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146 Chapter 5 Modeling and Forecasting with ARMA Processes
ˆρ(1) ˆθ1
1 + ˆθ 2 . (5.1.18)
1
If ˆρ(1) >.5, there is no real solution, so we define ˆθ1 ˆρ(1)/ ˆρ(1) .If ˆρ(1) ≤ .5,
then the solution of (5.1.17)–(5.1.18) (with |ˆθ| ≤1) is
ˆθ1 1 − 1 − 4 ˆρ 2 (1) 1/2
/ 2 ˆρ(1) ,
ˆσ 2
ˆγ(0)/ 1 + ˆθ 2
1 .
For the overshort data of Example 3.2.8, ˆρ(1) −0.5035 and ˆγ(0) 3416, so the
fitted MA(1) model has parameters ˆθ1 −1.0 and ˆσ 2 1708.
Relative Efficiency of Estimators
The performance of two competing estimators is often measured by computing their
asymptotic relative efficiency. In a general statistics estimation problem, suppose ˆθ (1)
n
and ˆθ (2)
n
are two estimates of the parameter θ in the parameter space based on the
observations X1,...,Xn.If ˆθ (i)
n is approximately N θ,σ 2
i (θ) for large n, i 1, 2,
then the asymptotic efficiency of ˆθ (1)
n relative to ˆθ (2)
n is defined to be
e θ, ˆθ (1) , ˆθ (2)
σ 2 2 (θ)
σ 2 1 (θ).
If e θ, ˆθ (1) , ˆθ (2) ≤ 1 for all θ ∈ , then we say that ˆθ (2)
n is a more efficient
estimator of θ than ˆθ (1)
n (strictly more efficient if in addition, e θ, ˆθ (1) , ˆθ (2) < 1 for
some θ ∈ ). For the MA(1) process the moment estimator θ (1)
n
5.1.2 is approximately N θ1,σ2 1 (θ1)/n with
σ 2
1 θ1) (1 + θ 2
4 6 8
2
1 + 4θ1 + θ1 + θ1 / 1 − θ1 2
discussed in Example
(see TSTM, p. 254). On the other hand, the innovations estimator ˆθ (2)
n discussed in
the next section is distributed approximately as N θ1,n−1 . Thus, e θ1, ˆθ (1) , ˆθ (2)
σ −2
1 (θ1) ≤ 1 for all |θ1| < 1, with strict inequality when θ 1. In particular,
e θ1, ˆθ (1) , ˆθ (2)
⎧
⎪⎨
.82, θ1 .25,
.37, θ1 .50,
⎪⎩
.06, θ1 .75,
demonstrating the superiority, at least in terms of asymptotic relative efficiency, of
ˆθ (2)
n over ˆθ (1)
n . On the other hand (Section 5.2), the maximum likelihood estimator ˆθ (3)
n
of θ1 is approximately N(θ1,(1 − θ 2 1 )/n). Hence,
e θ1, ˆθ (2) , ˆθ (3)
⎧
⎪⎨
.94, θ1 .25,
.75, θ1 .50,
⎪⎩
.44, θ1 .75.