On Weight Smoothing in the Current Employment Statistics Survey

Transcription

1 JSM Survey esearch Mehods Secion On Weigh Smoohing in he Curren Employmen Saisics Survey Julie Gershunskaya Michael Sverchkov Bureau of Labor Saisics 2 Massachuses Ave. NE Washingon DC Absrac The sampling weigh in he Curren Employmen Saisics Survey is deermined a he ime of sample selecion. I depends on a uni s Sae indusry and size class. However he populaion of businesses is highly dynamic. Esablishmens consanly grow or conrac; someimes hey also change heir indusrial classificaion or geographical locaion. Even he number of populaion unis is no fixed bu coninuously changes over ime. A uni may change is size class a he ime of esimaion or he conen of he original sraum may change. Under such circumsances applicaion of he original survey weighs may increase volailiy of survey esimaes. In his paper we invesigae if he survey esimaes can be improved by adusing he original weighs. Key Words: sampling weighs exreme observaion business survey sraum umper 1. Inroducion Under he classical design-based approach o inferences from survey sampling he sampling weighs are defined a he design sage of a survey and viewed as non-random quaniies a he esimaion sage. In conras Pfeffermann and Sverchkov (1999) and Beaumon (2008) view he sampling weighs as realizaions of a random vecor. This allows modeling he weighs and applying a new smoohed se of weighs esimaed from he model. The approach has he poenial for he improved efficiency compared o he esimaor based on he original weighs. Consider a version of he expansion esimaor for he populaion oal where a se of smoohed weighs is used in place of he original sampling weighs. The mehod has he poenial o give good resuls when he new weighs are beer relaed o he response variable. The usual layperson s inerpreaion of he sampling weighs in he expansion esimaor goes as follows: hink of sample uni s weigh as he number of corresponding unis in he populaion having he same value of he sample uni s response variable. If his correspondence would hold exacly for all sample unis hen he sample weighed esimaor provides a perfec esimaor of he populaion oal. I esimaes he oal wihou an error. Of course in realiy sampling weighs never exacly represen he number of such unis in he populaion. Each weigh may be considered as an esimae of he number of populaion unis wih like values. One can ry o improve his esimae by exploiing he relaionship beween weighs and sample responses and finding an average value of he weigh for unis wih similar measuremens. Thus one is smoohing he weighs. The heoreical idea is promising; however he mehod depends on finding an appropriae model for he weighs. In pracice he choice of a good model may be challenging and he model failure may lead o a 1139

2 JSM Survey esearch Mehods Secion bias in esimaion. Wih a good model also he model parameers need o be esimaed from he daa and his conribues o he variance of he resuling survey esimaor. Keeping in mind hese pracical difficulies applicaion of he mehod needs o be horoughly esed. In his paper we consider a nonparameric approach o esimaion of he smoohed weighs based on he values of response variables. The nonparameric approach does no require explicily formulaing a model; a drawback is ha he nonparameric esimaion generally is less efficien han he parameric approach. We apply he mehod o esimaion in he Curren Employmen Saisics (CES) survey and compare resuls wih he currenly used esimaor. CES is a large-scale esablishmen survey conduced by he U.S. Bureau of Labor Saisics. The survey produces monhly esimaes of employmen and oher imporan indicaors of he U.S. economy. The esimaes are published every monh a various levels of indusrial and geographical deail. Here we consider esimaion for he one monh relaive employmen change for indusrial divisions (supersecors) in meropolian saisical areas (MSA). In Secion 2 we give a brief descripion of relevan deails of he CES sample selecion and esimaion mehods and provide moivaion for considering he weighs smoohing mehod. In Secion 3 we adap he heoreical conceps developed by Pfeffermann and Sverchkov ( ) and Beaumon (2008) o he case of he CES esimaor of he relaive over-he-monh change. We describe he proposed esimaors for he CES survey he evaluaion crieria and provide he resuls in Secion 4. The las secion conains he summary. 2. Deails of he CES survey 2.1 CES Frame and Sample Selecion The CES sample is seleced once a year from a frame based on he Quarerly Census of Employmen and Wages (QCEW) daa file. This is he adminisraive daase ha conains records of employmen and wages for nearly every U.S. esablishmen covered by he Saes unemploymen insurance (UI) laws. The QCEW daa becomes available o BLS on a lagged basis and serves for he sampling selecion and for he benchmarking purposes; (see BLS Handbook of Mehods hp:// for more informaion abou QCEW). The QCEW based sampling frame is divided ino sraa defined by Sae indusrial supersecor based on he Norh American Indusrial Classificaion Sysem (NAICS) and on he oal employmen size of esablishmens wihin a UI accoun. A sraified simple random sample of UI accouns is seleced using opimal allocaion o minimize for a given cos per Sae a Sae level variance of he monhly employmen change esimae CES Esimaor The relaive growh of employmen from he previous o he curren monh is esimaed using a mached sample S of esablishmens ha is esablishmens reporing posiive employmen in boh adacen monhs: wy ; ˆ S (1) wy S ;

