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5.2 Maximum Likelihood Estimation 163
Example 5.2.3 An ARMA(1, 1) model
For a causal and invertible ARMA(1,1) process with coefficients φ and θ.
1 + φθ
V(φ,θ)
(φ + θ) 2
(1 − φ 2 )(1 + φθ) −(1 − θ 2 )(1 − φ 2 )
−(1 − θ 2 )(1 − φ 2 ) (1−θ 2
.
)(1 + φθ)
Example 5.2.4 The Dow Jones Utilities Index
Example 5.2.5 The lake data
For the Burg and Yule–Walker AR(1) models derived for the differenced and meancorrected
series in Examples 5.1.1 and 5.1.3, the Model>Estimation>Preliminary
option of ITSM gives −2ln(L) 70.330 for the Burg model and −2ln(L) 70.378
for the Yule–Walker model. Since maximum likelihood estimation attempts to minimize
−2lnL, the Burg estimate appears to be a slightly better initial estimate of φ.
We therefore retain the Burg AR(1) model and then select Model>Estimation>Max
Likelihood and click OK. The Burg coefficient estimates provide initial parameter
values to start the search for the minimizing values. The model found on completion
of the minimization is
Yt − 0.4471Yt−1 Zt, {Zt} ∼WN(0, 0.02117). (5.2.13)
This model is different again from the Burg and Yule–Walker models. It has
−2ln(L) 70.321, corresponding to a slightly higher likelihood. The standard
error (or estimated standard deviation) of the estimator ˆφ is found from the program
to be 0.1050. This is in good agreement with the estimated standard deviation
(1 − (.4471) 2 )/77 .1019, based on the large-sample approximation given in Example
5.2.1. Using the value computed from ITSM, approximate 95% confidence
bounds for φ are 0.4471 ± 1.96 × 0.1050 (0.2413, 0.6529). These are quite close
to the bounds based on the Yule–Walker and Burg estimates found in Examples 5.1.1
and 5.1.3. To find the minimum-AICC model for the series {Yt}, choose the option
Model>Estimation>Autofit. Using the default range for both p and q, and clicking
on Start, we quickly find that the minimum AICC ARMA(p, q) model with p ≤ 5
and q ≤ 5 is the AR(1) model defined by (5.2.13). The corresponding AICC value is
74.483. If we increase the upper limits for p and q, we obtain the same result.
Using the option Model>Estimation>Autofit as in the previous example, we find
that the minimum-AICC ARMA(p, q) model for the mean-corrected lake data, Xt
Yt − 9.0041, of Examples 5.1.6 and 5.1.7 is the ARMA(1,1) model
Xt − 0.7446Xt−1 Zt + 0.3213Zt−1, {Zt} ∼WN(0, 0.4750). (5.2.14)
The estimated standard deviations of the two coefficient estimates ˆφ and ˆθ are found
from ITSM to be 0.0773 and 0.1123, respectively. (The respective estimated standard
deviations based on the large-sample approximation given in Example 5.2.3 are .0788