Estimation and Comparison of Chained CPI-U Standard Errors With Regular CPI-U Results ( )

Transcription

1 2003 Join Saisical Meeings - Secion on Suvey eseach Mehods Esimaion and ompaison of hained PI-U Sandad Eos Wih egula PI-U esuls ( ) Owen J. Shoemake U.S. Bueau of Labo Saisics, 2 Mass Ave., NE, oom 3655, Washingon, D shoemake_o@bls.gov - Key Wods: onqvis; Saified andom Goups Any opinions expessed in his pape ae hose of he auho and do no consiue policy of he Bueau of Labo Saisics. In ealy 2002, he Bueau of Labo Saisics (BLS) began calculaing and publishing he fis -PI-U se of indexes. his new -PI-U (hained onsume Pice Index Uban), in is final fom, is calculaed and published evey yea, wih oughly a one-yea lag, using a onqvis fomula. Is se of weighs ae updaed yealy, so ha a unique se of monhly weighs ae available fo boh ime as well as fo ime n. hus, he new -PI-U index can be labeled supelaive. We biefly ouline he onqvis fomula and hen he mehodology fo esimaing a se of sandad eos fo hese new chained indexes. We hen compae, ove he 24- monh peiod of Januay 2000 hough Decembe 2001, hese new supelaive index esuls and hei sandad eos wih he egulaly published PI-U esuls and hei published sandad eos. 1. he hained Supelaive Index he official PI is no a supelaive index, and does no use a supelaive index fomula. he cuen official PI claims o know yeseday s pices, even oday s pices, and also claims o know yeseday s weighs. he PI is able o collec oday s pices bu no oday s weighs, a leas no in he same imely way. he PI calculaes and publishes, fo example, Apil s PI using Apil s pices in mid-may, while he weighs ae, a a minimum, wo yeas old. Bu o call an index supelaive equies oday s weighs as well as oday s pices. In ohe wods, fo an index, o an index fomul o be supelaive, all fou ingediens yeseday s pices, yeseday s weighs, oday s pices and oday s weighs mus be available. Wih he Final onqvis fomul albei wih a lag ime of a yea, he BLS has gaheed ogehe he fou necessay ingediens and, so, has been able o poduce a supelaive index. he Fishe Ideal fomul which is he geomeic mean of a Laspeyes index and a Paasche index, also migh have been used as a supelaive index fomula. Bu fo a vaiey of easons, afe much eseach and discussion, BLS has chosen he onqvis fomula ove he Fishe Ideal fomulaion. (In pacical ems, he choice is of lile consequence, since he onqvis and Fishe Ideal esimaes ae nealy indisinguishable in nealy all simulaions.) he onqvis fomula is simply he geomeic mean of wo Geomeans index fomulas, one a ime and he ohe a ime n (hee 1). (1) p = i, PEL I, [ 1; ] i I, a A 1 p i, a, s + s 1 i, a, i, a, 2 which is a monhly pice elaive, which is hen chained o poduce an index, (2) IX I, [0; ] = PEL I, [ 1; ] IX I, [0; 1] 3847

2 2003 Join Saisical Meeings - Secion on Suvey eseach Mehods whee, i = elemenay iem saa (any one of 211 such sa like Eggs) I = aggegae iem (fo example, All- Iems) a = elemenay index aea (any one of 38 such aeas, like Houson) A = aggegae index aea (fo example, All-US) = eplicae (eihe full o eplicae) = cuen calenda monh 1 = calenda monh pevious o calenda monh p i, = elemenay pice index fo calenda monh fo iem i in aea a fo eplicae p -1 i, = elemenay pice index fo calenda monh 1 fo iem i in aea a fo eplicae s i, = monhly expendiue shae in calenda monh fo iem i in aea a fo eplicae of oal monhly expendiues in aggegae iem I in aggegae aea A fo eplicae s -1 i, = monhly expendie shae in calenda monh 1 fo iem i in aea a of oal monhly expendiues in aggegae iem I in aggegae aea A. (3) s i, E i, = E i I, a A i, whee, E i, is he esimaed monhly expendiue in calenda monh fo iem i in aea a fo eplicae (4) 11 E n = n = i a E 0, E i, a i 11 E n a A n= 0 i, a whee, he new supelaive se of indexes also include Iniial and Ineim indexes, bu only he Final onqvis index, deailed above, can be hough of as supelaive. he official ile is hained PI-U. he Iniial -PI-U index is published in eal ime along wih he egula official PI-U. Boh PI-U and he Iniial -PI-U use he same se of lowelevel indexes. hese lowe-level indexes ae pice elaive updaes fom a Hybid sysem, whee mos of he pice elaive calculaions ae Geomeans and he emaining 30% ae Laspeyes. egula PI-U aggegaes hese lowe-level hybid indexes, using a Laspeyes fomulaion, o poduce all is highe-level indexes. Iniial -PI-U, on he ohe hand, aggegaes hese same lowe-level hybid indexes using a Geomeans fomula. he Ineim -PI-U indexes updae he Iniial indexes wih a simple faco o eflec he moe ecenly calculaed Final onqvis esuls. In ou sudy hee we ae only ineesed in he Final onqvis -PI-U esuls, in paicula as hey and hei sandad eos compae wih official PI-U esuls. he supelaive naue of he Final onqvis comes fom is use of a se of unique monhly weighs, deailed above a he iem-aea level, fo boh ime peiod and fo ime peiod 1. hese E i,a s ae smoohed weighs, bu hey do epesen a unique monhly weigh fo ha paicula monh fo ha paicula iem in one paicula aea. he smoohed aspec of hese weighs miigaes somewha he puiy of his uniqueness, bu he supelaive chaace of he Final onqvis fomally emains inac. he wo weighs (E i,a and E -1 i,a) ae unique, bu oughly 90% of he infomaion conen of he one is shaed by he ohe. he ohe obvious miigaing faco is he non- supelaive naue of he lowe-level indexes ha ae used in he Final onqvis fomula. E = expendiue = monh i = elemenay iem a = elemenay aea A = U.S. iy Aveage aggegae aea 3848

