Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-27T03:08:46.034Z Has data issue: false hasContentIssue false
  • English
  • Français

REGRESSION WITH SLOWLY VARYING REGRESSORS AND NONLINEAR TRENDS

Published online by Cambridge University Press:  25 April 2007

Peter C.B. Phillips
Peter C.B. Phillips
Affiliation:
Cowles Foundation, Yale University; University of Auckland; and University of York
Get access Rights & Permissions [Opens in a new window]

Abstract

Slowly varying (SV) regressors arise commonly in empirical econometric work, particularly in the form of semilogarithmic regression and log periodogram regression. These regressors are asymptotically collinear. Usual regression formulas for asymptotic standard errors are shown to remain valid, but rates of convergence are affected and the limit distribution of the regression coefficients is shown to be one dimensional. Some asymptotic representations of partial sums of SV functions and central limit theorems with SV weights are given that assist in the development of a regression theory. Multivariate regression and polynomial regression with SV functions are considered and shown to be equivalent, up to standardization, to regression on a polynomial in a logarithmic trend. The theory involves second-, third-, and higher-order forms of slow variation. Some applications to the asymptotic theory of nonlinear trend regression are explored.The author thanks two referees and Pentti Saikkonen for comments and suggestions, Sidney Resnick for references on second-order regular variation, and a Kelly Fellowship and the NSF for partial research support under grants SBR 97-30295 and SES 04-142254. An original draft of the paper was written in June 2000 and circulated under the title “Regression with Slowly Varying Regressors.”

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 4 , August 2007 , pp. 557 - 614
Copyright
© 2007 Cambridge University Press

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

Akonom, J. (1993) Comportement asymptotique du temps d'occupation du processus des sommes partielles. Annals of the Institute of Henri Poincaré 29, 57–81.Google Scholar
Barro, R.J. & X. Sala-i-Martin (2004) Economic Growth, 2nd ed. Prentice-Hall.
Bingham, N.H., C.M. Goldie, & J.L. Teugels (1987) Regular Variation. Cambridge University Press.
Csörgö, M. & L. Horváth (1993) Weighted Approximations in Probability and Statistics. Wiley.
de Haan, L. and S. Resnick (1996) Second order regular variation and rates of convergence in extreme value theory. Annals of Probability 24, 97–124.Google Scholar
Erdélyi, A. (1953) Higher Transcendental Functions, vol. 1. Krieger.
Grenander, U. & M. Rosenblatt (1957) Statistical Analysis of Stationary Time Series. Wiley.
Hannan, E.J. (1970) Multiple Time Series. Wiley.
Hurvich, C.M., R. Deo, & J. Brodsky (1998) The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series. Journal of Time Series Analysis 19, 19–46.Google Scholar
Knopp, K. (1990) Theory and Application of Infinite Series. Dover.
Phillips, P.C.B. (1999) Discrete Fourier Transforms of Fractional Processes. Cowles Foundation Discussion Paper no. 1243, Yale University.
Phillips, P.C.B. (2006) Unit root log periodogram regression. Journal of Econometrics (forthcoming).Google Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Phillips, P.C.B. & Y. Sun (2003) Regression with an evaporating logarithmic trend. Econometric Theory 19, 692–701.Google Scholar
Robinson, P.M. (1995) Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 1048–1072.Google Scholar
Shorack, G.R. & J.A. Wellner (1986) Empirical Processes with Applications to Statistics. Wiley.
Wang, Q., Y.-X. Lin, & C.M. Gulati (2003) Strong approximation for long memory processes with applications. Journal of Theoretical Probability 16, 377–389.Google Scholar
Wooldridge, J.M. (1994) Estimation and inference for dependent processes. In R.F. Engle & D.L. McFadden (eds.), Handbook of Econometrics, vol. 4. North-Holland.
Wu, C.-F. (1981) Asymptotic theory of nonlinear least squares estimation. Annals of Statistics 9, 501–513.Google Scholar