The Consequences of Non-Classical Measurement Error for Distributional Analysis.

The Consequences o Non-Classical Measu emen E o o Dis ibu ional Analysis Oc obe 29, 2004 Abs ac This pape analyzes he consequences o non-classical measu emen e o o dis ibu ional analysis. We show ha o a popula se o dis ibu ions nega i e co ela ion be ween he measu emen e o (u)and he ue alue (y)may educe he bias in he es ima ed dis ibu ion a e e y alue o y. Fo o he dis ibu ions he impac o non-classical measu emen di¤e s h oughou he suppo o he dis ibu ion. We illus a e he p ac ical impo ance o hese esul s using models o unemploymen du a ion and income. Keywo ds: Dis ibu ion unc ions, Non-classical measu emen e o 1 1 In oduc ion S a is ical analysis in ol es examining he ou comes o andom expe imen s in o de o make in e ences abou he dis ibu ion unc ion unde lying he ue da a gene - a ing p ocess. Measu emen e o may lead esea che s o d aw inco ec in e ences. The impac o speci…ca ion e o on means has been s udied ex ensi ely (e.g. Fulle (1987), Ca oll e al (1994) and Bound e al (2001)). Howe e , less is known abou he consequences o speci…ca ion e o o o he aspec s o he dis ibu ion unc ion. Ho owi z and Manski (1995) discuss ci cums ances in which we can use mismeasu ed da a o bound he dis ibu ion o he ue a iable. They conside si ua ions in which he a iable o in e es is in gene al well-measu ed hough some obse a ions may be subjec o po en ially la ge e o s. In con as he ypical ex book model o measu e- men e o e‡ec s a si ua ion o widesp ead mismeasu emen ( he e o dis ibu ion has no mass poin a ze o). Cheshe (1991) uses a small a iance app oxima ion o s udy he impac o his o m o measu emen e o on dis ibu ion unc ions and a - gues ha he sign o he bias a ising om he mismeasu ed da a can be de e mined by he cu a u e o he ue unde lying dis ibu ion. In pa icula in egions whe e he ue unde lying dis ibu ion is con ex we o e es ima e he dis ibu ion and in egions whe e i is conca e we unde es ima e. Howe e , Cheshe only conside ed classical measu emen e o , whe e he e o e m is dis ibu ed independen ly o he ue alue. In his pape we p o ide a simple geome ic exposi ion o he consequences o non-classical measu emen e o on dis ibu ion unc ions. In pa icula we show ha o a popula se o dis ibu ions, allowing o co ela ion be ween he e o and ue alue may o¤se he bias ha a ises wi h classical measu emen e o h oughou he dis ibu ion. We illus a e ou esul s by examining he impac o measu emen e o on models o unemploymen du a ions and income. 2 The Consequences o Measu emen e o o Dis- ibu ion Func ions 2.1 Theo e ical Resul s Le ybe a andom a iable, whose cumula i e dis ibu ion unc ion is gi en by Fy(y) wi h suppo y; y. Howe e , o some eason ycanno be measu ed accu a ely. Ins ead we obse e ywhich is de…ned as y=y+u, whe e uis measu emen e o wi h suppo [u; u]. The assump ion ha he e o e m is addi i e is less es ic i e han i appea s. In he case o mismeasu ed incomes i is o en assumed ha he e o e m en e s mul iplica i ely (see e.g. Cheshe and Schlu e (2001)). To apply he esul s es ablished in ou pape o hese models we simply conside a log ans o ma ion o he model. In his case we iew yas he log o obse ed income and yas he log o ue income, so ha he obse ed le el o income, I, may be w i en as I=IV, whe e Vexp(u). Fu he mo e, since FI(exp(y0)) = F(y0));ou esul s iden i y he anges o I o which which a mul iplica i e e o p ocess would cause o e o unde es ima ion o he ue unde lying dis ibu ion. 2 Measu emen e o can ake he misspeci…ed cd ou side he suppo o he ue cd . We deno e he ex ended dis ibu ion unc ion o a andom a iable wi h cd Fy(y);by e