Kooreman, P.

Melenberg, B.

English

Research Papers in Economics

C~;'~'~- DISCUSSlOri
IIiIiI.IIInIIIINI~INllullllGlllllll,liiillll
~,Q~~, , ~ a ~ , ~
1~~0J~i"No. 8948
MAXIMUM SCORE ESTIMATION IN THE
ORDERED RESPONSE MODEL
by Peter Kooreman
and Bertrand Melenberg
October, 1989Maximum Score Estimation in the Ordered Response Model
M
Peter Kooreman
.
Bertrand Melenberg
October 1989
r
Tilburg University
Department of Economics
P.o.B. 90153
5000 LE Tilburg
The Netherlands
Financial support has been provided by the Netherlands Organization for
Scientific Research (NWO) and the Royal Netherlands Academy of Arts and
Sciences (KNAW).2
AbsT.rect
We extend Manski's maximum score estimetor to the ordered response
case and prove that the generalized estimator is strongly consistent.
1. Introduction
This paper is concerned with semiparametric estimation of the
regression coefficients of a linear model when the dependent variable is
grouped, i.e. is only observed to fall in certain known intervals on a
continuous scale, its actual value remaining unobserved. Such a model is
usually referred to as the ordered response model. A typical example of a
variable for which often only grouped information is available is income.
Many respondents are often either unable or unwilling to provide a precise
measure of their income, and therefore many micro data sets only contain
bracketwise information on income; see, for example, Cremer ( i969),
Stewart ( 1983) and Kapteyn et al. (1988). The usual estimation method for
the ordered response model is maximum likelihood, under the assumption
that the error terms are independent, and identically normally
distributed. Stewart (1983) gives a convenient algorithm to obtain the
maximum likelihood estimates in this case.
As in many other models with limited dependent variables, the maximum
likelihood estimator is generally inconsistent if the underlying
distributional assumptions are not correct. Therefore, it is important to
search for alternative estimators which are consistent under weaker
distributional assumptions.
During the last decade several semiparemetric estimators have been
proposed for models with limited dependent endogenous variables; see
Robinson ( 1988) for e review. However, most of these methods are designed
for either the binary response model or the censored regression model, and
are not straightforwardly applicable to the ordered response case.
In this paper we extend Manski's Maximium Score Estimator (MSCORE;
see Manski (19~5, 1985)) to the ordered response case. An attractive3
property of MSCORE and its generalization is that it does not require the
error terms to be identically distributed. Our extension builds on the
fact that MSCORE can be interpreted as a]east absolute deviations
estimator. An important diT'T'erence with the binary case is that the
ordered response model does not require a normalization of the parameter
vector.
Section 2 introduces the generalized estimator and in section 3 we
prove its consistency.
2. Maximum Score estimation in the ordered response model
Consider the following binary response model
r
yi-xi~}ti
w
y.-1 if y.~0
1 0 otherwise
i-1,...,N (2.1)
It is assumed that the error term ei has a unique median Med(Ei~xi)-0. xi
is a K-vector of explanatory variables and p is a conformable vector of
true parameters.
Define, for some vector b, the score of observation i to be equal to
w
1 if yi and xib have the same sign and 0 otherwise. Manski has proved that
under mild conditions the (normalized) b which maximizes the sum of the
scores is a consistent estimator of the true parameter vector pl).
Our generalization is based on the fact that, as noted by Manski
(1985), the MSCORE estimator is equivalent to a Least Absolute Deviations
