UPDATE OF QUARTERLY NATIONAL ACCOUNTS MANUAL: CONCEPTS, DATA SOURCES AND COMPILATION 1 CHAPTER 7. SEASONAL ADJUSTMENT 2

Transcription

1 UPDATE OF QUARTERLY NATIONAL ACCOUNTS MANUAL: CONCEPTS, DATA SOURCES AND COMPILATION 1 CHAPTER 7. SEASONAL ADJUSTMENT 2 Table of Conens 1. Inroducion Main Principles of Seasonal Adjusmen Seasonal Adjusmen Procedure A. Preadjusmen Model Selecion Calendar Effecs Ouliers and Inervenion Variables B. Time Series Decomposiion Mehods The X-11 filer The SEATS filer Seasonal Adjusmen and Revisions A. Updae Sraegies B. Revision period Qualiy Assessmen of Seasonal Adjusmen A. Basic Diagnosics B. Advanced Diagnosics Paricular Issues A. Direc vs Indirec Seasonal Adjusmen of Aggregaes B. Relaionship among Price, Volume, and Value C. Temporal Consisency wih he Annual Accouns D. Lengh of he Series for Seasonal Adjusmen E. Seasonally Adjusing Indicaors or QNA series? F. Organizing Seasonal Adjusmen in he QNA Saus and Presenaion of Seasonally Adjused and Trend-Cycle QNA Esimaes Summary of Key Recommendaions Bibliography Boxes Box 7.1 Sofware for Seasonal Adjusmen The QNA manual is being updaed by he IMF Saisics Deparmen. For more informaion on he updae, please visi he websie hp:// 2 Draf posed for commens in Ocober 2014.

2 2 Box 7.2 Main Elemens of Seasonal Adjusmen Procedures Box 7.3 Tes for Calendar Effecs...48 Box 7.4 Tes for he Presence of Seasonaliy in he Original Series...49 Box 7.5 Tes for he Presence of Seasonaliy in he Seasonally Adjused Series...50 Box 7.6 The M Diagnosics...51 Box 7.7 Sliding Spans Tables...52 Box 7.8 Revisions Hisory Tables...53 Chars Char 7.1 Types of Ouliers...54 Char 7.2 A Simulaed Series wih Trend, Seasonal, Calendar, and Irregular effecs...55 Char 7.3 Seasonal Facors and Seasonal-o-Irregular Raios...56 Char 7.4 Presenaion of he Seasonally Adjused Series and Trend-cycle...57 Examples Example 7.1. Seasonally Adjused Series, Seasonal, Irregular and Trend-Cycle componens58 Example 7.2. Revisions o he Seasonally Adjused Series...59 Example 7.3. Revisions o he Trend-Cycle Componen...60 Example 7.4. Concurren Adjusmen vs. Curren Adjusmen...61

3 3 CHAPTER 7. SEASONAL ADJUSTMENT The purpose of seasonal adjusmen is o idenify and esimae he differen componens of a ime series, and hus provide a beer undersanding of he underlying rends, business cycle and shor-run movemens in he series. Seasonal adjusmen offers a complemenary view on he curren developmens of macroeconomic series, allowing comparisons beween quarers wihou he influence of seasonal and calendar effecs. This chaper inroduces he main principles of seasonal adjusmen. Nex, i oulines he main seps of he mos commonly used seasonally adjusmen procedures adoped by daa producer agencies. Pracical guidance is provided on how o evaluae and validae he qualiy of seasonally adjused daa. Finally, some specific issues arising from he applicaion of seasonal adjusmen in he framework of naional accouns are considered, such as he direc vs. indirec adjusmen of QNA aggregaes and he emporal consisency wih annual benchmarks. 1. INTRODUCTION 1. Seasonal adjusmen of he QNA allows a imely assessmen of he curren economic condiions and idenificaion of urning poins in key macroeconomic variables, such as quarerly GDP. Economic variables are influenced by sysemaic and recurren wihin-a-year paerns due o weaher- and social- facors, commonly referred o as he seasonal paern (or seasonaliy). When seasonal variaions dominae period-o-period changes in he original series (or seasonally unadjused series), i is difficul o idenify non-seasonal effecs, such as long-erm movemens, cyclical variaions, or irregular facors, which carry he mos imporan economic signals for QNA users. 2. Seasonal adjusmen is he process of removing seasonal and calendar effecs from a ime series. This process is performed by means of analyical echniques ha break down he series ino componens wih differen dynamic feaures. These componens are unobserved and have o be idenified from he observed daa based on a priori assumpions on heir expeced behavior. In a broad sense, seasonal adjusmen comprises he removal of boh wihin-a-year seasonal movemens and he influence of calendar effecs (such as he differen number of working days, or moving holidays). By removing he repeaed impac of hese effecs, seasonally adjused daa highligh he underlying long-erm rend and shor-run innovaions in he series. 3. In rend-cycle esimaes, he impac of irregular evens in addiion o seasonal variaions is removed. Adjusing a series for seasonal variaions removes he idenifiable, regularly repeaed influences on he series bu no he impac of any irregular evens. Consequenly, if he impac of irregular evens is srong, seasonally adjused series may no represen a smooh, easily inerpreable series. Sandard seasonal adjusmen packages provide an esimae of he rend-cycle componen, represening a combined esimae of he

