Estimation of Normal Mixtures in a Nested Error Model with an Application to Small Area Estimation of Poverty and Inequality

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1 Publc Dsclosure Authorzed Publc Dsclosure Authorzed Publc Dsclosure Authorzed Publc Dsclosure Authorzed Polcy Reserch Workng Pper 696 Estmton of Norml Mxtures n Nested Error Model wth n Applcton to Smll Are Estmton of Poverty nd Inequlty Chrs Elbers Roy vn der Wede WPS696 Development Reserch Group Poverty nd Inequlty Tem July 014

2 Polcy Reserch Workng Pper 696 Abstrct Ths pper proposes method for estmtng dstrbuton functons tht re ssocted wth the nested errors n lner mxed models. The estmtor ncorportes Emprcl Byes predcton whle mkng mnml ssumptons bout the shpe of the error dstrbutons. The pplcton presented n ths pper s the smll re estmton of poverty nd nequlty, lthough ths denotes by no mens the only pplcton. Monte-Crlo smultons show tht estmtes of poverty nd nequlty cn be severely bsed when the non-normlty of the errors s gnored. The bs cn be s hgh s to 3 percent on poverty rte of 0 to 30 percent. Most of ths bs s resolved when usng the proposed estmtor. The pproch s pplcble to both survey-to-census nd survey-to-survey predcton. Ths pper s product of the Poverty nd Inequlty Tem, Development Reserch Group. It s prt of lrger effort by the World Bnk to provde open ccess to ts reserch nd mke contrbuton to development polcy dscussons round the world. Polcy Reserch Workng Ppers re lso posted on the Web t The uthors my be contcted t rvnderwede@worldbnk.org. The Polcy Reserch Workng Pper Seres dssemntes the fndngs of work n progress to encourge the exchnge of des bout development ssues. An objectve of the seres s to get the fndngs out quckly, even f the presenttons re less thn fully polshed. The ppers crry the nmes of the uthors nd should be cted ccordngly. The fndngs, nterprettons, nd conclusons expressed n ths pper re entrely those of the uthors. They do not necessrly represent the vews of the Interntonl Bnk for Reconstructon nd Development/World Bnk nd ts fflted orgnztons, or those of the Executve Drectors of the World Bnk or the governments they represent. Produced by the Reserch Support Tem

3 Estmton of norml mxtures n nested error model wth n pplcton to smll re estmton of poverty nd nequlty Chrs Elbers nd Roy vn der Wede 1 Keywords: Norml mxtures, lner mxed models, smll re estmton, Emprcl Byes, poverty, nequlty JEL Clssfcton: I3, C31, C4 1 Chrs Elbers c.t.m.elbers@vu.nl) s t VU Unversty Amsterdm nd the Tnbergen Insttute. Roy vn der Wede rvnderwede@worldbnk.org) s t the World Bnk. We grtefully cknowledge fnncl support from VU Unversty Amsterdm nd the World Bnk's Knowledge for Chnge Progrm KCP II). A thnk you lso goes to Peter Lnjouw for provdng comments on erler versons of ths pper.

4 1 Introducton We propose method for estmtng dstrbuton functons tht re ssocted wth the nested errors n lner mxed models. The proposed estmtor s ccommodtng Emprcl Byes predcton nd mkes mnml ssumptons bout the shpe of the error dstrbutons. Our objectve s to ccurtely predct nonlner functons of the dependent vrble, where the shpe of the error dstrbuton functons potentlly plys n mportnt role. The pplcton we hve n mnd s the smll re estmton of poverty nd nequlty, lthough ths denotes by no mens the only pplcton. Ths prtculr pplcton hs been mde populr by the work of Elbers, Lnjouw nd Lnjouw 003; henceforwrd ELL). The pproch they put forwrd hs snce been ppled to obtn mps of poverty nd nequlty n over 60 countres worldwde. The hghly dsggregted estmtes of poverty nd nequlty hve n turn nspred rnge of other pplctons, for exmple: Demombynes nd Ozler 005) nvestgte whether nequlty t the smll re level hs n mpct on locl crme rtes, nd fnd tht t does usng dt from South Afrc; Elbers et l. 007) conduct n emprcl experment n order to estmte by how much one could potentlly lower the costs of gettng resources to the poor f one hd ccess to poverty mp, nd conclude tht the gns cn be substntl; Arujo et l. 008) exmne whether vllges wth hgher levels of nequlty re less lkely to nvest n publc goods tht would beneft the poor. Usng dt from Ecudor they fnd emprcl support for ths hypothess whch they ttrbute to elte cpture. Recently, Fuj 010) modfed the pproch to mke t better suted for the smll re estmton of chld mlnutrton outcomes wth n pplcton to Cmbod. 1.1 Problem sttement Consder the followng stndrd lner mxed model for log ncome of household h resdng n re : y h = x h β + u + ε h. Let y d denote vector of log ncomes for ll households from domn d where domn typclly refers to smll geogrphc re). We re nterested n estmtng W y d ), where W s some possbly nonlner) functon of y d. It s ssumed tht we hve dt on x for the entre populton, whle dt on both x nd y s vlble for smple S of households only. ELL propose to estmte W by: E[W y d ) x d; x, y) S], where the expectton s tken over the unobserved errors u nd ε h. Becuse W s nonlner, the shpe of the error dstrbutons wll mtter for the expected vlue of W. Estmton of β s obvously of prmry mportnce when estmtng E[W ], but n ths pper we wll concentrte on the error dstrbutons nd ssume except for the pplcton n secton 6 tht β s known. We rgue tht even f β s estmted perfectly, gettng the error dstrbuton wrong stll hs the