3 JSM Survey esearch Mehods Secion where denoes esablishmen is he curren monh curren and previous monhs repored employmen; y ; and y ; 1 w is he selecion weigh. denoe respecively a uni s The numeraor of he raio is he survey weighed sum of he curren monh repored employmen; similarly he denominaor is he survey weighed sum of he previous monh employmen. Once a year an esimae is benchmarked o a census level figure (he QCEW-based level ha becomes available on a lagged basis): Yˆ ˆ 1 Y0 1 ; monhly esimaes of he employmen level a subsequen monhs are derived by applicaion of esimae esimae of he employmen level: deails. Y ˆ Y ˆ ˆ 1 ˆ Y 0 of employmen rend o he previous monh. See he BLS Handbook of Mehods (Chaper 2) for furher 2.3 Moivaion for weighs adusmen and reamen of influenial observaions in CES In CES every monh we are essenially measuring employmen change in he populaion. Thus we should be looking a he relaionship beween he weighs and employmen changes. More precisely because we are using he raio esimaor and esimaing he relaive change we should consider he relaionship beween he weighs and residuals r y y 1. Indeed by he firs order Taylor decomposiion (1) can be approximaed as ˆ 1 w ( y y ) (2) S ; ; 1 Y 1 where Y 1 o monh. and are oal employmen in monh -1 and relaive growh of employmen from monh -1 As described in Secion 2.1 CES sraifies based on he employmen size (wihin indusry and geography) and allocaes opimally for a given cos. This sraegy is inended o produce an efficien sample weighed esimaor. Given how we sample larger weighs are usually associaed wih smaller businesses. The smaller businesses also end o have smaller changes in employmen and hus smaller unweighed residuals. However we ofen observe a relaively large weigh associaed wih a large change in employmen. This happens for various reasons. The general explanaion is he dynamic naure of he populaion of businesses (businesses may ump from one size class o anoher; he number of populaion unis may change; labels such as indusrial classificaion also change during he esimaion period). Even wih he opimal sampling design one canno accoun for he fuure changes in he populaion a he ime of planning and selecing he sample. Therefore he design weighs are hardly opimal for any given monh of he esimaion. The obus Esimaion procedure is he esimaion mehod currenly used in CES. I is designed o reduce he effec of he influenial observaions on he esimae of he relaive over-he-monh change. The obus esimaor idenifies a limied number of unis having exreme values of weighed residuals. These unis receive special reamen: heir weighs are reduced; in he mos egregious cases hey are considered self-represening aypical unis and are removed from he formula (1). 1141