Fy(y). We make no speci…c assump ions abou he ela ionship be ween yand u. Obse ed da a p o ide in o ma ion on e Fy(y). An exp ession o he di¤e - ence be ween he mismeasu ed and ue dis ibu ions F(y0) = e Fy(y0)e Fy(y0)is gi en in Theo em 1: Theo em 1 The bias (F(y0)) when yis measu ed wi h e o is gi en by Zu uZy0u y0e y;u (y; u)dydu: (1) P oo . Since y=y+u, we ha e e y(y0) = Ru ue y;u (y0u; u)du, such ha e Fy(y0) = Ru uRy0 ye y;u (yu; u)dydu. A he same ime, e Fy(y0) = Ru uRy0 ye y;u (y; u)dydu: Di¤e encing he las wo exp essions esul s in equa ion (1) o he heo em. Equa ion (1) can be gi en a simple g aphical in e p e a ion. This is shown in Figu e 1. Figu e 1 abou he e. In his Figu e uis on he x-axis and yis on he y-axis. The join densi y e y;u (y; u) is ep esen ed by he ellip ical con ou s. Equa ion (1) gi es us he p obabili y be ween he line y=y0uand y=y0. This in ol es sub ac ing he p obabili y mass in S2 om he p obabili y mass in S1. Since he co ela ion be ween yand ua¤ec s he shape o he con ou s, his g aph p o ides a geome ic illus a ion o he po en ial impo ance o his co ela ion in de e mining he size o he bias. To es ablish his ela ionship o mally, le u1=E(u)+,u2=E(u),y1=E(y)+,y2=E(y) and be he se o all dis ibu ions e y;u (y; u). Conside he ollowing de…ni ion. De…ni ion 1 Te y;u (y; u); "; ; :(R+)3!: e y;u (y; u); "; ; !gy;u (y; u) is a mean p ese ing co a iance inc easing ans o ma ion o a densi y unc ion y;u (y; u)i and only i o all u6=u1; u2and y6=y1; y2:gy;u (y; u) = e y;u (y; u), gy;u (y1; u1) = e y;u (y1; u1) + ", gy;u (y1; u2) = e y;u (y1; u2)", gy;u (y2; u1) = e y;u (y2; u1)", gy;u (y2; u2) = e y;u (y2; u2) + ". 3 Such a ans o ma ion inc eases he condi ional co a iance be ween yand u, bu does no a¤ec ei he he mean o a iance o he ma ginal dis ibu ions o uo y. We now es ablish he ollowing Theo em. Theo em 2 I gy;u (y; u)can be ob ained ou o e y;u (y; u)a e a sequence o mean p ese ing co a iance inc easing ans o ma ions, hen (a) I E(u)0and y0E(y) : e Gy(y0)e Gy(y0)e Fy(y0)e Fy(y0) (b) I E(u)0and y0E(y) : e Gy(y0)e Gy(y0)e Fy(y0)e Fy(y0) P oo . We only p o e he esul o case (a). The p oo o case (b) is simila . De…ne S1=(u; y)2[u; u]y;yju0; y0yy0u S2=(u; y)2[u; u]y;yju0; y0uyy0. These egions a e illus a ed in Figu e 1. A ans o ma ion Te y;u;a(y; u); "; ; will only a¤ec equa ion (1) i i changes he p obabili y mass in S1o S2. Unde he assump ion ha E(u)0 wo such cases exis . Fi s , i is possible ha and a e such ha he poin wi h coo dina es (u1; y1) 2S2. The p obabili y mass in S2inc eases, such ha G(y0)is smalle han F(y0). Second, and a e such ha he poin wi h coo dina es (u2; y1)2S1. In his case he p obabili y mass in S1dec eases, again esul ing in G(y0)being smalle han F(y0). Fo all o he alues o and , he p obabili y mass in S1and S2will no be a¤ec ed, o will be a¤ec ed in he same way, such ha G(y0)will equal F(y0) o hese alues. Since e Gy(y0) = e Fy(y0)Theo em 2 di ec ly ela es mean p ese ing co a iance inc easing ans o ma ions o he size o he bias in he mismeasu ed dis ibu ion unc ion. I he ans o ma ion inc eases he co a iance hen he bias becomes less posi i e (o mo e nega i e) p o ided y0is g ea e han he mean o yand E(u)0. I y0is less han he mean and E(u)0 he opposi e occu s. I is easy o see ha he esul s a e e e sed when he ans o ma ion dec eases he co a iance. In many applica ions i may be easonable o assume ha E(u) = 0:In his case Theo em 2 allows us o es ablish he impac o non-classical measu emen e o o e he en i e ange o y. 2.2 Examples Example 1. Conside a simple case whe e yand ua e independen and uis symme - ic a ound ze o. I y(y0)is no mal hen Cheshe (1991) shows ha we o e