4 4 underlying long-erm rend and he business-cycle movemens in he series. I should be noed, however, ha he decomposiion beween he rend-cycle and he irregular componen is subjec o large uncerainy a he end poin of he series, where i may be difficul o disinguish and allocae he effecs from new observaions. 4. A common soluion o deal wih seasonal paerns is o look a annual raes of change, ha is, compare he curren quarer o he same quarer of he previous year. Over-he-year comparisons presen he disadvanage, however, of giving signals of oudaed evens. 3 Furhermore, hese raes of change do no fully exclude all calendar-relaed effecs (e.g., Easer may fall in he firs or second quarer, and he number of working days of a quarer may differ beween subsequen years). Finally, hese year-on-year raes of change will be influenced by any evenual change in he seasonal paern caused by insiuional, climaic or behavioral changes. 5. Several mehods have been developed o remove seasonal paerns from a series. 4 Broadly speaking, hey can be divided ino wo groups: moving average mehods and modelbased mehods. Mehods in he firs group derive he seasonally adjused daa by applying a sequence of moving average filers o he original series and is ransformaions. These mehods are all varians of he X-11 mehod, originally developed by he U.S. Census Bureau (Shiskin and ohers, 1967). 5 The curren version of he X-11 family is X-13ARIMA-SEATS (X-13A-S), which will ofen be referred o in his chaper. Model-based mehods derive he unobserved componens in accord wih specific ime series models, primarily auoregressive inegraed moving average (ARIMA) models. The mos popular model-based seasonal adjusmen mehod is TRAMO-SEATS, 6 developed by he Bank of Spain (Gomez and Maravall, 1996). Box 1 illusraes he main characerisics of he X-13A-S and TRAMO- SEATS programs. Oher available seasonal adjusmen mehods include, among ohers, BV4, SABLE, and STAMP. 3 Quenneville and Findley (2012) found analyically ha year-on-year changes presen 5.5 monhs of delay wih respec o monh-o-monh growh raes. For quarerly series, his corresponds o a delay of almos wo quarers. 4 For a hisory of seasonal adjusmen mehods, see Ladiray and Quenneville (2001, chaper 1). 5 The X-11 program was he firs procedure designed for a large-scale applicaion of seasonal adjusmen. I emerged from decades of research originaed in he early 1930s by researchers a he NBER. Subsequen improvemens o he original X-11 program were implemened in he X-11-ARIMA program, developed by Saisics Canada (Dagum, 1980), and in he X-12-ARIMA program, developed by he U.S. Census Bureau (Findley and ohers, 1998). For furher deails on he hisory of X-11, see Box 1 and Ghysels and Osborn (2001). 6 TRAMO is he acronym for Time Series Regression wih Auoregressive inegraed moving average (ARIMA) Errors and Missing Observaions. SEATS sands for Signal Exracion for ARIMA Time Series.

5 5 6. Curren seasonal adjusmen packages offer buil-in funcionaliy o selec beween alernaive modeling opions in an auomaic manner (e.g., ARIMA model, calendar effecs, addiive vs. muliplicaive model). The selecion process mosly relies on saisical ess or heurisic rules based on he seasonal adjusmen resuls. These auomaic feaures are very helpful when seasonal adjusmen is o be applied o many ime series a a ime (hundreds, or even housands), avoiding a series-by-series, ime-consuming manual selecion process. However, compilers should use hese auomaic feaures wih care. Seps performed by he seasonal adjusmen procedure used in he QNA should be assessed and comprehended, as wih every oher mehod applied in he naional accouns. Seasonal adjusmen opions, a leas for he mos relevan QNA series, should always be esed for adequacy and moniored over ime. 7. Seasonally adjused daa should no replace he original QNA daa. Some users prefer o base heir economic analysis on unadjused daa, as hey rea seasonaliy as an inegraed par of heir modeling work. In his regard, seasonal adjusmen adoped by saisical agencies is someimes seen as a poenially dangerous procedure ha may compromise he inrinsic properies of he original series. In fac, here is always some loss of informaion from seasonal adjusmen, even when he seasonal adjusmen process is properly conduced. For his reason, producers of seasonally adjused daa should employ sound and inernaionally-acceped mehodology for seasonal adjusmen. More imporanly, hey should implemen a ransparen communicaion sraegy, indicaing he mehod in use and inegraing seasonally adjused figures wih appropriae meadaa ha allow he resuls o be replicaed and undersood by he general public. 8. Counries ha are ye o produce QNA in seasonally adjused form may follow an evoluionary approach o seasonal adjusmen. In a firs sage, seasonal adjusmen should be applied o he mos imporan aggregaes of he QNA (such as he GDP). For some ime, hese seasonally adjused series may be used inernally or published as experimenal daa. Nex, seasonal adjusmen could be expanded o he full se of QNA series once compilers gain more experience and confidence in he seasonal adjusmen work. Albei no published, seasonal adjusmen of QNA daa should a leas be done inernally; in effec, seasonally adjused daa ofen faciliae he idenificaion of issues in he unadjused daa as seasonaliy may hide errors and inconsisencies in he original esimaes. 9. This chaper is srucured as follows. The nex secion illusraes he main principles of seasonal adjusmen. Secion 3 oulines he wo sages of seasonal adjusmen procedures: preadjusmen and ime series decomposiion. I also provides a brief illusraion of he X-11 (moving average) filer and SEATS (model-based) filer. Secion 4 sresses he imporance of revisions in seasonal adjusmen, and how o handle and communicae hem properly in a producion conex. Qualiy assessmen ools for analyzing seasonal adjusmen resuls are described in secion 5. Secion 6 addresses a se of criical issues on seasonal adjusmen

6 6 specifically relaed o QNA issues, such as preservaion of accouning ideniies, seasonal adjusmen of balancing iems and aggregaes, and he relaionship beween annual daa and seasonally adjused quarerly daa. Finally, secion 7 discusses he presenaion and saus of seasonally adjused and rend-cycle daa. 2. MAIN PRINCIPLES OF SEASONAL ADJUSTMENT 10. For seasonal adjusmen purposes, a ime series is generally assumed o be made up of four main componens: he rend-cycle componen, he seasonal componen, he calendar componen, and he irregular componen. These componens are unobserved and have o be idenified (and esimaed) from he observed ime series using a signal exracion echnique. 11. The rend-cycle componen ( T ) is he underlying pah of he series. I includes boh he long-erm rend and he business-cycle movemens in he daa. The long-erm rend can be associaed wih srucural changes in he economy, such as populaion growh and progress in echnology and produciviy. Business cycle variaions are relaed o he periodic oscillaions of differen phases of he economy (i.e., recession, recovery, growh, and decline), which generally repea hemselves wih a period beween wo o eigh years. 12. The seasonal componen ( S ) includes hose seasonal flucuaions ha repea hemselves wih similar annual iming, direcion, and magniude. 7 Possible causes of seasonal movemens relae o climaic facors, adminisraive or legal rules, and social/culural radiions and convenions - including calendar effecs ha are sable in annual iming (e.g., public holidays, or oher naional fesiviies). Each of hese causes (or a combinaion of hem) can affec expecaions in such a way ha seasonaliy is indirecly induced. Similarly, changes in any of hese causes may change he properies of he seasonal paern. 13. The calendar componen ( C ) comprises effecs ha are relaed o he differen characerisics of he calendar from period o period. Calendar effecs are boh seasonal and nonseasonal. Only he nonseasonal par should be included in he calendar componen and reaed separaely, as he seasonal one is already caugh by he seasonal componen. 8 The mos used calendar effecs include he following: Trading-day or working-day effecs. The rading-day effec deecs he differen number of each day of he week wihin a specific quarer relaive o he sandard weekday composiion of a quarer. The working-day effec 7 Seasonaliy may be gradually changing over ime. This phenomenon is called moving seasonaliy. 8 For example, he effec due o he differen average number of days in each quarer is par of he seasonal effecs.