estimator, obtained by minimizing
N ~i~Yi-I(xib~0)~ (2.2)
w.r.t. b. Here I(E)-1 if event E occurs and 0 otherwise.4
r
Let t(.):R~R be a non-decreasing measurable function. If Med(t(y )~x)
is unique, there holds
Med(t(yk)~x)-t(Med(y~~x))
So we have
(2.3)
Med(yi~xi)-Med(I(yi)0)~xi)-I(Med(Yi~xi))0)-I(xi~~0) (2.4)
Thus I(xip)0), with p being the (not necessarily unique) maximum score
estimate, can be interpreted as an estimate of the median of Yi~xi'
This median interpretation of the maximum score estimator suggests
the following generalization to the ordered response case. Suppose that
the dependent variable is grouped in M non-overlapping known2) intervals
(AO,A1], (A1,AZ],...,(AM-1,AM) where Am-1CAm for m-1,...,M; possiblY,
A0--m and AM-~. Define the non-decreasing transformation
M-1 „
z. - ~ w I(y.~A ) ~ m ~ m m-o
(2.5)
where w, m-0,...,M-1, are non-negative weights. Again using (2.3) it
m
follows under certain conditions that
M-1
Med(zi~xi) - ~ wmI(xip)Am)
m-0
Now the generalized maximum score estimator is obtained by minimizing
M-1
N ~~zi- ~ wmI(xib)Am)~
i m-1
(2.6)
(2.7)
with respect to b.
In the next section we prove that this estimator is strongly
consistent.5
3. Strong consistency of the generalized maximum score estimator
Our proof will be along the lines of the proof for the binary case
given in Manski (19215 ) .
Let us impose
K ~`
Assumption 1. There exists a unique pER such that Med(yilxi)-xi'p'
Assumptton 2. a) The support of Fx is not contained in any proper linear
subspace of RK. b) There exists at least one kE{1,...,K} such that
Sk~O and such that, for almost every value of
x-(xl,x2,...,xk-1,xk.1,...,xK), the distribution of xk conditional on
x has everywhere positive Lebesgue density.
M
Asswrtptton 3. (yi,xi), i-1,...,N is a random sample from Fyx. For each i,
(zi,xi) is observed.
Assumptton 4. The bounds Am, m-0,1,...,M are known constants and M)2.
The weights wm are non-negative, with at least two of them being
strictly positive.
Assumptfon 5. The parameter space ~ is a compact subset of RK and
contains S.
Assumptions 1,2 and 3 are (almost3)) identical to assumptions made in
Manski (1985). Assumptions 4 and 5 distinguish the ordered response model
from the binary response model. Note that assumption 5 is less stringent
than in the binary case. There ~ is identified only up to scale so that
the parameter space is taken to be the unit hypersphere in RK.
Define B(r)-~m-OwmI(v)Am)-~m-1rm1(Am-lCv(Am) where rm-~m-lw for
~-0 j
m-1,...,M.
We first prove the following lemma.
Lemma 1 (Zdenttficatton). Let Xb-{xERK; g(x'b)~g(x'~)} and let
R(b)-JX dFx. Under assumptions 2 and 4, R(b))0 for all bEB.6
Proof. Let k-K satisfy part b) of assumption 2 and consider the case in
which yK)0 (the case SKCO is symmetric). Without loss of generality we may
assume that Am -0 and Am -Q)0 For some ml~m2; m1,mZE{1,...,M-1}. Let, for
~ ~
K -
any bER , b-(b1,...,bK-1)' so b-(b',bK)'. From Lemma 2 in Manski (1985) it
follows immediately that R(b))0 for all bERK such that bKCO.
Now consider the case bK)0. We can write
R(b) - P[g(x~b)~g(x'A)] -
P[x'bCO;OCx'pCQ] . P[x'bCO;x'p)Q] t
P[OCx'bCQ;x'~CO] t P[OCx'bCQ;x'~)Q] t
P[x'b)Q;x'gC0] ~ P[x'b)Q;OCx'~BCQ] -
- P[xKC-x'blbK:-x'PIAKCxKC(Q-x'P)~PK] t
P[xK~-x'b~bK:xK)lQ-x's)~PK] t
P[-x'b~bKCxKC(Q-x'b)~bK;xKC-x'A~9K] t
P[-x'b~bKCxKC(Q-x'b)~bK:xK)(Q-x'P)~AK] t
P[xK)íQ-x'b)~bK:xKC-x'P~AK] t
P[xK)(Q-x'b)~bK;-x'P~RKCxKC(Q-x'~)~PK] (3.1)
Under part b) of assumption 2, at least one term on the r.h.s. of (3.1) is
positive for almost any x such that -x'p~,BK~-x'b~bK and~or (Q-x'p)~~BK~(Q-
x'b)~bK. However, we have -x'~S~f3K--x'b~bK and (Q-x'~)~~K-(Q-x'b)~bK only
for x being in a(K-2)-dimensional subspace of RK-1. By part a) of
assumption 2, the probability that x is in such a subspace is less than
