Seasonality
Look actual time series data
Retail sales
NYS bonuses paid
CPI
The main causes of the seasonal swings are
Timing of public holidays
School vacations
Timing of payments (wages, dividend…)
Tax year or accounting periods
Effects of weather
There are many ways of smoothing the seasonal components of an economic
series, but they all basically aim to produce a series that appears to be nonseasonal. However, some of the proposed methods of seasonal adjustment are
not suitable in a forecast framework as they are designed only to remove a
seasonal component from a set of historical data and in doing so some
observations are lost.
Growth rate for not seasonally adjusted series
gt  100*(Yt / Yt s  1)
For monthly data, s=12, for quarterly data, s = 4. Note that this type of year-overyear growth rate tends to miss turning points.
Seasonal Adjustment (Smoothing)
Type of data series
Additive,
Yt  Tt  It  St  Ct
where Tt is trend, I t is irregular, St is seasonal, and Ct is cyclical.
Multiplicative
Yt  Tt  I t  St  Ct
Smoothing a time series using moving average
Moving average or centered moving average
m
1
Yt* 
Yt  j

2m  1 j  m
For example, taking m  2 , we have
1
Yt *  (Yt  2  Yt 1  Yt  Yt 1  Yt  2 )
5
Notice that 2 observations lost at the beginning and two at the end.
Also smoothing is centered and it is average (not real value).
Simple average vs. weighted average
The simple average is given above. The weighted average, for example, could
take the form
1
Yt *  (Yt  2  2Yt 1  4Yt  2Yt 1  Yt  2 )
10
Another example (3X3 moving average):
1
Yt *  (Yt 1  Yt  Yt 1 )
3
1
Yt**  (Yt*1  Yt*  Yt*1 )
3
then we have
1
Yt **  [(Yt  2  Yt 1  Yt ) / 3  (Yt 1  Yt  Yt 1 ) / 3  (Yt  Yt 1  Yt  2 ) / 3]
3
1
 (Yt  2  2Yt 1  3Yt  2Yt 1  Yt  2 )
9
Then define
Ratio:
Seasonal indices:
Seasonal factors:
rt  Yt ** / Yt
im  average (rt ) for month i
s j  i j /(i1 , i2 ,...im )
1
m
j  1, 2,...m
(scaling factor)
Seasonal adjusted series  Yt / s j
Issues:
1. Missing Observations. X11-ARIMA or X12_ARIMA
2.The estimation of seasonal factors is an iterative process. But how to separate
different components is very difficult.
3.Irregular (I) could distort the seasonal factors (S).
Modeling Seasonality
Regression with seasonal dummies:
s
Yt    i Dit   t
i 1
Dit is a series whose value = 1 when the season is i and = 0 otherwise for the
entire period t , t  1, 2,...T , which is called sometimes seasonal dummy variable.
Notice that if there is an intercept in the model then you only use s  1 seasonal
dummies.
Trend may be included
s
Yt   * t    i Dit   t
i 1
Other calendar effects: holiday and trading-day
Holiday: Easter dummy, which = 1 if the month contains Easter, otherwise = 0
Trading-day (different business days): dummy variable whose value equals to
the number of trading days for that month
Forecasting future values using the regression model.
Example