mation is accomplished by replacing
β
(0)
by
β
(1)
in (
3.126
). This process is
repeated by calculating, at iteration
j
= 2
,
3
, . . .
,
β
(
j
)
=
β
(
j

1)
+
∆
(
β
(
j

1)
)
until convergence.
Example 3.31
Gauss–Newton for an MA(
1
)
Consider an invertible MA(1) process,
x
t
=
w
t
+
θw
t

1
. Write the truncated
errors as
w
t
(
θ
) =
x
t

θw
t

1
(
θ
)
,
t
= 1
, . . . , n,
(3.127)
where we condition on
w
0
(
θ
) = 0. Taking derivatives,

∂w
t
(
θ
)
∂θ
=
w
t

1
(
θ
) +
θ
∂w
t

1
(
θ
)
∂θ
,
t
= 1
, . . . , n,
(3.128)
where
∂w
0
(
θ
)
/∂θ
= 0
.
Using the notation of (
3.123
), we can also write (
3.128
)
as
z
t
(
θ
) =
w
t

1
(
θ
)

θz
t

1
(
θ
)
,
t
= 1
, . . . , n,
(3.129)
where
z
0
(
θ
) = 0
.
Let
θ
(0)
be an initial estimate of
θ
, for example, the estimate given in
Ex
ample 3.28
. Then, the Gauss–Newton procedure for conditional least squares
is given by
θ
(
j
+1)
=
θ
(
j
)
+
∑
n
t
=1
z
t
(
θ
(
j
)
)
w
t
(
θ
(
j
)
)
∑
n
t
=1
z
2
t
(
θ
(
j
)
)
,
j
= 0
,
1
,
2
, . . . ,
(3.130)
where the values in (
3.130
) are calculated recursively using (
3.127
) and
(
3.129
). The calculations are stopped when

θ
(
j
+1)

θ
(
j
)

, or

Q
(
θ
(
j
+1)
)

Q
(
θ
(
j
)
)

, are smaller than some preset amount.
3.6 Estimation
135
0
5
10
15
20
25
30
35
−
0.4
0.0
0.4
0.8
LAG
ACF
0
5
10
15
20
25
30
35
−
0.4
0.0
0.4
0.8
LAG
PACF
Fig. 3.8.
ACF and PACF of transformed glacial varves.
Example 3.32
Fitting the Glacial Varve Series
Consider the series of glacial varve thicknesses from Massachusetts for
n
=
634 years, as analyzed in
Example 2.6
and in
Problem 2.8
, where it was
argued that a firstorder moving average model might fit the logarithmically
transformed and differenced varve series, say,
∇
log(
x
t
) = log(
x
t
)

log(
x
t

1
) = log
x
t
x
t

1
,
which can be interpreted as being approximately the percentage change in
the thickness.
The sample ACF and PACF, shown in
Figure 3.8
, confirm the tendency
of
∇
log(
x
t
) to behave as a firstorder moving average process as the ACF
has only a significant peak at lag one and the PACF decreases exponentially.
Using Table 3.1, this sample behavior fits that of the MA(1) very well.
The results of eleven iterations of the Gauss–Newton procedure, (
3.130
),
starting with
θ
(0)
=

.
10 are given in
Table 3.2
. The final estimate is
b
θ
=
θ
(11)
=

.
773; interim values and the corresponding value of the condi
tional sum of squares,
S
c
(
θ
) given in (
3.121
), are also displayed in the table.
The final estimate of the error variance is
b
σ
2
w
= 148
.
98
/
632 =
.
236 with
632 degrees of freedom (one is lost in differencing). The value of the sum of
the squared derivatives at convergence is
∑
n
t
=1
z
2
t
(
θ
(11)
) = 369
.
73, and con
136
3 ARIMA Models
−
1.0
−
0.8
−
0.6
−
0.4
−
0.2
0.0
150
160
170
180
190
200
210
S
c
Fig. 3.9.
Conditional sum of squares versus values of the moving average parameter
for the glacial varve example,
Example 3.32
. Vertical lines indicate the values of the
parameter obtained via Gauss–Newton; see
Table 3.2
for the actual values.
Table 3.2.
Gauss–Newton Results for
Example 3.32
j
θ
(
j
)
S
c
(
θ
(
j
)
)
∑
n
t
=1
z
2
t
(
θ
(
j
)
)
0

0
.
100
195.0010
183.3464
1

0
.
250
177.7614
163.3038
2

0
.
400
165.0027
161.6279
3

0
.
550
155.6723
182.6432
4

0
.
684
150.2896
247.4942
5

0
.
736
149.2283
304.3125
6

0
.
757
149.0272
337.9200
7

0
.
766
148.9885
355.0465
8

0
.
770
148.9812
363.2813
9

0
.
771
148.9804
365.4045
10

0
.
772
148.9799
367.5544
11

0
.
773
148.9799
369.7314
sequently, the estimated standard error of
b
θ
is
p
.
236
/
369
.
73 =
.
025;
7
this
leads to a
t
value of

.
773
/.
025 =

30
.
92 with 632 degrees of freedom.
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