Statistics New Zealand uses X-12-ARIMA (predominately version 0.2.6) to estimate the trend, seasonal and irregular component components of the seasonal model.
What is the underlying seasonal model?
The estimates produced are a reasonable approximation of what really happens.
Note:
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X-12-ARIMA is a seasonal adjustment program developed at the United States Bureau of the Census. The program is based on the bureau's earlier X11 program and the X-11-ARIMA/88 program developed at Statistics Canada.
http://www.census.gov/srd/www/x12a/
The following diagram and description is a simple summary of the key points in the X-12 process within X-12-ARIMA for a multiplicative model.
Key:
Sequence of steps illustrated above:
11-14 Steps 2-10 are repeated to produce the final set of seasonal factors (S), trend (C), and irregular components ( A/(S*C))
For more details see the downloadable file: "How the X-11 program implements a trend-seasonal decomposition" (Alistair Gray)
The particular set of weights used to calculate the moving average is the called the filter.
The preliminary trend is either a 5-term centred moving average for quarterly data, or a choice of 9-, 13-, or 23-term for monthly. X-12-ARIMA selects longer filters for series with larger irregular components.
X-12-ARIMA uses a Henderson filter for the final trend estimation.
For more details see the downloadable file: "The Surrogate Henderson Filters in X-11" (Doherty M 2001), The Australian and New Zealand Journal of Statistics , 43:4, 385-392.
The moving averages for the seasonal factors are more complex. Generally different filters are applied for different months/quarters, as some seasonal factors are more stable than others.
At Statistics New Zealand:
These 1.8 and 2.8 are sometimes referred to as the "sigma weights".
The defaults in X-12-ARIMA are (1.5, 2.5), but if we downweighted using these limits for New Zealand series, we would discard many observations. New Zealand's economy is very small compared to the United States or Canada where X-11/X-12-ARIMA was developed. This smaller economy leads to more volatility in the series, which we allow for by increasing our sigma limits to make greater use of real data.
The official decomposition is fully revised in all but two series.
The exceptions are:
It is preferable to make use of prior knowledge to remove extremes, rather than let X-12-ARIMA estimate it.
The program is likely to identify the extreme, but may not estimate its size accurately, thus slightly distorting the seasonal component.
For example, in the merchandise trade series Statistics New Zealand made a prior adjustment of $M 563 in the second quarter of 1997 for the purchase of a frigate.
The X-12-ARIMA seasonal adjustment package has an in-built option that performs this.
The X-12-ARIMA manual explains the process:
"The irregular component is the bit left over when the seasonal factor and trend have been removed from the data. What X-12-ARIMA does, in effect, is, for each of the seven days separately, plot the irregular component against the number of occurences of the days in the month."
The slope of the line dictates the weight:
A good seasonal adjustment can be done only if a time series has an identifiable seasonal pattern, which consist of peaks and troughs, occurring in the original series at approximately the same time every year. The more irregular the pattern, the harder it is to separate components for further extracting and removing the seasonal component from a time series.
A good seasonal adjustment can be judged by looking at a graph of the original series in comparison to the seasonally adjusted series. If the period to period changes in the original are not greatly reduced by the period to period seasonally adjusted series, then it is not a good adjustment.
Another way to judge a good seasonal adjustment is to look at the quality control statistics implemented into the X-12-ARIMA program. These statistics allow you not only to judge the quality of seasonal adjustment of a time series at a particular time, but also to monitor the quality of adjustment over time, and compare the strength and pattern of seasonal variations of different time series.
The United States Bureau of the Census state in their FAQ page(http://www.census.gov/mrts/www/faq.html):