The FORECAST Procedure

Specifying Seasonality

Seasonality of a time series is a regular fluctuation about a trend. This is called seasonality because the time of year is the most common source of periodic variation. For example, sales of home heating oil are regularly greater in winter than during other times of the year.

Seasonality can be caused by many things other than weather. In the United States, sales of nondurable goods are greater in December than in other months because of the Christmas shopping season. The term seasonality is also used for cyclical fluctuation at periods other than a year. Often, certain days of the week cause regular fluctuation in daily time series, such as increased spending on leisure activities during weekends.

Three kinds of seasonality are supported in PROC FORECAST: time-of-year, day-of-week, and time-of-day. The seasonal part of the model is specified by using the SEASONS= option. The values for the SEASONS= option are listed in Table 17.2.

Table 17.2: The SEASONS= Option

SEASONS= Value	Cycle Length	Type of Seasonality
QTR	yearly	time of year
MONTH	yearly	time of year
DAY	weekly	day of week
HOUR	daily	time of day

The three kinds of seasonality can be combined. For example, SEASONS=(MONTH DAY HOUR) specifies that 24 hour-of-day seasons are nested within 7 day-of-week seasons, which in turn are nested within 12 month-of-year seasons. The different kinds of intervals can be listed in the SEASONS= option in any order. Thus, SEASONS=(HOUR DAY MONTH) is the same as SEASONS=(MONTH DAY HOUR). Note that the Winters method smoothing equations might be less stable when multiple seasonal factors are used.

Multiple period seasons can also be used. For example, SEASONS=QTR2 specifies two semiannual time-of-year seasons. The grouping of observations into multiple period seasons starts with the first interval in the seasonal cycle. Thus, MONTH2 seasons are January–February, March–April, and so on. (There is no provision for shifting seasonal intervals; thus, there is no way to specify seasons December–January, February–March, April–May, and so on.)

For multiple period seasons, the number of intervals combined to form the seasons must evenly divide and be less than the basic cycle length. For example, with SEASONS=MONTHn, the basic cycle length is 12, so MONTH2, MONTH3, MONTH4, and MONTH6 are valid SEASONS= values (because 2, 3, 4, and 6 evenly divide 12 and are less than 12), but MONTH5 and MONTH12 are not valid SEASONS= values.

The frequency of the seasons must not be greater than the frequency of the input data. For example, you cannot specify SEASONS=MONTH if INTERVAL=QTR or SEASONS=MONTH if INTERVAL=MONTH2. You also cannot specify two seasons of the same basic cycle. For example, SEASONS=(MONTH QTR) or SEASONS=(MONTH2 MONTH4) is not allowed.

Alternatively, the seasonality can be specified by giving the number of seasons in the SEASONS= option. SEASONS=n specifies that there are n seasons, with observations 1, $n+1$ , $2 n+1$ , and so on in the first season, observations 2, $n+2$ , $2 n+2$ , and so on in the second season, and so forth.

The options SEASONS=n and SINTPER=m cause PROC FORECAST to group the input observations into n seasons, with m observations to a season, which repeat every nm observations. The options SEASONS=( n $_{1}$ n $_{2}$ ) and SINTPER=( m $_{1}$ m $_{2}$ ) produce $n_{1}$ seasons with $m_{1}$ observations to a season nested within $n_{2}$ seasons with $n_{1}m_{1}m_{2}$ observations to a season.

If the SINTPER=m option is used with the SEASONS= option, the SEASONS= interval is multiplied by the SINTPER= value. For example, specifying both SEASONS=(QTR HOUR) and SINTPER=(2 3) is the same as specifying SEASONS=(QTR2 HOUR3) and also the same as specifying SEASONS=(HOUR3 QTR2).