7 7 caches he difference beween he number of working days (e.g., Monday hrough Friday) and he number of weekend days (e.g., Saurday and Sunday) in a quarer. The rading-day effec assumes an underlying paern associaed wih each day of he week; he working-day effec posulaes differen behavior beween he groups of weekdays and weekend days. 9 Boh he rading-day and working-day effecs should incorporae he effecs of naional holidays (e.g., when Chrismas falls on Monday, ha Monday should no be couned as a rading/working day). Moving holiday effec. A moving holiday is associaed wih evens of religious or culural significance wihin a counry ha change dae from year o year (e.g., Easer or Ramadan). Leap year effec. This effec is needed o accoun for he exra day in February of a leap year, which may generae a four-year cycle wih a peak in he firs quarer of leap years. 14. The irregular componen ( I ) capures all he oher flucuaions ha are no par of he rend-cycle, seasonal, and calendar componens. These effecs are characerized by he fac ha heir iming, impac, and duraion are unpredicable a he ime of heir occurrence. The irregular componen includes he following effecs: Oulier effecs. These effecs manifes hemselves wih abrup changes in he series, someimes relaed o unexpeced weaher or social-economic effecs (such as naural disasers, srikes, or economic and financial crises). Such effecs are no par of he underlying linear daa generaion process assumed for he original series. For hese reasons, oulier effecs are also called nonlinear effecs. In he seasonal adjusmen process, ouliers should be removed by means of pre-defined inervenion variables. Three main ypes of ouliers are ofen used for economic ime series: i. Addiive oulier, which relaes o only one period; 9 Trading-day effecs are less imporan in quarerly daa han in monhly daa. Only quarer 3 and quarer 4 conain differen numbers of working days over ime (excluding he leap year effec). The working-day effec is mos commonly used for quarerly series.

8 8 ii. Level shif, which changes he level of a series permanenly; 10 iii. Transiory change, whose effecs on a series fade ou over a number of periods. Oher effecs are seasonal ouliers (which affec only cerain quarers/monhs of he year), ramp ouliers (which allow for a linear increase or decrease in he level of a series), or emporary level shifs. These effecs are modeled hrough specific inervenion variables. The differen ypes of ouliers are shown in Char 7.1. Furher deails on he reamen of oulier effecs will be given in Secion 3.A. Whie noise effecs. In he absence of ouliers, he irregular componen is assumed o be a random variable wih normal disribuion, uncorrelaed a all imes wih consan variance. In saisical erms, such a variable is called a whie noise process. Differenly from oulier effecs, a whie noise process is assumed o be par of he underlying linear daa generaion process of he series. 15. The purpose of seasonal adjusmen is o idenify and esimae he differen componens of a ime series, and hus provide a beer undersanding of he underlying rends, business cycle and shor-run movemens in he series. The arge variable of a seasonal adjusmen process is he series adjused for seasonal and calendar effecs (or seasonally and calendar adjused series). As menioned before, boh seasonal and calendar effecs should be removed from he original series o allow for a correc analysis of he curren economic condiions. 16. A fundamenal prerequisie for applying seasonal adjusmen procedures is ha he processed series should presen clear and sufficienly sable seasonal effecs. Series wih no seasonal effecs, or series wih seasonal effecs ha are no easy o idenify from he original series, should no be seasonally adjused. As discussed in he nex secion, he original series should always be esed for he presence of idenifiable seasonaliy. A he same ime, he series should also be esed for he presence of calendar effecs. Calendar effecs are usually less visible han seasonal effecs, herefore heir idenificaion relies on saisical ess ha reveal when heir conribuion o he series is saisically differen from zero. 10 For seasonal adjusmen purposes, ouliers producing srucural breaks in he series (such as a level shif or a seasonal oulier) may acually be allocaed o he rend or seasonal componens. Furher deails on he allocaion of ouliers will be given in Secion 3.A.

9 9 17. Two observaions on he limis of seasonal adjusmen are worh noing here. Firs, seasonal adjusmen is no mean for smoohing series. A seasonally adjused series 11 is he sum of he rend-cycle componen and he irregular componen. As a consequence, when he irregular componen is srong he seasonally adjused series may no presen a smooh paern over ime. To exrac he rend-cycle componen, he irregular componen should be furher removed from he seasonally adjused series. Trend-cycle exracion is a difficul exercise and subjec o greaer uncerainy han seasonal adjusmen, especially in he final period of a series. 18. Second, seasonal adjusmen and rend-cycle esimaion represen an analyical processing of he original daa. As such, he seasonally adjused daa and he esimaed rendcycle componen complemen he original daa, bu hey can never replace he original daa for he following reasons: Unadjused daa are useful in heir own righ. The non-seasonally adjused daa show he acual economic evens ha have occurred, while he seasonally adjused daa and he rend-cycle esimae represen an analyical elaboraion of he daa designed o show he underlying movemens ha may be hidden by he seasonal variaions. Compilaion of seasonally adjused daa, exclusively, represens a loss of informaion. No unique soluion exiss on how o conduc seasonal adjusmen. Seasonally adjused daa are subjec o revisions as fuure daa become available, even when he original daa are no revised. When compiling QNA, balancing and reconciling he accouns are beer done on he original unadjused QNA esimaes. While errors in he source daa may be more easily deeced from seasonally adjused daa, i may be easier o idenify he source for he errors and correc he errors working wih he unadjused daa. 19. Char 7.2 shows a quarerly ime series spanning 20 years of daa. This series has been simulaed using a well-known seasonal ARIMA model wih calendar effecs (see he char for deails). The series shows an eviden upward rend, sable seasonal effecs (high in quarers 3 and 4, low in quarers 1 and 2), plus oher non-sysemaic, random movemens. This series will be used hroughou his chaper o illusrae he differen sages of a seasonal adjusmen process, which are described in he following secion. 11 Unless oherwise specified, we indicae hereafer wih seasonally adjused series a series adjused for boh seasonal and calendar effecs (if hey are presen).