5 potentl to ntroduce sgnfcnt bs. ELL felt ther pproch would be most convncng f they mke mnml ssumptons bout the errors u nd ε h. They proceed by frst obtnng estmtes of the re errors u nd ε h, whch then llows them to smple from the emprcl errors û nd ˆε h. u s estmted s the smple re verge of the totl resduls pproprtely re-scled so tht the smple vrnce of û equls the estmte of σu tht corrects for the contrbuton of the re verge of ε h ). The estmte for ε h s obtned by subtrctng û from the totl resdul re-scled so tht ts smple vrnce mtches σε). 1 Ths non-prmetrc pproch to estmtng the error dstrbutons hs two lmttons. Frstly, ths procedure does not dequtely ccount for the fct tht û equls sum of u nd ε. Ths ltter term s often lrge enough for t to ffect the blty of the emprcl dstrbuton of û to reproduce the shpe of the ctul dstrbuton of u. Smply gnorng ths contmnton my result n sgnfcnt bs. If the emprcl dstrbuton of u s bsed, then ths bs wll lso hve mplctons for the emprcl dstrbuton of ε h. Ths s lso referred to s convoluton problem, where the objectve s to estmte the dstrbuton functon of rndom vrble tht s observed wth error. Secondly, the pproch dopted by ELL does not esly lend tself for Emprcl Byes EB) estmton where the dstrbuton of u s tghtened by condtonng on household dt y nd x vlble for domn d.e. n the event tht some households n domn d re ncluded n the smple). Workng out the condtonl dstrbuton s not trvl exercse wthout mkng further dstrbutonl ssumptons. Consequently ELL decded to forego EB estmton ltogether. In dong so, they hve ccepted certn loss n effcency by not fully utlzng ll vlble nformton. Moln nd Ro 010) recently pcked up on ths nd put forwrd n lterntve pproch tht does mplement EB estmton. They tke ELL s pont of deprture but then ssume tht both u nd ε h re norml dstrbuted, n whch cse the condtonl dstrbuton too wll be norml dstrbuted. Where ELL ccept loss n precson by not mplementng EB estmton, Moln nd Ro 010) ccept loss n precson tht mght stem from msspecfcton of the error dstrbuton functons. The dt t hnd wll ultmtely determne whch of the two wll be cceptng the lrger loss. ELL re rgubly most nterested n estmtng poverty nd nequlty n developng countres where the number of smll res or domns) tht re covered by the ncome surveys re often smll, thnk of 5 to 5 percent of ll domns n the populton. In ths cse the benefts of EB estmton wll be modest s survey dt re vlble for only few res). However, n more developed countres, or countres where trvel costs tht re ncurred when coverng ll smll res re mngeble, ncome surveys often cover much lrger number of the domns. In fct, there re numerous exmples where surveys cover between 50 nd 100 percent of ll domns n the country. In those nstnces, there my be cler benefts to doptng EB estmton. The pproch presented n ths pper mproves on both ELL nd Moln nd Ro 010). Lke ELL we mke no restrctve ssumptons bout the error dstrbutons. Our estmtor 1 Note tht ELL llows for heteroskedstcty, so tht σε cn be household-specfc. 3

6 for the dstrbuton functons wll generlly be more ccurte thn the estmtor dopted by ELL however, s we explctly ccount for the nested error structure tht s responsble for the convoluton problem ). Unlke ELL we lso ccommodte EB estmton. We cheve ths by fttng fnte norml mxtures NM) to the error dstrbuton functons. Norml mxtures re extremely flexble; they re ble to ft ny well-behved dstrbuton functon, nd re delly suted for ccommodtng EB estmton. If the mrgnl dstrbutons of u nd ε h cn be descrbed by norml mxtures, then the condtonl dstrbuton too cn be descrbed by norml mxture wth known prmeters tht re functons of these prmeters nd of the dt on whch s beng condtoned). Estmton of the norml mxtures for u nd ε h s complcted by the fct tht nether u nor ε h re observed. Our estmtor for the NM prmeters my be vewed s modfed verson of the EM lgorthm. Monte Crlo smultons ndcte tht estmtes of poverty nd nequlty cn be severely bsed when gnorng the non-normlty of the errors. The bs cn be s hgh s to 3 percent on poverty rte of 0 to 30 percent. Most of ths bs s resolved when mplementng our estmtor. Ths s confrmed by n emprcl pplcton to US dt. 1. Norml mxtures n nested-error models There re number of other studes tht hve explored dfferent wys of relxng the normlty ssumpton n mxed lner models. Verbeke nd Lesffre 1996) s n erly exmple tht lso consders norml mxtures s non-prmetrc representton of the error dstrbuton functon. However, they mpose number of mportnt restrctons. Frst, only the re rndom effects u re llowed to be non-norml; norml-mxture s ftted to the dstrbuton of u under the ssumpton tht ε h s normlly dstrbuted. Second, t s ssumed tht the component dstrbutons tht mke up the norml mxture shre common vrnce, whch notcebly smplfes estmton but t the sme tme sgnfcntly lmts the flexblty of the norml mxture to ft ny gven dstrbuton functon. Ths pproch hs lso been followed by Cordy nd Thoms 1997) who work wth the sme setup nd dopt the sme set of restrctons. There s nother strnd of the lterture tht permts both error terms to be non-norml dstrbuted by mposng n lterntve prmetrc fmly for the dstrbuton functons. See for exmple Zhou nd He 008) who ft skewed t-dstrbutons to the nested errors of the lner mxed model. Recently, there hve lso been efforts to explore the mpct of msspecfctons n the error dstrbutons for Emprcl Byes predctons derved from lner mxed models, see for exmple Skrondl nd Rbe-Hesketh 009) nd McCulloch nd Neuhus 011). They conclude tht the bs s resonbly smll. It should be noted however tht those studes focus on predcton of the dependent vrble tself, n whch cse the Emprcl Byes estmtes of the re rndom effects u denote the only source of bs. Msspecfctons n the error dstrbutons become consderbly more mportnt when predctng nonlner functons of the dependent vrble, such s mesures of poverty nd nequlty, s we wll show n ths pper. The remnder of ths pper s orgnzed s follows. In Secton we brefly dscuss how the error dstrbuton functons ssocted wth the errors n lner mxed model wll mtter for 4

7 predcton, wth n pplcton to poverty nd nequlty mesurement. In ths secton we wll lso ntroduce norml mxtures s flexble non-prmetrc representton of ny gven wellbehved dstrbuton functon, nd demonstrte the mplctons for EB estmton. Estmton of the norml mxture dstrbutons to both errors from the lner mxed model s presented n Secton 3. A modest Monte Crlo smulton study followed by n eqully modest emprcl pplcton s provded n Sectons 4 nd 5, respectvely. Fnlly, Secton 6 concludes. Estmton of poverty nd nequlty: Dstrbutons mtter.1 Lner mxed model for ncome Suppose tht t the household level the dt genertng process DGP) stsfes the equton lredy mentoned bove: y h = x T h β + u + ε h, 1) where x h denotes vector wth ndependent vrbles, nd where u nd ε h denote zero expectton error terms tht re ndependent of ech other. The subscrpts ndcte trget re or domn) nd household h. In ths pper we ssume tht errors re homoskedstc, so tht for ech household h nd re we hve: vr[y h x h ] = σ u + σ ε. Throughout the pper t s ssumed tht consstent estmtors for the vrnce prmeters re vlble, whch we shll denote by ˆσ u nd ˆσ ε. 3 dstrbutons. We wll not mke ny ssumptons bout the shpe of the error Let A denote the totl number of res covered by the ncome survey nd let n denote the number of households tht hve been smpled n re, so tht n = A =1 n denotes the totl smple sze of the survey. We shll denote the totl household error by: e h = y h x T hβ, nd ts re verge by ē = ȳ x T β where ē = h e h/n. We shll lso use the notton e = e,1,..., e,n ) whch denotes the vector of length n wth resduls for ll households from re. Wth slght buse of termnology we wll t tmes refer to the errors e h nd ē s dt s f we know the prmeter vector β). Let the probblty dstrbuton functons for u nd ε h be denoted by F u nd G ε. We wll propose computtonlly ttrctve method for estmtng these dstrbuton functons, where we mke no restrctve ssumptons concernng ther functonl form, n prtculr llowng these functons to be other thn norml dstrbuton functons. Another ppelng feture of our non-prmetrc estmtor for the error dstrbuton functons s tht t cn esly ccommodte Emprcl Byes estmton. These res my refer to geogrphc res such s dstrcts or muncpltes, but lso to non-geogrphc domns such s ethnc groups or ge groups, sy. 3 A commonly used estmtor for the vrnce prmeters from nested error model s Henderson s method III estmtor see Henderson, 1953; nd Serle et l., 199), whch my be vewed s method of moments estmtor whch does not requre ny ssumptons bout the shpe of the error dstrbutons. Alterntve estmtors re restrcted mxmum lkelhood, mnmum norm qudrtc unbsed estmton MINQUE; see e.g. Westfll, 1987; Serle et l., 199), nd so-clled spectrl decomposton estmton see e.g. Wng nd Yn, 00; nd Wu et l., 009). 5

