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.
Ashen el e and D. Ca d (eds.), Handbook o Labo Economics Vol.3A, Else ie .
Bound, J, C. B own and G. Duncan (1994), E idence on he Validi y o C oss-
Sec ional and Longi udinal Labo Ma ke Da a, Jou nal o Labo Economics, Vol.
12, No. 3, 345-368
Bound, J and C. B own and N. Ma hiowe z (2001), Measu emen E o in Su ey
Da a, in Handbook o Econome ics (edi o s J.J. Heckman and E. Leame ), Vol. 5,
3707-3843.
Ca oll, R.J. , D. Ruppe and L. S e anski (1994), Measu emen E o in Non-
linea Models, Chapman and Hall, London.
Cheshe , A. (1991), The E¤ec o Measu emen E o , Biome ika, Vol.78, 451-
462.
Cheshe , A. and Schlu e , C., (2001), Wel a e Measu emen and Measu emen
E o , Re iew o Economic S udies, 69(2), 357-378.
Cheshe , A. Dumangane, M. and Smi h, R., (2002). Du a ion Response Measu e-
men E o , Jou nal o Econome ics, 111(2), 169-194.
Cowell, F. (1995), Measu ing Inequali y, P enc ice-Hall.
Fulle , W (1987), Measu emen E o Models, New Yo k: Wiley and sons.
Ho owi z, J. and C. Manski (1995), Iden i…ca ion and Robus ness wi h Con ami-
na ed and Co up ed da a, Econome ica, Vol. 63, pp. 281-302.
Kie e , N. (1988), Economic Du a ion Da a and Haza d Func ions, Jou nal o
Economic Li e a u e, Vol. XXVI, June, pp. 646-679.
Lehmann, E.L. (1955), O de ed Families o Dis ibu ions, Annals o Ma hema ical
S a is ics, Vol. 26, 399-419.
8
O’Neill, D, O.Swee man and D. Van de gae (2004), The Consequences o Spec-
i…ca ion E o o Dis ibu ional Analysis, Economics Dep ., Na ional Uni e si y o
I eland Maynoo h.
Se neels, P. (2002), Explaining Non-Nega i e Du a ion Dependence Among he
Unemployed, The Cen e o he S udy o A ican Economies Wo king Pape Se ies.
Wo king Pape 172.
To elli, N and U. T i ella o (1989), You h Unemploymen Du a ion om he
I alian Labou Fo ce Su ey, Eu opean Economic Re iew, Vol. 33, no. 407-415.
Zimme man, D.J. (1992), Reg ession Towa ds Medioc i y in Economic S a u e,
Ame ican Economic Re iew, Vol.82, 409-429.
9