4 JSM Survey esearch Mehods Secion From (2) he influenial repors are hose having large posiive or negaive values of he weighed residuals w( y ; y ; 1) compared o he oher sample unis. The exreme residuals are reduced o specific cu-off values. The cu-off values depend on he disribuion of he weighed residuals in a given series and are deermined independenly for each monh and indusry series. Pushing he exreme residuals o he cu-off values is accomplished by using an appropriae weigh adusmen facor. The procedure used for he CES robus esimaion is a paricular variaion of a general mehod of weigh reducion known as Winsorizaion. See Kokic and Bell (1994) Gershunskaya (2011). The acual cu-off values are deermined by examining he relaive disances of unis wih exreme weighed residuals o he neares bu less exreme values in he same cell and monh. See he BLS Handbook of Mehods (Chaper 2) for furher deails. This procedure helps o reduce volailiy of he esimaor. Sill especially a he lower esimaion cell levels he esimaor remains unsable. A he lower levels weighs may be furher modified using he proposed mehod of weighs smoohing. 3. Sample weighs smoohing We consider he sampling process as a resul of hree sage procedure (see Pfeffermann and Sverchkov 2009). A he firs sage he finie populaion U { y : N z} is generaed from some unknown disribuion f ( y0... y Z) which is usually called he super-populaion disribuion or model here z denoes he se of design variables (frame). The sampling weighs w 1... N are defined on his realized final populaion. Then he sample S { y w : n} is seleced from he finie populaion. Finally since some unis do no respond S can be decomposed ino monhly ses conaining unis ha respond in monh and S 1. Under his model he oucome variables and he sampling weighs are random and follow he sample disribuion (for exac definiions see Pfeffermann and Sverchkov 2009). Therefore (2) can be approximaed as 1 1 ˆ w ( y ; y ; 1 ) [ ( ; ; 1) ] S E w y y S S Y Y E{ E[ w ( y y ) y y S ] S } S ; ; 1 ; ; 1 Y 1 1 E[ E( w y y S )( y y ) S ] S ; ; 1 ; ; 1 Y 1 v 1 v ( y y ) S ; ; 1 Y 1 where v E( w y ; y ; 1 S) are he smoohed weighs. Here in he firs line we approximae weighed residuals by heir expecaions and in he las line we do he opposie. 1142

5 JSM Survey esearch Mehods Secion This implies ha he relaive growh of employmen from he previous o he curren monh can be esimaed also as ˆ S where vˆ S S vy ˆ vy ˆ ; ; 1 (3) are esimaes of E( w y ; y ; 1 S ). emark 1. (General usificaion for smoohing weighs). Le s be a sample seleced from a final populaion wih inclusion probabiliies random variables and x w 1. Pfeffermann and Sverchkov (1999) show ha for any E( w x s) f ( x ) f ( x s) (4) E( w x s) where f is discree. is he probabiliy densiy funcion when is coninuous and he probabiliy funcion when Therefore for esimaing relaionships beween variables and x on he super-populaion from he observed sample daa he sampling weighs can be replaced by heir condiional expecaions E( w x s). For example if one is ineresed in regression of on hen by (4) E( w x s) E( x ) E[ x s] he laer implies ha one can use any weighs E( w x s) saisfying w E( w x s) E[ x s] E[ x s] in his case. E( w x s) E( w x s) * * x * w Example. (Esimaing an expecaion over populaion). E( w s) E( ) E[ s] which E( w s) suggess wo esimaors based on he smoohed weighs: a) esimaing he exernal expecaion and E( w s) s expecaion in he denominaor by respecive sample means Eˆ( ) (analog of E( w s) Haek esimaor); b) on he oher hand since for fixed size sampling schemes E( w s) N / n esimaing he exernal expecaion by he mean and subsiuing he laer equaliy one can ge E( w s) s Eˆ( ) (analog of Horviz-Thompson esimaor). N s 1143

