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HB31
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^
working paper
department
of economics
MISCLASSIFICATION OF A DEPENDENT VARIABLE
IN A DISCRETE RESPONSE SETTING
Jerry A. Hausman
Fiona M. Scott Morton
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
MISCLASSIFICATION OF A DEPENDENT VARIABLE
IN A DISCRETE RESPONSE SETTING
Jerry A. Hausman
Fiona M. Scott Morton
94-19
June 1994
M.I.T.
JUL
LIBRARIES
1 2 1994
RECEIVED
First Draft
December 1993
Misclassification of a
in a Discrete
Dependent Variable
Response Setting
Hausman'
Fiona M. Scott Morton
Jerry A.
and NBER
June 1994
MIT
We
would like to thank Y. Ait-Sahalia, G. Chamberlain, S. Cosslett, Z. Griliches, B. Honore,
W. Newey, P. Phillips, T. Stoker, and L. Ubeda Rives, for helpful comments. Thanks to Jason
Abrevaya for outstanding research assistance. Hausman thanks the NSF for research support.
Address for correspondence: E52-271A, Department of Economics, MIT, Cambridge, MA, 02139.
'
Introduction
A
dependent variable which
inconsistent in a probit or logit
we mean
variable
happen
a discrete response causes the estimated coefficients to be
is
model when misclassification
that the response is reported or recorded in the
recorded as a one when
is
it
present.
By
'misciassification'
wrong category;
should have the value zero.
for example, a
This mistake might easily
an interview setting where the respondent misunderstands the question or the
in
interviewer simply checks the wrong box.
measurement
error,
dependent variable
likelihood function
We
in the data.
misclassification
and
Other data sources where the researcher suspects
such as historical data, certainly exist as well.
is
we
We
show
that
when a
misclassified in a probit or logit setting, the resulting coefficients are
However,
biased and inconsistent.
distribution
is
the researcher can correct the
problem by employing the
derive below, and can explicidy estimate the extent of misclassification
also discuss a semi-parametric
method of estimating both the probability of
and the unknown slope coefficients which does not depend on an assumed error
is
also robust to misclassification of the dependent variable.
Each of these
departures from the usual qualitative response model specifications creates inconsistent estimates.
We apply our methodology to a commonly used data set,
the Current Population Survey,
where we consider the probability of individuals changing jobs. This type of question
known
for
its
potential misclassification.
is
well-
Both our parametric and semiparametric estimates
demonstrate conclusively that significant misclassification exists
the probability of misclassification
is
not the
in the
sample.
same across observed response
Furthermore,
classes.
A much
higher probability exists for misclassification of reported individual job changes than the
probability of misclassification of individuals
I.
Qualitative Response
We
we
present
Let
y
.
Model with
who
are reported not to have changed jobs.
Mlsclassiflcation
use the usual latent variable specification of the qualitative choice model; for the
consider the binomial response model,
c.f.
Greene (1990) or MacFadden (1984).
be the latent variable:
y."
=
X,/3
+
e,
(1)
The observed
data correspond
yi=l
ifX,/3
+ ei>0
=
if Xi/8
+
Yi
For now,
x be
let
and constant
the assumption that
where $(.)
is
6;
<
Assume
The model
sample for both types of responses.
that
t
is
distributed as N(0,1), the usual probit assumption:
=
T.pr(y,*>0)
=
x.*(Xi^)
+
+
=
1-T
+
independent of
specification follows
is
e,
X
from
(l-x).pr(y;<0)
(l-x).(l-'i'(Xi^))
the standard normal cumulative function.
E(yi|X)
(2)
the probability of correct classification.
in the
pr(yi=l!X)
to:
We
(3)
then calculate the expectation of
y/.
(2t-1)-*(X,/3)
This equation can be estimated consistently in the form,
=a
y,
where
q;
+
(l-2a).<i.(X,^) +7,,
= 1-t.
Alternatively,
we
could approach the problem using the other response,
pr(yi=0|X)
= T.pr(y;<0) +
(l-T).pr(yi->0)
=
T-(2x-l).4.(X./3)
=
(l-a)-(l-2a)-$(Xi/3)
In terms of a:
Note
(4)
that as
above we
find:
(5)
= l-{x +
E(yi|X)
If
(1-2t).<|.(X,^)}
one estimates a probit specification as
misclassification
if
(6)
were no measurement error when
there
present, the estimates will be biased and inconsistent.
is
contrast to a linear regression
where
classical
measurement error
in the
in fact
This result
is in
dependent variable leaves
the coefficient estimates consistent, but leads to reduced precision in coefficient estimates.
The
log likelihood for the probit specification with misclassification
L =
2i
a=0
In the case of
+
{yi-ln[a+(l-2a)-4»(Xi^)]
there
is
no
(1981) shows that
if
a=0
classification error
this log likelihood is
concave functions. Unfortunately, the
logit or probit functional forms.
result
as follows:
(l-yi)-ln[(l-a) + (2a-l)-#(X/?)]}
(7)
and the log likelihood will collapse
In order for the equation to be estimated,
usual case.
is
a must be
less than
one
to the
half.^
Pratt
everywhere concave, being the sum of two
does not hold for our log likelihood for either the
The appendix gives
details of the conditions for concavity.
also demonstrate in the appendix that the expected Fisher information matrix
is
We
not block
diagonal in the parameters (a,/3).^
Notice that the linear probability model does not allow separate identification of the
and x.
coefficients
pr(yi
= l|X)
Instead the estimated coefficients are linear combinations of
=
x.pr(y,->0)
= x-X^ +
^
+
x and
/3.
(l-x).pr(y;<0)
(8)
(l-x).(l-Xi/3)
In the case of Qf> .5, the data are so misleading that the researcher might
want
to
abandon the
project.
^
Thus, previous papers which assume they know
Summers (1993)
assumed a
is
inconsistent.
suffer
correct.
In certain special cases of multiple interviews,
identification of misclassification probabilities
However, generally they must make
identification.
a from exogenous
sources, e.g., Poterba and
from two defects. Their estimates are likely to be inconsistent unless their
But even with a correct a the standard errors of their coefficient estimates are
is
Chua and
Fuller (1987) demonstrate that
possible, given a sufficient
special assumptions
number of
interviews.
on the form of misclassification
to achieve
=
ECyJX)
[1-T
+
(2ir-l)./3o]
+
(2t-1).(X,^,)
This example show that identification of the true coefficients in probit and logit comes from their
To
non-linearity.
results
address this concern,
do not depend on
An
we
shall introduce a
distribution assumptions.
interesting feature of the misclassification
even for a small amount of misclassification.
below.
The
first
term
is
semiparametric approach so other
The
problem
is that
usual probit
first
observations where y =
summed over
inconsistency can be large
1
,
order condition
written
is
the second for those
where
y=0,
y.^t^
where
^,
=
and similarly for
^(Xj/J),
misclassification is as follows.
the likelihood
misclassified).
-
When
j.
(.-„.E^
The
=
(9)
intuitive reasoning for the inconsistency
misclassified observations are present, they are
due
added
into
and the score through the incorrect term (because the dependent variable
The
to
large inconsistency in the estimated parameters arises if probit (or logit)
is
is
used because misclassified observations will predict the opposite result from that actually
observed.
For example, an observation with a large index value
probability 4»(Xj/3) close to one.
value will be a zero.
However,
The observation
will
if that
observation
be included
in the
is
will predict a
misclassified,
second term
its
one with
observed y
equation (9); a
in
^(XJ3) close to one will cause the denominator to approach zero, so the whole term will
approach
The same problem
infinity.
Therefore, the
sum of
and the estimated
/3's
first
exists for observations with very
order conditions in the case of misclassification can
can be strongly inconsistent as a
result.
It is
somewhat
inconsistency will increase as the model gets "better" because there will be
are misclassified and therefore
Another way
that
maximum
*
low index values.
more
become
large,
ironic that the
more good
fits that
large terms in the log likelihood.'*
to think about the inconsistency caused
by misclassification
is to
use the fact
likelihood sets the score equal to zero under the situation of no misclassification.
See Table lb for a simulated demonstration of
5
this property.
Taking expectations and using the probabilities *; and 1-$;
in the case
However,
causes the denominators to cancel and the expectation to equal zero.
misclassification, the probabilities change.
y=l and
The "ones term"
[(l-a)'l'+a(l-*)]
-^
l-4»
-
$
misclassification
to the usual probit case.
is
If
present,
a
?i
a =
in the case
of
used for observations that have
are not misclassified and also for observations that have
^
When no
is
of the normal probit
y=0
and are misclassified:
[(l-a)(l-$)+a*]
=0
(10)
and the expectation of the equation above collapses
0, then the expectation
of the score
is,
in general, not equal
to zero.
We
are particularly interested in the inconsistency in probit or logit coefficients
only small amounts of misclassification are present, since this might be the most
We
facing a researcher.
when
common
case
can use the modified score above to evaluate the change in the
estimated coefficients with respect to the extent of misclassification at zero misclassification.