3 2003 Join Saisical Meeings - Secion on Suvey eseach Mehods 2. ompaison of hained vs. egula PI Wih he fis wo full yeas of Final onqvis esuls in hand, we can begin o ell if and how much he new hained PI indexes divege fom he coesponding egula PI indexes. FIG 1. egula INDEX vs. hained INDEX (Jan'00-Dec'01) Y % DIFF = % Pice hange DIFF = Y % DIFF = % Pice hange DIFF = 1.12 J F M A M J J A S O N D J F M A M J J A S O N D Pice hange (P) = (INDEX 1) * 100 Pice hange DIFF = P P % DIFF = (P / P 1) * 100 As ongoing indexes, he egula and he hained Indexes ae clealy diveging fom each ohe. he pecen diffeence of he inflaionay divegence seems o be holding seady a aound 29%, even as he pice change diffeence iself gows seadily wide. In he fou gaphs below, he pice change (no he pice elaive) diffeences ae gaphed fo he 1-, 2-, 6- and 12-monh esuls. 3849

4 2003 Join Saisical Meeings - Secion on Suvey eseach Mehods FIGs 2-5. HAINED vs. EGULA PI ALL-US ALL-IEMS 1-MO Pice hanges (Jan'00-Dec'01) 2-MO Pice hanges (Feb'00-Dec'01) = egula = hained % DIFF = +28.2% = egula = hained % DIFF = +26.6% J M M J S N J M M J S N F A J A O D F A J A O D 6-MO Pice hanges (Jun'00-Dec'01) 12-MO Pice hanges (Dec'00-Dec'01) = egula = hained % DIFF = +25.8% = egula = hained % DIFF = +27.8% J J A S O N D J F M A M J J A S O N D D J F M A M J J A S O N D he one summay saisic ha is displayed is a pecen diffeence (% DIFF) beween he mean of he egula PI pice changes and he mean of he hained PI pice changes. he consisency of hese pecen diffeences acoss he fou ses of pice changes ( 26%) is somewha noable and is compaable wih he 29% pice change diffeence in he index gaph. he magniude of he diffeences is clealy of inees and noe, and may need o be invesigaed moe closely in fuhe sudy. A chained index using a geomeic (albei onqvis) fomula houghou was expeced o poduce a consisenly lowe index han egula PI which uses a Laspeyes fomula a his highes level of aggegaion, bu he diffeence was no expeced o be quie his lage. Simple paied -ess fo he fou ses of paied pice changes find ha all fou of hese diffeences ae significan (p-values all less han 0.001, wih he diffeences gowing as he ess move fom 1- o 2- o 6- o 12-monh esuls). he acual mean diffeences of he fou ses ae: 1-monh pice change mean diffeence = monh pice change mean diffeence = monh pice change mean diffeence = monh pice change mean diffeence =

