# What is this method for seasonal adjustment calculation?

I got an example calculation for multiplicative model, which is shown as follows:

Quarter           1         2        3       4
Average         0.866    1.0005   1.403    0.660
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Seasonal factor  0.884    1.018   1.421    0.678


Then there is a note below:

Sum of averages = 3.9295. These should sum to 4, 4-3.9295=0.0705. Adding 0.0705/4=0.0176 to each average, to obtain the seasonal factors.

I saw from other resources that they are using "seasonal index" instead of "seasonal factor" by normalizing the values. Besides that, they also mentioned about X11, X12, ARIMA, and so on. I would like to know is, based on the example above, what is the method called?

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X11 is an older method of seasonally adjusting data. Here's a link:

http://www.census.gov/srd/www/sapaper/historicpapers.html

http://www.census.gov/srd/www/x12a/

You can download X12 and run it yourself. Part of the output will be similar to your data above. For example, in Quarter 1, it gave a "Seasonal Factor" of 0.884. That came from the "Average" of 0.866, with an "Adjustment" of 0.0176. In other words,

0.884 = 0.866 + 0.0176


Another way to look at this is, zero seasonality would give you a "Seasonal Factor" of 1.0 for each quarter (it would be "flat-line" for zero seasonality). If you add these four 1.0's together, you get a total of 4.0 for the year.

However, your data doesn't have zero seasonality. It is typically (or seasonally) down 11.6% in Q1 (0.884), up 1.8% in Q2 (1.018), up 42.1% in Q3 (1.421) and down 32.2% in Q4 (0.678) when compared to "flat-line". If you add those numbers up, again you'll get 4.0 for the year. In other words, the "Adjustment" was the evenly distributed adder that was required to get the "Average" to provide a "Seasonal Factor" that totals 4.0 for the year.

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