The
derivation
full
is in
the appendix.
In the case of the probit, the derivative will equal:
d&
Again, notice that
(f)(Xfi)
goes
to
in the case
zero
also.
1
where an observation has ^(Xfi) close
Therefore,
considerable inconsistency to the estimated
estimated
A
/3's
when
- 2^{Xfi)
misclassification
is
to either zero or one, then
even a single observation can theoretically add
jS's.
In general, the
amount of inconsistency
present will depend on the distribution of X's.^
Table
reports the value of this derivative for the job change dataset used later in this paper.
crucial feature of misclassification
the table
'
shows
that the bias in
is
/3
immediately apparent. The
"maximum
derivative"
in the
The
row of
caused by a specific misclassified observation can be very
Distributions such as the uniform lead the average derivative to be relatively small, whereas a
distribution with
unbounded support
will cause
it
to
be
larger.
Averaged across observations, the change
large.
in
is
/3
not nearly so large, indicating that
particular observations contribute heavily to the inconsistency of the estimated
jS.
Table A:
Derivative of estimated
jS
with respect to the fraction of misclassification in the
dataset, evaluated at zero misclassification using
Job change dataset from the CPS.*
N=5221
Married
Grade
Age
Union
Earnings
West
Average
519
65.5
19.5
535
0.961
839
179,666
27,640
7,486
179,666
360
359,332
-3.60
-0.200
-0.136
-3.60
-1.70
-3.48
Derivative
Max
Derivative
Min
Derivative
n. Simulation
In order to assess the empirical importance of misclassifications,
using
random number
The
generators.
lognormal distribution, the second, Xj,
third probability,
y' and
y;
and X3
is
is
hand side variable, X,,
first right
a
dummy
distributed uniformly,
we
create a sample
is
drawn from a
variable that takes the value one with a one
e is
drawn from a normal
(0,1) distribution.
are defined to be:
y;
=
yi=
/3o
+
1
X,^,
if
+
y;
X,fi,
+
X3J83
+
y;
called
=
€i
X;0
+
(12)
6;
>
otherwise
Next we create a misclassified version of
a = l-T
*
)
is
misclassified
The dummy
variables
compute the derivative.
on a random
were rescaled
y;.
A
proportion
a of
the sample (recall that
basis.
to equal
1
and 2 and the regressions rerun
in
order to
Table
I
reports the
the first
if
We
results.
For comparison, the
coefficients 146 times.
estimated
Monte Carlo
create the sample and estimate the
column reports
first
the parameters that
5%
misclassification; the second
row has 20%.
amount of
misclassification, ordinary probit can
problem
is
intensified as the
likely to
be a more severe problem when the model
function
we
amount of
propose does very well
Even
produce very inconsistent coefficients.
As discussed above,
misclassification grows.
fits
well, as
it
at estimating the "true" coefficients
assuming a
the case
is
random sample generation
already known.
where a
is
estimated.
Again,
MLE
However,
if
only.
includes a
It
The
does here.
The
this is
likelihood
The variance
and a.
column
in
of a small
in the case
of the estimates rises as the extent of misclassification in the sample increases.
estimates from one
would be
The sample design used
ordinary probit were run on these misclassified data.
row has
MLE
Table
11
reports
that estimates the
/3
a
does well, and standard errors are lower than in
is
not truly known, but treated as
if it is
known,
e.g. Poterba
and Summers (1993), these estimated standard errors will be downward biased.
In the fourth
column are
as estimates of
/3.
the
MLE estimates
of the misclassification error parameter, a, as well
Notice again that the estimates of
/3
are closer to the true parameters than the
estimates obtained without correcting for likely misclassification error (column 2).
As previously
noted, the standard errors in the probit specification using misclassified
data are relatively small.
The researcher depending on
model
the incorrectly estimated
think that the reported coefficients are precisely estimated, although misclassification
a problem in the data.
Although the
MLE
method
be able to estimate very precise coefficients
if
the
that accounts for misclassification
model does not
standard errors that reflect the true imprecision of the estimates.
quite precisely estimated in cases
absolute size of the
We
The
test
jS's,
where the data
fit
is
the
fit
well,
it
will
actually
may
still
Also, the estimates of
model well, as
in
Table
II.
will
not
give
a
are
Not just the
but their ratios vary from the true to the inconsistent case.^
our theoretical model two additional ways.
When
there
is
no misclassification
Xs are drawn from normal
The appendix includes a table which reports
the results of simulations run using normally distributed Xs. Ruud (1983) discusses reasons for this
result. The ratio of the estimated betas differs between probit and MLE when the Xs are distributed
^
ratios of the betas will stay constant if the simulated
distributions, even
when each
individual beta
is
biased.
uniformly or log normally.
8
in the sample, a regression
expected (see equation
to
of
By
4).
yj
on a constant and *(X/3) leads
contrast, the
altered
by nonlinear
somewhat from
the
least squares.
MLE case.
and
same regression with dependent variable
have 10% misclassification error gives coefficients
also be estimated
to the coefficients
We
.1
and
y;
1
as
created
The model can
.8, as predicted.
minimize a quadratic error term which
is
Instead of estimating a, the equation allows the constant
term and the coefficient on ^(X0) to differ from zero and one, respectively:
yi
We
=
+ dr^(X^) +
5o
(13)
Vi
use White standard errors here; they are calculated using the derivative of the nonlinear
The NLS method
function with respect to the parameters in place of the usual regressors, X.
results in the following estimates for 5o, 5,,
Table B:
One sample
The simulated X's
/3,
Design a =0.1
=
and the
/3's.
generated.
are
drawn from a standard normal
distribution.
1
Probit: correct y
NLS
y misclassified
N=2000
5,
5o
-1.99
(.141)
The
.099
(.059)
(.024)
results here are close to the
5,=(l-25o):
Thus,
test
0.995
NLS
.802
MLE results and
= l-(2-.099) =
/?!
/3o
.804
-2.05
1.03
(.028)
(.623)
(.278)
they satisfy the overidentifying restriction that
.804.
leads to consistent estimates in the presence of misclassification.
of the restriction of independent misclassification error
made
In a situation requiring misclassification treatment,
It
also permits a
in the basic specification.
the researcher
may
suspect the
probabilities of misclassification
from one category
to another are not
the basic model, modified to allow for this difference.
symmetric*
Below
is
Simulation results follow.
To = probability of correct classification of O's.
Let
T,
We can
=
probability of correct classification of I's.
then evaluate the probability of observing given responses under different probabilities
of misclassification:
pr(yi
= l|X)
pr(y;=0 X)
1
The
ir,.pr(y;>0)
=
(l-7ro)
=
To
-
+
(T,
this
hold,
To
-
1)
•
(14)
l)-#(X/3)
*(X/3).
is:
equation to be identified, Tj
it is
+
To must be greater than one.
likely that the data are sufficiently "noisy" that
practice, the condition should not
'
(l-TQ).pr(y;<0)
+ To-
(T,
+
+
log likelihood function for the specification which allows for different probabilities of
misclassification
For
=
be a
If the condition
model estimation
is
does not
not warranted; in
restrictive requirement.
This problem displays a loose relationship to the switching regression problem, e.g. Porter and
Lee (1984).
regime
However, the switching regression framework uses the
hand side variable.
shift as a right
10
discrete variable to represent a
One Sample Generated
Table C:
The simulated X's
^,
=
are
drawn from a standard normal
1
Design
Probit:
T'S
correct y
^0
00
^i
:
using
^0
/3.
MLE:
solving for Tq and x,
design x's
To
iS.
T,
/So
|8i
-2.11 1.03
-.770 .373
-2.18
1.02
.978
.975
(.107)(.044)
(.044)(.011)
(.155)
(.064)
(.007)(.004)(.145)(.061)
MLE
However, note
specification
It is
that
we
can no longer
when we allow
overidentifying restriction of the misclassification
for different probabilities of misclassification.
where he or she suspects
that
correlated with the yi*'s in a continuous fashion:
is
+
=
x(y-).pr(y.->0)
=
x(X,/3,)-<^(X2A)
=
Suppose
.969
(l-x(y-)).pr(y;<0)
+
(15)
(l-x(X,^.)).(l-<i.(X3A))
= E[l-x(X,A)|X,] + E[(2iriX,fi,yi)'^(XM\X,]
E(y;|X)
logit
l-x(X,ii3,)
E(y.|X)
=
X
1-A(X,5)
condition on the X's,
we do
+
(2x(X,^,)-l)-*(X2i/32)
were the monotonic function ranging from
the expected value of y; given
to estimate;
test the
also likely that a researcher could encounter a situation
pr(y.= l|X)
.970 -1.98
able to estimate the parameters both accurately and quite precisely.
is
the probability of misclassification
we
MLE
Probit: y
misclassified
.950
Thus, again
If
distribution.
all
to
1
that
determined x. Then
could be written,
+
(2A(X,5)-l)'4>(Xi)3)
the previous results hold.
not provide simulations here.