8 . Emprcl Byes estmton Emprcl Byes EB) estmton, lso known s Emprcl Best estmton, tghtens the error dstrbutons by condtonng on ll vlble dt n the survey for the purpose of predcton. The observton e provded tht re s covered by the survey) clerly crres nformton bout the re rndom effect u. In the extreme, for exmple, where n tends to nfnty ē wll perfectly revel u. In effect EB estmtes re obtned by ntegrtng out re errors u usng probblty densty pu e ),.e. the probblty densty of u condtonl on e observed from the survey. Non-EB estmtes re obtned by usng the uncondtonl densty pu ).) The chllenge s to work out pu e ) long wth the mrgnls pu ) nd pε h ) wthout mposng restrctons bout ther form so tht the resultng condtonl densty pu e ) cn lso tke on ny form. Currently, the lterture on EB estmton vods ths chllenge by ssumng normlly dstrbuted mrgnls, n whch cse the condtonl dstrbuton wll be norml too. For normlly dstrbuted errors t cn be shown tht pu e ) cn be wrtten s functon of ē, the men of smple errors from domn. Ths s not true for generl error dstrbutons. However, condtonng on the full vector e becomes computtonlly ntrctble. Therefore we wll condton on ē rther thn the full vector e even n the cse of non-norml errors. A second smplfcton concerns the known regresson resduls for smple households. When condtonng on e, we gnore tht for these households the totl error e h = u + ε h s known snce t s one of the components of e ). Insted we wll ssume tht ε h s ndependent of e even for smple households. The reson s tht n prctce smple household h cnnot be trced mong households n the trget domn households for whch only dt on x s vlble). In most prctcl settngs, the error thus ntroduced wll be neglgble. In fct, ths error tends to zero s the sze of the smple reltve to the populton sze tends to zero..3 Dstrbutons mtter Let y ) nd e ) denote vectors of length N wth elements y h nd e h for ll households from the populton. Smlrly, x ) wll denote mtrx wth rows gven by x T h for ll N households. y, e nd x wll denote the survey smple nlogues. As mentoned bove our objectve s to estmte: E[W y ) ) x ), y ] = W x ) β + e)pe e )de ) W x ) β + u + ε)pε)pu e )dεdu, 3) where the functon W wll generlly be non-lner. Non-normlty of the errors u nd ε wll ffect the estmtes of E[W ] v two dfferent chnnels. Frstly, when the functon W s ndeed non-lner, the expected vlue E[W ] wll be functon of the hgher moments of the dstrbutons of u nd ε. Gettng these moments wrong, n other words gettng the dstrbutons wrong, s then lkely to ntroduce bs. Secondly, f 6

9 the dstrbutons of u nd ε re wrong then the condtonl densty pu e ), whch concerns EB estmton, wll lso be wrong. Ths wll ffect ll moments of the condtonl dstrbuton of u, ncludng the frst, nd hence hs the potentl to ntroduce bs n the estmte of E[W ] even f W s lner. Note tht n the cse of lner W, non-eb estmtes of E[W ] re unbsed even f the dstrbutons for u nd ε re wrong.) Recently, Skrondl nd Rbe-Hesketh 009) nd McCulloch nd Neuhus 011) hve explored the mgntude of the bs tht s ntroduced by gettng the frst moment of pu e ) wrong, n the cse of lner W, nd found t to be modest. Our estmtor would therefore be most relevnt for the cse of nonlner W..4 Emprcl Byes estmton wth norml mxtures Let us ssume tht F u nd G ε cn be represented by mxture dstrbutons: F u = G ε = =m u =1 j=m ε j=1 π F 4) λ j G j, 5) where the F s nd G j s denote bss of dstrbuton functons whch we wll lso refer to s components or component dstrbuton functons. The π s nd λ j s denote unknown nonnegtve probbltes tht stsfy π = 1 nd j λ j = 1, whch we wll lso refer to s mxng probbltes. m u nd m ε denote the number of components used to represent F u nd G ε, respectvely. We wll denote the probblty densty functons ssocted wth F nd G j by respectvely f nd g j. Mxture dstrbutons re remrkbly well equpped to ft ny well-behved dstrbuton functon. For exmple, kernel densty estmtors re closely relted to mxture dstrbutons. We wll be workng wth norml component dstrbutons, so tht the mxture dstrbutons re norml mxtures. Assumpton 1 The components F re norml dstrbuton functons wth men µ nd vrnce σ. Smlrly, components G j re norml dstrbuton functons wth men ν j nd vrnce ωj. Note tht the modeler s t lberty to work wth dfferent bss of component dstrbutons. Ths choce does not hve rel mplctons for the blty of the mxture to ft gven dstrbuton functon. If pu ) nd pε h ) re norml-mxtures, then pu ē ) s norml mxture too. Ths s powerful result s the ntegrl n equton 3 wll generlly hve to be computed by smulton, nd smplng from norml mxtures s strghtforwrd. Lemm shows how the prmeters tht defne pu ē ) cn be obtned s functon of the prmeters of the norml-mxtures pu ) nd pε h ). Implementng EB estmton s thus s esy s smplng the re errors from the norml-mxture pu ē ) whenever ē s observed n the survey smple. 7

10 Lemm The probblty densty functon of u condtonl on ē, whch we denote by pu ē ), s norml-mxture wth known prmeters: pu ē ) = =m u =1 j=m ε j=1 k=m ε k=1 w jk ϕ u ; m jk ; s jk), 6) where: m jk = s jk = σ σ + ω j + ω k )/n ) ω j + ωk )σ ωj + ω k + n σ, ) ē ν j + ν k )/n ) + ω j + ω k ω j + ω k + n σ ) µ nd where w jk = w jk / jk w jk wth: w jk = π λ j λ k ϕē ; µ + ν j + ν k )/n ; σ + ω j + ω k )/n ), 7) where ϕ denotes the norml probblty densty functon, nd where π, µ, σ ) nd λ j, ν j, ω j ) denote the prmeters ssocted wth the norml-mxture dstrbutons F u nd G ε, respectvely. If we pply the Centrl Lmt Theorem to pproxmte the mrgnl dstrbuton of ε to norml dstrbuton wth men zero nd vrnce equl to σ ε/n, the expresson for pu ē ) smplfes consderbly: pu ē ) =m u =1 wth γ = σ /σ + σ ε/n ), nd where α = α / α wth: α ϕ u ; γ ē + 1 γ )µ ; 1/σ + n /σε ) ) 1, 8) α = π ϕē ; µ ; σ + σ ε/n ). 9) The expected vlue of u condtonl on ē, gven the densty functon pu ē ) from Lemm, s seen to solve: E[u ē ] α ē ) γ ē + 1 γ )µ ), 10) where γ = σ /σ + σ ε/n ), nd where α ē ) denotes the mxng probbltes of pu ē ). Note tht the stndrd ssumpton of norml errors s nested s specl cse, where there s just one component wth µ = 0 nd σ = σu. The frst nd second moment of the norml condtonl densty pu ē ) n ths cse re seen to solve: E[u ē ] = γ ē 11) vr[u ē ] = 1 γ )σu, 1) 8

