124
5 Time Series Modeling and Forecasting
given period
d
. For identifiability, the periodic component is assumed to be
centered, i.e.,
∑
d
t
=1
s
(
t
) = 0. If
d
is odd, let
q
= (
d

1)
/
2 and define
y
t
by the
moving average in (5.19), noting that the window size in this case is 2
q
+1 =
d
.
If
d
is even, let
q
=
d/
2 and define
y
t
=
1
2
q
y
t

q
2
+
y
t

q
+1
+
· · ·
+
y
t
+
q

1
+
y
t
+
q
2
,
q
+ 1
≤
t
≤
n

q.
(5.21)
For
k
= 1
, . . . , d
, let
¯
Δ
k
be the mean of the sample
{
y
k
+
jd

y
k
+
jd
:
q
+1

k
≤
jd
≤
n

q

k
}
. The
methodofmoments estimate
, which replaces population
moments by their sample counterparts, of the periodic function
s
(
t
) is
s
(
k
) =
¯
Δ
k

d

1
d
i
=1
¯
Δ
i
for 1
≤
k
≤
d,
s
(
t
) =
s
(
k
) for
t
=
k
+
jd,
(5.22)
in which
j
is some integer. From the
deseasonalized
series
y
t

s
(
t
),
m
(
t
) can be
estimated by parametric regression or moving average methods. More refined
nonparametric regression techniques than simple averaging, as in (5.20) and
(5.22), will be described in Chapter 7. The function
stl
in
R
(or
S
) uses these
more refined techniques to estimate the decomposition
y
t
=
m
(
t
) +
s
(
t
) +
w
t
of a time series into a trend
m
(
t
), a seasonal component
s
(
t
) and a stationary
disturbance
w
t
; see Venables and Ripley (2002, pp. 403–404) for details and
illustrations.
5.2.2 An empirical example
Figure 5.4 plots the monthly unemployment rates in Dallas County, Arizona,
from January 1980 to June 2005. The data are obtained from the Website
www.Economagic.com
. The ACF and PACF of the unemployment rates are
plotted in Figure 5.5. Note that all rates vary from 4% to 12% except those
in the first five months, which are above 15%. We split the time series into
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 Fall '09
 Regression Analysis, unemployment rates

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