An ARIMA(p,l,0) signal disturbed by MA(q) noise is an ARIMA(p,l, p+q+1) process restricted by nonlinear constraints on parameters. For this model with a unit root the restricted maximum likelihood estimator (RMLE) of the unit root is strongly consistent and it has the same limiting distribution as the ordinary least squares estimator of the unit root in an AR(1) model tabulated by Dickey and Fuller (1979). A modified RMLE is proposed which has the same limiting properties as the RMLE and is computationally much simpler. Simulation results show that our unit root tests based on the modified RMLE perform very well for small samples and compare favorably with the Said and Dickey (1985) tests with respect to both sizes and powers. An illustrative example from sample survey is given.
Bibliographical noteFunding Information:
The research of the first author was supported by Korea Research Foundation. The research of the second author was partly supported by the Dean's Incentive Grant from the College of Arts and Sciences and a Grant from the College of Arts and Sciences at Oklahoma State University. The authors thank Professor Wayne A. Fuller for kindly suggesting the problem and providing the two examples and the US unemployment dataset, and Dr. Abdoulaye Adam for providing the estimates of the autocovariances of the sampling error. The authors also thank a referee and an associate editor whose comments led to an improved version of this article.
- Monte Carlo study
- large sample
- maximum likelihood estimation
- measurement error
- restricted ARIMA model