11 where γ = σ u/σ u + σ ε/n ) see e.g. Moln nd Ro, 010). 4 For non-norml errors, we hve tht E[u ē ] s generlly non-lner functon of ē, nd vr[u ē ] wll generlly be functon of ē..5 Applcton to smll re estmton of poverty nd nequlty For our pplcton let y h mesure per cpt ncome or expendture) for household h resdng n re, nd let s h denote the number of household members for tht sme household. In vector notton, let y ) nd s ) be vectors wth elements y h nd s h for ll households from re. The objectve s to determne the level of welfre for re whch cn be expressed s functon of y ) nd s ) : W y ), s ) ). The welfre functon W s typclly non-lner. Populr exmples re the shre of ndvduls whose ncome flls below pre-specfed poverty lne lso known s the hed-count poverty rte), or the Gn ndex of ncome nequlty. Collectng dt on ncome or expendture) y h for ny gven household s generlly found to be expensve reltve to collectng dt on demogrphcs, educton, employment sttus, nd housng. Ths s prtculrly true for developng countres where much of the ncome does not come from wge employment. Consequently, ncome dt s often only vlble n the form of so-clled ncome surveys. The smple sze of these surveys s suffcent to estmte ntonl nd possbly sub-regonl welfre, but too smll to estmte welfre drectly t the level of much smller res.e. the trget res ). Elbers et l. 003; henceforwrd ELL) dvocte n pproch tht combnes the ncome survey wth unt record populton census dt. The census hs dt on the ndependent vrbles x h from equton 1), such s demogrphcs, educton, employment nd housng, but not the household ncome vrble y h. Cruclly, the dt on x h re lso collected by the ncome survey. The de s to use the ncome survey to estmte the prmeters from equton 1), nd then use the model to predct ncome for every household n the census. Wth these predcted ncomes we cn subsequently estmte welfre W for ech trget re. Stndrd errors cn be obtned by mens of smulton whch s delly suted for estmtng qunttes tht re non-lner functons of the rndom vrbles t hnd, s s the cse wth mesures of poverty nd nequlty. Let R denote the number of smultons. The estmtor then tkes the form: ˆµ = 1 R R W r=1 ) ỹ r) ), s ), 13) where ỹ r) ) denotes the r-th smulted or predcted) ncome vector wth elements ỹr) h = xt β h r) + ũ r) + ε r) h. Wth ech smulton, both the model prmeters β r) nd the errors ũ r) nd ε r) h re drwn from ther estmted dstrbutons. 5 In the end ths gves R smulted poverty rtes. The 4 Note tht the uncondtonl vrnce solves: vr[u ] = vr[γ ē ] + 1 γ )σ u = γ σ u + σ ε/n ) + 1 γ )σ u, whch equls vr[u ] = γ σ u + 1 γ )σ u = σ u, snce σ u + σ ε/n = σ u/γ. 5 Our preferred method s to drw β r) by re-estmtng the model prmeters usng the r-th bootstrp verson of the survey smple. Alterntvely, β r) my be drwn from ts estmted symptotc dstrbuton. The dfference between these two lterntves s expected to be modest, unless the survey smple s prtculrly smll so tht fnte smple effects my ply role. 9

12 pont estmtes nd ther correspondng stndrd errors re obtned by computng respectvely the verge nd the stndrd devton over these smulted vlues. It should be noted tht ELL drw the re error ũ from the estmted uncondtonl dstrbuton, whch s estmted non-prmetrclly. The dstrbuton for ε h too s estmted non-prmetrclly.) The dvntge of ths pproch s tht t s fully flexble n tht t does not restrct the shpe of the error dstrbutons. A possble shortcomng s tht t does not tke full dvntge of ll the vlble dt. Idelly one would wnt to drw ũ from dstrbuton tht s condtoned on ll relevnt dt tht hs been smpled from re. Moln nd Ro 010; henceforwrd MR) do exctly tht, they closely follow ELL but then drw the re error from the condtonl dstrbuton. Ths s referred to s Emprcl Byes EB) estmton. Importntly, MR lso dffer from ELL n tht they restrct the errors to be normlly dstrbuted. As we hve seen, under ths ssumpton the condtonl dstrbuton of ũ s norml too nd ts prmeter cn be esly determned. 6 When errors re non-norml, t s not obvous wht form the condtonl dstrbuton for ũ wll tke; t wll generlly be of dfferent form thn the uncondtonl dstrbuton. As s noted n secton.3, gettng the error dstrbutons rght s not merely mtter of effcency. When the welfre functon W s non-lner functon of the error terms, usng wrong error dstrbutons wll lso ntroduce bs n the welfre estmtes. Whether the mgntude of ths bs due to msspecfcton s mportnt n prctce s n emprcl queston. The choce between non-norml errors combned wth non-eb estmton whch s more flexble, but does not fully utlze ll vlble dt) or norml errors combned wth EB estmton whch s less flexble, but fully utlzes the vlble dt) my be determned/motvted by: ) the degree of non-normlty found n the dt, nd b) how much nformton one stnds to gnore/lose. The ltter depends lrgely on: ) how mny res hve been smpled by the ncome survey s for res not represented n the survey EB nd non-eb estmton re equvlent), nd ) the sze of the re error reltve to the totl error. The pproch developed n ths pper ms to combne the best of both worlds; we dopt EB estmton whle permttng non-norml dstrbuton functons. The non-prmetrc estmtor for the dstrbuton functons used by ELL does not lend tself for EB estmton. We propose non-prmetrc estmtor tht does. Remrk 3 Note tht n dded dvntge of representng the non-norml error dstrbutons by norml-mxtures s tht E[wy h )] cn be evluted s the sum of expecttons over norml errors,.e. E[wy h )] = E[wx T h β + u + ε h )] = j κ je[wx T h β + u) + ε j) h )], where u) nd ε j) h re norml dstrbuted wth mens µ nd ν j, nd vrnces σ nd ωj. Ths mens tht evlutng E[W y h )] nlytclly under norml-mxture errors s no more dffcult thn under norml errors. 6 Ths s rgubly why errors re generlly ssumed norml n the lterture on EB estmton. 10

