statistics. Econometrics Journal4, 7088.

Davidson, J. (2002) Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. Journal of Econometrics106, 243269.

De Jong, R.J. & J.Davidson (2000) Consistency of kernel estimators of heteroskedastic and autocorrelated covariance matrices. Econometrica68, 407424.

Elliott, G., T.J.Rothenberg, & J.H.Stock (1996) Efficient tests for an autoregressive unit root. Econometrica64, 813836.

Hansen, B.E. (1992) Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica60, 967972.

Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory18, 14491459.

Nelson, C.R. & C.I.Plosser (1982) Trends and random walks in macroeconomic time series. Journal of Monetary Economics10, 139162.

Ng, S. & P.Perron (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica69, 15191554.

Nicolau, J. (2002) Stationary processes that look like random walks—the bounded random walk process in discrete and continuous time. Econometric Theory18, 99118.

Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica55, 277301.

Phillips, P.C.B. (1987b) Toward a unified asymptotic theory for autoregression. Biometrika74, 535547.

Phillips, P.C.B. (2001) Descriptive econometrics for non-stationary time series with empirical illustrations. Journal of Applied Econometrics16, 389413.

Phillips, P.C.B. & S.Ouliaris (1990) Asymptotic properties of residual based tests for cointegration. Econometrica58, 165193.

Phillips, P.C.B. & V.Solo (1992) Asymptotics for linear processes. Annals of Statistics20, 9711001.

Saikkonen, P. & I.Choi (2004) Cointegrating smooth transition regressions. Econometric Theory20, 301340.

Sargan, J.D. & A.Bhargava (1983) Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica51, 153174.

Svensson, L.E.O. (1993) Assessing target zone credibility: Mean reversion and devaluation expectations in the ERM, 1979–1992. European Economic Review37, 763802.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 1
Total number of PDF views: 12 *
Loading metrics...

Abstract views

Total abstract views: 64 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th May 2017. This data will be updated every 24 hours.