6 JSM Survey esearch Mehods Secion The new weighs are smooher han he original sampling weighs and conain all necessary informaion on he relaionship beween he oucome variable and he sampling weigh. For example if is a deerminisic funcion of he oucome hen he smoohed weigh is he same as he original one def v E( w s) w on he conrary if he oucome and he sampling weighs are unrelaed hen he smoohed weigh is consan. Therefore he esimaes based on he smoohed weighs can be less variable (more efficien) han classical probabiliy weighed esimaors. The smoohed weighs were used in Pfeffermann and Sverchkov ( ) in parameric esimaion of linear and Generalized Linear Models. One can find anoher heoreical usificaion for using smoohed weighs in Beaumon (2008). Beaumon and ives (2009) suggesed he use of smoohed weighs o deal wih influenial observaions. emark 2. The previous remark is correc for heoreical smoohed weigh E( w x s). In pracice he laer expecaion has o be esimaed. If he esimae will be inaccurae hen he final esimaor can be biased and/or less efficien. We consider he following se of esimaors. 1) Unweighed aio esimaor informaive. 4. Proposed esimaors and heir evaluaion ˆ UNW 2) Probabiliy Weighed aio esimaor ˆ S S PW y y ; ; 1. This esimaor can be biased if sampling is S S wy wy ; ; 1 randomizaion disribuion can be no efficien if original sampling weighs y y are no srongly relaed see emark 1. ; ; 1 3) obus esimaor ˆ S S w y w y ; ; 1 wih weighs. This esimaor is unbiased over w and residuals w obained by obus Esimaion Procedure described in Secion 2. This esimaor is proeced agains influenial observaions. vˆ y; 4) obus smoohed esimaor ˆ S S where v ˆ are esimaed by regressing vˆ y S ; 1 ˆ PW w w agains esimaed residuals y ; y ; 1 by SAS Proc LOESS hp://suppor.sas.com/documenaion/cdl/en/saug/63033/html/defaul/viewer.hm#loess_oc.hm. emark 3. I is difficul o deermine he funcional form of he relaionship beween weighs and residuals (see Figure 1). Therefore we consider a nonparameric approach. This approach does no require explicily formulaing a model; he drawback is ha he nonparameric esimaion generally is less efficien han he parameric approach. We use he sandard SAS LOESS procedure wih defaul parameers. On he plo sars represen he values of smoohed weighs esimaed using his procedure. 1144

7 JSM Survey esearch Mehods Secion Figure 1: obus sample weighs w agains esimaed residuals y ; y ; 1 (black) and SAS LOESS procedure esimaes for regression E( w y ; y ; 1 S ) agains esimaed residuals for a paricular MSA (red). ˆ PW We made esimaes of employmen a he MSA supersecor level. The esimaes were consruced using daa repored monhly over hree years. For each year we sared he esimaion cycle from he corresponding Sepember QCEW employmen level as he benchmark. Each year s esimaion wen on for 12 consecuive monhs using he esimaion sequence described in Secion 2.2. The primary goal of his research is o find he esimaor ha improves he monhly volailiy as compared o he currenly used esimaor. A he same ime revisions afer 12 monhs of esimaion (he annual revisions) should be on average a leas as good as wih he curren esimaor. We make conclusions abou he relaive qualiy of he compeing esimaors based on he summary of he disances of corresponding esimaes from he QCEW figures ha serve as he ruh. The summaries are derived over he se of MSAs in each supersecor level and over he se of MSAs a he Toal Privae level. 1145

8 JSM Survey esearch Mehods Secion For cell m a monh is he difference beween he esimaed Yˆm and he rue Y m employmen levels d Yˆ Y. m m m The difference relaive o he level (imes 100) is rel _ d 100 Yˆ Y Y. m m m m The difference in he monhly changes is c Yˆ Yˆ Y Y. m m m 1 m m 1 The difference in he monhly changes relaive o he level (imes 100) is rel _ c 100 c Y. m m m 1 In his paper we publish resuls a he MSA Toal Privae level. The following summary saisics are presened in Table 1. 1 M Mean revision: d M dm M and 1 rel _ d rel _ d M m 1 a 1 M Mean absolue revision: d M dm M and a 1 rel _ d rel _ d M m 1 m 1 m m 1 m h percenile of he absolue revisions ( rel _ d. d m or rel _ d m ) over he se of M domains: d and The following summary saisics for he monhly changes are presened in Tables 2. Mean revision: c 12 M 12 M 1 cm 12 and 1 rel _ c rel _ c 12 M 1 m 1. M 1 m 1 m Mean absolue revision: c a 12 M 12 M 1 cm 12 and a 1 rel _ c rel _ c 12 M 1 m 1 M 1 m 1 m h percenile of he absolue revisions ( c m or rel _ c m ) over he se of M domains and 12 monhs: c and rel c. _ We obained encouraging resuls: he summary saisics look consisenly beer for he new esimaor (Tables 1 and 2). However here are examples where he new esimaor does no work as expeced. Of course i is no reasonable o expec ha one esimaor would work beer han anoher in every case. However we would like o be able o idenify and correc cerain cases where he new esimaor is egregiously wrong (as in Figure 3). 1146

