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5.2 Maximum Likelihood Estimation 161
searches systematically for the values of φ and θ that minimize the reduced likelihood
(5.2.12) and computes the corresponding maximum likelihood estimate of σ 2 from
(5.2.10).
Least Squares Estimation for Mixed Models
The least squares estimates ˜φ and ˜θ of φ and θ are obtained by minimizing the function
S as defined in (5.2.11) rather than ℓ as defined in (5.2.12), subject to the constraints
that the model be causal and invertible. The least squares estimate of σ 2 is
˜σ 2
S ˜φ, ˜θ
n − p − q .
Order Selection
In Section 5.1 we introduced minimization of the AICC value as a major criterion for
the selection of the orders p and q. This criterion is applied as follows:
AICC Criterion:
Choose p, q, φp, and θq to minimize
AICC −2lnL(φ p, θq,S(φ p, θq)/n) + 2(p + q + 1)n/(n − p − q − 2).
For any fixed p and q it is clear that the AICC is minimized when φ p and θq are
the vectors that minimize −2lnL(φ p, θq,S(φ p, θq)/n), i.e., the maximum likelihood
estimators. Final decisions with respect to order selection should therefore be made on
the basis of maximum likelihood estimators (rather than the preliminary estimators of
Section 5.1, which serve primarily as a guide). The AICC statistic and its justification
are discussed in detail in Section 5.5.
One of the options in the program ITSM is Model>Estimation>Autofit. Selection
of this option allows you to specify a range of values for both p and q, after
which the program will automatically fit maximum likelihood ARMA(p, q) values
for all p and q in the specified range, and select from these the model with smallest
AICC value. This may be slow if a large range is selected (the maximum range is from
0 through 27 for both p and q), and once the model has been determined, it should
be checked by preliminary estimation followed by maximum likelihood estimation
to minimize the risk of the fitted model corresponding to a local rather than a global
maximum of the likelihood. (For more details see Appendix D.3.1.)
Confidence Regions for the Coefficients
For large sample size the maximum likelihood estimator ˆβ of β : (φ1,..., φp,
θ1, ...,θq) ′ is approximately normally distributed with mean β and covariance matrix
n −1 V(β) which can be approximated by 2H −1 (β), where H is the Hessian