11
(16)
This case
is
much more complicated
Logit case
The
logit functional
form can also be analyzed
error in the dependent variable.
First,
we
in the case
of suspected measurement
calculate the probabilities of the observed data given
misclassification:
pr(yi=l|X)
pr(y-0|X)
+
=
T.pr(y,'>0)
=
T.[exp(X^)/(H-exp(X/8))]
= x-pr(y;<0) +
=
T-
(1-T).pr(y;<0)
+
(17)
(l-x).[l/(l+exp(X^))]
(1-T).pr(y;>0)
[1/(1 +exp(X/3))]
+
(1-x)
We then calculate the expectation of the left hand
•
[exp(X^)/(l +exp(X^))]
side variable conditional
on the X's and given
the probabilities of misclassification:
=
E(yi|X)
=
[(1-T)
[a
+
+
T.exp(X/3)]/[l+exp(X/3)]
(l-a)-exp(X/3)]/[l+exp(X/3)]
We now demonstrate that the case of misclassification leads
to a
case of endogenous sampling, e.g., the papers in Manski and
very different outcome than the
MacFadden
of endogenous sampling the slope coefficients are estimated consistently by
constant
is
incorrectly estimated.
(1981).
In the case
MLB logit;
only the
Thus, estimation refinements are concerned with increasing
asymptotic efficiency. Here, in the situation of misclassification,
estimates of both the slope coefficients and the constant term.
MLE logit leads to inconsistent
This result follows easily from
a comparison of the (Berkson) log odds form of the logit in the case of measurement error and
endogenous sampling. Without misclassification we have:
+
X)
=
exp(x,/3)/(exp(X<^
pr(yi=0|X)
=
exp(X,^)/(exp(X(^)+exp(X^)).
pr(yi=
1
1
Thus, the expectation of the log odds
exp(X,/S))
is:
12
(18)
EOn(s,/So))
=
Enn(exp(X,/3)/exp(X<;3))]
With endogenous sampling where we redefine X
+
to
=
(X,
=
Xexp(X,/3)/(exp(X<^)
pr(yi=OjX)
=
(l-X).exp(X<^)/(expCM)
E(ln(s,/So))
=
ln(Xexp(X,^)[(l-X) • exp(X<y3)]-')
=
(ln(X/(l-X))
Thus, only the constant term
is
+
/3o)
-
Xo)^.
be the weighting constant we
pr(yi= 1 X)
1
-
find:
exp(X,^))
+
(19)
exp(X,^))
(XrXo)^,
inconsistent with
(20)
endogenous sampling. Using the expressions
derived above for the case of misclassification, where as before
t
is
the probability of correct
classification:
Ean(s,/So))
If
we
=
+
In [(x
(l-T).exppC,^))/(l-T
express the problem in a log odds form,
it is
+ r'txp(XMl
(21)
easier to see that the misclassification will
induce inconsistency in both the constant and the slope coefficient estimates.
of misclassification
feature
inconsistency.
is
The dependent
that
it
prevents one observation
variable for any one observation can
small as the proportion of ones approaches one or zero.
a terms prevent
When
An
interesting
from causing unbounded
become
arbitrarily large or
misclassification
is
present, the
the limit from reaching infinity.
Simulation results analogous to those using a probit functional form appear in Tables in
and IV.
Similar results hold
responses.
Furthermore,
the case of three or
if
more
if
we
the probabilities of misclassification are not equal across both
use the log odds specification,
we
categories of qualitative responses
probabilities differ both across response category
holds a simple treatment of this topic.
13
can extend our approach to
where the misclassification
and within response categories. The appendix
m.
Semiparametric Analysis
The assumption of normally
probit (logit) specifications
distributed (or extreme value) disturbances required
not necessary to solve for the
is
amount of
by the
One
misclassification.
can use semiparametric methods which estimate the extent of misclassification in the dependent
We
variable and estimate the slope coefficients consistentiy.
technique here.
The
kernel regression uses an Epinechnakov kernel, as
The window width,
efficiency properties.'
weighted average, operates;
we
h*
One could
employ a kernel regression
also determine h*
h, defines the intervals
over which the kernel, or
use Silverman's rule-of-thumb method as a
=
1.06ffn
has desirable
it
first
approximation:
•1/5
by cross-validation, a computationally intensive procedure which
minimizes the integrated square error over
h.
However, past experience leads us
to expect
very
similar results.
Our
kernel regression
kernel one-dimensional.
is
of the simplest form as
We calculate the index
I=X/3, where the
procedure which will be discussed subsequentiy.
range of the index
is
chosen.
The
kernel estimator
with each observation falling in the
window
uses an index function to
it
Then a
is
set
/3's
make
the
are found in a first-stage
of evaluation points covering the
the weighted average of the y's associated
centered at the point of evaluation:
/-/.
/(04e^:(I^)
where
I is
the index, h
is
the
window
Epinechnakov kernel.
The cumulative
dividing the weighted
sum of
'
in the
width, n
is
the
(22)
number of
observations, and K(.)
distribution of the response function (cdf)
y's by the total
number of
points falling in the
is
is
the
formed by
window of
the
Silverman (1986), p. 42. describes how the Epinechnakov kernel is the most "efficient" kernel
sense of minimizing the mean integrated square error if the window width is chosen optimally.
14
kernel.
We plot the cdf using
the probit coefficients
from above
to construct the index.
We use
simulated data with a misclassification probability, a, of 0.07, which approximately corresponds
to the estimated probability
Figure
la.
Standard
we
find in our empirical
CDF
example below with the CPS
Figure lb.
MM !—
a
o
CDF
data:'°
with Misclassification
.w
•
«
o
o
o
o
«
o
o
6
o
a
o
o
o
^^ooooooooo
o
o
a -0
-0,2
S
fi
10
1
4
index
index
.07 y misclassified: simulated data
true y: simulated data
Above
0.2
are two graphs, one using misclassified data and one using true data.
The method
produces the smooth curves of Figures la and lb, where the misclassification problem shows
up
clearly.
For example, a .07 misclassification
rate
would cause the cdf of the dependent
variable to begin at .07 and asymptote at a value of only .93.
there
is
always a seven percent chance
y will be one.
in the
cdf s. Note
increases.
'°
A
that the observation will
No
matter
how
small
X^
is,
be misclassified and observed
kernel regression performed on simulated data, demonstrates the difference
that in Figure la the cdf
However,
The simulated
in
goes between the values of 0.0 and 1.0 as the index
Figure lb the beginning value
is
data consist of 2,000 observations of the
es.
15
about .07 while the cdf asymptotes to
sum of normally
distributed X's and
about .93.
Using the cdf
in
this
manner allows
for identification of the probability of
misclassification using semiparametric or nonparametric techniques so long as the probability
of misclassification
is
independent of the X's.
We experienced
problem of how
two major problems with the kernel regression. The
to handle the tails
critical to the discussion
regression as far out into the
the
if
is
as possible because the
tails
window width becomes very
obscured
Yet,
low.
the classic
In this case, the behavior of the tails is
of the distribution.
of misclassification.
first is
it
misleading to continue the kernel
number of observations
falling within
In fact, the asymptotic behavior of the cdf
estimation continues to the extreme points.
In Figure lb
above
we
becomes
use the points
one bandwidth from the extreme index values as truncation points for the regression.
Other
solutions include variable kernel and nearest neighbor methods.
We
use the nearest neighbor algorithm as an alternative method to investigate the
behavior of the cdf in the
tails.
The
nearest neighbor
closest points to the evaluation point.
become
further
away
use
k=100 and
tail
exhibit similar
considered above.
is
As observations begin
rather than fewer in
methods have on our particular
method
number.
We
looked
a weighted average of the k
to thin out, nearest
at the effect nearest
neighbors
neighbor
problem using several different k values. The graphs below
tail
behavior as the truncated kernel regressions which
Again, the probit coefficients are used to construct the index.
16
we
Nearest Neighbor Methcxl
Figure 2a:
Standard
CDF
CDFs
Figure 2b:
Misclassified
CDF
o
*.«^JJUUUPULW.«^.«.»
OooO°°ooooooooo
oO
o
oo
«
o
f,^
a
o°
»
o
•n
a
o
o
o
^
a
o
o
t^
o
o°
-
oooooooOo
o
.
-0.6
o
o
nnnnfinnnPOO
-0.2
0-2
10
6
1
-O.e
is
more
severe:
how
to find the starting jS's.
to use the probit regression coefficients in the construction
we
are clearly
0.6
1.0
l.»
1
a
.07 y misclassified: simulated data
true y: simulated data
problem,
0.2
index
index
The second problem
-O.J
not freeing
the data
of the index.
One method
While
from an assumed error
importantly, while the results will demonstrate that misclassification
is
is
simply
illustrating the
structure.
More
present because of the
behavior of the cdf, the estimates of misclassification will not, in general, be consistent
estimates.
rv.
Application to a Model of Job Change
We now
to
consider an application where misclassification has previously been considered
be a potentially serious problem.
