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Department of Economics
Working Paper Series
INFERENCE
ON PARAMETER SETS
IN
ECONOMETRIC MODELS
& Confidence Regions for Parameter Sets
Econometric Models," id# 963451)
(See recent revision called "Estimation
in
Chemozhukov
Han Hong
Elie Tamer
Victor
Working Paper 06-1 8A
June 16,2004
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Networl< Paper Collection at
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^
'•^'^^^^^i^f^LSI^'™''!
FEB
2 2 2007
"libraries
INFERENCE ON PARAMETER SETS IN ECONOMETRIC MODELS*
VICTOR CHERNOZHUKOVf HAN HONG^ ELIE TAMER*
Abstract. This paper provides confidence regions for minima of an econometric criterion function
Q{d). The minima form a set of parameters, Q/, called the identified set. In economic applications,
0/ represents a class of economic models that are consistent with the data. Our inference procedures are criterion function based and so our confidence regions, which cover 0/ with a prespecified
probability, are appropriate level sets of Qn{S), the sample analog of Q{9). When 0/ is a singleton,
our confidence sets reduce to the conventional confidence regions based on inverting the likelihood or
We show that our procedure is valid under general yet simple conditions,
and we provide feasible resampling procedure for implementing the approach in practice. We then
show that these general conditions hold in a wide class of parametric econometric models. In order
to verify the conditions, we develop methods of analyzing the asymptotic behavior of econometric
criterion functions under set identification and also characterize the rates of convergence of the confidence regions to the identified set. We apply our methods to regressions with interval data and
set identified method of moments problems. We illustrate our methods in an empirical Monte Carlo
study based on Current Population Survey data.
other criterion functions.
Key Words:
Set estimator, level sets, interval regression, subsampling bootsrap
Date: Original Version: May 2002. This
*
Note: This paper
and rates
is
currently undergoing
2004
version: June,
MAJOR
.
revision following referee
of convergence are added. Convexity assumptions
comments.
New
results
on estimation
on the parameter space are being replaced by more general
conditions.
The most
*We thank
Alberto Abadie, Gary Chamberlain, Jiuyong Hahn, Bo Honore, Sergei Izmalkov, Guido Imbens, Christian
recent, yet preliminary
and incomplete, revision can be requested via e-mail. September, 2005.
Hansen, Yuichi Kitamura, Chuck Manski, Whitney Newey, and seminar participants at Berkeley, Brown, Chicago,
Harvard-MIT, Montreal, Northwestern, Penn, Princeton, UCSD, Wisconsin, and participants
of Econometric Society in San-Diego for
'
Department of Economics, MIT, vchernSmit
0241810)
§
is
Winter Meetings
.
edu.
Research support from the National Science Foundation (SES-
gratefully acknowledged.
Department
of
Economics, Duke University, han.hong8duke.edu. Research support from the National Science Foun-
dation (SES-0335113)
J
at the
comments.
Department
of
is
gratefully acknowledged.
Economics, Princeton University, tamerSprinceton.edu. Research support from the National Science
Foundation (SES-0112311)
is
gratefully acknowledged.
1
Introduction and Motivation
1.
Parameters of interest
econometric models can be defined as
in
minimize a population objective or criterion function.
at a particular
If this criterion
tJiose
parameter vectors that
function
parameter vector, then one can obtain valid confidence regions
is
minimized uniquely
(or intervals) for this
parameter using a sample analog of this function. Likelihood and method of moments procedures are
two commonly used and well studied methods
in this setting.
inference to econometric models that are set identified,
minimized on a
identified set
set of parameters, the identified set.
i.e.,
Our
This paper extends this criterion-based
models where the objective function
goal
is
to
make
is
inferences directly on the
and to provide a method of obtaining confidence regions with good properties (such as
consistency and equivariance) that cover the identified set with a prespecified probability.
Our
point of departure
analog Qn{0) where 9 e
where every 9
paper
this
is
in
is
@
a nonnegative population criterion function Q{0) and
czW^. The identified
9/ indexes an economic model that
lim
for a prespecified confidence level
Qn
a e
for
can be defined as 0/
A
9/ from the
level set
{6>
consistent with the data.
is
^(9/ C C„) =
(0, 1).