es ima e (unde es ima e) Fy(y0)a poin s below (abo e) E(y), while he bias is ze o a E(y).1 1Cheshe ’s esul s a e based on app oxima ions. O’Neill e al (2004) es ablish exac esul s o cd s unde which he sign he bias can be es ablished. These depend on modi…ed cu a u e condi ions o he ue unde lying dis ibu ion. Fo he no mal dis ibu ion wi h symme ic measu emen e o Cheshe ’s app oxima e condi ions a e equi alen o he exac cu a u e condi ions. 4 Theo em 2 shows ha , in his case, in oducing nega i e co ela ion be ween yand u(which is wha we end o see in ea nings da a (Bound e al. (1994)) may in ac educe he ex en o o e es ima ion (unde es ima ion) h oughou he dis ibu ion. Indeed his is ue o he wide class o dis ibu ions wi h cd s ha a e con ex be- low he mean and conca e abo e he mean. These include dis ibu ions such as he -dis ibu ion and he logis ic dis ibu ion . Example 2. Conside he powe dis ibu ion F(y; ) = y o 0y1;  > 0. This dis ibu ion is con ex o all ybe ween ze o and 1p o ided  > 1. Cheshe ’s (1991) esul s imply ha classical measu emen e o will lead us o o e es ima e he dis ibu ion a each o he poin s in he o iginal suppo . Theo em 2 shows ha he consequences o co ela ed measu emen e o di¤e s depending on he alue o y. Fo alues o ybelow he mean, nega i ely co ela ed measu emen e o may esul in he bias becoming smalle , while o alues o yabo e he mean he bias mus inc ease. Example 3a. Conside he exponen ial dis ibu ion F(y; )=1ey; y > 0;  > 0. The exponen ial dis ibu ion a ises na u ally in many s a is ical p oblems associa ed wi h wai ing imes. Fo ins ance, i he occu ence o an e en is go e ned by a Poisson p ocess hen i can be shown ha he sequence o in e -a i al imes a e independen iden ically dis ibu ed exponen ial andom a iables. In applied esea ch he exponen ial dis ibu ion is widely used as a s a ing poin o he analysis o un- employmen du a ion and s ike du a ion da a (Kie e (1988)). I is easy o show ha he exponen ial dis ibu ion unc ion is conca e o all y. In his case classical mea- su emen e o will lead us o unde es ima e he dis ibu ion h oughou he o iginal suppo .2Howe e , he e is some e idence ha longe spells o unemploymen a e mo e likely o be subjec o unde epo ing, implying a nega i e co ela ion be ween he ue le el o unemploymen du a ion and measu emen e o (To elli and T i - ella o (1989)). As in Example 2 he consequences o co ela ed measu emen e o di¤e s depending on he alue o y, hough in his case he e¤ec goes in he opposi e di ec ion. Fo alues o ybelow he mean, nega i ely co ela ed measu emen e o will accen ua e he bias ( he bias becomes mo e nega i e); howe e o alues o y abo e he mean he bias may all in absolu e alue (become less nega i e), hough i he co ela ion is su¢ cien ly nega i e he mismeasu ed dis ibu ion could mo e abo e he ue dis ibu ion, hus inducing a posi i e bias. Thus measu emen e o in unem- ploymen du a ions, ha is nega i ely co ela ed wi h he u h, may be p e e able o independen measu emen e o i ou ocus is on long unemploymen spells bu will compound he p oblem o independen measu emen e o when conside ing sho e spells.3 Example 3b. The exponen ial dis ibu ion is es ic i e in ha he implied haza d a e is cons an . Howe e he conclusions om Example 3a gene alise o less es ic i e cases wi h non-cons an haza ds. The Weibull dis ibu ion, gi en by 2Fo a de ailed discussion o classical measu emen e o in du a ion eponse da a see Cheshe e al (2002). 