We
estimate a model of the probability of individuals
changing jobs over the past year. Because of the possible confusion of changing positions versus
17
changing employers, respondents
that job
we
well
make a mistake
in their answers.
We
would expect
changers have a higher probability of misclassification than do non-job changers. Thus,
consider models both with the same and different probabilities of misclassification, and
test for equality
the January 1987 Current Population Survey of the Census Bureau.
extracted the approximately 5,000 complete personal records of
25 and 55 whose wages are reported.
respondent's job tenure.
Among
may have
men between
the questions in the survey
This type of question
confuse position with job or
is
well-known for
its
is
the ages of
one asking for the
misclassification; people
Those respondents who give
re-entered the labor force.
tenure as 12 months or fewer are classified as having changed jobs in the last year.
individuals
who answer more
means of the data are
We
in
V
estimate a probit specification on the variable for changing jobs, which
is
of Freeman (1984).
misclassification.
We
t-statistic
Thus,
of about 8.3.
misclassification of the response
Next,
is
allowed
for.
change also becomes much greater
we
maximize the log likelihood
we
estimate the probability to be
we
find quite strong evidence of
is
Likewise, the effect of higher earnings in
in the misclassification specification.
allow for different probabilities of misclassification as in equation (15) since
In the third
the probability of misclassification if the observed response
is
CK,,
For
found to be much higher when possible
non-job changers are less likely to misreport their status.
from
Note
of the probit coefficients also change by substantial amounts.
instance, the effect of unions in deterring job changes
affecting job
We
present the results in Table VI.
for the probability of misclassification
Many
call
same explanatory variables but allowing a,
the probability of misclassification, to be non-zero.
0.058, with an asymptotic
we
quite similar to specifications previous used in the applied labor
literature, c.f. the specification
when we allow
The
below.
function presented initially in equation (7) using the
that
Those
than one year are classified as not having changed jobs.
Table
Jobchange. Our specification
economics
we
of the probabilities.
Our data come from
We
may
column of
no job change,
the probability of misclassification if the observed response
allow the misclassification probabilities to differ across the responses.
is
is
the table, oq,
allowed to differ
a job change. Thus,
we
Freeing the a parameters
produces a markedly different value for a, than in the case where the two are constrained to be
equal,
a,
jumps
to 0.31 while ao
remains
at
0.06 The difference between the two estimates of
18
a
in
1.51,
column 3
which
is
is
.248 and has a standard error of .164, giving
it
an asymptotic
t-statistic
of
not quite significant. However, the effect of both union and earnings in deterring
job change are even greater than previously estimated. Thus, allowing for different probabilities
of misclassification again leads to quite different results than in the original probit specification.
We now
perform kernel regression
misclassification exists in the responses.
the situation clearly.
We
find the
0.6.
These
The graph below
minimum
determine
to
The
if
it
also demonstrates that response
kernel regression results in Figure 3 demonstrate
uses the probit coefficients to create an index function.
probability of job change to be around .06 with an increase up to about
results are extremely close to the
MLE probit results of Table VI
where we estimate
the respective probabilities to be .06 and 0.7 with an asymptotic standard error of 0.17.
Semiparametric estimation of the cdf can reveal the a' s by showing different asymptotes
at
each end of the function. In our case, the lower asymptote
much
less well defined, is in the
50-60%
to
the upper one, though
not totally satisfactory.
is
We cannot achieve
model parameters nor can we estimate the sampling
Thus,
distribution of the misclassification probabilities.
methodology which allows us
6-7% and
However, semiparametric estimation of the
range.
probability of misclassification by visual inspection
consistent estimates of the job change
is
overcome
we now develop
a semi-parametric
shortcomings of our previous approach.
the
Nevertheless, our results up to this point do demonstrate quite convincingly that misclassification
of job changes does exist in the
Figure
f»»^.
>»•.
17
3:
TI.M
n
CPS
data.
Probit Coefficients; Jobchange Data
IMJ
O
r
o
.
in
o
o
rn
O
o
(N
o
O
o
o
o
OOOOq
^oooooooO
o
o -26
-22
-'8
-11
-'0
-06
-0-2
2
V. Consistent Semiparametric Estimates
We now derive a methodology which permits consistent estimates of the misclassification
We first use the Maximum Rank
probabilities and the unknown slope coefficients."
(MRC)
Correlation
Han
estimator described in
This binary response estimator
(1987).
straightforward to calculate even in the instance of multiple explanatory variables and
MRC
variables.
operates on the principle that for observations where the index Xj3
a given estimate of
MRC
)3,
we would
expect that
indicator variable a "one" in cases
Thus, the
MRC
function which
An
is
over
all distinct
advantage of using
required.
> yj)nx]0 >
1(.)
used.
The
MRC
=
Xj0) +
(i,j)
to I, but instead is
problem remains monotonic
in the "correct" order.
may
1(.)
i
=
=
<
is
Ij
=
(1,...,N).
the normality assumption is not
/3's in
when
just such a case; the cdf
Xj/3,
probit
the index in the case of
any
no longer
However,
Thus the
Manski (1985) discusses use of the maximum score estimator when a single misclassification
probability exists for the data.
20
the
so long as the probabilities of
correct classification, Xq and t,, exceed 0.5 each and do not depend on the X's.
"
(23)
otherwise and the
probabilities of misclassification.
in the indicator variable,
y)]
lead to inconsistent results
allow estimation of the
bounded by the
yj)l{x.^
while
is that
monotonic transformation of the index. Misclassification
runs from
constructed by giving an
is
from the sample
instead of probit (or logit)
to
for
The
see equation (14).
1;
<
/(>'..
if (.) is true,
1
pairs of elements
was designed
Xj/3 for
is:
Violation of the distributional assumption
is
(or logit)
MRC
>
x,
Xfi and then a sum
maximized
an indicator function with
1(.) is
summation
E[^(>'.
+
where the dependent variables are also
is
''
S^ (0) =
=
dummy
The necessary monotonicity property
yj.
holds under misclassification so long as Tq
observations are ordered by the index ^
where
>
y;
>
is
/3's
MRC
from
by Han (1987) and asymptotic normality of the
and
MRC
coefficients
Cavanaugh and Sherman (1992). However, we
in
The consistency
estimation are consistent in the case of misclassification.
proved
is
proved by Sherman (1992)
is
need a semiparametric, consistent
still
estimate of the misclassification probabilities, ao and a,.
Misclassification probabilities are given
The
technique gives estimates of those asymptotes.
would be kernel regression on each
we can
construct such a cdf.) However,
of a distribution.
last
tail
Where does
by the asymptotes of the
first
of the cdf.
it is
method
that
not obvious
how
to
the estimated
data
become very
thin
so
tails
and the
observation becomes disproportionately important.
Instead, in order to solve for the
as
semiparametrically,
we
use the technique of isotonic
regression (IR). IR nonparametrically estimates a cdf using ranked index values.
is
|8's
weight observations in the
The
the "tail" begin, for example.
a researcher might choose
we know
(Recall that
the chosen
cdf;
useful for our problem because the final estimate
is in
The technique
the form of a step function; the lowest
and highest "steps" provide consistent estimates of oq and 1-a,. The combination of
MRC
and
IR techniques achieves consistent estimates of both the coefficients on the explanatory variables
and an estimate of the amount of misclassification
in
The
the data.
Isotonic Regression
technique involves the following basic steps.
First,
N"^
we
consistent.
use the coefficient estimates from the
We
want
p
.
MRC
Next, the indices are ordered so that
respect to the index v
is
procedure of Han (1983) which are
v,
estimated
<
Vj
<
The index
/3's.
Vfj.
An
any non-decreasing function defined on the
isotonic regression of y if
of the dependent variable
to estimate the probability distribution
conditional on an index constructed from the
Xj
MRC
is
defined
Vj
=
isotonic function, F, with
N
index values.
F
is
an
minimizes
it
N
Jliy, -^(v,))^
i-l
In addition,
decreasing,
F
so
(v)
F
=
(Vj.,)
if
<
Groeneboom (1993) proves
v
<
F
v,
(Vi).
and
F
Under
(v)
=
this
that the point estimates,
21
1
if
v
>
Vn.
specification for
F
(v)
F(.)
must also be non-
known index
are N"^ consistent for
values
v,
F
€
(v)
(0,1).
We
Han's estimator of v leads
will demonstrate that using
Performing isotonic regression
index values into pools, where each pool
upon the index being
to the index values
V;
distinct index value.
the algorithm
indexed pool. If the guess for the
are left intact; the second pool
first
is
is
just the average of the
initial set
pool
is
less than the
guess for the second pool, the pools
used for the next comparison.
(Vj) is
Otherwise, the pools are
This process
is
Otherwise, the regression
v,
at the
However,
it is
critical to realize that the
is
is
complete.
end of the algorithm.
