Efficient Tests for an Autoregressive Unit Root

46 Pages Posted: 27 Jun 2007 Last revised: 12 May 2023

See all articles by Graham Elliott

Graham Elliott

University of California, San Diego (UCSD) - Department of Economics

Thomas J. Rothenberg

affiliation not provided to SSRN

James H. Stock

Harvard University - Department of Economics; National Bureau of Economic Research (NBER); Harvard University - Harvard Kennedy School (HKS)

Date Written: December 1992

Abstract

This paper derives the asymptotic power envelope for tests of a unit autoregressive root for various trend specifications and stationary Gaussian autoregressive disturbances. A family of tests is proposed, members of which are asymptotically similar under a general 1(1) null (allowing nonnormality and general dependence) and which achieve the Gaussian power envelope. One of these tests, which is asymptotically point optimal at a power of 50%, is found (numerically) to be approximately uniformly most powerful (UMP) in the case of a constant deterministic term, and approximately uniformly most powerful invariant (UMPI) in the case of a linear trend, although strictly no UMP or UMPI test exists. We also examine a modification, suggested by the expression for the power envelope, of the Dickey-Fuller (1979) t-statistic; this test is also found to be approximately UMP (constant deterministic term case) and UMPI (time trend case). The power improvement of both new tests is large: in the demeaned case, the Pitman efficiency of the proposed tests relative to the standard Dickey-Fuller t-test is 1.9 at a power of 50%. A Monte Carlo experiment indicates that both proposed tests, particularly the modified Dickey-Fuller t-test, exhibit good power and small size distortions in finite samples with dependent errors.

Suggested Citation

Elliott, Graham and Rothenberg, Thomas J. and Stock, James H., Efficient Tests for an Autoregressive Unit Root (December 1992). NBER Working Paper No. t0130, Available at SSRN: https://ssrn.com/abstract=995460

Graham Elliott (Contact Author)

University of California, San Diego (UCSD) - Department of Economics ( email )

9500 Gilman Drive
La Jolla, CA 92093-0508
United States
858-534-4481 (Phone)
858-534-7040 (Fax)

Thomas J. Rothenberg

affiliation not provided to SSRN

No Address Available

James H. Stock

Harvard University - Department of Economics ( email )

Littauer Center
Cambridge, MA 02138
United States
617-496-0502 (Phone)
617-496-5960 (Fax)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Harvard University - Harvard Kennedy School (HKS) ( email )

79 John F. Kennedy Street
Cambridge, MA 02138
United States