Hostname: page-component-89b8bd64d-j4x9h Total loading time: 0 Render date: 2026-05-08T12:41:25.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Expansions for Random Walks with Normal Errors

Published online by Cambridge University Press:  11 February 2009

J.L. Knight  and
S.E. Satchell
J.L. Knight
Affiliation:
University of Western Ontario
S.E. Satchell
Affiliation:
Trinity College, Cambridge University, and Birkbeck College, University of London
Get access Rights & Permissions [Opens in a new window]

Abstract

The asymptotic distribution of the least-squares estimators in the random walk model was first found by White [17] and is described in terms of functional of Brownian motion with no closed form expression known. Evans and Savin [5,6] and others have examined numerically both the asymptotic and finite sample distribution. The purpose of this paper is to derive an asymptotic expansion for the distribution. Our approach is in contrast to Phillips [12,13] who has already derived some terms in a general expansion by analyzing the functionals. We proceed by assuming that the errors are normally distributed and expand the characteristic function directly. Then, via numerical integration, we invert the characteristic function to find the distribution. The approximation is shown to be extremely accurate for all sample sizes ≥25, and can be used to construct simple tests for the presence of a unit root in a univariate time series model. This could have useful applications in applied economics.

Information

Type
Articles
Information
Econometric Theory , Volume 9 , Issue 3 , June 1993 , pp. 363 - 376
Copyright
Copyright © Cambridge University Press 1993

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.)

Article purchase

Temporarily unavailable

References

1.Abramowitz, M. & Stegum, I.. Handbook of Mathematical Functions, 9th ed.New York: Dover, 1972.Google Scholar
2.Dickey, D.A. Estimation and hypothesis testing for nonstationary time series, Ph.D. Thesis, Iowa State University, Ames, Iowa, 1976.Google Scholar
3.Dickey, D.A. & Fuller, W.A.. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74 (1976): 427–431.Google Scholar
4.Dickey, D.A. & Fuller, W.A.. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 (1981): 1057–1072.CrossRefGoogle Scholar
5.Evans, G.B.A. & Savin, N.E., Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
6.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 2. Econometrica 52 (1984): 1241–1269.CrossRefGoogle Scholar
7.Hisamatsu, H. & Maekawa, K.. The distribution of the D-W statistic in integrated and near integrated models. Journal of Econometrics (forthcoming).Google Scholar
8.Perron, P.The calculation of the limiting distribution of the least squares estimator in a near-integrated model. Econometric Theory 5 (1989): 241–255.CrossRefGoogle Scholar
9.Perron, P.The adequacy of limiting distributions in the AR(1) model with dependent errors. Econometrics Research Program Memorandum #349, Princeton University, 1990.Google Scholar
10.Perron, P.A continuous time approximation to the unstable first-order autoregressive process: The case without an intercept. Econometrica 59 (991): 211–236.CrossRefGoogle Scholar
11.Phillips, P.C.B.Approximations to some finite sample distributions associated with first order stochastic difference equation. Econometrica 45 (1977): 463–485.CrossRefGoogle Scholar
12.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
13.Phillips, P.C.B.Asymptotic expansion in nonstationary vector autoregressions. Econometric Theory 3 (1987): 45–68.CrossRefGoogle Scholar
14.Phillips, P.C.B.Regression theory for near-integrated time series. Econometrica 56 (1988): 1021–1044.CrossRefGoogle Scholar
15.Rao, M.M.Asymptotic distribution of an estimator of the boundary parameter of an unstable process. Annals of Statistics 6 (1978): 185–190.CrossRefGoogle Scholar
16.Satchell, S.E.Approximation to the fintie sample distribution for nonstable first order stochastic difference equations. Econometrica 52 (1984): 1271–1289.CrossRefGoogle Scholar
17.White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29 (1958): 1188–1197.CrossRefGoogle Scholar