In order to observe the lower asymptote of the cdf, the
Similarly for the upper
there are no observations at high index values, the estimated cdf will end before
those values are reached.
value
continued
estimated a's can only be
researcher must have observations with very low index values.
final
is
described completely by the guesses associated
the guess for the pool containing
identified using nonparametric methods.
if
values corresponding
highest steps resulting from the isotonic regression are consistent
estimates of Uq and ai.
asymptote;
>»;
compares the lowest-indexed pool with the next lowest-
isotonic regression of y with resi>ect of v
The lowest and
to organize the
Finally, if any combinations of pools occurred during the last
pools are exhausted.
with the final set of pools;
is
of pools has each pool corresponding to a
sequential comparison of the pools, repeat the process.
The
idea
result.
assigned a "best guess" for the value of y conditional
used for the next comparison.
is
combined, and the combined pool
until the
The
in the pool.'^
Then
is
The "guess"
in the pool.
The
quite straightforward.'^
is
same
to the
The researcher must ask
the question, does the cdf end because the
the "true" asymptote or because there are
no data points
in that area?
In order to
solve the problem, the data would have to contain observations with index values which
approach negative or positive
However,
infinity.
approximately constant at some non-zero a, that
be a problem
in
the data.
Certainly
is
if the
data display a long
good evidence
further investigation
tail
that is
that misclassification could
would be warranted; perhaps
resampling techniques could be employed to find out more about the problem.
Once we have
'^
See Barlow
the
et. al.
MRC
estimates and the IR estimate of the cdf,
(1972) or Robertson
et. al.
we
find the asymptotic
(1988) for detailed discussions of IR and the
algorithms used for estimation.
'^
The non-decreasing property
constrains pools to contain only adjacent index values.
22
distribution of the
IR estimator. To date no asymptotic distribution has been found for methods
which estimate both the unknown
slope
coefficients
from the
insight follows
we
of the form
as
known
fact that
We
use
MRC
use with step functions
the
However, we are able
semiparametric models of discrete response.
of both sets of coefficients here.
and
/3
known
results
cdf simultaneously
to derive the distribution
combined with a basic
insight.
Groeneboom's (1993)
is
N"' consistent, so we can
(The proof
results.
The
estimates are N"^ consistent while isotonic regression
treat the
MRC
deriving the asymptotic distribution for the isotonic regression
in
in
is
coefficients
applying
in
given in the Appendix V.).
Estimation of the Asymptotic Distribution for the Isotonic Regression Step Function
We
a step function where
parameters which
I
is
the index,
come from
I
=
We
X/3.
estimating the model by the
estimator (or other N"'^ estimator),
MRC
we would be
technique in
from
x,,
...^ and
F^
estimate
.
of the steps of
known v
s.t.
the
We
F„
F(v)
.
E
The asymptotic
where /is the derivative of the
its
the last time
maximum. Thus,
/3),
we
we
(1987) and
did not use the
//,...,/„
find the
distribution
(derived
IR functional
was found by Groeneboom (1993): For any
the
( 1
- F(v))
-F(v)
)
^^ TH
^^
^^
(24)
^(v)j
true distribution F,
g
is
the derivative of the distribution of the
where two-sided Brownian motion minus the parabola
Groeneboom technique
to estimate the
However, our approach which
23
utilizes
v^
reaches
asymptotic distribution cannot
be applied to the Cosslett-type IR estimator because the slope coefficients
along with the cdf.
Han
(0,1),
a
is
with
are interested in the asymptotic standard errors associated with the height
^- F(v)
Z
|7)
unable to derive the asymptotic
N"^ consistent parameter estimates of
n'^iFAv)
index, and
If
Using the index observations
distribution of the misclassification probabilities.
Pr()'=l
use the N"^ consistent estimates of these
then do an isotonic regression using these estimates for the index value.
MRC
=
use the method of isotonic regression to estimate the c.d.f. F(/)
/3
are being estimated
an N"^ estimator for
jS
allows the
parameter vector
The
to
be treated as known for purposes of estimating the cdf.
distribution of
Z can
be written (Groeneboom (1984)) as
= l.s(v)s(-v),
h(v)
^
V €
St
(25)
2
where the function
s has a Fourier transform
9 1/3
H^)
^
=
(26)
i4j(2-"^H'i)
and where A/
is
the Airy Function and
Z will
Solving for
symmetry of
integration,
h^.
we
Using
I
= V-
1
allow us to estimate the variance of F^
HAT. We have E(Z) =
from
Also, via numerical approximation of the Fourier transform and numerical
estimate Var(Z)
(23),
we
=
0.26.
then have
^
^.(v)(l--^n(^))/(v)
VariF(v)
- F(v))
= 1.04
f
(27)
2ngiv)
where we use the numerical estimate of the index density for
g(v).
use the numerical derivative of F„(v) for/(v) since the derivative of a step function
at
a
fmue number of
points (where
"slope approximation" arrived at by looking at the
jumps from or
defined to be the slope of the straight line from the point half
current step and the point half
way up
The
density of the index
sensitive to the choice of
the step.
The process
is
window
we
calculate
estimated using a
width.
The
it
assuming
of the
/3's
we
use a
this definition
of the
that the next step level is one.
point at which g{v)
24
zero except
the vertical rise to the
Because
window width of
that estimates the standard errors
cannot
to the adjacent steps. J{v) is
way up
the rise to the next step.
slope does not exist for the highest step,
is
Instead, for a given step level,
is infinity).
it
we
Unfortunately,
is
0.
1
;
the results are not
estimated
uses a kernel
is
the middle of
window of
a*n'
"^
where a
number of
observations.
IX use a kernel window of 0.05. Again,
the estimates
the standard deviation of the index function and n
is
The standard
errors for the a's in Table
window
are not greatly affected by the choice of
width.
Consistent Semiparametric Estimates of the Job
We now
data from the
MRC
apply the combined
GPS. The
results
Change Model
and isotonic regression estimators
MRC/IR
length one.
Thus, the absolute magnitude of the coefficients
MRC/IR
MRC/IR.
In order to
compare
MRC/IR
Western stays fixed across methods;
it is
which will convert the
MRC/IR
by the same
is
indeterminate; only the ratios
coefficients.
We
also the coefficient
assume
which
In this
is
assumed
way we
obtain a scaling factor
MRC/IR
was seen before
in
Thus, when
we
procedure gives results that always
use the combined
MRC/IR
can
significantly
it
between the
correcting for
we
find that the
Both are estimated quite
reject the hypothesis that the probabilities
of misclassification are
changes the estimate of a^; only 3.5% of observed non-jobchangers have a
misclassified response.
As mentioned above,
amount of data we have
range
lie in
Freeing the misclassification model from the assumption of a normal error distribution
equal.
the
now
However,
MLE
approach
estimated probabilities of misclassification are 0.035 and 0.395.
MLE produces
Table VI.
estimates obtained from ordinary probit, ignoring misclassifications, and
we
be fixed in
to
factor.
estimating the model using the
and
on
coefficient into the probit coefficient and multiply the remaining
point estimates which differ from those of probit as
precisely,
with
that the coefficient
Table VIII restates the earlier results (Table VI) for ease of comparison.
misclassification.
MLE
from probit and
the results
calculating the asymptotic semiparametric standard errors.
coefficients
job change
produces coefficients which are scaled so that each coefficient vector has
necessary to scale the
it is
to the
of using the estimators on the job change data are reported in
Table VIII.
can be found from
the
is
is
seems
in that
the accuracy of the upper asymptote depends
index range.
Since the number of datapoints in the upper
small, the estimates of a, should be perhaps be viewed with
to us that the
lower asymptote
is
well established.
low values of the index and the lowest step
is
some
caution.
However,
There are plenty of observations
relatively long.
25
on
at
Figure 4
In
we
misspecification and also the
Figure 5 shows the
in either direction.
the cdf
is
the
plot
MRC/IR
The confidence
we smooth
comparison of the
MRC/IR
when
becomes non-monotonic
is
at the
upper
of our data.
tail
MRC/IR
quite different from the
One can
also include a
Figure 6b
tries
Figure 6a
estimate.
an alternative
Again, the upper
Kernel regression techniques are
where data become
sparse.
assumption of normally distributed errors were correct for our data
expect to see the
In
see that the kernel
to construct the kernel estimate.
basically not suited to estimating the asymptotes of a cdf
If the
We
small.
problematic for observations at the ends of the distribution
approach by using the 200 nearest neighbors
of the cdf
becomes
the size of the step function
because there are only observations on one side of the point.
tail
IR step function estimate of
estimated cdf with a standard kernel estimate of the cdf.
is
for
cdf with a confidence interval of two standard errors
the estimation of the confidence interval.
uses a fixed-window kernel; this
estimate
model which allows
results are reasonably similar.
interval demonstrates that the
estimated accurately, except
these situations,
The
estimate of the cdf.
MRC/IR estimate of the
MLE
cdf from the
estimated
MRC coefficients come very close to the MLE coefficients.
set,
Some
we would
coefficients,
such as Union, match closely, but others such as Last Grade Attended and Married, do not. The
differences could well arise from a failure of the underlying probit
model assumptions of
normality.