5 2003 Join Saisical Meeings - Secion on Suvey eseach Mehods hese mean diffeences ae nealy exacly linea wih espec o he lengh of he (ime) chain: he 2- monh diffeence is wice he 1-monh, he 6-monh is six imes he 1-monh, and he 12-monh diffeence is jus ove 12 imes as lage. hus, he cuen esuls show ha he yealy inflaion ae using he Official PI is unning moe han one-half of a pecenage poin highe han he hained onqvis Supelaive Index. One impoan way o deemine how significan ae hese diffeences is o calculae sandad eos fo he new onqvis indexes. We aleady have published sandad eo esuls fo egula PI, bu we need o know if and how much moe vaiable ae hese new supelaive indexes, and if ha vaiabiliy alone migh accoun fo hese obseved diffeences beween he wo indexes a he All-US All-Iems level. 3. Sandad eos fo he hained PI Official BLS sandad eos fo he new Final onqvis Indexes have no ye been calculaed fo publicaion, bu he geneal mehodological equiemens have been wien and appoved. he sandad eos fo he new -PI-U ha will be calculaed and analyzed hee follow hese equiemens. he egula PI-U sandad eos ae calculaed evey monh alongside he PI-U iself, and hen published, a he end of he yea each yea fo he whole peceding yea, in he PI Deailed epo (usually he Febuay issue). he sandad eos fo he PI-U uilize a Saified andom Goups mehodology. he sandad eos fo he new -PI-U boow his mehodology and faily closely mimic i. Essenially, he Saified andom Goups Mehod employs eplicae pice change values a he Iem-Aea level, which ae hen subaced off fom hei coesponding full-sample pice change value. Each eplicae P value, in is simples fom, is he full-sample P wih one of he aea s componen value conibuing a eplicae value in place of is egula full-sample value. he diffeences ae hen squaed and summed up and finally appopiaely divided hough by he N a (N a 1) numbe of eplicaes in each Iem-Aea goup. hus, (5) N a = 1 VA( I,, k) = a A N a ( N 1) ( P( I,, k, ) P( I,, k,0) ) his fomula applies equally o he new -PI-U vaiances and o he egula PI vaiances. he diffeences aise in he Pice hange (P) fomulas used fo each. Fo he PI-U, P PI-U = ( W / W -k 1) * 100 whee W = IX * AGGW Fo he -PI-U, Equaion (1) applies, and P -PI-U = (PEL 1) * 100 he same IX s ae used in boh fomulas, bu he aggegaion weighs (AGGWs) in he egula PI-U fomula ae a fixed se of weighs, which ae anywhee fom wo o fou yeas old. he monhly onqvis weighs ae fom he exac monh ha is specified. Boh ses of weighs do come fom he same collecion of E (onsume Expendiue) da bu he wo ses of weighs ae pocessed diffeenly and of couse epesen diffeen ime peiods. he diffeences beween BLS egula PI sandad eos and he sandad eos fo he new onqvis index ae gaphed below. 1 a

6 2003 Join Saisical Meeings - Secion on Suvey eseach Mehods FIGs 5-8. HAINED vs. EGULA PI ALL-US ALL-IEMS 1-MO Sandad Eos (Jan'00-Dec'01) 2-MO Sandad Eos (Feb'00-Dec'01) = hained = egula % DIFF = +12.7% = hained = egula % DIFF = +11.6% J M M J S N J M M J S N F A J A O D F A J A O D 6-MO Sandad Eos (Jun'00-Dec'01) 12-MO Sandad Eos (Dec'00-Dec'01) = hained = egula % DIFF = +14.5% = hained = egula % DIFF = +27.6% J J A S O N D J F M A M J J A S O N D D J F M A M J J A S O N D One summay saisic, simila o he one in he ealie Pice hange gaphs, is displayed hee: he pecen diffeence (% DIFF) beween he especive means of he 1-,2-, 6- and 12-monh sandad eos. he new onqvis sandad eos do un highe on aveage han he egula PI sandad eos, bu he diffeences ae no appeciable. A he 1-, 2- and 6-monh levels he diffeences ae oughly 13% (compaing hained o egula esuls). he 12-monh onqvis sandad eos ae unning abou 28% highe han hei egula PI sandad eo counepas. he chaining effec in he onqvis, he geomeic fomula iself, and he moe volaile weighs, all conibue o his inceased vaiabiliy. he supelaive weighs ae monhly expendiue daa values which ae smoohed, fis ove all he IEMs in he gouping and hen back acoss 12 monhs of ime wihin he IEM, which effecively means ha he supelaive weighs ae caying a yea s woh of E daa compaed o he wo o hee yeas woh of E daa which he egula (AGGW) weighs eflec. hese slighly highe sandad eos fo he onqvis wee boh expeced and ye well wihin accepable levels (a leas a he All-US All-Iems level). 3852

7 2003 Join Saisical Meeings - Secion on Suvey eseach Mehods 4. onclusions In applying hese new sandad eos o ou onqvis esuls, we ae pincipally ineesed in noing whehe he onqvis esuls ae significanly diffeen fom he egula PI esuls. We can esablish confidence bounds a an σ =.05 level by simply muliplying ou sandad eos by wo and hen seeing if he egula PI-U esuls fall wihin ha 2-sigma bound. Fo he 24 1-monh onqvis pice changes, only five ae significanly diffeen (i.e., highe). Fo he 23 2-monh onqvis pice changes, we find ha only six ae significanly diffeen. Howeve, when we move up o he 6- and 12-monh pice changes, he significance esuls evese. All of he monh onqvis pice changes ae clealy significanly diffeen han hei PI-U counepas, and all bu fou of he 19 6-monh onqvis pice changes ae significanly diffeen han egula PI. As he ime uni is lenghened he sandad eos do incease (in a faily saighfowadly linea fashion, as was menioned peviously), bu he diffeences in he pice changes hemselves, popoionaely, have widened even fuhe. And since i is he 12-monh pice changes (ou yealy inflaion numbes) ha we ae mos ineesed in, hese clealy significan diffeences need o be noed. 5. Acknowledgmens I would like o hank Jane Williams fo he caeful eading of his pape and he helpful commens. In addiion, I would like o hank my colleagues Bill Johnson and Yeon hee. 3853