=
level sets of
e
9
The
sample
Q{e)
:
=
0}
objective of
Qn{9) such that
a,
Cn{c) of the
finite
sample criterion function
defined as
Cn{c) := iO
for
C„
to construct confidence sets
(1.1)
is
set
its finite
:
Qn{d) ~ Qn
some appropriate normalization
Cn = Cn{c) with
the cut-off level c
<
a„.
—
Ca,
c/a-n \,
where g„ :=
inf
Qn{0) or g„ :=
0,
In order to obtain the correct coverage (1.1),
we choose
where c^ equals the asymptotic a-quantile of the coverage
statistic:
Cn := sup an{Qn{d)
which
is
a quasi-likelihood-ratio type quantity.
properties, such as consistency
situations,
where the
and equivariance
identified set
is
-
g„),
The constructed confidence
to reparameterization.
defined as the
minimand
sets possess
Our approach
important
covers general
of an objective function. In addition,
our confidence regions are sets that are robust to the failure of point-identifying assumptions in that
they cover the unknown identified set - whether
These confidence
where the
We
it is
a set or a point - with a prespecified probability.
sets collapse to the usual confidence intervals
identified set
9/
is
show that the above
based on the likelihood ratio
in cases
a singleton.
level set
Cn[ca) has correct asymptotic coverage, where c^
is
the ap-
propriate quantile of a well defined coverage statistic C, the nondegenerate large sample limit of
Cn-
Then, we provide general resampling methods to consistently estimate the percentile
Cq.
The
coverage and resampling results hold under general (high
level) conditions.
econometric models, we then show that these conditions are
satisfied. In
Focusing on a class of
the process,
acterizations of the stochastic properties of the criterion functions process, an{Qn{-)
the fact that the population criterion
Q
is
minimized on a
set rather
we provide
— g„),
char-
exploiting
than a point. Furthermore, we
obtain the asymptotics of the coverage statistic C„ and of the level sets Cn{c), focusing on coverage
and speed
of convergence of Cn{c) to 9/.
The paper then
considers two important applications:
We
regression with interval censored outcomes and set-identified generalized method-of-moments.
finally illustrate
our methods in a Monte Carlo study based on data from the Current Population
Survey.
In the last decade, a growing body of literature has considered the problem of inference in
partially identified models,
(2003).
i.e.,
models where parameters of interest are
While most of the work,
e.g.
is
paper
is
cf.
Manski
Horowitz and Manski (1998) and Imbens and Manski (2004),
exclusively focuses on the case where there
this
set identified,
is
a scalar parameter of interest that
lies in
an
interval,
concerned with inference on vector parameters in problems where the identified set
defined via a general optimization problem, as in the economic problems described below.
multivariate set
is
This
usually not an interval (or cube) and can be a set of isolated points or manifolds.
Moreover, the methods used in the interval case are based on estimating the two end-points of
the interval. Hence they are not applicable in the general set-identified case, for which a different
approach
is
needed.
The main motivation
examples.
for posing the identified set as the object of inference
In Hansen, Heaton, and Luttmer (1995), the identified set
models that obey the pricing-error and
models with multiple
is
equilibria supported across markets or industries.
(2003).
Another example
is
E
through a
birth).-'
Z
flexible,
is
There
is
no single "true" equilibrium that
is
example where potential income
Y=
ao
+ aiE
-\-
a-iE'^
Y
+
is
related to
e.
Although
not point-identified in the presence of the standard quarter-
suggested in Aiigrist and Krueger (1992) (indicator of the
In the absence of point identification,
some
In
the set of parameters that describe different
quadratic functional form,
the instrumental orthogonality restriction
In
market returns.
vary across observational units; see Ciliberto and Tamer
in the following
parsimonious, this simple model
of-birth instrument
many
the structural instrumental variable estimation of returns to schooling.