To
MRC/IR
explore further the accuracy of the
conducted another simulation analysis.
We generated
approach compared
to probit,
we
5,000 observations of simulated data with
three explanatory variables (none distributed normally), a constant, and a normally distributed
error.
The
Ten percent of
results
the binary response dependent variable
was purposefully
misclassified.
MLE
correcting for
of the three estimation techniques are reported in Table IX.
misclassification should return the coefficients used to construct the data, as should
no special correction. The conclusion of
are not
It is
met by our job change
also interesting to note
data,
this exercise is that the
and
that the
from Table VIII
that the
misclassification probabilities quite accurately
formed by the
earlier semiparametric
and
MRC
and
MLE
with
assumptions required for probit
estimator provides superior estimates.
MRC/IR
estimator
is
able to estimate the
their values are close to
analysis.
26
MRC
our expectations
VI. Conclusions
The
discussion above shows that ignoring potential misclassification of a dependent
when
variable can result in substantially biased and inconsistent coefficient estimates
The researcher can use our maximum
or logit specifications.
above
likelihood procedure described
of misclassification and estimate consistent coefficients.
to estimate the extent
misclassification probabilities are asymmetric across groups, they
However, should the
errors in the data not
may
be estimated
amount of
misclassification
Furthermore, the isotonic regression techniques detailed above provide a procedure
will give consistent estimates of the misclassification percentages in the data.
to derive
easily.
MRC technique of Han (1987)
will yield consistent estimates of the coefficients regardless of the
which
still
If the
be normally distributed, these coefficients may
nevertheless be inconsistent. Semiparametric regression using the
in the data.
using probit
We are able
and estimate the asymptotic distribution for both the slope parameters and for the
misclassification probabilities.
Other model specification problems may exist besides misclassification: e.g. nonnormality or heteroscedasticity. Misclassification in particular need not be the problem in a case
where the probit model does not
fit.
However,
the types of results
misclassification can be a serious problem; results that suggest
we
its
achieve here suggests that
existence certainly justify
looking more closely at the data to determine what error structure does exist.
We
apply our econometric techniques to job change data from the CPS.
misclassification exists in the data.
depending on the response.
We
We
find that
Furthermore, the probabilities of misclassification differ
are able to estimate the parameters by the
MLE
parametric
approach and by the distribution free semi-parametric approach; the estimated parameters are
reasonably similar although they are estimated with only moderate levels of precision.
approach
is
quite straightforward to use on discrete response models that are
applied research.
Thus,
we hope our approach
data —especially in probit and logit models.
27
will
Our
commonly used
in
be useful to others working with discrete
References
Barlow, R.
et.al.
(1972), Statistical Inference under Order Restrictions
.
(New York, John
Wiley).
Cavanagh, Chris and Sherman, Robert (1992), "Rank Estimators for Monotone Index Models,"
Bellcore mimeo.
Chua, Tin Chiu and Wayne Fuller (1987), "A Model for Mulitnomial Response Error Applied
Labor Flows," Journal of the American Statistical Association 82:397:46-51.
Maximum
Cosslett, Stephen, R. (1983), "Distribution-Free
to
Likelihood Estimator of the Binary Choice
Model," Econometrica 51:3:765-782.
Freeman, Richard, (1984), " Longitudinal Analyses of the Effects of Trade Unions" Journal of Labor
Economics, vol. 2, no. 1, 1-26.
Greene, William (1990), Econometric Analysis
Groeneboom,
Piet (1985), "Estimating
.
(New York, Macmillan).
Monotone Density", in L.M. Le Cam and R.A. Olshen, eds.,
in Honor of Jerzy Neyman and Jack Keifer vol H,
Proceedings of the Berkeley Conference
.
(Hay ward, CA: Wadsworth).
Groeneboom,
Censored Observations", mimeo.
Piet (1993), "Asymptotics for Incomplete
Han, Aaron (1987), "Non-Parametric Analysis of a Generalized Regression Model: The
Rank Correlation Estimator," Journal of Econometrics 35:303-316.
Maximum
Han, Aaron, and Hausman, Jerry A., (1990), "Flexible Parametric Estimation of Duration and
Competing Risk Models," Journal of Applied Econometrics, 5, 1-28.
Hardle, W., (1990), Applied Nonparametric Regression
Klein, R., and Spady, R., (1993),
"An
.
Cambridge University
Press, Cambridge.
Efficient Semiparametric Estimator for Binary
Response
Models," Econometrica 61:2.
Manski, Charles F., (1985), "Semiparametric Analysis of Discrete Response: Asymptotic Properties
of the Maximum Score Estimator," Journal of Econometrics 27:313-333.
Manski, Charles and Daniel McFadden (1981), Structural Analysis of Discrete Data with Econometric
A pplications (Cambridge, MIT Press).
.
MacFadden, Daniel (1984), "Econometric Analysis of Qualitative Response Models,", in Z. Griliches
and M. Intrilligator, eds.. Handbook of Econometrics vol. 2, (Amsterdam: North Holland).
,
and Lee, Lung Fei (1984), "Switching Regression Models with Imperfect Sample
Separation Information with an Application on Cartel Stability" Econometrica 52(2):391-418.
Porter, Robert
28
Poterba,
J.
and L. Summers (1993), "Unemployment Benefits, Labor Market Transitions, and
A Multinomial Logit Model with Errors in Classification" MIT mimeo.
Spurious Flows:
Pratt,
John W., (1981), "Concavity of the Log Likelihood," Journal of the American
Association, 76, No. 373, 103-106.
Robertson, T.
et. al.,
(1988), Order Restricted Statistical Inference
.
(New York, John
Statistical
Wiley).
Ruud, Paul, (1983), "Sufficient Conditions for the Consistency of MLE Despite Misspecifications of
Distribution in Multinomial Discrete Choice Models," Econometrica 51:1:225-28.
Sherman, Robert P., (1993), "The Limiting Distribution of the
mimeo.
Stock, James,
Thomas
Coefficients,"
Stoker, and James Powell (1989),
Maximum Rank Correlation
Estimator"
"Semiparametric Estimation of Index
Econometrica 57:6:1403-30.
Silverman, B.W., (1986), Density Estimation for Statistics and Data Analysis
London,
29
.
Chapman
&
Hall,
Table
la:
Simulations
drawn from a lognormal distribution
Xj is a dummy variable drawn from a uniform
Xjis drawn from a uniform distribution
X,
is
Sample
Repetitions
=
Design
146
n=5000
Probit: y
misclassified
a
0.02
—
00
-1.0
-0.787
^1
0.20
02
1.5
/?3
-0.60
distribution
ratio to
constant
—
MLE:
solving
for
a
ratio to
constant
0.0192
(.0054)(.0051)
-0.990
1
(.068)(.077)
(.069)(.066)
0.158
0.20
(.001)(.005)
1.49
1.61
0.66
—
00
-1.0
-0.567
/3,
0.20
0.0497
—
1
(.0076)(.0075)
-1.007
1
0.20
1.06
1.87
0.76
-0.431
(.019)(.019)
n=5000
a
0.20
—
00
-1.0
-0.163
0^
0.20
02
1.5
0,
-0.60
0.20
1.504
1.50
(.082)(.089)
(.051)(.048)
-0.60
0.201
(.OlOK.Oll)
(.010)(.004)
03
0.60
(.084)(.090)
0.114
1.5
-0.598
—
(.073)(.058)
02
1.51
(.026)(.028)
(.023)(.021)
0.05
0.20
(.075)(.073)
-0.518
a
0.199
(.008)(.009)
1.27
(.063)(.054)
n=5000
1
-0.599
0.60
(.032)(.034)
—
—
0.198
(.014)(.013)
-0.991
1
(.168)(.159)
(.061)(.050)
0.037
0.23
0.554
3.40
0.20
1.48
1.49
(.182)(.169)
(.045)(.041)
-0.228
1.40
-0.592
0.60
(.072)(.064)
(.018)(.016)
left
0.198
(.023)(.022)
(.005)(.002)
(The
1
hand parentheses contain the standard deviation of the coefficient across 146 Monte Carlo
The right hand parentheses contain the average (n= 146) standard error calculated from
simulations.
MLE. The
similarity of the
two show
that the
maximum
30
likelihood asymptotics
work
well.)
