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162 Chapter 5 Modeling and Forecasting with ARMA Processes
matrix ∂2 p+q ℓ(β)/∂βi∂βj . ITSM prints out the approximate standard deviations
i,j1
and correlations of the coefficient estimators based on the Hessian matrix evaluated
numerically at ˆβ unless this matrix is not positive definite, in which case ITSM instead
computes the theoretical asymptotic covariance matrix in Section 8.8 of TSTM. The
resulting covariances can be used to compute confidence bounds for the parameters.
Example 5.2.1 An AR(p) model
Large-Sample Distribution of Maximum Likelihood Estimators:
For a large sample from an ARMA(p, q) process,
ˆβ ≈ N β,n −1 V(β) .
The general form of V(β) can be found in TSTM, Section 8.8. The following are
several special cases.
The asymptotic covariance matrix in this case is the same as that for the Yule–Walker
estimates given by
V(φ) σ 2 Ɣ −1
p .
In the special cases p 1 and p 2, we have
AR(1) :V(φ) 1 − φ 2
1 ,
AR(2) :V(φ)
1 − φ 2
2
−φ1(1 + φ2)
−φ1(1 + φ2)
1 − φ 2
2
.
Example 5.2.2 An MA(q) model
Let Ɣ ∗ q be the covariance matrix of Y1,...,Yq, where {Yt} is the autoregressive process
with autoregressive polynomial θ(z), i.e.,
Yt + θ1Yt−1 +···+θqYt−q Zt, {Zt} ∼WN(0, 1).
Then it can be shown that
V(θ) Ɣ ∗−1
q .
Inspection of the results of Example 5.2.1 and replacement of φi by −θi) yields
MA(1) :V(θ) 1 − θ 2
1 ,
MA(2) :V(θ)
1 − θ 2
2
θ1(1 − θ2)
θ1(1 − θ2)
1 − θ 2
2
.