one. Therefore, S is also identified relative to all b for which bK)O.oThe crucial difference with the binary case is that there R(b))0 as
long as P[-x'pjpK--x'b~bK](1. Since P[-x'pjpK--x'b~bK]-1 if b is a scalar
multiple of p, p is then identified only up to an arbitrary scale factor.
In the ordered response case, however, R(b))0 as long as F[-x'pjpK--x''b~bK
and (Q-x'p)jpK-(Q-x''b)jbK]C1. Clearly, this is true even if b is a scalar
multiple of p(except for 'x being in a(K-2)-dimensional subspace of
RK-1).
Theorem 1 (Strong consfstencyj. Let assumptions 1 through 5 hold. Then
the estimator defined by minimizing (2.7) with respect to b is strongly
consistent for p.
Proof: The sample score function can be written as
1
SN(b) - N ~~zi- g(xib)~-
i
M-1 M ~
-~ ~ ~rm-r I
{PN(Am-1Cy
CAm;A, iCx'bCA.) t
m-1 j-mt1 j ~- ~
M
. PN(Aj-1Cy CAj'Am-1Cx'b(Am)} (3.2)
r
where PN denotes the empirical probability distribution of (yi,xi).
Similarly, we write the population score function as
5(b) - EIg(Yr)-g(x~b)~-
M-1 M N
- ~ ~ ~rm-rjI{P(Am-1Cy
(Am;Aj-lCx'bCAj) f
m-1 j-m.1
K
t P(Aj-1Cy CAj'Am-lCx'bCAm)} (3.3)
A. Under assumption 3, it follows by application of a Glivenko-Cantelli
type result, similarly as in Lemma 4 in Manski (1985), that each right8
hand side term in the above decomposition oF SN converges to the
corresponding term in the decomposition of S, uniformly in b. Hence,
SN(b)a'-~ ' 'S(b)
uniformly over bEBCRK.
B. Applying similar arguments as in Lemma 5 in Manskí (1985), it follows
under assumptions 1 and 2, that S(b) is continuous at all b for which
bkt0.
C. We have to show that S(b) attains a unique global minimum at p. Using
the fact that
M
E~g(Y )-g(x~b)~-
J ( J [1-2Fzlx(t)]dt t J Fzlx(t)dt ~ dFx
all x l{t)g(x'b)} R
(3.4)
this is equivalent to showing that the right hand side of (3.4) exceeds
f I f [1-2F I(t)]dt 4 J F I(t)dt
J
dF
all x lll{t~g(x's)} z x R z x x
for all b~p.
Consider, for a given x, the case where x'gsAm for all m, and
g(x'b)tg(x'y). If g(x'b))g(x'g)), then
f [1-2Fzlx(t)]dt ) f [1-2Fzlx(t)]dt
{t)g(x'b}} {t)g(x'g)}
(3.5)
will hold since 1-Fzlx(t)(0 for t)g(x'p). But if g(x'b)Cg(x'p), (3.5) will
also hold since 1-Fzlx(t))0 for t(g(x'p).
From assumption 2, it follows that, for all m, P(x'g-Am) has
K
probability zero, whereas Lemma 1 guarantees that Xb-{xER ; g(x'b)tg(x'g)}
has positive probability. So, we conclude that S(p)(S(b) for all b~p, bEB.9
Combining parts A, B and C and applying theorem 2 in Manski (1983).
it Ibllows that
'r{iimN~sup jb-P'-0} - 1.
bEBN
where B is the set of solutions to min S(b).
N bEB N o10
Footnotes
1) In Manski's 1975 article, the maximum score esti.mator was introduced
for the more general multinomial discrete response model. riawever, the
results in that paper do not carry over to the ordered response case. For
example, tiie assumption in Manski (1975) of independence of the error
terms across alternatives is clearly not appropriate here.
2) If the interval bounds are not known, p is identified only up to scale.
3) Manski (1985) makes the additional assumption OCP(y)Olx)~1, a.e. Fx.
This assumption (c.q. its analogue in the ordered response case) does not
M
seem to be necessary. Suppose P(Am-lCy ~Am~x)-1 a.e. Fx (Am-lt-m and
r
A~m). Since Med(y ~x)-x'p, it must be the case that P(Am-1(x's(Am~x)-1.
m
But this contradicts part b) of assumption 2(following the same arguments
as in the proof of Lemma 1).11
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Bibliotheek K. U. Brabant
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Some methodologicai issues in the implementation of subjective poverty definitions&quot;,

Maximum score estimation in the ordered response model

Maximum score estimation in the ordered response model

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