Suppose that we are interested
education
may
motivated by
a subset of asset-pricing
volatility constraints implicit in asset
equilibria, the identified set
played, since particular equilibria
is
is
all
first
quarter of
of the parameter values (qqi 0^11^2) consistent with
E{Y —
ao
+
Qi-E
-f a2£^'^)(l,
Z^ =
are of interest for
cases the indicators of other quarters of birth are used as instruments. However, these instruments are not
correlated with education (correlation
is
extremely small) and thus bring no additional identification information.
3
moment and
purposes of economic analysis. Similar partial identification problems arise in nonlinear
instrumental variables problems, see
e.g.
Literature. To our knowledge, the
models can be found
in the
Demidenko (2000) and Chernozhukov and Hansen
earliest
work
in econometrics
paper of Marschak and Andrews (1944).
on parametric
(2001).
set identified
There, the identified set
is
the collection of parameters representing different production functions that are consistent with
the data and functional restrictions the authors consider.^ Gisltein and Leamer (1983) provide set
consistent estimation in a class of likelihood models where
0/
as the set of parameters that are
robust to misspecification. Klepper and Leamer (1984) generalize the Frish bounds to multivariate
measurement
regression models with
errors.
On
the other hand, Hansen, Heaton, and Luttmer (1995)
provide consistent set estimates of means and standard deviations in a class of asset pricing methods.
Manski and Tamer (2002) provided conditions under which an appropriately defined
estimates the identified
about the
identified set.
However, these consistency results do not contain a method
set.
To the best
of the paper
is
for inference
of our knowledge, there are no results in the literature that deal
with the general problem of obtaining confidence regions
The remainder
set consistently
for
organized as follows.
parameter
sets.^
Section 2 provides a general theory of
inference on the identified sets based on general criterion functions. Section 3 focuses on a class of
regular parametric models, provides verification of the regularity conditions posed in Section
provides additional results that pertain to regular cases. Section 4 provides a
of the methods,
and Section
2.
forms the basis
General Set Inference
for the rest of the analysis.
We
first
in
Large Samples
this section
The
Wn), where data {Wi,
criterion function
0/, the identified
which
0/ and formalize some
...,
W„}
are defined
is
based on a criterion function Qn{S)
on some common probabihty space
Qn{0) converges to a continuous criterion function Q{9), that
(Basic Setup). Criterions (5„
K*^
:
— M+
»
and
For a good description of Marschak and Andrews (1944), see Chapter
The only other paper known
to us
is
that can be estimated.)
The problem
is
interval-identified
of inference
Imbens and Manski (2004) and
Q
III of
:
E'^
—
>
E"*"
and
is
(fi,
=
F, P).
minimized at
in
about
9'
Appendix G.
4
is
(i.e.
satisfy
Nerlove (1965).
Imbens and Manski (2004). Their paper considers a
inference about the a real parameter 6' that
in
result
set.
Assumption A.l
shown
we present our main
define the identified set
For given data, the inference about the parameter set 0/
...,
Monte Carlo evaluation
be used throughout.
definitions that will
Qn{d, Wi,
and
5 concludes.
Generic Inference on Identified Sets. In
2.1.
2,
different
problem of
contained between some upper and lower bounds
fundamentally different from inference about G/, as
e
i.
is
Q{9)
ii.
continuous and Qn{d)
is
=
iii.
0/
iv.
QiOj)
=
V.
Qn{0)
- Q{B) =
argminege[Q(0)]
a
is
when 0/
hyperplanes)
is
eQ.
states a standard compactness
finite
It also
as the minimizer of the limit criterion function Q.
0/
defines
and convexity assumption, which are important
union of compact connected
a
finite
sets,
is
a
an assumption that serves to organize
union of isolated points and the
finite
union of compact sets with boundaries defined by manifolds (nonlinear
The assumption
.
a finite union of connected compact subsets ofQ,
is
This covers both the case when 0/
the presentation.
case
lower-semicontinuous,
is
Op(l) for each 9
to the subsequent analysis.
region 0/
BE RELAXED],
to
for each Oj £ 0/,
Assumption A.l
The
[CONVEXITY
o compact and convex subset ofR''
that Q{di)
Q
convergence condition serves to relate
=
The pointwise
serves as a convenient normalization.
as the limit of the finite
sample objective function. The
pointwise convergence will be strengthened later on.
