A time series is a variable that is observed
repeatedly at regular
intervals;
for example, car sales at Monopoly Motors each
year or
sales of Rocky Gold's jewelry sales each
month.
Time series forecasting means using just this data to try to forecast future values.
When a time series does not show strong seasonal
variation, we can
apply
the assumption that the future will resemble the past using strategies
such
as naive forecasting, lagged moving averages, exponential smoothing,
or regression with time as the "X" variable. Monopoly Motors may
or
may not experience seasonal variations in car sales, but since their
time
series is annual rather than monthly or quarterly, seasonality is not
an issue.
See MonoMot.htm
When a time series shows strong seasonal variations,
the overall
approach
is:
A.  Estimate the seasonal factors "seasonal indexes" 
(See below for simple and complex methods for doing this)  
B.  Use division to take out the seasonality, leaving a deseasonalized time series 
(See column F of sheet "Deseas Fcst" in Jewelry)  
C.  Use least squares regression to model and Forecast the deseasonalized time series 
(See column G of sheet "Deseas Fcst," paying special attention to cells G25 and G26 in Jewelry)  
D.  Use multiplication to put the seasonality back in to get a model and forecast of the actual time series. 
(See column H of sheet "Reseas Fcst" in Jewelry) 
A.1  Find the average for each period (month or quarter) averaged across all years 
(See columns E2 through K6 of tab "Indexes3" in Jewelry)  
A.2  Calculate the ratio between the period average from A1 and the grand average; use this as the approximate seasonal index. 
(See cell K7 and column L in Jewelry) 
The textbook method is more complicated:
A.1  Create a centered moving average time series. 
The centered moving average for July 1996 is
the average monthly sales (or
whatever) for the year whose middle month is July 1996; this is the year from January 15, 1996
through January 14, 1997
 5½ months before July 1996, July 1996 itself, and 5½
months
after July 1996. 

A.2  Calculate the ratio between each month's actual data and the same month's centered moving average. 
A.3  Calculate the average of the January ratios, the average of the February ratios, et ceters 
A.4  Adjust the average ratios so that the average of the averages is 1.00 
A.5  Use adjusted mean ratios as seasonal indexes of each month in data base and forecast horizon 
Once we have a model of the historical data, we can use it to assess the historical accuracy of the model. Under the assumption that the future resembles the past, we expect if one method gives a more accuratehistorical model than another, it is likely to also give a more accurate forecast of the future.