Traditional Smoothing Techniques

Transcription

1 Tradoal Smoohg Techques Smple Movg Average: or Ceered Movg Average, assume s odd: 2 ( 2 ( Weghed Movg Average: W W (or, of course, you could se up he W so ha hey smply add o oe. Noe Lear Movg Averages (MAs of MAs: Cosder a sysem of weghs for a 7-po weghed movg average {,,,,,,}. Aoher 4-po movg average wh weghs {,,,}. The he 7 4 movg average would have weghs {,2,3,4,4,4,4,3,2,} ad s esseally he covoluo of he wo ses of weghs.

2 (Sgle Epoeal Smoohg: ( ( ( or Adapve Respose Rae Sgle Epoeal Smoohg (ARRSES. The advaage here s ha s dyamc.: ( where: varable 0.2,.., (, ( choce a M E smoohed error of val abs M e M smoohed error E e E e β β β β β Chow s Adapve Corol Mehod: Ca be used for osaoary daa. s adaped by small cremes so as o mmze he MSE. b S ad b S S b S S ( ( ( 2

3 Wers Lear ad Seasoal Epoeal Smoohg: S b m where ( S b m I L m ( ( S b I L γ( S S ( γ b I β ( β I L S L: legh of he seasoaly I: seasoal adjusme facor S : smoohed value for he seres m: forecas perod β : a wegh o suppress radomess (ofe 0.05 : epoeal facor for smoohg (ofe 0.2 γ : parameer (ofe 0. DECOMPOSITION METHODS f ( I, T, C, E : acual a me I : seasoal compoe (or de a T : s red compoe a C : cyclcal compoe a E : error or radom compoe a Ths fuco f( ca be addve or mulplcave, yeldg a addve decomposo or a mulplcave decomposo. 3

4 Eample, Addve Mehod:. Compue a movg average of legh N, where N s he legh of he seasoaly. Ths elmaes he seasoaly by averagg seasoally hgh perods wh seasoally low perods, ad reduces radomess as well. 2. Subrac he movg average from he seres. The MA s he red plus cycle. The error s he seasoal compoe. 3. Isolae he seasoal compoe by averagg hem for each of he perods makg up he complee legh of he seasoaly. 4. Idefy he approprae form of he red lear, epoeal, S-curve, ec., ad calculae s value a each perod. 5. Subrac he esmaed red from he deseasoalzed seres o oba he cyclcal facor. 6. Subrac he seasoal, red, ad cycle compoes from he orgal seres o yeld he radom compoe. 4

5 CENSUS II X- Decomposo/Seasoal Adjusme Mehod The Cesus Mehod I bega 954, followed by welve epermeal programs, amed X-0, X-, ec., of Mehod II. Ths culmaed X-. U. S. Deparme of Commerce, Bureau of he Cesus. Julus Shsk (955, based upo he rao-o-movg average classcal decomposo. Shsk, J., A. H. Youg, ad J. C. Musgrave. The X- vara of he Cesus mehod II seasoal adjusme program. Techcal Paper 5, Bureau of he Cesus, U.S. Deparme of Commerce, 967. Shsk, Julus. Seasoal Adjusme of Sesve Idcaors, 978. I A. Zeller, edor, Seasoal Aalyss of Ecoomc Tmes Seres, pages U. S. Deparme of Commerce, Bureau of he Cesus. X- was popular because: I was relavely easy o use. I dd o requre resag pas values whe ew daa was released. I hadled ereme values well. I used well-kow movg averages mehods for esmag red ad seasoal compoes. The asymmerc movg averages used ear he eds of he me seres were hough o be red ad rue. I had a clear-cu way of esmag radg day effecs. Sascs Caada eeded he mehod as X--ARIMA. Ths mehod cluded he full X- mehod, bu used ARIMA backcass ad forecass o provde opmal esmaes of daa ousde he daa wdow o mprove esmaes a he eds of movg averages. 5

6 X--ARIMA resuls seasoal adjusmes whose revsos are smaller, o average, whe hey are recalculaed afer fuure daa becomes avalable. I he addve decomposo case, eeso wh opmal forecass ad backcass for he half-legh of he symmerc seasoal fler mmzes revsos he mea square sese. Bobb, L, ad M. C. Oo. Effecs of forecass o he revsos of seasoally adjused values usg he X- seasoal adjusme procedure. I Proceedgs of he Busess ad Ecoomcs Sascs Seco, pages , Aleadra, Vrga, 990. Amerca Sascal Assocao. X--ARIMA also added dagoscs for comparg drec ad drec seasoal adjusmes of seres ha are aggregaes of mulple compoe seres. WE START WITH THE ORIGINAL X- METHOD: Sep. Tradg Day Adjusme. Deerme he umber of acve days each moh for he years of eres. 2. Compue he average umber of radg days for each moh. 3. Dvde he umber of days each moh by hs average o ge a adjusme facor. 4. Use he adjusme facor o adjus he mohly fgures. 5. Ths creaes a value called orgal daa adjused for radg days. 6