10 10 3. SEASONAL ADJUSTMENT PROCEDURE 20. A seasonal adjusmen procedure follows a wo-sage approach (see he diagram in Box 7.2). The firs sage is called preadjusmen. The objecive of preadjusmen is o selec a regression model wih ARIMA errors ha bes describes he characerisics of he original series. The chosen model is used o adjus he series for deerminisic effecs (from which he name preadjusmen is aken), and o exend he series wih backcass and forecass o be used in he ime series decomposiion process. The preadjusmen sage comprises mainly he choice of (i) how he unobserved componens are relaed o each oher (addiive, muliplicaive, or oher mixed forms); (ii) he order of he ARIMA model; (iii) calendar effecs 12 ; and (iv) ouliers and oher inervenion variables. 21. The second sage performs a decomposiion of he preadjused series ino unobserved componens. The series adjused for deerminisic effecs is decomposed ino hree unobserved componens: rend-cycle, seasonal and irregular. This secion will illusrae he wo mos applied decomposiion mehods for seasonal adjusmen: he X-11 filer and he SEATS filer. Afer unobserved componens are esimaed, he adjusmen facors idenified in he firs sage (calendar effecs, ouliers, ec.) are allocaed o heir respecive componen so o end up wih a full decomposiion of he original series ino final rend-cycle, seasonal (including calendar effecs), and irregular componens. The seasonally adjused series is obained as he series wihou seasonal and calendar effecs. 22. Alhough he wo seps are reaed separaely, hey should be considered fully inegraed in any seasonal adjusmen procedure. Differen choices in he preadjusmen phase lead o differen decomposiion resuls. Also, resuls from he ime series decomposiion may poin o changes in he preadjusmen sage. A careful analysis of he diagnosics in he wo sages (as discussed in secion 5) is fundamenal o deermine wheher he seasonal adjusmen resuls are of accepable qualiy. 23. The X-13A-S program implemens his wo-sage procedure. This sofware allows he user he choice beween he X-11 and he SEATS filers wihin he same environmen. 13 Because he same diagnosics are produced for he wo filers, a comparaive assessmen beween he wo mehods is now feasible for any series. Thanks o his flexibiliy, X-13A-S 12 The X-13A-S program offers an alernaive mehod o esimae rading-day effecs from he irregular componen, inheried from he original X-11 mehod. However, he regression framework is he preferred approach for idenifying and esimaing calendar effecs. 13 X-13A-S implemens a version of he SEATS procedure originally developed by Gómez and Maravall (1998). The X-13A-S manual cauions ha possible delays in he updaing of versions may cause sligh differences beween he X-13A-S version of SEATS and he one a he Bank of Spain websie. (U.S. Census Bureau, 2013).

11 11 is (a he ime of wriing) he recommended seasonal adjusmen procedure for producing seasonally adjused QNA daa The res of his chaper briefly presens he main elemens of he preadjusmen and decomposiion seps. Model Selecion A. Preadjusmen 25. The firs sep in he preadjusmen sage is o deermine he decomposiion model assumed for he series. For he X-11 decomposiion, wo main models are usually seleced: he addiive model and he muliplicaive model. 15 In he addiive model he original series X can be hough of as he sum of unobserved componens, ha is X T S C I, (1) where T is he rend-cycle componen; S is he seasonal componen; C is he calendar componen; and I is he irregular componen. The addiive model assumes ha he unobserved componens are muually independen from one anoher. The seasonal and calendar adjused series for he addiive model is derived by subracing he seasonal and calendar componens from he original series a X X ( S C ) T I. (2) 14 This chaper provides hins on how o se opions in he X-13A-S inpu specificaion file. The inpu specificaion file conains a se of specificaions (or specs ) ha give informaion abou he daa and desired seasonal adjusmen opions. For more deails on he heory and pracice of seasonal adjusmen, see he X-13A- S guide (U.S. Census Bureau, 2013) and he lieraure herein cied. 15 In some cases, a mixed model may be chosen. In paricular, X-13A-S includes a pseudo-addiive model X T( S I 1) for series ha show a muliplicaive decomposiion scheme bu whose values are zero in some periods.

12 In he muliplicaive model, he series X is decomposed as he produc of he unobserved componens: X T S C I. (3) The muliplicaive model assumes ha he magniude of he unobserved componens is proporional o he level of he series. For he seasonal componen, for example, a muliplicaive model implies ha seasonal peaks increase as he level of he series increases. Because he rend-cycle componen deermines he overall level of he series, he oher unobserved componens are expressed as percenages of T (usually called facors). The seasonal and calendar adjused series for he muliplicaive model is he raio beween he original series and he seasonal and calendar facors: a X X /( S C ) T I. (4) 27. Wih SEATS, he muliplicaive model canno be used direcly because he modelbased decomposiion assumes ha he unobserved componens are addiive. The muliplicaive adjusmen is approximaed hrough he log-addiive model log( X ) log( T S C I ) log( T) log( S ) log( C ) log( I ). (5) Afer an addiive decomposiion of he logged series is compleed, he seasonally adjused series is derived aking he exponenial of he logged rend-cycle and irregular componens a X exp[log( T) log( I )] T I. (6) 28. Someimes, a graphical inspecion of he series can give some clues abou he bes decomposiion model for he series. If he seasonal paern increases over ime, hen he relaionship beween componens is expeced o be muliplicaive and a muliplicaive (or a log-addiive) adjusmen is recommended. Such a ransformaion allows sabilizing he evoluion of he seasonal paern and conrolling for possible heeroschedasiciy in he irregular componen (and in he regression residuals). Alernaively, if he seasonal paern appears o be sable over ime and does no evolve in accordance wih he movemens of he rend, hen no ransformaion is o be performed and he decomposiion should follow an addiive approach. 29. A visual inspecion of he series may no be enough o deermine he underlying relaionship beween componens. In addiion o he exper knowledge abou he series, X-

