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{{Short description|Statistical algorithms for time series data analysis}}
{{Infobox software |
name = X-13ARIMA-SEATS |
logo = |
screenshot = |
caption = |
developer = [[U.S. Census Bureau]] |
latest_release_version = 3.0 (Windows) |
latest_release_date = {{release date and age|2020|6|15}} |
operating_system = [[Microsoft Windows|Windows]], [[Linux]]/[[Unix]] |
genre = [[List of statistical packages|Statistical software]] |
license = [[Public ___domain]]<ref name="census.gov">{{Cite web |url=https://www.census.gov/srd/www/x13as/x13down_unix.html |title=X-13ARIMA-SEATS Download Page (Unux/Linux Version) - X-13ARIMA-SEATS Seasonal Adjustment Program - US Census Bureau |access-date=2014-02-13 |archive-date=2014-02-22 |archive-url=https://web.archive.org/web/20140222020757/https://www.census.gov/srd/www/x13as/x13down_unix.html |url-status=dead }}</ref> (in the US; and copyright granted elsewhere)<ref name="copyrightable">{{cite web |url=https://www.census.gov/srd/www/disclaimer.html |url-status=dead |archive-url=https://web.archive.org/web/20000816012546/http://www.census.gov/srd/www/disclaimer.html |archive-date=2000-08-16 |title=Disclaimer}}</ref> |
website = {{URL|https://www.census.gov/data/software/x13as.html}}
}}
'''X-13ARIMA-SEATS''', successor to '''X-12-ARIMA''' and '''X-11''', is a set of statistical methods for [[seasonal adjustment]] and other descriptive analysis of [[time series]] data that are implemented in the U.S. Census Bureau's software package.<ref>{{cite web |title=X-13ARIMA-SEATS Seasonal Adjustment Program
|url=https://www.census.gov/data/software/x13as.html |publisher=United States Census Bureau |access-date=March 24, 2021}}</ref> These methods are or have been used by [[Statistics Canada]], [[Australian Bureau of Statistics]], and the statistical offices of many other countries.<ref name=X1.Australia>{{cite web |url=https://www.abs.gov.au/websitedbs/d3310114.nsf/51c9a3d36edfd0dfca256acb00118404/c890aa8e65957397ca256ce10018c9d8!opendocument|title=Time Series Analysis: Seasonal Adjustment Methods|website=abs.gov.au |date=November 14, 2005}}</ref><ref>{{cite web |author1=Susie Fortier |author2=Guy Gellatly |title=Seasonally adjusted data – Frequently asked questions|date=19 May 2015 |url=https://www.statcan.gc.ca/eng/dai/btd/sad-faq |website=Statcan.gc.ca|access-date=March 24, 2021}}</ref>
X-12-ARIMA can be used together with many statistical packages, such as [[SAS (software)|SAS]] in its econometric and time series (ETS) package, [[R (programming language)|R]] in its (seasonal) package,<ref>{{Cite web|title=seasonal: R Interface to X-13-ARIMA-SEATS version 1.8.2 from CRAN|url=https://rdrr.io/cran/seasonal/|access-date=2021-05-25|website=Rdrr.io|language=en}}</ref> [[Gretl]] or [[EViews]] which provides a [[graphical user interface]] for X-12-ARIMA, and NumXL which avails X-12-ARIMA functionality in Microsoft Excel.<ref>{{Cite web|url=https://v8doc.sas.com/sashtml/ets/chap21/sect21.htm|title=Implementation of the X-11 Seasonal Adjustment Method|website=V8doc.sas.com|access-date=24 January 2022}}{{Dead link|date=July 2025 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> There is also a version for [[MATLAB]].<ref>{{Cite web|title=X-13 Toolbox for Seasonal Filtering|url=https://www.mathworks.com/matlabcentral/fileexchange/49120-x-13-toolbox-for-seasonal-filtering|access-date=2021-05-25|website=Mathworks.com|language=en}}</ref>
Notable statistical agencies presently{{When|date=July 2021}} using X-12-ARIMA for seasonal adjustment include [[Statistics Canada]],<ref>{{Cite web|url=https://www150.statcan.gc.ca/n1/pub/12-539-x/2009001/seasonal-saisonnal-eng.htm|title=Seasonal adjustment and trend-cycle estimation|website=150.statcan.gc.ca|access-date=24 January 2022}}</ref> the U.S. [[Bureau of Labor Statistics]]<ref>{{cite web |url=http://www.bls.gov/cpi/cpisahoma.htm |url-status=dead |archive-url=https://web.archive.org/web/20030226204819/http://www.bls.gov/cpi/cpisahoma.htm |archive-date=2003-02-26 |title=BLS Handbook of Methods, Appendix A. Seasonal Adjustment Methodology at BLS}}</ref> and [[Census and Statistics Department (Hong Kong)]].<ref>{{Cite web|url=https://www.censtatd.gov.hk/hkstat/sub/sc30.jsp|title = Statistics - by Subject|website=Censtatd.gov.hk}}</ref> The [[Brazilian Institute of Geography and Statistics]] uses X-13-ARIMA.<ref>[https://ftp.ibge.gov.br/Contas_Nacionais/Contas_Nacionais_Trimestrais/Ajuste_Sazonal/X13_NasContasTrimestrais.pdf Nas Contas Trimestrais] ibge.gov.br</ref>
