Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-28T11:44:22.471Z Has data issue: false hasContentIssue false
  • English
  • Français

The Calculation of the Limiting Distribution of the Least-Squares Estimator in a Near-Integrated Model

Published online by Cambridge University Press:  18 October 2010

Pierre Perron
Pierre Perron
Affiliation:
Princeton University and Centre de Recherche et Développement en Économique
Get access Rights & Permissions [Opens in a new window]

Abstract

We tabulate the limiting cumulative distribution and probability density functions of the least-squares estimator in a first-order autoregressive regression when the true model is near-integrated in the sense of Phillips. The results are obtained using an exact numerical method which integrates the appropriate limiting moment generating function. The adequacy of the approximation is examined for various first-order autoregressive processes with a root close to unity.

Type
Articles
Information
Econometric Theory , Volume 5 , Issue 2 , August 1989 , pp. 241 - 255
Copyright
Copyright © Cambridge University Press 1989

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

1.C., Cavanagh, Roots local to unity. Mimeo, Harvard University, 1986.Google Scholar
2.Chan, N.H.The parameter inference for nearly nonstationary time series. Journal of the American Statistical Association 83 (1988): 857–862.CrossRefGoogle Scholar
3.Chan, N.H. & Wei, C.Z.. Asymptotic inference for nearly nonstationary AR(1) processes. The Annals of Statistics 15 (1987): 1050–1063.CrossRefGoogle Scholar
4.Dickey, D.A. Estimation and hypothesis testing for nonstationary time series. Ph.D. Thesis, Iowa State University, Ames, IA, 1976.Google Scholar
5.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 (1979): 427–431.Google Scholar
6.Dickey, D.A. & Fuller, W.A.. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 (1981): 1057–1072.CrossRefGoogle Scholar
7.Evans, G.B.A. & Savin, N.E.. The calculation of the limiting distribution of the least-squares estimator of the parameter in a random walk model. Annals of Mathematical Statistics 5 (1981): 114–118.Google Scholar
8.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
9.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 2. Econometrica 52 (1984): 1241–1269.CrossRefGoogle Scholar
10.Fuller, W.A.Introduction to statistical time series. New York: Wiley, 1976.Google Scholar
11.J., Gurland, Inversion formulae for the distribution of ratios. Annals of Mathematical Statistics 19 (1948); 228–237.Google Scholar
12.Imhof, P.J.Computing the distribution of quadratic forms in normal variables. Biometrika 48 (1961): 419–426.CrossRefGoogle Scholar
13.Koerts, J. & Abrahamse, A.P.J.. On the theory and application of the general linear model. Rotterdam: Rotterdam University Press, 1969.Google Scholar
14.Mann, H.B. & Wald, A.. On the statistical treatment of linear stochastic difference equations. Econometrica 11 (1943): 173–220.CrossRefGoogle Scholar
15.Nabeya, S. & Tanaka, K.. Asymptotic distribution of the estimator for the nonstable first-order autoregressive model. Mimeo, Hitotsubashi University, 1987.Google Scholar
16.Perron, P., Tests of joint hypotheses in time series regression with a unit root. Forthcoming in Fomby, T.B. & Rhodes, G.F. (eds.), Advances in Econometrics: Cointegration, Spurious Regression, and Unit Roots. Greenwich, CT: JAI Press, 1989.Google Scholar
17.Perron, P., Test consistency with varying sampling frequency. Cahier de recherche no 4187, C.R.D.E., Université de Montréal, 1987.Google Scholar
18.Phillips, P.C.B.Approximations to some finite-sample distributions associated with a first-order stochastic difference equation. Econometrica 45 (1977): 463–485.CrossRefGoogle Scholar
19.Phillips, P.C.B.Regression theory for near-integrated time series. Econometrica, 56 (1988): 1021–1044.CrossRefGoogle Scholar
20.Phillips, P.C.B.Towards a unified asymptotic theory for autoregression. Biometrika 74(3) (1987): 535–547.CrossRefGoogle Scholar
21.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
22.Phillips, P.C.B.Multiple regression with integrated time series. Cowles Foundation Discussion Paper No. 852, Yale University, 1987.Google Scholar
23.Phillips, P.C.B. & Perron, P.. Testing for a unit root in time series regression. Biometrika 75 (1988): 335–346.CrossRefGoogle Scholar
24.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
25.Rubin, H., Consistency of maximum likelihood estimates in the explosive case. In Koopmans, T.C. (ed.), Statistical Inference in Dynamic Economic Models, 356–364. New York: Wiley, 1950.Google Scholar
26.Said, S.E. & Dickey, D.A.. Testing for unit roots in autoregressive moving average of unknown order. Biometrika 71 (1984): 599–607.CrossRefGoogle Scholar
27.Satchell, S., Approximation to the finite-sample distribution for nonstable first-order stochastic difference equations. Econometrica 52 (1984): 1271–1289.CrossRefGoogle Scholar
28.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