Table lb: Simulations from Table la in Ratios
X,
is
Xj
X3
is
is
Repetitions
drawn from a lognornial distribution
a dummy variable drawn from a uniform
drawn from a uniform distribution
=
Design
146
n=5000
00
a
solving for
1
small
e
large
1
1
e
small
1
e
large
1
0.2
0.20
0.20
0.20
0.20
02
1.5
1.61
1.58
1.51
1.48
/3,
0.6
0.66
0.63
0.60
0.58
.02
n=5000
a =
MLE:
Probit: y misclassified
Ratios
e
a =
distiibution
^0
1
1
1
1
0.2
0.20
0.20
0.20
0.19
02
1.5
1.87
1.67
1.50
1.47
03
0.6
0.76
0.66
0.60
0.57
n=5000
00
1
a = .20
0:
0.2
0.23
0.24
0.20
0.21
02
1.5
3.40
2.49
1.49
1.50
03
0.6
1.40
1.04
0.60
0.63
.05
1
1
31
1
Simulation
Table
11:
vector
=
Simulations:
One Sample Generated
(-3, 0.3, 0.2)
X, drawn from a normal distribution
Xj a
true
dummy
a
variable
drawn from a uniform
True Probit
X and y
sample size
distribution
Probit: y
misclassified
MLE
using
true
a
MLE:
for
solving
a
a
a = .025
^0
n=2000
/32
a = .05
/3o
n=2000
^2
a = .l
/3o
n=2000
02
a = .2
00
n=2000
02
Note:
The
-2.958
-2.443
-2.937
-3.170
(.138)
(.119)
(.166)
(.289)
.295
.240
.290
.314
(.012)
(.010)
(.015)
(.028)
.204
.208
.224
.236
(.080)
(.075)
(.088)
(.095)
-3.048
-2.396
-3.266
-3.210
.046
(.137)
(.114)
(.200)
(.291)
(.014)
.296
.233
.316
.311
(.012)
(.009)
(.018)
(.027)
.296
.168
.290
(.079)
(.074)
(.097)
(.100)
-3.007
-1.901
-3.191
-2.961
.085
(.137)
MOl)
(.243)
(.359)
(.020)
(.013)
.282
.296
.187
.316
.293
(.012)
(.008)
(.022)
(.054)
.235
.143
.238
.218
(.077)
(.069)
(.110)
(.105)
-3.270
-1.186
-3.480
-3.282
.190
(.151)
(.086)
(.401)
(.532)
(.019)
.328
.122
.370
.348
(.013)
(.006)
(.040)
(.056)
.164
.051
-.0379
-.020
(.082)
(.065)
(.173)
(.163)
coefficients are different in each
generated for each row; thus
.0397
it is
row of column one because only one random sample
/3's would match exactly.
unlikely that the
32
is
Table
ill:
vector =(-2,
Logit Simulations
:
One Sample Generated
1)
Xi drawn from a normal distribution
True a
Sample
size
True Logit
X and y
00
Logit: y
misclassified
^0
/3i
.05
-1.990 1.012
-.967
n=
2000
(.153) (.054)
(.095)(.023)
n=
a=
n=
.1
2000
.2
2000
true
^.
a=
a=
MLE
.504
with
a known
/3o
-1.897
^1
.933
/3o
solving for
jS
/3,
-1.830 .901
a
.045
(.207)(.080)(.008)
-2.100
-2.115 1.079 .101
-.612
(.159) (.058)
(.081) (.017)
(.257) (.107)
-2.057 1.045
-.399
-2.338 1.146
(.154)(.056)
(.071) (.012)
.201
a and
(.192) (.070)
-2.131 1.065
.342
MLE
1.071
(.401) (.178)
Standard errors are in parentheses.
33
(.287)(.125)(.010)
-2.425 1.192 .205
(.479)(.223)(.012)
^=s=s^^^^^^=^=^^^^^^s=^=^=^^=^==
1
Table IV: Logit Simulations:
Simulation
true
/3
vector
=
X,
is
drawn from a normal
is
a
a
distribution
variable
True Logit
X and y
sample size
One Sample Generated
(-2, 0.5, 0.2)
X,
dummy
=s=:^=^^^^^=^^=s^=^^==
Logit: y
misclassified
MLE
using
true
a
MLE:
for
solving
a
a
a = .025
n=2000
^0
-2.08
-1.63
-2.14
-2.18
0.029
(.151)
(.133)
(.172)
(.207)
(.011)
0.511
0.520
(.027)
(.038)
0.407
0.504
(.019)
(.023)
/32
a = .05
n=2000
^0
02
a = .l
00
n=2000
02
a = .2
00
n=2000
0^
02
Note:
The
.238
0.128
0.232
0.235
(.137)
(.127)
(.149)
(.152)
-1.895
-1.56
-2.03
-1.917
0.037
(.146)
(.130)
(.184)
(.215)
(.015)
0.506
0.398
0.530
0.497
(.023)
(.018)
(.032)
(.047)
0.117
0.142
0.120
0.129
(.136)
(.125)
(.163)
(.153)
-2.26
-1.17
-2.08
-2.40
0.124
(.155)
(.117)
(.229)
(.389)
(.019)
0.509
0.281
0.494
0.565
(.023)
(.014)
(.038)
(.076)
0.420
0.110
0.292
0.373
(.136)
(.115)
(.183)
(.220)
-2.25
-0.903
-2.271
-2.29
0.201
(.158)
(.107)
(.330)
(.540)
(.027)
0.526
0.189
0.509
0.515
(.024)
(.012)
(.058)
(.118)
0.400
0.194
0.398
0.401
(.140)
(.107)
(.259)
(.268)
row of column one because only one random sample
that the /3's would match exactiy. Standard errors are
coefficients are different in each
generated for each row; thus
it is
unlikely
1
parentheses.
34
is
in
-
-
Table V: Means of the
CPS job change
data
The first row of each cell is the sample statistic.
The second row contains the statistic ior jobchange=Q observations
The third row contains the statistic for jobchange = I observations
mean
std
min
dev
max
obs
.7293
.4443
1
.7468
.4349
1
.6253
.4844
1
14.38
2.823
1
19
14.40
2.834
1
19
14.30
2.760
1
19
37.43
8.526
8.535
34.17
7.712
25
25
25
55
37.98
.2454
.4303
1
.2668
.4424
1
5221
4471
.1173
.3220
1
750
earn per
488.9
240.2
week
507.9
235.7
375.1
married
grade
age
union
west
55
55
5221
4471
750
5221
4471
750
5221
4471
750
5221
4471
235.6
999
999
999
.2015
.4012
1
.1946
.3960
1
5221
4471
.2427
.4290
1
750
2
35
750
Table VI: Determinants of Job Change
"o
Probit
MLE ao=ai>0
—
0.0579
0.061
(.007)
(.007)
0.0579
0.309
(.007)
(.174)
-0.108
-0.073
-0.103
(.049)
(.077)
(.100)
—
«1
married
MLE
ao
^
0.026
0.063
0.080
attended
(.009)
(.015)
(.026)
age
-0.022
-0.028
-0.033
(.003)
(.005)
(.007)
-0.434
-0.707
-0.811
(.061)
(.148)
(.199)
-0.001
-0.003
-0.004
(.0001)
(.0004)
(.0009)
0.214
0.301
0.367
(.054)
(.086)
(.127)
0.051
0.171
0.581
(.162)
(.259)
(.495)
-1958.1
-1941.4
-1940.9
last
grade
union membership
earnings per
week
western region
constant
log likelihood
number of
obs.
5221
5221
Standard errors are in parentheses.
36
5221
tt]
Table Vila: Marginal Effects
at the
mean,
MLE ao=ai>0
-.13
-.04
-.05
(.08)
(.07)
(.12)
-.20
-.04
-.04
(.12)
(.10)
(.23)
.41
.86
.93
(.21)
(.25)
(.35)
.61
.73
.79
(.34)
(.31)
(.60)
.80
.54
.49
(.51)
(.38)
(.97)
1st
-.81
-.88
-.87
(.13)
(.17)
(.36)
mean
-1.35
-.85
-.83
(.29)
(.25)
(1.16)
3rd
-1.76
-.62
-.51
(.51)
(.37)
(2.07)
-.18
-.14
-.14
(.06)
(.07)
(.14)
1st
-.49
-1.05
-1.09
(.11)
(.16)
(.34)
mean
-1.05
-1.32
-1.37
(.31)
(.25)
(1.05)
3rd
-1.49
-1.04
-.92
(.54)
(.34)
(1.89)
.07
.05
.05
(.02)
(.02)
(.07)
3rd
1st
mean
3rd
age
union
MLE
tto 5^
1st
mean
grade
and third quartiles of the data
regular probit
quartile
married
first,
1st
mean
3rd
earnings per
week
western
region
1st
mean
3rd
(
Asymptotic standard errors are
in parentheses.
37
)
CK,
II
at the
married
mean,
first,
-.0032
-.0012
(.0196)
(.0013)
-.0080
(-.0073
(.0261)
(.0077)
-.0389
-.0184
(.0415)
(.0200)
1st
3rd
1st
mean
3rd
union
1st
mean
3rd
earnings per
1st
week
Oo
?i
.0010
.0009
(.0065)
(.0005)
.0026
.0057
(.0090)
(.0032)
.0127
.0143
(.0122)
(.0069)
-.0007
-.0004
(.0042)
(.0002)
-.0018
-.0023
(.0055)
(.0011)
-.0085
-.0058
(.0026)
(.0021)
-.0156
-.0092
(.0965)
(.0042)
-.0395
-.0575
(.1281)
(.0216)
-.1920
-.1447
(.1040)
(.0553)
-.0001
-.00005
(.0003)
-.0001
(.00002)
-.0003
(.0004)
(.0001)
-.0007
-.0007
(.0003)
(.0003)
.0072
.0042
(...)