13 13 13A-S implemens an auomaic selecion procedure o decide wheher he series should be log ransformed or no. 16 This auomaic ool should be used when seasonal adjusmen is applied for a large number of ime series. However, he auomaic choice from he X-13A-S should always be validaed individually for imporan ime series. 30. If he muliplicaive approach is chosen, final componens have he naure of muliplicaive facors, ha is, he seasonal and irregular componens will be raios cenered around 1. On he oher hand, if he addiive approach is seleced, seasonal and irregular componens will have he form of addends and will be cenered around 0 (addiively neural). 31. The nex sep in he preadjusmen phase is o idenify an ARIMA model for he series. The ARIMA selecion process should be seen in conjuncion wih he choice of regression effecs. In effec, using cerain regression variables may change he order of he ARIMA model. An ARIMA wih regression effecs is called regarima model. For he sake of clariy, however, ARIMA model and regression effecs are reaed separaely in his presenaion. 32. Using he same noaion of he X-13A-S manual, an ARIMA model for seasonal ime series can be wrien as ( B) ( B s )(1 B) d (1 B s ) D Y ( B) ( B s ) (7) where Y is he original series (possibly preadjused for deerminisic effecs); B is he lag operaor, which is defined by Y 1 BY ; s is he seasonal frequency, 4 for quarerly series and 12 for monhly series; p ( B) 1 B B is he regular auoregressive (AR) operaor of order p; 1 p ( B) 1 B B is he seasonal AR operaor of order P; 1 4 4P P q ( B) 1 B B is he regular moving average (MA) operaor of order q; 1 q Q ( B) 1 B B 1 4Q Q is he seasonal MA operaor of order Q; and 16 The spec TRANSFORM enables he auomaic ransformaion selecion procedure. TRANSFORM can also be used o selec auomaically beween he addiive and muliplicaive models for X-11.

14 14 is a whie noise process. 33. Idenifying an ARIMA model consiss in deermining he orders of he AR operaors (p for nonseasonal and P for seasonal), he MA operaors (q and Q), as well as esablishing he non-seasonal and seasonal inegraion orders (d and D). 17 A seasonal ARIMA model for a quarerly series is usually indicaed as ( pdq,, )( PDQ,, ) 4. In X-13A-S, he following auomaic selecion procedure is implemened o idenify he ARIMA order: 18 a defaul model is esimaed. The defaul model for boh monhly and quarerly series is (0,1,1)(0,1,1) 4, also known as he airline model. This model is parsimonious (only wo parameers are esimaed) and usually fis very well economic ime series. Regression effecs are also idenified and removed using he defaul model; he differencing orders d and D are esimaed by performing a series of uni roo ess; he ARMA 19 order ( pq, )( PQ, ) 4 is seleced by comparing values of a saisical informaion crierion 20 of a number of models, up o a maximum order for he regular and seasonal ARMA polynomial which can be specified by he user; diagnosics on he residuals for he chosen ARIMA model are compared wih hose from he defaul model. Based on hese ess, he final model is seleced and validaed. 34. Selecing he correc ARIMA model has imporan consequences in he seasonal adjusmen process. Firs, he chosen order of he ARIMA model may affec he auomaic selecion process of calendar effecs and ouliers. Second, he ARIMA model is used o produce forecass and backcass ha are necessary o apply symmeric filers a he endpoins of he series. Third, he ARIMA order chosen for he original series is cenral o he SEATS 17 Inegraed series refers o non saionary underlying processes ha mus be differeniaed in order o urn he series ino a saionary process. The order of inegraion (i.e., number of uni roos in he AR polynomial) reflecs he need for differeniaion of he ime series o become saionary. 18 The auomaic ARIMA model idenificaion procedure implemened in X-13A-S is based on he one available in he TRAMO program (Gomez and Maravall, 1996). In X-13A-S, he reference spec is AUTOMDL. For more deails on he procedure, refer o he X-13A-S manual (U.S. Census Bureau, 2013). 19 Auoregressive moving average (ARMA). In conras wih ARIMA models, ARMA processes are saionary and do no require differeniaion. 20 X-13A-S uses he Bayesian Informaion Crierion (BIC) for selecing he ARMA order. The bes model is he one wih he minimum BIC value.

15 15 decomposiion, because he seasonal and rend filers are derived from he coefficiens of he esimaed ARIMA model (as discussed laer in his secion). 35. In general, users should accep he ARIMA order auomaically seleced by he X- 13A-S program. However, acions should be aken when diagnosics on he residuals signal misspecificaion of he idenified model. Parsimonious models should always be preferred o complex models. The differencing orders d and D should never be larger han one. Mixed ARMA models, which are models where AR and MA operaors are boh presen in he seasonal or nonseasonal pars, should usually be avoided. For mos series, he defaul model (0,1,1)(0,1,1) 4 ofen works very well wih seasonal economic ime series and should be considered when no oher model provides saisfacory resuls. Calendar Effecs 36. Calendar effecs should be removed from he series because hey could affec negaively he qualiy of decomposiion ino unobserved componens. For example, consider he effec from a differen number of working days in wo periods. When a monh conains more working days han usual, series measuring economic aciviies may presen a spike in ha paricular monh due o he fac ha here is more ime for producion. This effec canno be capured by any linear represenaion of he series (like an ARIMA model), and will be allocaed in he ime series decomposiion process mosly o he irregular componen. As a resul, he seasonally adjused series will presen an increase ha is merely aribuable o he differen number of working days in he wo periods compared. To avoid such disorions, calendar effecs should be esimaed and eliminaed from he original series before he ime series decomposiion process. 37. All calendar effecs are capured hrough specific deerminisic effecs ha are mean o reproduce he changes in he calendar srucure over ime. These deerminisic effecs are called calendar regressors, as hey are used as independen variables in he regarima model specified in he seasonal adjusmen process. The mos frequenly used calendar regressors are formalized below. 38. The rading-days effec is defined by he following six regressors The following presenaion is based on he Monday-o-Friday workweek in place in mos Wesern counries (he defaul one in X-13A-S). However, oher counries have differen workweeks (in paricular Sunday-o- Thursday in many Muslim counries). The groups of weekdays and weekend days should be defined according o he legal workweek in he counry.