Lemma 2.1
shows how to construct a
level set of the
sample objective function that
The
provide the proper inferential statement (1.1) about 0/ in a generic setting.
objective function
Q„
is
later.
is
Note that choosing g„
Example
e.g.
-.an (Q„((9)
-
g„)
<
c[, where g„ := inf Q„(6') or g„ := 0,
defined in Assumption A. 2 below.
non-empty, though
1
in
c-level set of
given by
Cnic) := \e
where a„
will eventually
=
Typically a„ equals n in the regular cases studied
infeeeQn(^) guarantees that the confidence region Cn{c)
some cases one has inf^ge Qn{9)
=
is
always
with probability converging to one, see
in Section 3.
Consider the following coverage index
fo
P{c)=
1")
^
c
- snp an {Qn{9) -
The
0/ C Cn(c)
sign of the index p(c) indicates whether
such that 9 ^ Cn{c), we have a„ {Qn{9)
index
is
also linear in
property
is
c,
which
—
>
q-n)
will allow us to
c
or not.
For example,
which implies that p{c)
have data-dependent
<
0,
if
there
and vice
cut-off' levels c.
£ 0/
is
versa.
The
Another desirable
invariance of the index p to parameter transformations which implies equivariance of Cn{c)
For instance, a one-to-one transformation of parameters 9
to parameter transformations.
changes the level set
in the equivariant
{t{9)
The
Qn)
eee,
'
following
:
a„ {Qn{T-\T{e)))
lemma summarizes
For definition of manifold, see
e.g.
way
the discussion.
Milnor (1964)
.
-qn)<c}=T {Cn{c))
—
»
t{9)
Lemma
p{c)
Assumptions A.l-(ii) and
2.1. Let
^
<
0/ 2 Cn{c)
cut-off level;
and p
the coverage index
III.
>
(c)
Then,
A.l-(iii) hold.
•^
9/ C Cn{c);
II.
the coverage property holds:
I.
the coverage index
is
linear in the
invariant to re-parameterization; and IV. the level sets are
is
equivariant to bijective parameter transformations.
Next we
state the
Assumption A. 2
main condition that enables
(Coverage
large sample inference
Suppose that there
Statistic).
on 0/.
sequence of constants a„
exist a
^
oo
such that
Cn=
where C
is
a nondegenerate
we explain how
In the sequel
interest,
random
where a„
=
n.
We
sup an{Qn{Sl)- Qn) -^d C,
variable.
this
main assumption
methods
also provide
new asymptotic methods
Assumption A. 2 leads us
attained in suiRciently regular models of
for verification of this
the limit variable C. Verification of Assumption A. 2
set of
is
is
a
difficult
assumption and
for finding
matter and requires developing a
of dealing with criterion functions under set identification.
main theorem, which provides a generic
to the following
on
result
inference in set-identified models.
Theorem
2.1 (Generic Inference on Sets). Suppose Assumptions A.l
continuity point of distribution function of C such that
p{Cn{ca))
I.
The
result follows
Op(l) -^d
C —
Ca-
P{ei C
—»d Ca-
immediately from
C and
lim
II.
Lemma
2.1
P{C <
and A. 2.
C„(Ca)}
=
P[p{Cn{Ca))
Then for any Cq -^p
P{ 0/ C C„(Cq)
First, p{Cn{c))
\
=
=
a
Cq,,
a.
€„
> O} = P{ca -
C„
>
= -P{C„ <
c„
+ Op(l)} = P{c <
some
criterion function.
to a singleton {9i} so that the coverage statistic
=
a.
is
—
Ca
=
Cn
—
Ca
+
O}
c„}
+ o(l),
C.
from Theorem 2.1 that our method of constructing valid confidence sets builds on the
classical principle of inverting
Cn
=
that Ca
Second,
provided Cq a continuity point of distribution function of
It is clear
Cq}
and A. 2 hold and
On [Qni^l)
—
Qn)
,
which foUows well known
involved quantity, being the
supremum
Indeed, in point identified cases 0/ reduces
becomes a standard likelihood
ratio type quantity
limit laws. In the general case, the statistic
over the elements of 0/: C„
=
supg^^Q^ a„ {Qni^l)
is
a more
—
qn)
In practice, our procedure for constructing the confidence regions critically depends on being
able to consistently estimate Cq, the a-quantile of the limit variable C. In
6
all
examples that we study,
C
non-standard and
is
we
distribution depends on 0/. Despite this problem,
its
show how
will
to
obtain a consistent estiniate of Cq by the following method.