3Since we a e basing he sign o he bias wi h independen measu emen e o on small- a iance app oxima ions his p e e ence anking o e ypes o measu emen e o need no apply o e y sho o e y long du a ions. 5 F(y; ; )=1ey; y > 0;  > 0;  > 0;is a wo pa ame e gene alisa ion o he exponen ial dis ibu ion, which allows o a non-cons an haza d. I is easy o show ha he conclusions eached in Example 3a ega ding non-classical measu emen e o emain alid p o ided  < 1. Howe e ,  < 1is equi alen o speci ying nega i e du a ion dependence, which is ypical in many s udies o unemploymen .4 Example 4. When modelling he consequences o measu emen e o in non- nega i e a iables, such as unemploymen du a ion, one may p e e o adop a mul- iplica i e o m o he e o p ocess. As no ed ea lie his is easily inco po a ed wi hin ou speci…ca ion. To see his econside he exponen ial dis ibu ion. Assume ha F( ; ) = 1 e , > 0;  > 0and deno e he measu emen e o by V. In his case we may wish o model obse ed du a ion as S=TV . To apply ou amewo k o his model we simply ake a log ans o ma ion o he mul iplica i e model, so ha ln(S) = ln(T) + ln(V).5This model is now in he o ma speci- …ed in ou ea lie heo ems. To es ablish he impac o measu emen e o in his model we need o be able o desc ibe he dis ibu ion o ln(T). Howe e , i Tis exponen ially dis ibu ed hen ln(T)has a Type 1 ex eme alue dis ibu ion wi h densi y gi en by g(y) = exp(y) exp(exp(y)). The mean o his andom a i- able is gi en by E(ln(T)) = ln(), whe e is Eule ’s cons an (app oxima ely .5772). Fu he mo e i is easy o show ha his dis ibu ion is con ex p o ided ln(T)<ln(1=) = ln()and conca e o he wise. Using Cheshe ’s esul s we con- clude ha wi h classical measu emen e o we o e es ima e p o ided ln(T)<ln(1=) and unde es ima e o he wise. In e ms o he ac ual unemploymen du a ions, T, his implies ha he dis ibu ion o So e es ima es he dis ibu ion o Tp o ided T < 1= E(T)and unde es ima es p o ided T > E(T). F om Theo em 2 we can deduce ha allowing ln(T)and ln(V) o be nega i ely co ela ed will cause he bias o become less posi i e p o ided ln(T)< E(ln(T)) = ln(); and causes he bias o become less nega i e when ln(T)>ln() . Combining his wi h ou ea lie analysis we see ha o sho unemploymen du a ions, speci…cally hose such ha ln(T)<ln(), nega i e co ela ion may help o¤se he o iginal posi i e bias esul ing om classical measu emen e o . Fo long unemploymen du a ions, such ha ln(T)>ln();non-classical measu emen e o may help o¤se he nega i e bias in oduced by unco ela ed measu emen e o . Howe e , since E(ln(T)) <ln(E(T)), he e is now also an in e media e ange o log du a ions, om [ln(); ln()], o which he o iginal endency o o e es ima e wi h classical measu emen e o is compounded by co ela ed measu emen e o . In e ms o he aw du a ions T; he ange o which non-classical e o compounds he o iginal biases is gi en by [E(T)=1:78; E(T)]. This shows how he amewo k we ha e in oduced can be easily ex ended so as o yield p ac ical insigh s in o he consequences o non-classical measu emen e o wi h al e na i e e o s uc u es.6A simila analysis can also be conduc ed in cases 4Fo a ecen o e iew o he li e a u e on du a ion dependence in unemploymen see Se neels (2002). 5See Kie e (1988), Sec ion IV, o a mo e de ailed discussion o he po en ial use o log-linea models o du a ion analyses. 6In his example we ha e assumed ha E(ln(V)) = 0. This need no imply ha E(V)=1 which 6 whe e he o iginal dis ibu ion o du a ions is Weibull, since he na u al loga i hm o a andom a iable wi h a Weibull dis ibu ion also has a Type 1 ex eme alue dis ibu ion. 2.3 Consequences o Es ima ed Po e y Ra es To explo e he magni ude o non-classical measu emen e o in p ac ice we conside a calib a ed model o he dis ibu ion o income o whi e couples in he U.S in he ea ly 1990’s. Fo simplici y we assume ha income is dis ibu ed as log no mal7. Le ing ydeno e he log o income