13 3 Estmton of norml mxtures when errors re nested 3.1 Estmton of unrestrcted norml mxtures The objectve s to estmte the prmeters π, µ, σ ) nd λ j, ν j, ωj ), whch determne the norml mxtures NM) F u u) = =m u =1 π F u; µ, σ ) nd G εε) = =m ε j=1 λ j G j ε; ν j, ωj ), respectvely, where m u nd m ε denote the number of components. Estmton of NM to observed dt s routne tsk, mostly usng the EM lgorthm, see e.g. Dempster et l. 1977) nd McLchln nd Peel, 000). Suppose tht some dt y s drwn from NM wth m components nd prmeter vector θ. The lgorthm ntroduces ltent rndom vrble z j tht equls 1 f observton y j hs been drwn from component, nd 0 otherwse. Let the log-lkelhood functon tht trets both y nd z s dt be denoted by Lθ; y, z), nd ts dervtve wth respect to θ by L θ θ; y, z). The correspondng mxmumlkelhood ML) estmtor solves the followng moment condton: E[L θ θ; y, z)] = E[E[L θ θ; y, z) y]] = 0. 14) The EM lgorthm essentlly solves the sme moment condton but tertvely: E[E[L θ θ; y, z) y, ˆθ ]] = 0, 15) where ˆθ denotes the terton-k estmte for θ. Solvng equton 15) yelds ˆθ k+1). The procedure results n non-decresng sequence of lkelhoods Ly; ˆθ k ). The dvntge of the EM lgorthm s tht solvng equton 15) s often consderbly eser thn solvng equton 14). Note tht the unobserved dt z s ntegrted out condtonl on the observed dt y nd the current estmte of θ. The nested error structure however poses chllenge tht prevents strghtforwrd pplcton of the EM lgorthm. Conventonlly, mxture dstrbutons re estmted to dt tht re observed drectly. We wsh to estmte the dstrbutons for u nd ε h but we do not observe ether of them. In some sense u nd ε h re both observed wth error even f the totl error e h s known wth certnty), whch mkes ths convoluton problem. We propose modfcton of the EM lgorthm tht s ble to del wth ths convoluton problem. In nutshell, we tret the mesurement error s ltent vrble nd ntegrte t out the sme wy s the ltent vrble z s ntegrted out by the EM lgorthm. When estmtng F u the NM for u ), the observble dt ncludes ē = u + ε, where ε = h ε h/n wll ct s mesurement error. The modfed EM lgorthm wll then ntegrte out ε long wth z. When estmtng G ε the NM for ε h ), the observble dt ncludes e h = u + ε h. Now u tkes on the role of mesurement error nd wll hence be ntegrted out jontly wth z. Note tht we would need some ntl estmte of G ε when estmtng F u ths wy s we hve to ntegrte out ε). Smlrly, we need some ntl estmte of F u when estmtng G ε to ntegrte out u). Ths by tself s very much n the sprt of the EM lgorthm whch requres n ntl estmte of θ to determne to dstrbuton of z condtonl on the observed dt. 11

14 Our modfed EM lgorthm solves the followng moment condton when estmtng F u : E[L θu θ u ; ē, ε, z ) ē, ˆθ u ] = 0, 16) whch s equvlent to: E[E[L θu θ u ; ē, ε, z ) ē, ε, ˆθ u ] ē ] = 0. 17) Smlrly, t solves the followng moment condton when estmtng G ε : E[L θε θ ε ; e, u, z ) e, ˆθ ε ] = 0, 18) whch s equvlent to: E[E[L θε θ ε ; e, u, z ) e, u, ˆθ ε ] e ] = 0, 19) where e = e 1,..., e,n ). Specfclly, n order to ntegrte out ε condtonl on the observed ē see equton 17), we need n ntl estmte of the probblty dstrbuton functon for ε ē. It cn be verfed tht p ε ē ) s gn norml-mxture dstrbuton. Lemm 4 The probblty densty functon of ε condtonl on ē, whch we denote by p ε ē ), s norml-mxture wth known prmeters: p ε ē ) = =m u =1 j=m ε j=1 k=m ε k=1 w jk ϕ ε ; m jk ; s jk), 0) where: m jk = s jk = ωj + ) ) νj ω k n σ ) + ν k ωj + ω k + n σ ē µ ) + ωj + ω k + n σ n ) ω j + ωk )σ ωj + ω k + n σ, nd where w jk = w jk / jk w jk wth: w jk = π λ j λ k ϕē ; µ + ν j + ν k )/n ; σ + ωj + ωk )/n ), 1) where ϕ denotes the norml probblty densty functon, nd where π, µ, σ ) nd λ j, ν j, ωj ) denote the prmeters ssocted wth the norml-mxture dstrbutons F u nd G ε, respectvely. Smlrly, to ntegrte out u condtonl on e see equton 19), we need the probblty dstrbuton for u e. As mentoned erler however, we use the dstrbuton of u ē, ssumng 1

15 tht pu e ) pu ē ). It follows tht u ē too s norml-mxture dstrbuted. Lemm 5 The probblty densty functon of u condtonl on ē, whch we denote by pu ē ), s norml-mxture wth known prmeters: pu ē ) = =m u =1 j=m ε j=1 k=m ε k=1 w jk ϕ u ; m jk ; s jk), ) where: m jk = s jk = σ σ + ω j + ω k )/n ) ω j + ωk )σ ωj + ω k + n σ, ) ē ν j + ν k )/n ) + ω j + ω k ω j + ω k + n σ ) µ nd where w jk = w jk / jk w jk wth: w jk = π λ j λ k ϕē ; µ + ν j + ν k )/n ; σ + ω j + ω k )/n ), 3) where ϕ denotes the norml probblty densty functon, nd where π, µ, σ ) nd λ j, ν j, ω j ) denote the prmeters ssocted wth the norml-mxture dstrbutons F u nd G ε, respectvely. It should be noted tht by workng wth pu ē ) nsted of pu e ) we wll be solvng modfed moment condton, nmely: E[E[L θε θ ε ; e, u, z ) ē, u, ˆθ ε ] ē ] = 0. 4) Ths denotes genune deprture from the orgnl EM lgorthm where one condtons on ll the dt tht fetures n the log-lkelhood functon. We wll refer to the resultng estmtor s pseudo-em estmtor. We expect the loss n precson to be mnor, whle the gn n prctclty s substntl. The resultng estmtors, n the form of tertve equtons, re presented below see Annex 7.3 for dervton). Gven some ntl estmte of p ε ē ), we my mplement the estmtor for F u. Subsequently, gven ths estmte for F u, we my mplement the estmtor for G ε. The newly obtned estmtes cn n turn be used to updte our estmtes for p ε ē ) nd pu ē ), fter whch we my obtn new round of estmtes for F u nd G ε. Ths s contnued untl convergence. In prctce, one terton s found to be suffcent to obtn ccurte estmtes.) Estmtor for F u The fxed-pont soluton to the followng set of tertve equtons yelds the estmtor ˆπ, ˆµ, ˆσ ) 13

