Maximum Likelihood Estimation of Regression Models with First Order Moving Average Errors when the Root Lies on the Unit Circle

Econometrica ◽  
1983 ◽  
Vol 51 (3) ◽  
pp. 799 ◽  
Author(s):  
J. D. Sargan ◽  
Alok Bhargava
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Unit Circle ◽  
Regression Models ◽  
Moving Average ◽  
Likelihood Estimation ◽  
First Order ◽  
Moving Average Errors

Maximum Likelihood Estimation for MA(1) Processes with a Root on or near the Unit Circle

1996 ◽  
Vol 12 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Richard A. Davis ◽  
William T.M. Dunsmuir
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Unit Circle ◽  
Asymptotic Distribution ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Small Sample ◽  
Practical Significance ◽  
True Parameter ◽  
Small Sample Sizes

This paper considers maximum likelihood estimation for the moving average parameter θ in an MA(1) model when θ is equal to or close to 1. A derivation of the limit distribution of the estimate θLM, defined as the largest of the local maximizers of the likelihood, is given here for the first time. The theory presented covers, in a unified way, cases where the true parameter is strictly inside the unit circle as well as the noninvertible case where it is on the unit circle. The asymptotic distribution of the maximum likelihood estimator subMLE is also described and shown to differ, but only slightly, from that of θLM. Of practical significance is the fact that the asymptotic distribution for either estimate is surprisingly accurate even for small sample sizes and for values of the moving average parameter considerably far from the unit circle.

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