16 16 d d... d (# Mondays #Sundays) (# Tuesdays #Sundays) (#Saurdays #Sundays) (8) which calculae he difference beween he number of each day of he week (#Mondays, #Tuesdays,...) and he number of Sundays 22 (#Sundays) in monh. The assumpion is ha each day of he week may influence he underlying phenomenon wih differen magniude and direcion. 39. The working-days effec is caugh via a single regressor ha compares he group of working days (e.g., Monday o Friday) wih he group of weekend days (e.g., Saurday and Sunday) hrough he equaion wd 5 # Weekdays # Weekend days 2. (9) The 5 facor is needed o make he working-days regressor nil over a regular seven-day 2 week composiion. Any monhly deviaion from he sandard week will be refleced in he regressor (e.g., when wd is larger han zero, i means ha monh/quarer has more working days han a sandard week). This approach assumes ha weekdays have similar effecs (in sign and value) and are differen from weekend days effecs. 40. The Easer dae moves beween March (q1) and April (q2). 23 The Easer regressor calculaes he proporion of days before Easer falling in March (q1) and April (q2). Afer defining he lengh of he Easer effec, he regressor is calculaed as e W w W (10) where W is he number of w days falling in monh/quarer ; and 22 Naional holidays should be considered non-working days. Therefore, he number of non-working days should be increased by he number of naional holidays and he number of working days should be decreased accordingly. The same holds rue for he working-days regressor. 23 Only he Caholic Easer is considered here. Orhodox Easer falls beween April and May, hus i does no affec quarerly series.

17 17 W is he long-erm proporion of days in monh/quarer. Usually W can be approximaed wih 0.5 for boh March (q1) and April (q2) 24, ha is he number of days of he Easer effec is equally disribued beween he wo periods. In X-13A- S, he lengh w of he Easer effec can be provided by he user (from 1 o 25) or seleced auomaically by he program (lenghs of 1, 8, 15 are compared). 41. Finally, he leap year effec is capured as follows: ly 0.75, if is February of a leap year 0.25, if is February of a non-leap year 0, oherwise. (11) The regressor ly reproduces a deerminisic four-year cycle wih a peak in February of leap years; over a four-year period, he leap year effec is fully compensaed by he negaive effecs in he subsequen non-leap years. 42. The adjusmen for calendar effecs should be performed only for hose series for which here is boh saisical evidence and economic inerpreaion of calendar effecs. This assessmen should be based on he saisical and economic significance of heir regression coefficiens. Saisically, a regression coefficien is said o be significanly differen from zero when he associaed -saisic is higher (in absolue value) han a cerain hreshold (usually 2, bu lower hresholds may be accepable). Furhermore, he sign of he regression coefficien should be inerpreable from an economic sandpoin. For example, he leap year effec should always be posiive; he working-days effec for economic aciviies where producion is organized on a five-day week should be posiive; he Easer effec should be posiive for consumpion of ourism-relaed services 25 and negaive for oher producing aciviies, ec. When he esimaed coefficien for a calendar effec is no saisically significan (i.e., -saisic lower han a chosen hreshold) or is difficul o inerpre in economic erms (i.e., implausible size or sign of he coefficien), he series should no be adjused for ha calendar effec. As an example, Box 7.3 shows he oupu reurned by X- 13A-S o evaluae he resuls on calendar effecs. 43. Two compilaion aspecs concern he frequency of calculaion of calendar effecs. Firs, calendar effecs are saisically more eviden on monhly series han on quarerly 24 The Easer regressor may also be nonzero in February, bu his happens very rarely. 25 In some counries, Easer creaes a peak in reail rade aciviy due o he increase in household spending; in oher counries, however, mos shops are closed during he Easer holiday.

18 18 series. The quarerly aggregaion reduces (and someimes eliminaes) he variabiliy of calendar regressors up o a level ha makes hem hardly deecable in he esimaion process. For his reason, he adjusmen for calendar effecs should be preferably performed on monhly indicaors and hen he resuling effec aggregaed a he quarerly level. 44. On he oher hand, rading-days and working-days effecs may also be relevan a he annual frequency. Adjacen years may conain up o 3-4 working days of difference, which may disor he comparison beween annual observaions. When such effecs are significan on an annual basis, i may be necessary o calculae annual aggregaes adjused for calendar effecs (mosly rading/working-days and leap year) and use hese as annual benchmarks for he quarerly seasonally and calendar adjused esimaes. 26 When calendar effecs are negligible on an annual basis, quarerly seasonally and calendar adjused esimaes can be benchmarked o he original ANA aggregaes. 45. X-13A-S provides predefined calendar effecs. Furhermore, he program allows userdefined regressors o be included in he regarima model. The user can prepare any specific calendar effec and es is economical and saisical significance from he resuls reurned by he program. This funcionaliy is imporan for adjusing QNA daa for counry-specific effecs no included as a buil-in opion of X-13A-S (e.g., Chinese New Year, Ramadan, ec.). An auomaic selecion procedure of calendar effecs (boh buil-in and user-defined) is available. 27 Similarly o he ARIMA order, he auomaic selecion procedure should always be used when seasonal adjusmen is applied o a large number of ime series. However, he sign of each calendar effec acceped by X-13A-S should always be evaluaed in economic erms. Moreover, regression coefficiens associaed wih calendar effecs should remain sable as new observaions are incorporaed in he series. Esimaed calendar effecs ha are no suppored by economic raionale should no be included in he adjusmen Annual daa adjused for calendar effecs can be obained by aggregaing he quarerly calendar adjused series reurned by X-13A-S. When calendar adjusmen is applied o monhly indicaors, annual daa of naional accouns adjused for calendar effecs should be derived proporionally o he adjusmen derived on he indicaor or via a regression approach (Di Palma and Marini, 2004). 27 In X-13A-S, he spec name for he auomaic idenificaion of calendar effecs is REGRESSION. The es is based on he Akaike Informaion Crierion (AIC). 28 X-13A-S also provides anoher opion for esimaing rading-day and Easer effecs based on ordinary leassquare (OLS) regression analysis from he final irregular componen. The spec name for his opion is X11REGRESSION. However, regarima models for esimaing calendar effecs are preferred because hey usually provide beer resuls han OLS regressions.

