Statistics for Social and Behavioral Sciences Estela Bee Dagum Silvia Bianconcini Seasonal Adjustment Methods and Real Time Trend- Cycle Estimation Statistics for Social and Behavioral Sciences Serieseditor StephenE.Fienberg CarnegieMellonUniversity Pittsburgh Pennsylvania USA Statistics for Social and Behavioral Sciences (SSBS) includes monographs and advancedtextbooksrelatingtoeducation,psychology,sociology,politicalscience, publicpolicy,andlaw. Moreinformationaboutthisseriesathttp://www.springer.com/series/3463 Estela Bee Dagum • Silvia Bianconcini Seasonal Adjustment Methods and Real Time Trend-Cycle Estimation 123 EstelaBeeDagum SilviaBianconcini DepartmentofStatisticalSciences DepartmentofStatisticalSciences UniversityofBologna UniversityofBologna Bologna,Italy Bologna,Italy ISSN2199-7357 ISSN2199-7365 (electronic) StatisticsforSocialandBehavioralSciences ISBN978-3-319-31820-2 ISBN978-3-319-31822-6 (eBook) DOI10.1007/978-3-319-31822-6 LibraryofCongressControlNumber:2016938799 MathematicsSubjectClassification(2010):62G08,62M10,62P20,62P25 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland To mysons,Alex, Paul,andLeo, with their lovelyfamiliesfortheirstrongvaluable supportandlivelydiscussionsduringthe writingofthisbook To Paolowho alwaysinspiresmetotrymy best Preface In order to assess the current stage of the business cycle at which the economy stands, real time trend-cycle estimates are needed. The basic approach is that of assessing the real time trend-cycle of major socioeconomic indicators (leading, coincident, and lagging) using percentage changes, based on seasonally adjusted data,calculatedformonthsandquartersinchronologicalsequence.Themaingoal is to evaluate the behavior of the economic indicators during incomplete phases bycomparingcurrentcontractionsorexpansionswithcorrespondingphasesinthe past. This is done by measuring changes of single time series (mostly seasonally adjusted) from their standing at cyclical turning points with past changes over a series of increasing spans. This differsfrom business cycle studies where cyclical fluctuationsare measured arounda long-termtrend to estimate complete business cycles. The real time trend corresponds to an incomplete business cycle and is stronglyrelated to whatis currentlyhappeningonthe businesscycle stage. Major changes of global character in the financial and economic sector have introduced high levels of variability in time series making difficult to detect the direction of the short-termtrend by simply lookingat seasonally adjusted data, and the use of trend-cycledataorsmoothedseasonallyadjustedserieshasbeensupported.Failure in providing reliable real time trend-cycle estimates could give rise to dangerous drift of the adopted policies. Therefore, a reliable estimation is of fundamental importance. Thisbookincludestwoopeningchapters,Chap.1whichisageneralintroduction and Chap.2 on time series components. The remaining nine chapters are divided in two parts, one on seasonal adjustment and the other on real time trend-cycle estimation. Sincetheinputfortrend-cycleestimationisseasonallyadjusteddata,PartIofthis book thoroughlydiscusses the definitionsand concepts involvedwith three major seasonaladjustmentmethodsasfollows: Chapter3.Seasonaladjustment,meaning,purpose,andmethods. Chapter4.Linearsmoothingormovingaverageseasonaladjustmentmethods. Chapter 5. Seasonal adjustment based on ARIMA model decomposition: TRAMO-SEATS. vii viii Preface Chapter6.Seasonaladjustmentbasedonstructuraltimeseriesmodels. Two of the seasonal adjustment methods are officially adopted by statistical agencies,namely,X12ARIMAandTRAMO-SEATS,andtheirrespectivesoftware default options are illustrated with an application to the US Orders for Durable Goods series. The third method, structural time series models and its software STAMP,isalsoillustratedwithanapplicationtotheUSUnemploymentMales(16 yearsandover)series. PartIIofthebookcomprises: