Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-09T02:15:11.026Z Has data issue: false hasContentIssue false
  • English
  • Français

THE PROPERTIES OF KULLBACK–LEIBLER DIVERGENCE FOR THE UNIT ROOT HYPOTHESIS

Published online by Cambridge University Press:  01 December 2009

Patrick Marsh
Patrick Marsh*
Affiliation:
University of York
*
*Address correspondence to Patrick Marsh, Department of Economics, University of York, Heslington, York, United Kingdom, YO10 5DD; e-mail: pwnm1@york.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

The fundamental contributions made by Paul Newbold have highlighted how crucial it is to detect when economic time series have unit roots. This paper explores the effects that model specification has on our ability to do that. Asymptotic power, a natural choice to quantify these effects, does not accurately predict finite-sample power. Instead, here the Kullback–Leibler divergence between the unit root null and any alternative is used and its numeric and analytic properties detailed. Numerically it behaves in a similar way to finite-sample power. However, because it is analytically available we are able to prove that it is a minimizable function of the degree of trending in any included deterministic component and of the correlation of the underlying innovations. It is explicitly confirmed, therefore, that it is approximately linear trends and negative unit root moving average innovations that minimize the efficacy of unit root inferential tools. Applied to the Nelson and Plosser macroeconomic series the effect that different types of trends included in the model have on unit root inference is clearly revealed.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 6: Newbold Conference Special Issue , December 2009 , pp. 1662 - 1681
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

Critchley, F., Marriott, P., & Salmon, M. (1994) Preferred point geometry and the local differential geometry of the Kullback-Leibler divergence. Annals of Statistics 22, 1587–1602.CrossRefGoogle Scholar
Dejong, D.N. & Whiteman, C.H. (1991) The case for trend-stationarity is stronger than we thought. Journal of Applied Econometrics 6, 413–421.CrossRefGoogle Scholar
Dufour, J.-M. & King, M.L. (1991) Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors. Journal of Econometrics 47, 115–143.Google Scholar
Durlauf, S.N. & Phillips, P.C.B. (1988) Trends versus random walks in time series analysis. Econometrica 56, 1333–1354.Google Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Gibbs, A.L. & Su, F.E. (2002) Choosing and bounding probability metrics. International Statistical Review 70, 419–435.CrossRefGoogle Scholar
Granger, C.W.J. & Newbold, P. (1974) Spurious regressions in econometrics. Journal of Econometrics 2, 111–120.Google Scholar
Harris, D., Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2009) Testing for a unit root in the presence of a possible break in trend. Econometric Theory 25, 1548–1592 (this issue).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
Kullback, S. (1997) Information Theory and Statistics. Dover.Google Scholar
Leybourne, S.J., Mills, T.C., & Newbold, P. (1998) Spurious rejections by Dickey–Fuller tests in the presence of a break under the null. Journal of Econometrics 87, 191–203.Google Scholar
Magnus, J.R. & Neudecker, H. (1999) Matrix Differential Calculus, with Applications in Statistics and Econometrics, (rev. ed.) Wiley.Google Scholar
Marsh, P. (2007) The available information for invariant tests of a unit root. Econometric Theory 23, 686–710.Google Scholar
Nelson, C.R. & Plosser, C.I. (1982) Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics 10, 139–162.CrossRefGoogle Scholar
Perron, P. (1989) The Great Crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361–1401 (Erratum, 61, 248–249).CrossRefGoogle Scholar
Perron, P. & Ng, S. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.Google Scholar
Phillips, P.C.B. (1986) Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311–340.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6, 333–364.CrossRefGoogle Scholar
Phillips, P.C.B. (1998) New tools for understanding spurious regressions. Econometrica 66, 1299–1325.CrossRefGoogle Scholar
Phillips, P.C.B. (2007) Regression with slowly varying regressors and nonlinear trends. Econometric Theory 23, 557–614.CrossRefGoogle Scholar
Phillips, P.C.B. & Ploberger, W. (2003) Empirical limits for time series econometric models. Econometrica 71, 627–673.Google Scholar
Phillips, P.C.B. & Xiao, Z. (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423–470.Google Scholar
Würtz, A. (1997) A universal upper bound on power functions. University of New South Wales, Discussion paper 97/17.Google Scholar
Zivot, E. & Andrews, D.W.K. (1992) Further evidence on the Great Crash, the Oil Price Shock, and the unit root hypothesis. Journal of Business & Economic Statistics 10, 251–270.CrossRefGoogle Scholar