C„
We
Feasible Inference.
2.2.
=
sup^gQ a„ {Qn{0)
—
first
Since
Qn)
6/ =
(2.2)
where
A;
we can
is
need to obtain an approximation to the sampling distribution of
we do not observe 9/, we
where
C„(fc),
G
fc
[ci,C2]
•
wp —
Inn
=
0.)
The
result proven
initial estimate'
first class
of our examples,
it
1,
>
a possibly data-dependent starting value of the cut-off. (In the
use th starting cut-off k
by an
will replace
below suggests that the asymptotic vahdity
of the procedure will not depend on the starting value. In finite samples, the choice of the starting
value
may
be important, and we discuss
in Section 4.
it
Consider the following subsampling algorithm:
For cases
1.
cases
when data {Wi}
when {Wt} form a
of the
form {Wi,
are
construct
iid,
For each j
=
...,Wi+i,-i}.
1, ...,
(Bn) subsets of size
stationary time series, construct
5„ = n —
6
^ n of the data.
b
+
I
For
subsets of size b
one can use a smaller number B„ of randomly
(In practice,
B„ —
chosen subsets under the condition that
2.
all
»
oo as n
—
>
oo.)
Bn, compute
Cj,b,n
=
sup
-
ab {Qj,biO)
qj,b)
eeCn{c)
where Qj^ :=
if
and Qj^ := inf^ee
g„ :=
(3j,6((9)
otherwise; Qj,b{d) denotes the criterion
function defined using the j-th subset of the data only.
3.
Let Cq be the a-th quantile of the sample {Cjfi^n,j
4.
(Optional)
As commented
times using c
=
in Section 4.3,
Ca Inn.^ (Any finite
=
l,
, Bn}.
one could repeat steps 2 and 3
number
finite
number
of
of repetitions produces the consistent estimate
of Cq.)'^
We
require that as n
>
b/n
(2.3)
The
—
choice of
b
oo,
—
>
0,
The adjustment
as
oo for
all
o
—
>
oo,
and other practical aspects
To guarantee asymptotic vahdity
n —
i?„
>
factor
fe
—
oo
>
at polynomial rates.
of the procedure are discussed in Section 4.
of the above procedure, the following assumption
Inn can be replaced by
In
Inn
or
is
needed.
any other m{n) such that ni(n) —> oo and m{n)/n''
—
»
0.
The adjustment by In n is not needed in the first class of our examples.
In fact, we found in Monte-Carlo experiments described in Section 4 that
just
two repetitions worked the best
terms of coverage and computational expense. This was also confirmed by Bajari, Benkard, Levin (2003).
7
in
Assumption A. 3 (Sandwich
n—
>
oo.
For k €
[co,ci]
9/ C
-Inn,
C„(fc)
C
Property). Suppose that h/n
lup
—
»
—
+
+
:= {dj
t
<
||i||
:
en,0i £
ab{ sup Qb{9)
6;", such that
Cn ~*
0/} and
Assumption A. 3 guarantees
b
—
oo ai polynomial rates as
>
1
is
- sup
Qb{0))
=
Op(l),
See,
eeQ]"
QY
and
a sequence of positive constants.
inferential validity but also allows us to establish consistency
characterize the rate of convergence of the level sets C„(A;) to the identified set 0/, as
in
Theorem
The
2.2.
know 0/, hence we
intuition behind A. 3
replace
it
is
by an estimate Cn{c). The replacement should have only a
impact on the distribution of the coverage
Wald
it
3
Wald
9j,
subsamples.