X-12-ARIMA was the successor to X-11-ARIMA; the current version is X-13ARIMA-SEATS.<ref>{{Cite web |url=https://www.census.gov/srd/www/x13as/ |title=X-13ARIMA-SEATS Seasonal Adjustment Program - US Census Bureau |access-date=2014-02-13 |archive-date=2014-02-22 |archive-url=https://web.archive.org/web/20140222004858/http://www.census.gov/srd/www/x13as/ |url-status=dead }}</ref>
X-13-ARIMA-SEATS's [[source code]] can be found on the Census Bureau's website.<ref name="census.gov"/>
==Methods==
The default method for seasonal adjustment is based on the X-11 algorithm. It is assumed that the observations in a time series, <math>Y_{t}</math>, can be decomposed additively,
:<math>
\begin{align}
\textit{Y}_{t} &= {T}_{t} + {S}_{t} + {I}_{t}
\end{align}
</math>
or multiplicatively,
:<math>
\begin{align}
\textit{Y}_{t} &= {T}_{t} \times {S}_{t} \times {I}_{t}.
\end{align}
</math>
In this decomposition, <math>T_{t}</math> is the trend (or the "trend cycle" because it also includes cyclical movements such as business cycles) component, <math>S_{t}</math> is the seasonal component, and <math>I_{t}</math> is the irregular (or random) component. The goal is to estimate each of the three components and then remove the seasonal component from the time series, producing a seasonally adjusted time series.<ref>{{citation |last1=Findley |first1=David F. |last2=Monsell |first2=Brian C. |last3=Bell |first3=William R. |last4=Otto |first4=Mark C. |last5=Chen |first5=Bor-Chung |date=1998 |title=New Capabilities and Methods of the X-12-ARIMA Seasonal Adjustment Program |journal=Journal of Business and Economic Statistics |volume=16 |url=https://www.census.gov/ts/papers/jbes98.pdf}}</ref>
The decomposition is accomplished through the iterative application of centered moving averages. For an additive decomposition of a monthly time series, for example, the algorithm follows the following pattern:
# An initial estimate of the trend is obtained by calculating centered moving averages for 13 observations (from <math>t-6</math> to <math>t+6</math>).
# Subtract the initial estimate of the trend series from the original series, leaving the seasonal and irregular components (SI).
# Calculate an initial estimate of the seasonal component using a centered moving average of the SI series at seasonal frequencies, such as <math>t-24, t-12, t, t+12, t+24</math>
# Calculate an initial seasonally adjusted series by subtracting the initial seasonal component from the original series.
# Calculate another estimate of the trend using a different set of weights (known as "Henderson weights").
# Remove the trend again and calculate another estimate of the seasonal factor.
# Seasonally adjust the series again with the new seasonal factors.
# Calculate the final trend and irregular components from the seasonally adjusted series.
The method also includes a number of tests, diagnostics and other statistics for evaluating the quality of the seasonal adjustments.
==Copyright and conditions==
The software is US government work, and those are in the public ___domain (in the US); for this software copyright has also been granted for other countries; the "User agrees to make a good faith effort to use the Software in a way that does not cause damage, harm, or embarrassment to the United States/Commerce."<ref name="copyrightable" />
==See also==
* [[Autoregressive integrated moving average|ARIMA]]
* [[CSPro]]
* [[Seasonality]]
==References==
{{reflist|30em}}
==External links==
* [https://www.census.gov/data/software/x13as.html X-13ARIMA-SEATS Seasonal Adjustment Program]{{snd}} documentation on website of the US Census Bureau
{{Statistical software}}
[[Category:Free econometrics software]]
[[Category:Free statistical software]]
[[Category:Public-___domain software with source code]]
[[Category:Science software for Windows]]
[[Category:Science software for Linux]]
[[Category:United States Census Bureau]]
[[Category:Time series software]]
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