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Seasonal adjustment in Stats NZ
How Statistics New Zealand computes the components of X13-ARIMA-SEATS


How does Statistics New Zealand compute the components of the seasonal model?

Statistics New Zealand uses X-13-ARIMA-SEATS (predominately Version 1.1 Build 9) 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.


  • There is no single "best" decomposition.
  • The decomposition depends on the quality and amount of consistent data available.
  • Changes in the series, such as changes in data collection methodology, cause difficulties in quantifying the components.


What is X-13-ARIMA-SEATS?

X-13-ARIMA-SEATS 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.


How does X-13-ARIMA-SEATS work?

The following diagram and description is a simple summary of the key points in the X-13 process within X-13-ARIMA-SEATS for a multiplicative model.

Graph, X-12 Process (multiplicative model).


  • MA - moving average
  • A - actual (observed) series
  • C - trend series
  • S - series of seasonal factors
  • I - irregular series
  • S*I - detrended series obtained by removing the trend from the actual series

Sequence of steps illustrated above:

  1. A preliminary trend cycle (C') is obtained using weighted moving means of the actual (A).
  2. This trend is removed from the actual series to obtain a "detrended" series (S x I = A/C').
  3. Seasonal factors (S') are estimated from the detrended series by taking weighted moving means for each group of months/quarters separately.
  4. A new estimate of irregular components is obtained by removing the seasonal component from the detrended series (I'=A / (C'*S')).
  5. Outliers are identified from the estimate of irregular components (I'), and the actual value is temporarily replaced by an imputed value (A').
  6. A new trend cycle (C') is obtained from the modified actuals (A').
  7. A new detrended series is obtained by removing the trend estimate from the modified actuals ( A'/C')
  8. A new set of seasonal factors (S') are obtained from the detrended series by taking weighted moving means for each group of months/quarters separately.
  9. A seasonally adjusted series is obtained by removing the seasonal factors from the modified actuals (SA'=A'/S').
  10. A new trend cycle (C') is obtained from the seasonally adjusted series (SA') using weighted moving means.

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)


What moving averages (filters) are used by X-13-ARIMA-SEATS?

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-13-ARIMA-SEATS selects longer filters for series with larger irregular components.

X-13-ARIMA-SEATS 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.


What are Statistics New Zealand's default parameter settings for decomposing series?

Sigma weights

At Statistics New Zealand:

  • Observations with values more than 2.8 standard deviations away from the mean are given a weight of 0.
  • Observations between 1.8 and 2.8 standard deviations are given a weight between 0 and 1.

These 1.8 and 2.8 are sometimes referred to as the "sigma weights".

The defaults in X-13-ARIMA-SEATS are (1.5, 2.5), but if we down-weighted 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-13-ARIMA-SEATS 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.

Key series Type of adjustment Transform option X-11 regression options
All economic series sigmalim=(1.8 , 2.8)
Electronic Card Transactions (ECT) - Monthly Indirect multiplicative adjustment None variables=( td)
Retail Trade - Quarterly sales Indirect multiplicative adjustment of 24 storetypes Prior adjustment for leap year - apply a divisor of 91/90 to leap year observed values.
Quarterly Gross Domestic Product (QGDP) Direct multiplicative adjustment None seasonalma=(s3x3 s3x9 s3x3 s3x3)
Merchandise Trade - Exports Multiplicative adjustment tdprior
Merchandise Trade - Imports Multiplicative adjustment tdprior
Merchandise Trade - Balance Balance trend = exports trend - imports trend
Consumer Price Index (CPI) Direct adjustment using either an additive, multiplicative, or log additive model Cigarettes and tobacco series are adjusted seasonalma = (MSR s3x3 s3x5 s3x9 stable)
Balance of Payments (BOP) - Current Account Balance Indirect additive adjustment of 4 credit and 4 debit series Goods debit: prior adjustment for two frigates
Transfer Credits: split adjustment
Goods Credit only: seasonalma=s3x9
Household Labour Force Survey (HLFS) - Unemployment Indirect multiplicative adjustment of male and female unemployed Males Part-time Empl: Prior Adjustment. seasonalma=(msr)



Does Statistics New Zealand freeze any of its decompositions?

The official decomposition is fully revised in most series.

The two examples of exceptions are:

  • The CPI is never revised.
  • BOP decomposition is frozen once it is one year old.


Should known extremes be removed from the data before seasonal adjustment?

It is preferable to make use of prior knowledge to remove extremes, rather than let X-13-ARIMA-SEATS 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 $563 million in the second quarter of 1997 for the purchase of a frigate.


How are the trading day factors calculated by X-13-ARIMA-SEATS?

The X-13-ARIMA-SEATS seasonal adjustment package has an in-built option that performs this.

The X-13-ARIMA-SEATS manual explains the process:

  • "Seven daily weights are estimated by regressing the irregular series upon the number of times each day of the week occurs in each particular month."

"The irregular component is the bit left over when the seasonal factor and trend have been removed from the data. What X-13-ARIMA-SEATS does, in effect, is, for each of the seven days separately, plot the irregular component against the number of occurrences of the days in the month."

Graph,Trading Day 1.

Graph,Trading Day 2.

The slope of the line dictates the weight:

  • / indicates a weight greater than 1
  • \ indicates a weight less than 1
  • - indicates a weight = 1.


What indicates a good quality seasonal adjustment?

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-13-ARIMA-SEATS 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

  • There should be no residual seasonal effect in the seasonally adjusted series.
  • The seasonally adjusted series is the combination of the trend cycle and the irregular component.
  • Neither of these components should contain seasonality.
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