Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-11T03:56:10.887Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIT ROOT AND COINTEGRATING LIMIT THEORY WHEN INITIALIZATION IS IN THE INFINITE PAST

Published online by Cambridge University Press:  01 December 2009

Peter C.B. Phillips  and
Tassos Magdalinos
Peter C.B. Phillips*
Affiliation:
Yale University, University of Auckland, University of York and Singapore Management University
Tassos Magdalinos
Affiliation:
University of Nottingham
*
*Address correspondence to Peter C.B. Phillips, Department of Economics, Yale University, P.O. Box 208268, New Haven, CT 06520-8268; e-mail: peter.phillips@yale.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory, producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient, and a limit normal distribution for the t-ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continue to hold in spite of the degeneracy in the limit theory and are therefore robust to initial conditions that extend to the infinite past.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 6: Newbold Conference Special Issue , December 2009 , pp. 1682 - 1715
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abadir, K.M. & Magnus, J.R. (2005) Matrix Algebra. Econometric Exercises, vol. 1. Cambridge University Press.CrossRefGoogle Scholar
Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676–687.CrossRefGoogle Scholar
Andrews, D.W.K. (1987) Asymptotic results for generalised Wald tests. Econometric Theory 3, 348–358.CrossRefGoogle Scholar
Andrews, D.W.K. & Guggenberger, P. (2008) Asymptotics for stationary very nearly unit root processes. Journal of Time Series Analysis 29, 203–210.CrossRefGoogle Scholar
Elliott, G. (1999) Efficient tests for a unit root when the initial observation is drawn from its unconditional distribution. International Economic Review 40, 767–783.CrossRefGoogle Scholar
Elliott, G. & Müller, U.K. (2006) Minimizing the impact of the initial condition on testing for unit roots. Journal of Econometrics 135, 285–310.CrossRefGoogle Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2009) Unit root testing in practice: Dealing with uncertainty over the trend and initial condition. Econometric Theory 25, 587–636.CrossRefGoogle Scholar
Kim, T.-H., Leybourne, S., & Newbold, P. (2004) Behaviour of Dickey-Fuller unit root tests under trend misspecification. Journal of Time Series Analysis 25, 755–764.CrossRefGoogle Scholar
Leybourne, S.J., Kim, T.-H., & Newbold, P. (2005) Examination of some more powerful modifications of the Dickey-Fuller test. Journal of Time Series Analysis 26, 355–369.CrossRefGoogle Scholar
Magdalinos, T. & Phillips, P.C.B. (2009) Limit theory for cointegrated systems with moderately integrated and moderately explosive regressors. Econometric Theory 25, 482–526.CrossRefGoogle Scholar
Moon, H.R. & Phillips, P.C.B. (2000) Estimation of autoregressive roots near unity using panel data. Econometric Theory 16, 927–998.CrossRefGoogle Scholar
Müller, U. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4, 468–497.CrossRefGoogle Scholar
Perron, P. (1991) A continuous time approximation to the unstable first order autoregressive process: The case without an intercept. Econometrica 59, 211–236.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–302.CrossRefGoogle Scholar
Phillips, P.C.B. (1988a) Multiple regression with integrated processes. In Prabhu, N.U. (ed.), Statistical Inference from Stochastic Processes, Contemporary Mathematics 80, 79–106.CrossRefGoogle Scholar
Phillips, P.C.B. (1988b) Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Econometric Theory 4, 528–533.CrossRefGoogle Scholar
Phillips, P.C.B. (1988c) Weak convergence to the matrix stochastic integral ∫BdB’. Journal of Multivariate Analysis 24(2), 252–264.CrossRefGoogle Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181–240.CrossRefGoogle Scholar
Phillips, P.C.B. (2006) When the Tail Wags the Unit Root Limit Theory. Mimeo, Yale University.Google Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 4, 473–495.CrossRefGoogle Scholar
Phillips, P.C.B. & Hansen, B.E. (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125.CrossRefGoogle Scholar
Phillips, P.C.B. & Lee, C.C. (1996) Efficiency gains from quasi-differencing under nonstationarity. In Robinson, P.M. and Rosenblatt, M. (eds.), Athens Conference on Applied Probability and Time Series: Essays in Memory of E.J. Hannan. Springer–Verlag.Google Scholar
Phillips, P.C.B. & Magdalinos, T. (2007) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115–130.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2008) Limit theory for explosively cointegrated systems. Econometric Theory 24, 865–887.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Uhlig, H. (1995) On Jeffreys’ prior when using the exact likelihood function. Econometric Theory 10, 633–644.CrossRefGoogle Scholar
White, J.S. (1958) The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29, 1188–1197.CrossRefGoogle Scholar