19 19 Ouliers and Inervenion Variables 46. Unusual evens canno be prediced ex-ane, bu once hey have manifesed in he series, hey should be undersood and modeled in he seasonal adjusmen process hrough specific regression variables. The reason is ha leaving abnormal values in he series may lead o significan disorion in he decomposiion of QNA series such as producion, consumpion, invesmen, ec. For insance, unexpeced exreme weaher condiions (droughs, floods, ec.) can seriously affec oupu of agriculural crops. The sudden fall in he agriculural aciviy should be allocaed o he irregular componen, wihou influencing he long-erm rend or seasonaliy in agriculure. Oher unusual evens may be allocaed o he rend (e.g., level shif) or seasonaliy (e.g., seasonal break). To achieve his, abnormal values (commonly known as ouliers) should be aken ou he original series and re-inroduced in he final componens afer he decomposiion sep has been applied o he series preadjused for such evens. Oher known evens ha are supposed o have a significan impac on he series should be reaed in he preadjusmen sep by means of inervenion variables (e.g., srikes, emporal shudowns, quaranines). 47. X-13A-S conains a procedure for auomaic idenificaion of addiive ouliers, emporary change ouliers, and level shifs (see Char 7.1). 29 This procedure consiss of including dummy-ype variables in he regression model for all possible periods wihin a specified ime span. The program calculaes regression coefficiens for each ype of oulier specified and adds o he model all ouliers wih absolue -saisics exceeding a criical value. 30 Furhermore, X-13A-S allows he use of predefined inervenion variables. 31 As menioned in Secion 2, hree common inervenion variables are emporary level shifs, seasonal ouliers, and ramps (see Char 7.1). Oher inervenion variables can be creaed from he user exernally and given as inpu o he program. 48. Seasonal adjusmen resuls are severely affeced by ouliers and inervenion variables. A differen combinaion of regression effecs can produce significan changes in he esimaion of rend and seasonal componens. As for any oher regression effecs, ouliers should be evaluaed based on he saisical significance of heir regression coefficiens (hrough -saisics) and heir robusness. Ouliers are paricularly difficul o deec and inerpre in real-ime, especially during periods of srong economic changes such as recessions. 32 When abnormal values firs arise in a series, hey should eiher be adjused as 29 In X-13A-S, he spec name for he auomaic idenificaion of ouliers is OUTLIER. 30 Defaul criical values for he oulier -saisics depend on he series lengh. 31 In X-13A-S, he spec name for including inervenion variables in he regarima model is REGRESSION. 32 For a discussion on how o handle recession effecs on seasonal adjusmen, see he experimens in Ciammola and ohers (2010) and Lyras amd Bell (2013).

20 20 addiive ouliers or lef unadjused. Level shifs or oher ransiory effecs involving more han one period should be aken ino consideraion only when fuure observaions of he series make clear he naure of he even. B. Time Series Decomposiion Mehods 49. For seasonal adjusmen of QNA daa, a choice should be made beween wo alernaive mehods: he X-11 filer and he SEATS filer. These mehods are welldocumened and have become sandard mehods for seasonal adjusmen of official saisics. Furhermore, heir use increases comparabiliy of seasonally adjused ime series across counries. 50. Boh mehods give saisfacory resuls for mos ime series and are equally recommendable. Counries should choose heir preferred mehod according o saisical and pracical consideraions. The fac ha X-13A-S offers boh filers in he same program allows an easy comparison on series wih differen characerisics using a common se of diagnosics. 33 However, he choice may also be grounded on pas experience, inernal experise and subjecive judgmen. Once he choice is made, he same mehod should be used o seasonally adjus all he QNA series (indicaors or final resuls) and clearly communicaed o he public. Mixing differen seasonal adjusmen mehods in he same saisical domain may reduce he level of comparabiliy of seasonally adjused series and cause confusion in he users. 51. Boh X-11 and SEATS apply symmeric filers o he preadjused series o derive esimaes of rend-cycle, seasonal, and irregular componens. However, he naure of such filers differs significanly from one anoher. The following provides a brief descripion and highlighs he main differences beween he wo mehods. The X-11 filer 52. The X-11 filer is derived as an ieraive process, which consiss in applying a sequence of predefined moving average filers. Afer he series is preadjused and exended wih backcass and forecass, i goes hrough hree rounds of filering and exreme value adjusmens called B, C, and D ieraions. 53. The moving average filering procedure implicily assumes ha he irregular effec is approximaely symmerically disribued around heir expeced value (1 for a muliplicaive and 0 for an addiive model) and hus can be fully eliminaed by using a symmeric moving 33 The original version of SEATS, developed by Gomez and Maravall (1996), is implemened in he program TRAMO-SEATS (available a he Bank of Spain websie). X-13A-S provides a close approximaion of he SEATS decomposiion resuls.