9 JSM Survey esearch Mehods Secion Table 1: Differences from QCEW afer 12 monhs of esimaion. Summary over all MSAs a he Toal Privae Level Esimaor N d 12 rel _ d 12 a d 12 d 12 rel _ d a 12 Based on Sepember 2009 benchmark LOESS obus Unwaio Waio Based on Sepember 2010 benchmark LOESS obus Unwaio Waio Based on Sepember 2011 benchmark LOESS obus Unwaio Waio rel _ d 12 Table 2: Monhly differences from QCEW for 12 monhs of esimaion. Summary over all MSAs a he Toal Privae Level and all 12 monhs Esimaor N c rel _c a c c rel _c a rel _ c Based on Sepember 2009 benchmark LOESS obus Unwaio Waio Based on Sepember 2010 benchmark LOESS obus Unwaio Waio Based on Sepember 2011 benchmark LOESS obus Unwaio Waio

10 JSM Survey esearch Mehods Secion We presen wo examples of esimaion over 12 monhs a an MSA supersecor level (see Figures 2 and 3.) There are hree lines on each plo. The black line corresponds o he rue employmen level a each of he 13 monhs including he saring Sepember. The blue line shows he obus esimaor (currenly used esimaor) and he magena line is for he new LOESS based esimaor. The firs example (Figure 2) shows he siuaion where he new esimaor resuls are smooher han he obus esimaor. Look especially a he change in employmen beween February and March and noice how he volailiy of he obus esimaor was correced in he new esimaor. Figure 2: Example of esimaion where new esimaor works well. esuls from wo compeing esimaors (obus: blue sars; LOESS-based: magena spides) and he employmen levels from QCEW. 1148

11 JSM Survey esearch Mehods Secion The second example (Figure 3) demonsraes ha here exis insances where he new mehod is no working as expeced. Noice April o May and May o June changes. Figure 3: Example of esimaion where new esimaor is no working well. esuls from wo compeing esimaors (obus: blue sars; LOESS-based: magena spides) and he employmen levels from QCEW. In Figure 4 we plo weighs agains he residuals for he problem monh (April o May change). Noice ha he smoohed weighs for he 4 poins on he righ are obained by nearly linear inerpolaion. As a resul we have hugely exaggeraed smooh weighs for wo of hese poins. This ells us ha here is room for improvemen in he nonparameric mehod we use. 1149

12 JSM Survey esearch Mehods Secion Figure 4: Illusraion for he April-May change in Figure 3. obus sample weighs ˆ PW w agains esimaed residuals y ; y ; 1 (black) and SAS LOESS procedure esimaes for regression E( w y y S ) agains esimaed residuals for a paricular MSA (red). ; ; 1 Summary: The overall resuls are promising bu here are cerain cases where he new mehod is no working properly. Some uning is needed of he nonparameric mehod we used. One minor pracical inconvenience is ha he smooh weighs change every monh. eferences Beaumon J.-F. (2008). A new approach o weighing and inference in sample surveys. Biomerika 95 3 pp

13 JSM Survey esearch Mehods Secion Beaumon J.-F. and ives L.-P (2009). Dealing wih ouliers in survey daa. Chaper 11 in D. Pfeffermann and C.. ao (eds.) Handbook of Saisics. No. 29A Sample Surveys: Inference and Analysis pp Bureau of Labor Saisics (2011). Employmen hours and earnings from he esablishmen survey. Chaper 2 of BLS Handbook of Mehods U.S. Deparmen of Labor hp:// Gershunskaya J. (2011) Treamen of influenial observaions in he Curren Employmen Saisics survey. Unpublished docoral disseraion Universiy of Maryland College Park. Kokic P. N. and Bell P. A. (1994) Opimal Winsorizing Cuoffs for a Sraified Finie Populaion Esimaor Journal of Official Saisics Pfeffermann D. and Sverchkov M. (1999). Parameric and semi-parameric esimaion of regression models fied o survey daa Sankhya B 61 P Pfeffermann D. and Sverchkov M. (2003). Fiing generalized linear models under informaive sampling. Chaper 12 in. L. Chambers and C. Skinner (eds.) Analysis of Survey Daa Chicheser: Wiley pp Pfeffermann D. and Sverchkov M. (2009). Inference under informaive sampling. Chaper 39 in D. Pfeffermann and C.. ao (eds.) Handbook of Saisics. No. 29B Sample Surveys: Inference and Analysis pp