We
show how
negligible
Romano, and Wolf
know
(1999), p 44, in the
the true 9j and replaces
effect
on the distribution
to verify A. 3 for parametric models in Section
The next theorem summarizes our
3.2).
we do not
similar in nature to the
is
hence requiring that this replacement has negligible
statistic in
(Theorem
Politis,
inference in point identified cases. There, one does not
with an estimate
of the
subsamples. A. 3
statistic in
polynomial rate of convergence assumption used by
case of
shown below
In the subsampling bootstrap,
as follows.
and to
results for general inference in set identified
models.
Theorem
are iid or
I.
2.2 (Consistency and Validity of Inference). Suppose that the estimation data {Wi,i
form
a stationary strongly -mixing sequence
Then for
Ca
—
*
and
that A. 2
the subsampling algorithm defined above, provided
Ca,
and A. 3
P{C <
c}
<
n}
=
Ca,
hold.
continuous atc
is
and
P(0/eC„(ca))-^a.
II.
The
set
Cn{k) for
/c
=
Calnn
is
consistent under the Hausdorff metric:
dH{Cn{k),Qj) <
Recall that the Hausdorff metric between two sets
dH{A,B) :=ms^[h{A,B),h{B,A)],
rate e„ will
set at the rate Cn
be shown to be essentially l/^/n
Under the given
The conventional
level of generality,
is
for
^>
1
0.
defined as:
is
where
Thus, under general conditions our inference method
Cn{k) that conv^erge to the identified
^
en
wp
h{A, B) := sup
mi
\\a
-
b\\.
asymptotically valid and delivers level sets
with respect to the Hausdorff distance (the
parametric models).
subsampling appears to be the only valid resampling method.
(n-out-of-n) bootstrap will not be generically consistent in the present settings.
One counterexample
is
as follows: In Section 3,
one of the leading examples, the partially identified
linear
IV
regression, necessarily involves a parameter
bootstrap
fails in
on the boundary problem.
parameter-on-the boundary problems,
we
Andrews
known
that the
(2000).
Regular Parametric Case and Applications
3.
In this section
cf.
It is
establish the
methods
for verifying the
main conditions required
for
implement-
ing the approach, namely that existence of the limit distribution for the coverage statistic (A. 2) and
the sandwich condition (A. 3).
and
illustrate the the
We
develop these methods to cover a variety of parametric examples,
approach with applications to regression models with missing outcome data
and generalized method of moments under
we
partial identification. For parametric models,
establish
further properties of the confidence regions, such as the speed of convergence of the level sets to the
identified set,
and the stochastic properties of objective functions
Examples
3.1.
of Parametric
Problems with Set
in partially identified cases.
Example
Identification.
I.
Regression
with Interval-Censored Outcomes.
Consider the linear conditional expectation models
Ep^[Y\X]
The models
are not
assumed
=
X'9, where
observation on
eR"^.
to be correctly specified, in the sense that there
£'pg[y|X] agrees with £'p[y|X] under the actual law
Observed data consists
9ee,X
of
i.i.d.
P
may be no
6 such that
of the data.
observations (Yu, Y2i, Xi) where
Yu and
Y2i represents the interval
Yf.
Yi
(3.1)
e [Yu,Y2i] given Xi,
a.s.
In the absence of further information, the set
ei = {eeR'^
(3.2)
is
the object of interest, as
consistent with data.
it
:
E[Yi \X]
< X'9 <
E[Y2\X]
a.s.
}
represents the set of hnear conditional expectation models that are
We assume
that
9/ C 0, where
9
is
a compact subset of
R"^.
Observe that 9/
minimizes the objective function
(3.3)
where [u)%
= {uf
Q{9)
=
>
0)
x l[u
Using a sample analog of
(3.4)
Qn{s)
j [{E[Y,\x]-x'9)l +
and
(u)?.
= {uf
x l[u
<
{E[Y2\x]-x'0)l]dPix),
0].
Notice also that 9/
=
(3.3),
= IY1 {MyiA^i] - K0)l +
i<n
{e[Y2^\x,]
- x[9y_
,
{61
G
:
Q{9)
=
0}
.
