Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T17:55:59.330Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Distributions of the Least-Squares Estimators and Test Statistics in the Near Unit Root Model with Non-Zero Initial Value and Local Drift and Trend

Published online by Cambridge University Press:  11 February 2009

Seiji Nabeya  and
Bent E. Sørensen
Seiji Nabeya
Affiliation:
Tokyo International University
Bent E. Sørensen
Affiliation:
Brown University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers the distribution of the Dickey-Fuller test in a model with non-zero initial value and drift and trend. We show how stochastic integral representations for the limiting distribution can be derived either from the local to unity approach with local drift and trend or from the continuous record asymptotic results of Sørensen [29]. We also show how the stochastic integral representations can be utilized as the basis for finding the corresponding characteristic functions via the Fredholm approach of Nabeya and Tanaka [16,17], This “link” between those two approaches may be of general interest. We further tabulate the asymptotic distribution by inverting the characteristic function. Using the same methods, we also find the characteristic function for the asymptotic distribution for the Schmidt-Phillips [26] unit root test. Our results show very clearly the dependence of the various tests on the initial value of the time series.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 5 , December 1994 , pp. 937 - 966
Copyright
Copyright © Cambridge University Press 1994

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

1Bhargava, A.On the theory of testing for unit roots in observed time series. Review of Economic Studies 53 (1986): 369–384.CrossRefGoogle Scholar
2Cavanagh, C. Roots Local to Unity. Discussion paper #1259. Harvard Institute of Economic Research, Cambridge, 1986.Google Scholar
3Cavanagh, C. The Fragility of Unit Root Tests. Mimeo, Harvard University, Cambridge, 1989.Google Scholar
4Chan, N.H.The parameter inference for nearly nonstationary time series. Journal of the American Statistical Association 83 (1988): 857–862.CrossRefGoogle Scholar
5Chan, N.H. & Wei, C.Z.. Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15 (1987): 1050–1063.CrossRefGoogle Scholar
6Courant, R. & Hilbert, D.. Methods of Mathematical Physics 1. New York: Interscience Publishers, 1953.Google Scholar
7Dickey, 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
8Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
9Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 2. Econometriea 52 (1984): 1241–1269.CrossRefGoogle Scholar
10Fuller, W.A.Introduction to Statistical Time Series. New York: John Wiley & Sons, 1976.Google Scholar
11Haldrup, N. & Hylleberg, S.. Integration, Near-Integration and Deterministic Trends. Working paper, University of Aarhus, Aarhus, 1991.Google Scholar
12Hochstadt, H.Integral Equations. New York: John Wiley & Sons, 1973.Google Scholar
13Ito, K.Multiple Wiener integral. Journal of the Mathematical Society of Japan 3 (1951): 157–169.CrossRefGoogle Scholar
14Nabeya, S. Application of the Theory of Integral Equations to the Derivation of Asymptotic Distributions in Time Series Analysis. Dissertation, Hitotsubashi University, Tokyo, 1987.Google Scholar
15Nabeya, S. & Perron, P.. Local asymptotic distributions related to the AR(1) model with dependent errors. Journal of Econometrics 62 (1994): 229–264.CrossRefGoogle Scholar
16Nabeya, S. & Tanaka, K.. Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative. Annals of Statistics 16 (1988): 218–236.CrossRefGoogle Scholar
17Nabeya, S. & Tanaka, K.. A general approach to the limiting distribution for estimators in time series regression with nonstable autoregressive errors. Econometrica 58 (1990): 145–163.CrossRefGoogle Scholar
18Nabeya, S. & Tanaka, K.. Limiting power of unit-root tests in time-series regression. Journal of Econometrics 46 (1990): 247–271.CrossRefGoogle Scholar
19Perron, P.The calculation of the limiting distribution of the least-squares estimator in a near-integrated model. Econometric Theory 5 (1989): 241–256.CrossRefGoogle Scholar
20Perron, P.A continuous time approximation to the stationary first-order autoregressive model. Econometric Theory 7 (1991): 236–253.CrossRefGoogle Scholar
21Perron, P.A continuous time approximation to the unstable first-order autoregressive process: The case without an intercept. Econometrica 59 (1991): 211–237.CrossRefGoogle Scholar
22Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
23Phillips, P.C.B.Towards a unified asymptotic theory for autoregression. Biometrika 74 (1987): 535–547.CrossRefGoogle Scholar
24Phillips, P.C.B.Regression theory for near integrated time series. Econometrica 56 (1988): 1021–1045.CrossRefGoogle Scholar
25Protter, P.Stochastic Integration and Differential Equations. A New Approach. New York: Springer-Verlag, 1990.CrossRefGoogle Scholar
26Schmidt, P. & Phillips, P.C.B.. LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54 (1992): 257–288.CrossRefGoogle Scholar
27Shepp, L.A.Radon-Nikodym derivatives of Gaussian measures. Annals of Mathematical Statistics 37 (1966): 321–354.CrossRefGoogle Scholar
28Sims, CA.Stock, J.H. & Watson, M.W.. Inference in linear time series models with some unit roots. Econometrica 58 (1990): 113–145.CrossRefGoogle Scholar
29Sørensen, B.E.Continuous record asymptotics in systems of stochastic differential equations. Econometric Theory 8 (1992): 28–51.CrossRefGoogle Scholar
30Tanaka, K.The Fredholm approach to asymptotic inference on nonstationary and noninvertible time series models. Econometric Theory 6 (1990): 411–433.CrossRefGoogle Scholar
31West, K.D.Asymptotic normality when regressors have a unit root. Econometrica 56 (1988): 1397–1419.CrossRefGoogle Scholar
32White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29 (1958): 1188–1197.CrossRefGoogle Scholar