16 for π, µ, σ ) for = 1,..., m u: wth: ˆπ k+1) = E ˆµ k+1) = ˆσ k+1) = [ ˆτ ε ) ē ; ˆp ε ē ) ] /A 5) [ ] E ˆτ ε )ē ε ) ē ; ˆp ε ē ) [ ] 6) E ˆτ ε ) ē ; ˆp ε ē ) [ ) E ˆτ ε ) ē ε ˆµ k+1) ē ; ˆp ε ē )] [ ], 7) E ˆτ ε ) ē ; ˆp ε ē ) ˆτ ε ) = ) ˆπ ϕ ē ε ; ˆµ, ˆσ ), 8) ē ε ; ˆµ, ˆσ ˆπ ϕ where ϕ denotes the norml probblty densty functon, nd where the expecttons re tken over ε condtonl on ē usng the terton-k estmte of the condtonl densty functon ˆp ε ē ). Lemm 4 shows how ˆp ε ē ) cn be obtned s functon of ˆπ, ˆµ, ˆσ ). Estmtor for G ε The fxed-pont soluton to the followng set of tertve equtons yelds the estmtor ˆλ j, ˆν j, ˆω j ) for λ j, ν j, ω j ) for j = 1,..., m ε: wth: ˆλ k+1) j = E ˆν k+1) j = ˆω k+1) j = ˆτ hj u ) = [ h ˆτ hj u ) ē ] /n 9) [ ] E h ˆτ hj u )e h u ) ē [ ] 30) E h ˆτ hj u ) ē [ ) ] E h ˆτ hj u ) e h u ˆν k+1) j ē [ ], 31) E h ˆτ hj u ) ē j ˆλ j ϕ ˆλ j ϕ e h u ; ˆν j e h u ; ˆν j ), ˆω j ), 3), ˆω where ϕ denotes the norml probblty densty functon, nd where the expecttons re tken over u condtonl on ē usng the probblty densty functon pu ē ). See Lemm 5 for the probblty densty functon for u ē. j 14

17 In order to get the tertve scheme strted, we would of course need sutble ntl estmtes for p ε ē ) nd pu ē ). Note tht even wthout ny pror knowledge bout the dstrbutons for u nd ε h, we should be ble to obtn resonble estmte of the dstrbuton for ε by ppelng to the Centrl Lmt Theorem CLT). Under the ssumpton tht ε h re ndependend cross households, we hve tht the dstrbuton of ε tends to norml dstrbuton wth men zero nd vrnce equl to σ ε/n. In typcl household ncome survey, n wll be n the rnge of 10 to 100, whch s suffcently lrge for the CLT to tke effect. The corollres below derve the ntl estmtes for p ε ē ) nd pu ē ) under the ssumpton of norml dstrbuted ε wth known vrnce). Corollry 6 The probblty densty functon of ε condtonl on ē, whch we denote by p ε ē ), s norml-mxture wth known prmeters: p ε ē ) = =m u =1 ) σ w ϕ ε ; ε /n σ σε/n + σ ē µ ); σε/n )) σ +, 33) σ ε/n where w = w / w wth: w = π ϕē ; µ ; σ + σ ε/n ), 34) where ϕ denotes the norml probblty densty functon, nd where π, µ, nd σ prmeters ssocted wth the norml-mxture dstrbuton F u. denote the Corollry 7 The probblty densty functon of u condtonl on ē, whch we denote by pu ē ), s norml-mxture wth known prmeters: pu ē ) = =m u =1 wth γ = σ /σ + σ ε/n ), nd where α = α / α wth: α ϕ u ; γ ē + 1 γ )µ ; 1/σ + n /σε ) ) 1, 35) α = π ϕē ; µ ; σ + σ ε/n ), 36) where ϕ denotes the norml probblty densty functon, nd where π, µ, nd σ prmeters ssocted wth the norml-mxture dstrbuton F u. denote the The followng propostons provde some propertes of the estmtors. Proposton 8 At every terton-k, the men nd vrnce for the probblty densty functon ĝ ε ε) = j ˆλ j ϕε; ˆν j, ˆω j ) solve: E[ε; ĝ ε ] = 1 n vr[ε; ĝ ε ] = 1 n e h E[u ē ; ˆp u ē )] h h E[e h u ) ē ; ˆp u ē )] E [ε; ĝ ε ]. 15

18 Proposton 9 At every terton-k, the men nd vrnce for the probblty densty functon u) = ˆπ ϕu; ˆµ, ˆσ ) solve: ˆf u E[u; vr[u; ˆf u ] = 1 A ˆf u ] = 1 A ē E[ ε ē ; ˆp ε ē )] 3. Some prctcl prmeter restrctons E[ē ε ) ē ; ˆp ε ē )] E [u; ˆf u ]. Let us lso consder the cse where the component dstrbutons re ssumed to be gven, so tht only the mxng probblty vectors π nd λ wll hve to be estmted. Ths s obvously specl cse of the more generl pproch where the probbltes re jontly estmted wth the prmeters of the component dstrbutons. Keepng the ltter fxed mrkedly benefts the numercl convergence, ndctng tht the pproch deserves to be consdered dstnct opton n nd of tself. Ths s precsely the pproch dvocted by Cordy nd Thoms 1997). The estmtors for π nd λ re obtned s solutons to the followng tertve equtons: π k+1) = 1 A π ϕ ē ; µ, σ + ) σ ε/n π ϕ ē ; µ, σ + ), 37) σ ε/n nd: λ k+1) j = 1 n h j λ j π ϕ λ j )) e h ; µ + ν j, σ + ω j π ϕ e h ; µ + ν j, σ + ω j )), 38) where π denotes the estmtor for π. Note tht Cordy nd Thoms 1997) only consder the estmtor for F u.e. the estmtor for π for = 1,..., m u ), s they re not nterested n the dstrbuton functon for ε h. Snce the prmeters of the component dstrbutons re not beng estmted, the modeler wll hve to set the vlues for the mens µ nd ν j, nd vrnces σ nd ω j beforehnd. Cordy nd Thoms 1997) recommend to set common vrnce s s lso done n kernel densty estmtors where the common vrnce s refered to s the bndwdth prmeter). A nturl choce s to set ths common vrnce equl to: σ = ˆσ u m u for ll nd hence ωj = ˆσ ε m ε ). Note tht for ny gven common vrnce σ we hve: vr[u] = σ + π µ, wth p µ 0. Ths mens tht t mnmum the common vrnce must be chosen so tht: vr[u] σ 0. Obvously ths s stsfed for σ = σ u m u, whch yelds: π µ = mu 1 m u )σu. A convenent choce for the men prmeters s to tke eqully spced µ nd ν j ) wth rnge tht respects the vlues ttned by the observed dt. Note tht the estmted mxng probbltes re expected to stsfy: E[u] = ˆπ µ = ē/a mtchng frst moment), s well s: ˆπ µ = mu 1 m u )ˆσ u mtchng second moment), lthough ths s not gurnteed by the estmton pproch. Ths s n contrst wth the pproch where the component dstrbuton prmeters re lso beng estmted, n whch cse the frst nd second moments re gurnteed. 16