Chapter7.Trend-cycleestimation. Chapter8.Recentdevelopmentsonnonparametrictrend-cycleestimation. Chapter9.Aunifiedviewoftrend-cyclepredictorsinreproducingkernelHilbert spaces. Chapter10.Realtimetrend-cycleestimation. Chapter11.Theeffectofseasonaladjustmentmethodsonrealtimetrend-cycle estimation. Chapter7systematicallydiscussesthedefinitionsandconceptsofthetrend-cycle componentofthevariousseasonaladjustmentmethodspreviouslyintroduced. Chapter8concentratesonthelast20years’developmentsmadetoimprovethe Hendersonfilterusedtoestimatethetrend-cycleinthesoftwareoftheUSBureau of CensusX11 and its variants,the X11/X12ARIMA.The emphasishasbeen on determining the direction of the short-term trend for an early detection of a true turningpoint.Itintroducesindetailthreemajorcontributions:(1)anonlineartrend- cycleestimatoralsoknownasNonlinearDagumFilter(NLDF),(2)acascadelinear filter (CLF) thatclosely approximatesthe NLDF, and (3) an approximationto the HendersonfilterviathereproducingkernelHilbertspace(RKHS)methodology. Chapter 9 presents a unified approach for different nonparametric trend-cycle estimatorsbymeansofthereproducingkernelHilbertspace(RKHS)methodology. Thesenonparametrictrend-cycleestimatorsarebasedondifferentcriteriaoffitting and smoothing, and they are (1) density functions, (2) local polynomial fitting, (3) graduation theory, and (4) smoothing spline regression. It is shown how nonparametric estimators can be transformed into kernel functions of order two, whichareprobabilitydensitiesandfromwhichcorrespondinghigher-orderkernels arederived.Thiskernelrepresentationenablesthecomparisonofestimatorsbased ondifferentsmoothingcriteriaandhasimportantconsequencesinthederivationof theasymmetricfilterswhichareappliedtothemostrecentseasonallyadjusteddata forrealtimetrend-cycleestimation. Chapter 10 is dedicated to real time trend-cycle estimation. Official statistical agencies generally produce estimates derived from asymmetric moving average techniques which introduce revisions as new observations are incorporated to the series as well as delays in detecting true turning points. This chapter presents a reproducingkernelapproachtoobtainasymmetrictrend-cyclefiltersthatconverge fastandmonotonicallytothecorrespondingsymmetricones.Thisisdonewithtime- varying bandwidth parameters because the asymmetric filters are time-varying. It showsthatthepreferredoneisthebandwidthparameterthatminimizesthedistance betweenthegainfunctionsoftheasymmetricandsymmetricfilters.Thetheoretical Preface ix results are empirically corroboratedwith a set of leading, coincident, and lagging indicatorsoftheUSeconomy. Chapter11dealswith theeffectsof theseasonaladjustmentmethodswhenthe realtime trendis predictedwithnonparametrickernelfilters. Theseasonaladjust- mentscomparedarethetwoofficiallyadoptedbystatisticalagencies,X12ARIMA and TRAMO-SEATS, applied to a sample of US leading, coincident,and lagging indicators. Theelevenchaptershavebeenwrittenascompleteaspossible,andeachonecan bereadratherindependently.Wehavealsointroducedauniformnotationallalong thechapterstofacilitatethereadingofthebookasawhole. This book will prove useful for graduate and final-year undergraduate courses in econometrics and time series analysis and as a reference book for researchers and practitioners in statistical agencies, other government offices, and business. The prerequisites are a good knowledge of linear regression, matrix algebra, and knowledgeofARIMAmodeling. We are indebted to participants and students who during many seminars and presentations raised valuable questions answered in this book. Our most sincere gratitudegoestoourcolleagueswhoencouragedustowritethisbookandallthose who providedmany valuableand useful suggestionsthroughlively discussionsor written comments. Our thanks also go to Veronika Rosteck, associate statistics editorofSpringer,forhersustainedandvaluablesupportwhilewritingthebook. Wearesolelyresponsibleforanyerrorsandomissions. Bologna,Italy EstelaBeeDagum February2016 SilviaBianconcini
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