Time Series Forecasting

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 Blockhead Concrete's foundation blocks 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  Mono-Mot.htm

When a time series shows strong seasonal variations, the overall approach
A. Estimate the seasonal factors "seasonal indexes"

(See below for simple and complex methods for doing this)
B. Take out the seasonality, leaving a deseasonalized time series

(See column F of sheet "Found-fcst" in Foundation.xls)
C. Model and Forecast the deseasonalized time series

(See columns G and H of sheet "Found-fcst" in Foundation.xls)
D. Put the seasonality back in to get a model and forecast of the actual time series.

(See column I of sheet "Foundation" in Foundation.xls)

The simplified version of calculating seasonal indexes is as follows:
A.1  Find the average for each period (month or quarter) averaged across all years
A.2 Calculate the ratio between the period average from A1 and the grand average; use this as the approximate seasonal index.

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.  (See columns D,E,F of sheet "Found-Seas" in Foundation.xls)
A.2 Calculate the ratio between each month's actual data and the same month's centered moving average.

(See column G of sheet "Found-Seas" in Foundation.xls)
A.3 Calculate the average of the January ratios, the average of the February ratios, et ceters

(See columns H through T of sheet "Found-Seas" in Foundation.xls)
A.4 Adjust the average ratios so that the average of the averages is 1.00

(See columns U and V of sheet "Found-Seas" in Foundation.xls)
A.5 Use adjusted mean ratios as seasonal indexes of each month in data base and forecast horizon

(See column W of sheet "Found-Seas" in Foundation.xls)

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.