19 Remrk 10 One possble extenson to the set of component dstrbutons proposed bove wth dfferent µ but common σ ) cn be obtned by ddng components wth common zero men.e. µ r = 0) but dfferent vrnces σ r. 4 A smll smulton study Ths secton presents modest Monte-Crlo smulton experment. We wll focus our ttenton to the followng two questons: 1) How effectve s our pproch n fttng non-norml error dstrbutons tht re not necessrly norml-mxtures)?, nd ) Wht re the mplctons for the estmton of poverty nd nequlty? Do dstnct devtons from normlty hve the potentl to ntroduce menngful bs n estmtes of poverty nd nequlty f ths nonnormlty s gnored? We mke the followng ssumptons. The smulted country conssts of 500 domns or trget res), whch denotes the level t whch mesures of poverty nd nequlty wll be estmted. Ech domn s home to 3000 households. Household per cpt ncomes wll be generted by mens of the followng model: y h = βx h + u + ε h, 39) where y h denotes the log of household ncome, nd where x h represents sngle covrte. For the model prmeters we consder the vlues: vr[u + ε h ] = σ u + σ ε = 0.3, β = 1, E[x h ] = 0, nd where vr[x h ] s chosen so tht R = 0.4 whch represents goodness-of-ft tht s typcl for emprcl household ncome models). It wll be convenent to ntroduce prmeter tht mesures the sze of the rndom re effect reltve to the totl error term. Let us denote ths prmeter by ρ = σ u/σ u + σ ε). We wll be consderng both the cse of smll re effect ρ = 0.05) nd medum/lrge re effect ρ = 0.5). The non-norml errors u nd ε h wll be drwn from Log-Dgum dstrbuton wth probblty densty functon: fx) = pe px logb)) 1 + e x logb)) ) p+1, 40) where p > 0 s the prmeter tht determnes the shpe or skewness) of the dstrbuton. Smller vlues of p wll result n lrger devtons from normlty. The remnng two prmeters nd b wll be fxed by mposng zero men nd settng the vrnce to σu or σε. Note tht the Dgum dstrbuton s not n uncommon choce when modelng ncome dt see e.g. Kleber 007), nd the references theren). The covrte x h wll be drwn from norml dstrbuton. Fnlly, our rtfcl survey wll smple 15 households from ech domn. 17

20 4.1 Estmton of F u nd G ε The frst test wll be whether we cn successfully uncover the probblty densty functons for u nd ε h. We wll keep the shpe prmeter for F u fxed t p u = 0.5, but wll consder three dfferent vlues of the shpe prmeter for G ε : p ε = 0.10, 0.5, 0.50). The benchmrk densty functons re obtned s kernel densty estmte ppled to the ctul relztons of the errors n the census. Our estmtors for F u nd G ε re bsed on norml-mxtures wth 3 nd component dstrbutons, respectvely. Fgure 1 presents our estmtes of the probblty densty functon for u. The estmtes n the rght pnel show remrkbly good ft. These correspond to the cse where the rndom re effect s lrge, wth ρ = 0.5, n whch cse the contrbuton of u mtters. The mperfect ft shown n the left pnel suggests tht t s hrder to estmte F u when the rndom re effect s smll n ths cse ρ = 0.05). Ths s rgubly due to the poor sgnl-to-nose rto n ths cse; u wll only mke up smll frcton of the totl resdul whch s wht s observed. However, ths lck of precson wll lso hve lttle to no mplcton for estmtes of poverty nd nequlty, precsely becuse of the smllness of u. In sum, n ths exmple estmtes of F u re precse when they mtter, nd less precse when they mtter less. Probblty densty functon for u dens.uhtx) dens.uhtx) x x Fgure 1: Norml-mxture densty Blck) versus benchmrk densty Red): ) ρ = 0.05 left pnel); b) ρ = 0.5 rght pnel) Our estmtes for the probblty densty functons for ε h re presented n Fgure. Wht s pprent s tht, despte usng no more thn two components, our estmtes provde remrkbly ccurte fts regrdless of the sze of the rndom re effect. When ρ s smll, we hve tht ε h s lrge reltve to u, nd so the sgnl-to-nose rto s n our fvour when estmtng G ε. It mtters less n ths cse tht we do not hve precse estmte for F u. On the other hnd, when ρ s lrge nd hence ε h s of more modest mgntude reltve to u, our estmton of G ε my be helped by hvng precse estmte of F u. Note tht we frst estmte F u, nd then use ths estmte to subsequently estmte G ε.) 18

21 Probblty densty functon for ε dens.epshtx) dens.epshtx) x x Fgure : Norml-mxture densty Blck) versus benchmrk densty Red): ) ρ = 0.05 left pnel); b) ρ = 0.5 rght pnel) 4. Implctons for estmtes of poverty Next we nvestgte whether hvng more precse estmtes of F u nd G ε wll lso gve us more precse estmtes of poverty, nd whether ny gn n precson s economclly menngful. For ese of exposton we wll focus on the percentge of households wth ncomes below the poverty lne s the mesure of poverty whch we wsh to estmte. Tbles 1 nd provde estmtes of the bs for dfferent vlues of the shpe prmeter for G ε the shpe prmeter for F u s fxed t p u = 0.5), nd for dfferent vlues of the log) poverty lne. The poverty rtes ssocted wth the dfferent log) poverty lnes re roughly 15, 0, 30, nd 45 percent. It should be noted tht the bs n ths cse s estmted s the dfference between the estmted nd the true poverty rte verged over the 500 trget res for one gven replcton of the census. At lest two observtons stnd out. Frst, estmtes obtned under the ncorrect) ssumpton of norml errors cn be severly bsed, wth bs of 3 to 4 percent on poverty rte tht rnges between 0 nd 30 percent dependng on the shpe prmeter for G ε. Our pproch provdes fr superor estmtes of poverty n these cses wth bs of less thn percent. Second, the beneft of ccommodtng non-norml errors s opposed to ssumng norml errors chnges wth the vlue of the poverty lne. At the fr left tl of the log) ncome dstrbuton.e. for prtculrly low vlues of the poverty lne), t wll be hrd to emprclly seprte the two dstrbutons. The dfference wll be more pronounced where the dstrbutons hve more mss,.e. for ntermedte to hgher vlues of the poverty lne. 4.3 Implctons for estmtes of nequlty Tble 5 compres the bs for estmtes of ncome nequlty for dfferent vlues of the shpe prmeter nd dfferent vlues of the re locton effect. We focus on the Men-Log-Devton MLD) s the mesure of nequlty whch we wsh to estmte. Note tht by defnton we hve tht the relztons of the re error u wll drop out of the true mesure of MLD for tht re, 19

