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Based on the multiplicative or additive model, the SEASON procedure decomposes the existing series into three components: trend-cycle, seasonal, and irregular.
Model
Multiplicative Model
X (^) t = TC S It t t, t = 1 , K,n
Additive Model
X (^) t = TC (^) t + S (^) t + I (^) t, t = 1 , K,n
where TCt is the “trend-cycle” component, S (^) t is the “seasonal” component, and I (^) t is the “irregular” or “random” component. The procedure for estimating the seasonal component is: (1) Smooth the series by the moving average method; the moving average series reflects the trend-cycle component. (2) Obtain the seasonal-irregular component by dividing the original series by the smoothed values if the model is multiplicative, or by subtracting the smoothed values from the original series if the model is additive. (3) Isolate the seasonal component from the seasonal-irregular component by computing the medial average (average) of the specific seasonal relatives for each unit of periods if the model is multiplicative (additive).
Moving Average Series
Based on the specified method and period p, the moving average series Z (^) t for X (^) t is defined as follows: p is even, weight all points equally
Z
X p t t^ p^ n^ p
j j t p
t p
= =^ +^ −^ +
%
&
K K KK
'
K K K K
= −
2 1
, K, 1
SYSMIS otherwise
p is even, weights unequal
Z
X X p X p t p^ n p t
t p^ t p^ j j t p
t p
+
%
&
K K K
'
K K K
− + = − +
2 1 2 , 2 1 , K, 2
SYSMIS (^) otherwise
p is odd
Z
X p t t^ p^ n^ p
j j t p
t p
(^) = !
" $#
" $#
%
&
K K KK
'
K K KK
= − ! "$#
+ ! "$# ∑ 2
2
, K,
SYSMIS otherwise
SAF F p F
t t t^ p t t
= (^) p =
=
∑
1
, , K,
Additive Model
Ft is defined as the arithmetic average of the series shown above. Then
SAFt = Ft − F,
where
F Ft p t
p
=
∑ 1
Seasonally Adjusted Series (SAS)
SAS X^ SAF
t (^) X SAF t m t m
% &
K 'K
1 6 100, ,
if model is multiplicative if model is additive
where
m = t −t p p
Smoothed Trend-Cycle Series
The smoothed trend-cycle series (STC) is obtained by applying a 3 × 3 moving average on seasonally adjusted series (SAS). Thus,
STC SAS SAS SAS SAS SAS
t n
t =^ t +^ t +^ t +^ t + t = −
− − + +
1 6 2 1 6 1 1 6 1 6 1 1 6 2 , , K,
and for the two end points on the beginning and end of the series
STC SAS SAS SAS
STC (^) n SAS (^) n SAS (^) n SASn
0 5 0 5 0 5 0 5
0 5 0 5 0 5 0 5
2 1 2 3
1 2 1
− =^ − +^ − +
STC STC STC STC
STC (^) n STC (^) n STC (^) n STCn
1 2 2 3
1 1 2
Irregular Component
For (^) t = 1, K,n
I
SAS STC
t SAS STC t t t t
% & '
0 5 0 5 0 5 0 5
if model is multiplicative if model is additive
