Constructing seasonal factors from centered moving averages

Sequential observations still carry calendar bias

Time series used in technical calculations are sequential observations, typically sampled at fixed intervals, and each new period refreshes the indicators built from them.

Harvest-linked agricultural prices and other calendar-sensitive economic series can embed a recurring seasonal bias that conceals nonseasonal movement. Seasonal analysis isolates that recurring within-year variation so a single series can be read in a calendar-aware regime context instead of as an unfiltered path.

Center the twelve-month average

The construction begins with a trailing 12-month arithmetic moving average of the unadjusted series. The December 1993 window equals 18196.

With an even 12-month span, the current month is centered by averaging two overlapping 12-month means that straddle it. The July 1993 centered level is 18281. A moving average is an arithmetic smoother of ordered observations: that trailing twelve-period mean and the pair of overlapping twelve-period means supply the local level used to form monthly trend ratios.

Turn ratios into standardized seasonal factors

A monthly trend ratio is the unadjusted observation divided by that period's centered moving average. The July 1993 ratio is 0.971.

Calendar-month averages of those ratios are rescaled by 12 divided by their sum so the twelve standardized factors total 12. January moves from 1.025 to 1.022 when the raw factor sum is 12.030. A standardized seasonal factor is that calendar-month average of trend ratios after the rescaling.

The ratio to moving average construction turns raw observations into monthly trend ratios, averages those ratios by calendar month, standardizes the twelve factors, and divides them back out of the original series.

Read the seasonally adjusted series

Seasonally adjusted values equal the unadjusted observation divided by the matching standardized monthly factor. January 1993 converts from 17184 to 16809. That seasonally adjusted series is the original observation divided by its month's standardized seasonal factor, used to make nonseasonal movement easier to inspect.

In the worked volume example, both the raw and adjusted series decline in June 1998, but the adjusted path falls faster, which the construction presents as a clearer view of the nonseasonal pattern.

Score the forecast on the residual path

Exponential smoothing is a quantitative forecast baseline that updates from ordered price, volume, or breadth observations over a defined sampling interval and lookback. The teaching use is to score that baseline on the seasonally residual series rather than on raw calendar-biased observations.

Editorial: once the seasonally adjusted series is in hand, a moving-average forecast can be scored the same way. The construction does not itself make a market call.