*Time Series Analysis and Control Examples* |

Seasonal phenomena are frequently observed
in many economic and business time series.
For example, consumption expenditure might have strong
seasonal variations because of Christmas spending.
The seasonal phenomena are repeatedly
observed after a regular period of time.
The number of seasons within a period is defined as
the smallest time span for this repetitive observation.
Monthly consumption expenditure shows a strong increase
during the Christmas season, with 12 seasons per period.
There are three major approaches to seasonal
time series: the regression model, the moving
average model, and the seasonal ARIMA model.

*Regression Model*

Let the trend component be
and the seasonal component be
.Then the additive time series can
be written as the regression model

In practice, the trend component can be written as the
th order polynomial, such as

The seasonal component can be approximated
by the seasonal dummies (*D*_{jt})

where *L* is the number of seasons within a period.
The least squares method is applied to
estimate parameters and .The seasonally adjusted series is obtained by subtracting
the estimated seasonal component from the original series.
Usually, the error term is assumed to
be white noise, while sometimes the autocorrelation
of the regression residuals needs to be allowed.
However, the regression method is not robust to the regression
function type, especially at the beginning and end of the series.

*Moving Average Model*

If you assume that the annual sum of a seasonal
time series has small seasonal fluctuations, the
nonseasonal component can
be estimated by using the moving average method.

where *m* is the positive integer and
is the symmetric constant such that and .When the data are not available, either an
asymmetric moving average is used, or the forecast
data is augmented to use the symmetric weight.
The X-11 procedure is a complex
modification of this moving average method.

*Seasonal ARIMA Model*

The regression and moving average approaches
assume that the seasonal component is deterministic
and independent of other nonseasonal components.
The time series approach is used to handle
the stochastic trend and seasonal components.
The general ARIMA model can be written

where *B* is the backshift operator and

and if
*d*_{i} = 0; otherwise, .The power of *B*, *s*_{i}, can be considered as a seasonal factor.
Specifically, the Box-Jenkins multiplicative
seasonal ARIMA(*p*,*d*,*q*)(*P*,*D*,*Q*)_{s} model is written as

ARIMA modeling is appropriate for particular
time series and requires burdensome computation.
The TSBAYSEA subroutine combines the simple characteristics
of the regression approach and time series modeling.
The TSBAYSEA and X-11 procedures use
the model-based seasonal adjustment.
The symmetric weights of the standard X-11 option
can be approximated by using the integrated MA form

With a fixed value , the
TSBAYSEA subroutine is approximated as

The subroutine is flexible enough to handle
trading day or leap year effects, the shift
of the base observation, and missing values.
The TSBAYSEA-type modeling approach has some advantages:
it clearly defines the statistical model of the time series;
modification of the basic model can be an efficient method of
choosing a particular procedure for the seasonal adjustment
of a given time series; and the use of the concept of the
likelihood provides a minimum AIC model selection approach.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.