22 Poverty ρ = 0.05 ρ = 0.5 lne/shpe ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Tble 1: Bs for EB estmtes of hed-count poverty: ) Norml-Mxture dstrbuton left number); b) Norml dstrbuton rght number) Poverty ρ = 0.05 ρ = 0.5 lne/shpe ) ) ) ) ) ) ) 0..73) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Tble : Bs for NON-EB estmtes of hed-count poverty: ) Norml-Mxture dstrbuton left number); b) Norml dstrbuton rght number) Poverty ρ = 0.05 ρ = 0.5 lne/shpe ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Tble 3: RMSE for EB estmtes of hed-count poverty: ) Norml-Mxture dstrbuton left number); b) Norml dstrbuton rght number) Poverty ρ = 0.05 ρ = 0.5 lne/shpe ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Tble 4: RMSE for NON-EB estmtes of hed-count poverty: ) Norml-Mxture dstrbuton left number); b) Norml dstrbuton rght number) whch vres round 0 for our smulted dt we multpled the numbers by fctor 100). Note however tht the sze of the locton effect my stll hve berng on our estmtes of nequlty, to the extent tht t mpcts on our estmtes of the dstrbuton for ε h. Smlrly to wht we observed for poverty, the potentl gn of ccommodtng non-norml errors s opposed to ncorrectly ssumng norml errors cn be qute lrge. We observe bses of to 3 wth true nequlty round 0 when ssumng norml errors, compred to bses of roughly 0.5 when usng our pproch. 0

23 Shpe prmeter MLD ρ = ) ) ) ρ = ) ) ) Tble 5: Bs for estmtes of nequlty: ) Norml-Mxture dstrbuton left number); b) Norml dstrbuton rght number) 5 A smll emprcl study For ths smll emprcl pplcton we wll tret the 010 mcro census from the Unted Sttes, 1 percent smple, s new country.e. s f the dt consttutes full populton census). The dt s publclly vlble t IPUMS USA. There re totl of 1.5 mllon households 7 resdng n 4 countes, whch s the level t whch we wll be estmtng poverty nd nequlty.e. the smll re level ). The census ncludes ndvdul ncome dt s well s dt on wde rnge of ndvdul chrcterstcs. We use the ncome dt to compute household ncome per cpt. All other ndvdul level vrbles re lso ggregted t the household level s ths denotes the level t whch we wll buld the ncome model. We defned the hed of the household s the member of the household wth the hghest level of ndvdul ncome. In mny emprcl pplctons much of the vlble dt s t the household level.) Tble 6 provdes some descrptve sttstcs of the dt. Vrble Descrpton Men Std. Dev. lnycp Log of household ncome per cpt metro suburb In sub-urb of mjor cty dummy) metro none Awy from mjor cty dummy) hh sze 1 Household of sze one dummy) hh sze Household of sze two dummy) hh sze 3 Household of sze three dummy) shre chld15 Shre of chldren ged younger thn hh hedu At lest one member wth msters degree dummy) hh employed Number of employed household members hd femle Hed of household s femle dummy) hd logge Log of ge of household hed hd ledu Household hed hs lower educton only dummy) hd hedu1 Household hed hs college degree dummy) hd hedu Household hed hs msters degree dummy) hd employed Household hed s employed dummy) hd hsp Household hed s Hspnc dummy) hd blck Household hed s Afrcn-Amercn dummy) Tble 6: Descrptve sttstcs of vrbles used n emprcl pplcton derved from 010 IPUMS USA) We drw blnced smple of 6330 households 15 households per county). The log of 7 The exct number of households n the census s 1, 4,

24 household per cpt ncome wll be our dependent vrble. Our ndependent vrbles nclude: urbnty, household sze, ge composton, gender, ethncty, educton, nd employment. There re totl of 16 regressors excludng the constnt). The regresson model s shown n Tble 7. We obtned n djusted R-squred of 0.43, whch denotes typcl level for ths type of ncome or consumpton) regresson model. The locton effect s estmted t 0.07, whch s not unusul for developed ntons but rther smll for developng countres where one mght fnd locton effects n the rnge of 0.05 to 0.5. lnycp metro suburb ) metro none ) hh sze ) hh sze ) hh sze ) shre chld ) hh hedu ) hh employed ) hd femle ) hd logge ) hd ledu ) hd hedu ) hd hedu ) hd employed ) hd hsp ) hd blck ) cons ) N 6330 dj. R 0.43 t sttstcs n prentheses p < 0.10, p < 0.05, p < 0.01 Tble 7: Regresson model wth log household ncome per cpt s dependent vrble smple from 010 IPUMS USA) Fgure 3 shows the estmtes of the probblty densty functon ssocted wth the re error u nd the household error ε h, respectvely. The red lne denotes the norml mxture densty estmte, blue lne denotes the norml densty functon tht best fts the dt. As we re delng wth emprcl dt, not smulted dt, we do not know the dstrbutons from whch the errors re drwn, nd hence re not ble to nclude the true densty functon s benchmrk. The estmtes suggest tht the household errors re rgubly drwn from dstrbuton tht devtes vsbly from norml dstrbuton. We do not see smlr devton from the norml densty for the re error. The smllness of the vrnce of the re error reltve to the totl error however, suggests tht the sgnl to nose rto s low mkng t dffcult to dentfy the underlyng densty functon. The smll vrnce lso mens tht the re error mkes only mnor contrbuton to the totl error nd so ts densty functon s less mportnt. It s clerly more mportnt tht our estmte of the densty ssocted wth the

25 Probblty densty functons for u nd ε dens.uhtx) dens.epshtx) x 4 0 x Fgure 3: Norml-mxture densty Red) versus norml densty Blue) estmtes for: ) re error u left pnel); b) household error ε h rght pnel) household error s ccurte. The log poverty lne s set t 9.5 whch yelds n ggregte poverty rte of bout 8.3 percent. County level estmtes of poverty nd nequlty re obtned usng both EB nd non- EB estmtes under the ssumpton of norml errors nd under the less restrctve ssumpton of norml mxture errors. For ech of the specfctons we obtn 4 estmtes of poverty nd nequlty whch we then compre to the true estmtes derved from the full census. We wll judge the ccurcy of the estmtes on the bss of two summry sttstcs: the ggregte bs nd the RMSE both obtned by ggregtng the county level dfferences between our estmtes nd the true rtes over the 4 countes). The results re presented n Tble 8. Bs RMSE EB ) ) NON-EB ) ) Tble 8: Bs nd RMSE for estmtes of county-level poverty usng 010 dt from the Unted Sttes: ) Norml-Mxture dstrbuton left number); b) Norml dstrbuton rght number) We observe rther lrge bs when estmtng county level poverty under the ssumpton of norml dstrbuted errors; lmost 5 percentge ponts gven ntonl poverty rte of bout 8 percent. By relxng the normlty ssumpton, nd ssumng norml mxture dstrbutons nsted, we re ble to reduce ths bs to just below 3 percent whch denotes non-trvl mprovement. We see smlr reductons n the RMSE. Workng wth EB or non-eb estmtes mkes lttle to no dfference n ths emprcl pplcton. Ths ws to be expected gven the smllness of the locton effect. Note tht the fct tht some bs stll remns suggests tht degree of model msspecfcton stll perssts; these could nclude ms-specfctons n the structurl model but lso forms of heteroskedstcty tht re currently beng gnored. 3