we assume ha yN(10:72,:24).8We assume ha uN(0; 2 u)whe e 2 uis chosen so ha 2 u 2 y=.33. This co esponds o a eliabili y a io o .75 when measu emen e o is classical. This is wi hin he ange o es ima es p esen ed in ecen s udies (Zimme man (1992), Ang is and K uege (1999)). We conside wo cases. Fi s we assume ha yand ua e independen . We hen compa e his o he case whe e yand uha e a co ela ion equal o -.3.9;10 Figu e 2 p esen s he ue dis ibu ion and bo h he misspeci…ed dis ibu ions (wi h and wi hou co ela ion)11. The …ndings wi h independen measu emen e o a e consis en wi h Cheshe (1991); we o e es ima e in he egion whe e he ue dis ibu ion is con ex, unde es ima e whe e i is conca e and he bias is ze o a he mean. Gi en he calib a ion o ou model he size o he bias a ising om indepen- den measu emen e o is ela i ely small. As p edic ed by Theo em 2, in oducing nega i e co ela ion be ween he e o and he ue alue causes he bias o become less posi i e o alues o ybelow he mean and less nega i e o alues abo e he mean. Indeed, o ou calib a ed model he dis ibu ion wi h co ela ed measu emen is i ually iden ical o he ue model. To summa ise he impac o co ela ed e o e ms we conside measu es o he po e y a e based on mismeasu ed da a, bo h wi h and wi hou co ela ion be ween he e o and he ue income. We choose 1/2 median income as he measu e o po e y. Unde ou assump ions he po e y line is cons an ac oss all 3 dis ibu ions. The es ima ed po e y a e is 8% o bo h he ue and co ela ed models and 11% o he independen speci…ca ion. I is wo h emphasising ha i he co ela ion becomes mo e nega i e (i.e less han -.3) he co e- la ed income dis ibu ion alls below he ue dis ibu ion o y0sbelow he mean and ises abo e i o y0sabo e he mean. The co esponding po e y a e wi h co ela ed e o s would hen unde es ima e he ue po e y le els. Fo ins ance i we pick a co ela ion o -.69 (Code (1992) as e e enced by Bound e al (2001) Table 1) he may be desi able in mul iplica i e e o models. Howe e , his will be app oxima ely ue gi en he small a iance app oxima ions adop ed in his pape . 7Fo a discussion o he sui abili y o his speci…ca ion see Cowell (1995). 8See Al onji and Do aszelski (2005). 9This is wi hin he ange o es ima es epo ed by Bound e al (Sec ion 6). 10Fo simplici ly in his la e case we also assume ha uand ya e bi a ia e no mal. This allows us o ob ain analy ical exp essions o he dis ibu ions o conce n. Theo em 2 does no equi e any such pa ame ic assump ions. Mo e gene al dis ibu ions could be inco po a ed in o ou example using Mon e-Ca lo me hods. 11The e a e ac ually 3 cu es in Figu e 2. Howe e , gi en he alues used in calib a ing ou model he ue dis ibu ion and he dis ibu ion wi h co ela ed measu emen e o a e indis inguishable. 7 es ima ed po e y a es a e 8% o he ue model, 11% o he independen case and 3% o he co ela ed case. 3 Conclusion In his pape we p esen a simple geome ic exposi ion o he impac o non-classical measu emen e o o he de i a ion o dis ibu ion unc ions. Fo a popula se o dis ibu ions we show ha posi i ely co ela ed e o s will unambiguously wo sen he bias h oughou he dis ibu ion, while nega i ely co ela ed e o may help o¤se he bias ha a ises wi h independen e o s. Fo o he dis ibu ions he consequence o co ela ed e o s di¤e s h oughou he dis ibu ion in a way ha depends on he cu a u e o he ue unde lying dis ibu ion. Re e ences Al onji, J. and Do aszelski, U (2005), The Role o Pe manen Income and De- mog aphics in Black-Whi e Di¤e ences in Weal h, o hcoming Jou nal o Human Resou ces. Ang is , J. and A. K uege (1999), Empi ical Me hods in Labo Economics, in O. 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