(.0026)
mean
.0182
.0260
(...)
(.0162)
3rd
.0887
.0655
(...)
(.0299)
me^n
3rd
western region
MLE
1st
mean
age
quartiles
MRC/IR
3rd
grade
and third
quartile
mean
(
11
Table Vllb: Semiparametric Marginal Effects
1st
Asymptotic standard errors are
in parentheses.
38
)
a,
Table VIU.
Comparison of Probit, MLE, and MRC/IR Results
The dependent variable is Jobchange.
MLE ao=a,>0
Probit
"o
"l
MLE
Qto
^
«i
MRC/IR
.058
.061
.035
(.007)
(.007)
(.015)
.058
.309
.395
(.007)
(.174)
(.091)
married
-.108
-.073
-.103
-.161
(0 if unmarried)
(.049)
(.077)
(.100)
(.191)
.026
.063
.080
.052
attended
(.009)
(.015)
(.026)
(.043)
age
-.022
-.028
-.033
-.035
(.003)
(.005)
(.007)
(.021)
-.434
-.707
-.811
-.794
(.061)
(.148)
(.199)
(.503)
-.001
-.003
-.004
-.003
(.0001)
(.0004)
(.001)
(.0015)
last
grade
union membership
earnings per
week
western region
constant
log likelihood
number of
obs.
.214
.301
.367
(.054)
(.086)
(.127)
.051
.171
.581
(.162)
(.259)
(.495)
-1958.1
-1941.4
-1940.9
5221
.367
(...)
5221
5221
Notes:
1.
MRC/IR
comparable
2.
coefficient estimates
and
their standard errors
have been rescaled so as
to
be
to the other coefficients.
The standard
error for the variable Western can not be estimated because one coefficient
must be held fixed when estimating the standard errors for the
39
MRC
estimator.
Table IX: Simulation
e
drawn from a standard normal
X,
is
drawn from a lognormal
X2
X3
is
a
is
drawn from a unifonn
dummy
distribution
variable
distribution
Sample generated with 10% misclassi fied observations
Sample Design
-0.80
0.20
1.50
-0.70
0.106
0.096
/3o
0^
/3,
^3
ao
«i
N
MRC/IR
Probit
MLE
scaled
-0.539
-1.01
—
(.052)
(.137)
0.139
0.235
(.011)
(.026)
0.952
1.69
1.44
(.046)
(.164)
(.036)
-0.446
-0.729
-0.594
(.023)
(.060)
(.203)
assume q;o=0
0.117
0.116
(.015)
(.020)
assume a,=0
5000
40
0.200
0.111
0.128
(.037)
(.108)
5000
5000
A ppendix
Concavity
I.
The conditions
for a probit log likelihood function, with
argument
z, to
have a negative second
derivative are:
a(z^-l)
-
(l-2a)exp(-.5zV2T <
and,
(l-a)(z^-l)
For the
-
(2a-l)exp(-.5f)v/27r
<
logit function:
a-1
+
<
aexp(-2z)
and,
-a
+
(l-a)exp(-2z)
<
Derivative of Estimated Coefficients with respect to the Misclassification
II.
Parameter
The
log likelihood can be transformed:
E[E[l{^^,y)\X]
]
=
E[\nFix0)-pr{y = l\x,0„a,)
- \nil-F(x0))
-
pr{y=O\x,0„a,)]
Notice that the probability of observing a particular y value depends on the true parameters of
the model, ^q and ao. By collecting terms, the expression above can be rewritten,
= E^„^l{0^,y) - aE[(\rxF{x0) - lnil-F(x0)](l-2F(x0^)
The
derivative with respect to
Since
we want
to
with respect to a.
know how
Call the
is
then set equal to zero to form the usual
the optimal estimated
first
/3
term of equation (2)
varies with a,
J,,
we
first
(2)
order condition.
take another derivative
and the second term,
aJj,
and use the
product rule to calculate,
i^l
81380
Since
we
'^''^
.!l\
8a°"°
.^1 „.a^^l
d0da"'°
a^
are interested in evaluating this expression at
isolate the expression
of
„
'"•"
interest:
41
a=0,
(3)
=
the last term drops out
and
we
can
aa'-"
Note
term
that the first
likelihood without
3/3 3/3
equation (4)
in
misclassification.
is
3/3
simply the inverted information matrix for the
In the case
where
F(.)
is
the probit functional form,
equation (4) will equal:
which can be evaluated for different
X
distributions.
bounded if X has bounded second moments, a standard condition
Use a condition that (F7F) is bounded by c(l + z ) to show
change at a finite rate as a increases from zero.
Additionally, this expression
is
for consistency of probit estimates.
that the estimates
ni.
of
/3
will
|
j
Non-Block Diagonality of the Probit Information Matrix
(symmetric misclassification probabilities)
The
true log likelihood is equation (7)
L
from page four of the
(l-j)ln[(l-a)+(2a-l)#(X/3)]
= >'ln[a+(l-2a)<l'(X^)] +
The information matrix
d^L
pap)er:
(1)
off-diagonals are:
<i>{X&)X
^
a/3aa
<i>{X&)X
_
[a *
[(l-a)+(2a-l)#(A:/3)]^
(2)
{\-2a)^{X^)f
In expectation these off-diagonals are not equal to zero:
E\J!L_^
d&da
<^(A:g)X
-
(l-a)
+ (2a-l)«^(X/3)
42
0(X/3)X
_
a
+ (l-2a)*(X/3)
(3)
IV.
Identification of
Berkson Log Odds
with Three Classes, Misclassification, and the Logit Functional
Form
Define the following:
To
=
the probability of correct classification of an observed 0.
^20
a,o
X,
=
aoi
=
is
misclassified to be a 0.
1 is
misclassified to be a 0.
probability a 2
probability a
the probability of correct classification of an observed
Uii
T2
=
=
=
=
probability a 2
is
misclassified to be a
1.
probability a
is
misclassified to be a
1.
1
the probability of correct classification of an observed 2.
cKo2
a,2
=
=
probability a
probability a
The probabilities of observing any
1
is
misclassified to be a 2.
is
misclassified to be a 2.
particular
outcome must add
to one, resulting in the following
restriction:
T;
Then
+
Qjj
+
ajk
=
V
1
the probability of observing the
i
?i j
?i
outcome
k
(1)
can be expressed:
Y.e"^
Y.e"'
Y.'"'
;=0
;>0
;»0
Similarly, the probabilities of observing outcomes
Pr(y=l) = _!
+
1
and 2 can be expressed:
^'
°'
(3)
y-o
J-o
TT,.^^
Pr(y=2)
=
^
a^''
>-0
^
a^f^
(4)
.V
52
J'O
The
probabilities can be reinterpreted as shares
and any two can be written
43
in ratio
form.
The
and
log odds of ups
It is
1
will be:
straightforward to write out the expression for other pairs of ups, so they are not included
here.
The condition
results in overidentification
coefficients
of seeing any particular outcome must add to one
that the probabilities
of equation
one would estimate
(5).
Condition (1) places two restrictions on the six
in equation (5).
Proof of N"^ Convergence for Isotonic Regression when Index Parameters are
V.
Estimated with N"^ Convergence
Assume
V define
In(i')
of the jump.
we
Groeneboom
as the positive distance to the closest
We now
property of the
because
the regularity conditions of
MRC
jump and
s
define
use two facts to establish the proof: {\) p
']J,v)
-
estimator and (2) the total cumulative size of the
are estimating a cumulative distribution by IR.
Pr
For each n and index value
(1993).
(
^„(v) - F,(v)
I
I
^—-
f
y,(v)
-
n'l'
We
\
as the (vertical) size
=
Op(N'"^ from the
jumps cannot exceed
1.0
can then calculate:
^ e n-'^)
wiv)dv
(1)
^
^
n-'l'
where w(v)
is
the density of v
and
[
>'
44
—
—
/n(v)
w(y)dv
^ K.
Thus, because the
total
cumulative size of the jumps cannot exceed 1.0, either jumps are
relatively large but far apart, or
the bound.
To complete
„i/3
The
first
jumps are frequent but
the proof
[F(1^rv)-F(v)]
V
^j
'
we
Groeneboom
(1993).
-
Thus,
F{v)]
= n'/3
we
is
The
net result
is
we
can derive
calculate
[F(v)-F(v)]
•
)!' '
n^ '»
term on the right hand side, n"^ [F„( v
the second term n"^ [F„(v)
small.
) -
n^[Fn rv)-F(v)]
^
*
F„{i')] is
(2)
order Op(l) by equation (1) while
order Op(l) with the asymptotic distribution as proven by
can treat the estimated
the asymptotic distribution of the IR.
45
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