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A quick review of some of the more common terms in seasonal adjustment

If you have more questions, please see the Seasonal Adjustment FAQ.


ARIMA models

ARIMA (which stands for Auto-Regressive Integrated Moving Average) models are a group of models for describing and forecasting a time series. When talking about moving averages in ARIMA models, the moving average component refers to lagged forecast errors and not the moving averages used in X-11-type seasonal filters. A seasonal ARIMA model would be designed as ARIMA(p,d,q)(P,D,Q) where

  • p is the number of nonseasonal autoregressive terms,
  • d is the number of nonseasonal differences,
  • q is the number of nonseasonal moving average terms,
  • P is the number of seasonal autoregressive terms,
  • D is the number of seasonal differences, and
  • Q is the number of seasonal moving average terms.
When some of the terms are zero, we can drop part of the ARIMA label. For example, the model ARIMA(0,1,1) is sometimes written as IMA(1,1).

Irregular Component

Anything not included in the trend-cycle, the seasonal effects, or other calendar-related effects (i.e., trading day or moving holiday effects). Its values are unpredictable when it comes to timing, impact, and duration. Irregular can arise from sampling error, non-sampling error, unseasonable weather, natural disasters, strikes, etc.

Moving Holiday Effects

Effects from holidays that are not always on the same day of a month, such as Labor Day or Thanksgiving in the United States or New Year in China. The most important moving holiday in the U.S. is Easter, and it not only moves between days, but can also move between months since it can occur in March or April.

RegARIMA models

RegARIMA models combine a regression model with an ARIMA model for the error terms from the regression model. In the seasonal adjustment, we include variables for moving holiday effects, trading day effects, and/or outliers as linear regression variables and the remaining errors from the regression are modeled with a seasonal ARIMA model.

Seasonal Adjustment

The process of estimating and removing the seasonal effects (and other calendar-related effects, such as trading day) from a time series. The basic goal of seasonal adjustment is to decompose a time series into a several different components including a seasonal component and an irregular component.

Seasonal Effects

Effects that are reasonably stable in terms of annual timing, direction, and magnitude. Possible causes include natural factors (the weather), administrative measures (starting and ending dates of the school year), and social/cultural/religious traditions (fixed holidays such as Christmas). Effects associated with the dates of moving holidays like Easter are not seasonal in this sense, because they occur in different calendar months depending on the date of the holiday.


SEATS stands for Signal Extraction in ARIMA Time Series and is a seasonal adjustment software package developed by Agustin Maravell at Banco de Espana (Bank of Spain). It uses ARIMA models to estimate the different componeents of a time series. This implimentation of a signal extraction approach to seasonal adjustment is used within both TRAMO/SEATS and X-13ARIMA-SEATS.

Time series

A sequence of measures of a given phenomenon taken at regular time intervals. For the information to be useful for time series analysis, that data should be comparable over time. That means 1) that the measurements should be taken over discrete consecutive periods, i.e., every month or every quarter, and 2) that the definition of the concept and the way it is measured should be consistent over time.

Trading Day Effects

Recurring effects associated with individual days of the week. This occurs because only non-leap-year Februaries have four of each day - four Mondays, four Tuesdays, etc. All other months have an excess of some types of days. If an activity is higher on some days compared to others, then the series can have a trading day effect. For example, building permit offices are usually closed on Saturday and Sunday. Thus, the number of building permits issued in a given month is likely to be higher if the month contains a surplus of weekdays and lower if the month contains a surplus of weekend days. Trading day effects also often include an adjustment for a longer February in leap years.


TRAMO (Time Series Regression with ARIMA noise) and SEATS are integrated seasonal adjustment software from Banco de Espana (Bank of Spain). TRAMO automatically selects a regARIMA model for a time series, and SEATS uses that regARIMA model to estimate components and seasonally adjust the series.

Trend or Trend-Cycle

An estimate of the local level of the series for each month (quarter) derived from the surrounding recent (a year or two) observations It includes long-term increases and/or decreases and cycles longer than a year (such as "business cycles").


Seasonal adjustment software that uses iteration and linear filters. It was originally developed by United States Census Bureau beginning in the 1960's.


Seasonal adjustment software developed by Statistics Canada that used ARIMA models to forecast the series and improve estimates.

X-12-ARIMA (or X-12)

Seasonal adjustment software developed at the U.S. Census Bureau. Used for official adjustments at the Census Bureau from the early 1990's until 2015. Still used by many government agencies in the U.S. and around the world for official seasonal adjustments. Besides the inclusion of regARIMA models, this program also includes ARIMA model diagnostics and several seasonal adjustment diagnostics developed at the U.S. Census Bureau.

X-13ARIMA-SEATS (or X-13AS or X-13)

The newest in the X-11 family of seasonal adjustment software. Developed and maintained by Brian Monsell at the U.S. Census Bureau in collaboration with the Bank of Spain. Used for official seasonal adjustments at U.S. Census Bureau, the Bureau of Labor Statistics, and many other government agencies in the U.S. It integrates an enhanced version of X-12-ARIMA with an enhanced version of SEATS to provide both non-parametric X-11-type seasonal adjustments and ARIMA-model-based SEATS-type adjustments, combined with the diagnostics available in X-12-ARIMA.

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Last modified: 17 Jan 2019

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