2006). Fitting MA(q) Models in the Closed Invertible Region, Statistics and Probablity Letters, 76, 1331-1334.
AbstractThe use of reparameterization in the maximization of the likelihood function of the MA (q) model is discussed. A general method for testing for the presence of a parameter estimate on the boundary of an MA(q) model is presented. This test is illustrated with a brief simulation experiment for the MA(q) for q = 1, 2, 3, 4 in which it is shown that the probability of an estimate being on the boundary increases with q.
A symbolic method which can be used to obtain the asymptotic bias and variance coefficients to order O(1/n) for estimators in stationary time series is discussed. Using this method, the large-sample bias of the Burg estimator in the AR(p) for p = 1, 2, 3 is shown to be equal to that of the least squares estimators in both the known and unknown mean cases. Previous researchers have only been able to obtain simulation results for the Burg estimator's bias because this problem is too intractable without using computer algebra. The asymptotic bias coefficient to O(1/n) of Yule-Walker as well as least squares estimates is also derived in AR(3) models. Our asymptotic results show that for the AR(3), just as in the AR(2), the Yule-Walker estimates have a large bias when the parameters are near the nonstationary boundary. The least squares and Burg estimates are much better in this situation. Simulation results confirm our findings. Copyright 2005 Blackwell Publishing Ltd.
This algorithm is easily implemented in high level QuantitativeProgramming Environments (QPEs) such as Mathematica, MatLab and R.In order to obtain reasonable speed, previous ARMA maximum likelihood algorithms are usually implemented in C or some other machine efficient language. With our algorithm it is easy to do maximum likelihood estimation for long time series directly in the QPE of your choice. The new algorithm is extended to obtain the MLE for the mean parameter.Simulation experiments which illustrate the effectiveness of the new algorithm are discussed. Mathematica and R packages which implement the algorithm discussed in this paper are available (McLeod and Zhang, 2007).Based on these package implementations, it is expected that the interested researcher would be able to implement this algorithm in other QPE's.
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