The TIMESERIES Procedure

Seasonal Decomposition

Seasonal decomposition/analysis can be performed on the working series by specifying the OUTDECOMP= option, the PRINT=DECOMP option, or one of the PLOTS= options associated with decomposition in the PROC TIMESERIES statement. The DECOMP statement enables you to specify options related to decomposition. The TIMESERIES procedure uses classical decomposition. More complex seasonal decomposition/adjustment analysis can be performed by using the X11 or the X12 procedure of SAS/ETS.

The DECOMP statement MODE= option determines the mode of the seasonal adjustment decomposition to be performed. There are four modes: multiplicative (MODE=MULT), additive (MODE=ADD), pseudo-additive (MODE=PSEUDOADD), and log-additive (MODE=LOGADD) decomposition. The default is MODE=MULTORADD which specifies MODE=MULT for series that are strictly positive, MODE=PSEUDOADD for series that are nonnegative, and MODE=ADD for series that are not nonnegative.

When MODE=LOGADD is specified, the components are exponentiated to the original metric.

The DECOMP statement LAMBDA= option specifies the Hodrick-Prescott filter parameter (Hodrick and Prescott, 1980). The default is LAMBDA=1600. The Hodrick-Prescott filter is used to decompose the trend-cycle component into the trend component and cycle component in an additive fashion. A smaller parameter assigns less significance to the cycle; that is, LAMBDA=0 implies no cycle component.

The notation and keywords associated with seasonal decomposition/adjustment analysis are defined in Table 32.3.

Table 32.3: Seasonal Adjustment Formulas

Component

Keyword

MODE= Option

Formula

original series

ORIGINAL

MULT

${ O_{t} = TC_{t}S_{t}I_{t} }$

   

ADD

${ O_{t} = TC_{t}+S_{t}+I_{t} }$

   

LOGADD

${ log(O_{t}) = TC_{t}+S_{t}+I_{t} }$

   

PSEUDOADD

${ O_{t} = TC_{t}(S_{t}+I_{t}-1) }$

trend-cycle component

TCC

MULT

centered moving average of ${O_{t}}$

   

ADD

centered moving average of ${O_{t}}$

   

LOGADD

centered moving average of ${log(O_{t})}$

   

PSEUDOADD

centered moving average of ${O_{t}}$

seasonal-irregular component

SIC

MULT

${ SI_{t} = S_{t}I_{t} = O_{t}/TC_{t} }$

   

ADD

${ SI_{t} = S_{t}+I_{t} = O_{t}-TC_{t} }$

   

LOGADD

${ SI_{t} = S_{t}+I_{t} = log(O_{t})-TC_{t} }$

   

PSEUDOADD

${ SI_{t} = S_{t}+I_{t}-1 = O_{t}/TC_{t} }$

seasonal component

SC

MULT

seasonal Averages of ${SI_{t}}$

   

ADD

seasonal Averages of ${SI_{t}}$

   

LOGADD

seasonal Averages of ${SI_{t}}$

   

PSEUDOADD

seasonal Averages of ${SI_{t}}$

irregular component

IC

MULT

${ I_{t} = SI_{t}/S_{t} }$

   

ADD

${ I_{t} = SI_{t}-S_{t} }$

   

LOGADD

${ I_{t} = SI_{t}-S_{t} }$

   

PSEUDOADD

${ I_{t} = SI_{t}-S_{t} + 1 }$

trend-cycle-seasonal component

TCS

MULT

${ TCS_{t} = TC_{t}S_{t} = O_{t}/I_{t} }$

   

ADD

${ TCS_{t} = TC_{t}+S_{t} = O_{t}-I_{t} }$

   

LOGADD

${ TCS_{t} = TC_{t}+S_{t} = O_{t}-I_{t} }$

   

PSEUDOADD

${ TCS_{t} = TC_{t}S_{t} }$

trend component

TC

MULT

${ T_{t} = TC_{t}-C_{t} }$

   

ADD

${ T_{t} = TC_{t}-C_{t} }$

   

LOGADD

${ T_{t} = TC_{t}-C_{t} }$

   

PSEUDOADD

${ T_{t} = TC_{t}-C_{t} }$

cycle component

CC

MULT

${ C_{t} = TC_{t}-T_{t} }$

   

ADD

${ C_{t} = TC_{t}-T_{t} }$

   

LOGADD

${ C_{t} = TC_{t}-T_{t} }$

   

PSEUDOADD

${ C_{t} = TC_{t}-T_{t} }$

seasonally adjusted series

SA

MULT

${ SA_{t} = O_{t}/S_{t} = TC_{t}I_{t} }$

   

ADD

${ SA_{t} = O_{t}-S_{t} = TC_{t}+I_{t} }$

   

LOGADD

${ SA_{t} = O_{t}/exp(S_{t}) = exp(TC_{t}+I_{t}) }$

   

PSEUDOADD

${ SA_{t} = TC_{t}I_{t} }$


When ${s}$ is odd the trend-cycle component is computed from the ${s}$-period centered moving average as follows:

\[  TC_{t} = \sum _{k=-{\lfloor s/2 \rfloor }}^{{\lfloor s/2 \rfloor }}{y_{t+k}/s}  \]

When ${s}$ is even the trend-cycle component is computed from the ${s}$-period centered moving average as follows:

\[  TC_{t} = \sum _{k=-{s/2}}^{{s/2}-1}{(y_{t+k}+y_{t+1+k})/2s}  \]

The seasonal component is obtained by averaging the seasonal-irregular component for each season.

\[  S_{k+js} = \sum _{t = k \bmod s}\frac{SI_ t}{T/s}  \]

where ${0 {\le } j {\le } T/s }$ and ${1 {\le } k {\le } s }$. The seasonal components are normalized to sum to one (multiplicative) or zero (additive).