21 21 average filer. Therefore, seasonaliy and rend cycle componen are isolaed from he irregular componens by means of a successive applicaion of ad-hoc moving average filers. 54. The main seps of he X-11 muliplicaive adjusmen 34 for quarerly daa in he B, C, and D ieraions are below reproduced: Ieraion B. Iniial Esimaes 1 a. Iniial rend-cycle ( T ). The original series Y is filered using a weighed 5- erm (2 x 4) 35 cenered moving average, which exracs an iniial rend componen from he series. 1 b. Iniial seasonal- irregular raios ( SI ). The original series is divided by give an iniial (join) esimae of he seasonal and irregular componens 1 T o c. Iniial preliminary seasonal facors. Irregular effecs from he iniial SI raios are removed by applying a weighed 5-erm (3 x 3) cenered seasonal moving average so as o derive an iniial preliminary esimae of he seasonal facors. 1 d. Iniial seasonal facors ( S ). Preliminary seasonal facors are hen normalized o ensure ha he annual average of he iniial seasonal facors is close o 1. e. Iniial seasonally adjused series ( adjused series 1 seasonal facors S, ha is SI 1. 1 A ). The iniial esimae of he seasonally 1 A is derived by dividing he original series by he iniial A Y TI 1 S Ieraion C. Seasonal Facors and Seasonal Adjusmen 34 Muliplicaive decomposiion is he defaul X-11 mehod in X-13A-S. However, he decomposiion mehod should be consisen wih he ype chosen in he preadjusmen sage (auomaically or manually). In he case of addiive decomposiion, subracions are used insead of divisions. The spec o modify opions of he sandard X-11 filer is X An N x M cenered moving average is obained by applying simple moving averages of lengh N and M in succession. See Ladiray and Quenneville (2001) for a discussion of he properies of moving averages used in X-11.

22 22 2 a. Inermediae rend-cycle ( T ). A revised esimae of he rend-cycle is derived by applying a Henderson filer 36 1 o he iniial seasonally adjused series A. The Henderson filer is a (2h+1)-erm symmeric filer whose values are designed o exrac a rend componen from he inpu series. For quarerly series, X-13A-S selecs auomaically a 5- or a 7-erm Henderson moving average based on he saisical characerisics of he daa. b. Revised SI raios ( SI ). Revised SI raios are derived by dividing he original series Y by he inermediae rend-cycle 2 2 T. c. Revised preliminary seasonal facors are derived by applying a (3 x 5) cenered seasonal moving average o he revised SI raios SI. 2 d. Revised seasonal facors ( S ). As in sage B, revised preliminary facors are normalized o produce revised seasonal facors. e. Final seasonally adjused series ( revised seasonal facors Ieraion D. Final Trend and Irregular 2 A ). The original series is divided by he 2 S o derive he final seasonally adjused series. 3 a. Final rend-cycle ( T ). A final esimae of he rend-cycle componen is derived by applying a Henderson moving average o he final seasonally adjused series A. 2 b. Final irregular ( I 3 ). A final esimae of he irregular componen is derived by dividing he final seasonally adjused series 2 3 A by he final rend-cycle T. 55. In addiion o he hree-sage procedure described above, he X-11 filer implemens an algorihm o reduce he impac of exreme values in he adjusmen process. Based on a saisical analysis of SI raios, exreme values are idenified and emporarily replaced wih average values in sages B and C, so as o eliminae heir effecs from seasonal facors. 56. One quick way o analyze he X-11 resuls is o look a he final SI raios. X-13A-S produces a char comparing he final seasonal facors wih he SI raios (see he example in Char 7.3). Seasonaliy is expeced o be sable over ime. If SI raios are oo volaile relaive 2 36 A Henderson filer is a cenered moving average whose weighs are designed o exrac a smooh rend-cycle from series wih noise.

23 23 o he seasonal facors, his indicaes ha he series conains a srong irregular componen and he seasonal effecs may absorb oo much volailiy. Shorer filers for exracing seasonal effecs from SI raios are warraned when he irregular componen is large relaive o he seasonal effecs; longer filers are beer o exrac sable seasonal facors. 37 The SEATS filer 57. The SEATS filer is based on he ARIMA model-based (AMB) approach for seasonal adjusmen. This approach consiss of esimaing an ARIMA model for he original (possibly preadjused) series, deriving consisen ARIMA models for he unobserved componens (rend-cycle, seasonaliy, and irregular), and esimaing he componens using an opimal signal exracion echnique. An imporan propery of he AMB approach is ha he seasonal adjusmen filer adaps iself o he paricular srucure of he series. Conversely, X-11 is an ad-hoc seasonal adjusmen filer ha applies o every single series in he same manner regardless of he srucure of he seasonal and nonseasonal componens (alhough he filer lengh may be changed o beer sui differen characerisics). 58. The AMB approach implemened by SEATS is briefly illusraed below. 38 The following presenaion of SEATS is informal, as a comprehensive illusraion of he AMB approach for seasonal adjusmen requires he use of advanced conceps of ime series analysis (such as specral analysis and signal exracion heory) ha go beyond he scope of his manual. The advanage of using seasonal adjusmen programs such as X-13A-S 39 or TRAMO-SEATS is ha hey have been designed and equipped wih auomaic feaures ha faciliae he selecion of seasonal adjusmen opions, making his ask easy even for less experienced seasonal adjusmen users. However, compilers who are ineresed in applying SEATS in he QNA should acquire a sound knowledge of he mehod, which will be necessary o help hem evaluae and validae he resuls and be able o handle he adjusmen of problemaic series. 59. The ARIMA model (7) (see paragraphs 31-35), idenified and esimaed on he inpu series, is decomposed ino ARIMA models for he rend-cycle, seasonal, and irregular componens. A number of assumpions are made o derive an opimal decomposiion (among infinie ones) of he esimaed ARIMA model. Firs, componens are assumed o be muually independen (componens are said o be orhogonal). This is no a harmless assumpion, as i requires, for example, ha he rend and seasonal componens are independen from one anoher. However, his assumpion is radiionally acceped in model-based seasonal 37 In X-13A-S, he spec name for modifying he lengh of he sandard X-11 filer is X For an inroducion o he AMB decomposiion of ime series and furher reference, see Kaiser and Maravall (2000). 39 In X-13A-S, he spec name for running SEATS is SEATS.