7 Sep 2. Prelmary Seasoal Adjusme. Seasoaly Adjusme. Apply a 2-moh MA o elmae seasoaly. 2. Average he MAs of 2 successve mohs o form he 7 h moh value. Ths addresses he ceerg problem. 3. Form he rao of he orgal seres o he MA seres. Ereme Values 4. Calculae he 33 moh movg average (3-moh average of a 3-moh average. a. Ths s roughly equvale o a 5-moh movg average. b. Srcly speakg, hs should resul he loss of 2 mohs a he begg ad ed of he seres, bu Cesus esmaes replacemes for hese. 5. Calculae he sadard devao of he ceered raos from he 33 MA. a. Ths s used o cosruc corol lms o defy ereme values. b. If he ceered MA > 33 MA ± 2s 2, he replace wh he average of prevous ad followg perod. Prelmary Seasoal Facor Esmao & Applcao 6. Replace he 6 moh a he begg ad ed of he raos by he eares values a eghborg year. 7. Normalze years so ha he raos each year add o 2. (Average rao s. 8. Dvde he prelmary seasoal facors o he orgal daa o oba he prelmary adjused seres. 7

8 Sep 3. Refe Seasoal Adjusmes.. Apply Specer s 5 moh weghed movg average o he seasoally adjused daa. Ths s a 5544 movg average (quadruple MA a. Isolaes he red-cycle compoe. 2. Dvde he orgal daa by he red-cycle compoe a. Seasoal ad radom facors rema. These are called he fal seasoal rregular raos. b. Normally Specer s Mehod would cause he loss of 7 pos a he begg ad he ed of he seres, so Cesus replaces he los daa pos wh esmaes. 3. Replace he ereme values as above. 4. Esmae mssg values. 5. Adjus (ormalze raos. 6. Take 5-year averages of hese fal seasoalrregular raos 7. These are he sable facors (seasoal dces. Sep 4.. Apply a 33 movg average (or 55 f sll looks oo radom o he fal seasoal rregular raos. 2. Esmae values for he 2 perods a he begg ad ed of he seres ha would be los. 3. Take he las 2 values for each moh, ad form a epeced value. For eample, for 992, 3 / [( ] Dvde hese fal seasoal facors o he orgal daa o form he seasoally adjused seres. 8

9 Sep 5. Fal adjusme.. Calculae a 5 moh MA o creae he fal seasoally adjused daa. a. Ths s a esmae of he red cycle compoe. Sep 6. Creae a moua of summary sascs. The Cesus X-2-ARIMA cludes X-, bu eeds he modelg ad dagosc capables. X- ARIMA RegARIMA Models (Forecass, Backcass, ad Preadjusmes Modelg ad Model Comparso Dagoscs SEASONAL ADJUSTMENT (Ehaced X- DIAGNOSTICS 9

10 The major mehodologcal mprovemes of X-2-ARIMA are: New X- adjusme opos New dagoscs New modelg capables emphaszg regarima modelg. (RegARIMA s a lear regresso model wh ARIMA me seres errors. NEW X- ADJUSTMENT OPTIONS New fler opos, cludg: o loger seasoal movg average, o user specfcao of Hederso flers o modfcaos o asymmerc movg averages Opo for pseudo-addve decomposo, somemes useful for seres wh perodcally small or zero values. Improvemes radg day adjusmes ad opos for user-defed effecs based upo prelmary esmaes of he rregular compoe. NEW DIAGNOSTIC CAPABILITIES Specral esmaes for deeco of seasoal ad radg day effecs Revsos hsory dagoscs for assessg he sably of seasoal adjusmes. Beer dagoscs for decdg wheher o use drec or drec adjusmes for aggregae seres. New RegARIMA CAPABILITIES Capably o add regresso effecs o he models for forecas eeso. Use of RegARIMA models ca poeally mprove forecass ad backcass, ad provde earler ouler deeco capables. 0

11 TYPES OF DECOMPOSITIONS THAT MAY BE SELECTED WHEN USING X- Mulplcave Decomposo o Usually approprae for seres of posve values whch he sze of he seasoal oscllaos creases wh he level of he seres. o The seasoally adjused seres s obaed by dvdg he orgal seres by he esmaed seasoal compoe. Addve Decomposo o More approprae o saoary seres. o The seasoally adjused seres s obaed by subracg he esmaed seasoal compoe. Log-addve Decomposo o The addve decomposo of he logarhms of he seres beg adjused s epoeaed. o Maly used for research purposes. Requres a bas correco. Pseudo-addve decomposo I he updaed X- ( X-2, he Specer MA s replaced by he Hederso fler. Ths s eher 9, 3, or 23 pos ad s symmerc. I s desged o appromae a cubc f o saoary daa. Specral aalyss of he Hederso fler reveals ha has subsaal power afer he frs seasoal frequecy (leakage?. As a resul, Schps ad Ser (995 argue ha he Hederso fler eaggeraes shor-erm cyclcal behavor. The 7-erm Hederso fler s he shores

12 oe ha does o resul a sgfca peak beyod he frs seasoal frequecy. 2