Manski and Tamer (2002) characterize consistent estimates
no method
of 0/.* However,
We
ence about 0/ in the sense of providing the inferential statements was given.
confidence approach to inference about
Example
Moment
Structural
II.
terested in deducing the set of
all
moment equation computed with
data.
The data
{fl,J^,P).
<
{Xi,i
0/
and similar examples.
in this
Equations. In method
moments
we
settings,
R'^
are in-
that satisfy the
respect to the probability sampling distribution of the economic
dj of interest are
assumed to
E{mi{e])]
m{9, Xi)
of
economic models indexed by a parameter 9 £
(3.5)
=
provide below a
n) are stationary and strongly mixing, defined on the probability space
The economic models
where mi{d)
for infer-
is
=
satisfy
0,
a lower-semi-continuous function
in 9 a.s.
The
entire set of
models 0/
C
that solve (3.5) also minimize the criterion function
Q{9)
(3.6)
where Q{9)
to nonlinear equations (3.5)
Demidenko
conditions
E[mi{9)]'W{9)E[mi{9)],
continuous for each ^ € 0, a compact subset of W^, and W{9)
is
positive definite matrix for each 9
e.g.
=
general be minimized on a
where Wn{9)
is
In nonlinear models, the existence of multiple solutions
more of a
rule rather than an exception, except in special cases, see
mentioned
set.
Qn{e)
(3.7)
=
The
may be
multiple solutions as
in the introduction.
inference on
Thus, the
GMM
weU
0/ may be based on the usual
n[gn{e)]'Wn{e)[gn{9)],
5n(^)
for inference
about 0/
if
the usual rank
function (3.6) will in
GMM function
= - f^m^(e),
a lower-semi-continuous, uniformly positive definite matrix.^
knowledge, no methods
a continuous and
Q.
E.
(2000). In linear models there
to hold, as
fail
is
is
in the sense of providing
To the
best of our
Neyman-Pearson confidence
statements has been yet given in such settings.
3.2.
General Asymptotics of Criterion Functions in Parametric Models. The prime
of this section
is
to provide primitive conditions
a wide variety of parametric models.
and
tools that help verify conditions A. 2
goal
and A. 3
in
The
tools will be illustrated using the examples stated in the
{5 £
9
next section.
Specifically,
if
£n
>
they show that the set
0„ =
;
Qn(d) < mine Qn{0)
+
in} converges almost surely to the set
9/
converges to zero at a rate strictly slower than the rate at which Qti(-) converges uniformly to Q(-)- They use
a Hausdorff metric as a distance function between
For example,
it is
sets.
convenient to use the continuous updating type weight matrix Wn{6),
consistent estimate of asymptotic variance of n~''^
^"_j mi (9).
10
i.e.
to let
Wn{0) equal a
Since Qn(^) approaches Q{0),
borhood
of 0/.
The
we should be
able to
that 9
tell
^ Qj
\i
6
is
outside
some neigh-
neighborhood depends on the rate at which the boundary of 9/ can
size of this
be learned. In the remainder of the paper, the rate we consider
the parametric rate and hence the
is
points of uncertainty are those 6j that are within a 1/ \/n neighborhood of the true set 0/.
Such
points are of the form
+
Oi
In order to keep track of such points
central interest
is
The
over a suitable domain.
dQj. Given
6j
eQi,Xe
for Oi
M'^.
will suffice to record all pairs of the
it
form {9j,\). Hence of
the local empirical process
(^/,A)
set
XI Vn,
^ in{9i,\) ~ n(Qn(.6i + \l^)
domain
pertinent
Vn(ei) := {A e R''
:
for
A
+ A/Vn'
6>/
Q{0j)),
be shown to be the boundary of identified
for dj will
£ dQi, the pertinent domain
-
is
€ 0,
for all n'
€ [n,oo)}.
In addition, the following limit version of Vn{dj) will play an important role:
V^{9l) := {A G
R'^
:
6'/
+ A/Vn
G
for all sufficiently large n},
(3.8)
where when 9i^ int(0),
V^{9i)
=
R'^.
Thus, V^{9j) plays the role of the limit local parameter space relative to
interior of the
parameter space,
parameter space,
i.e.
9j
i.e.
9i
£ int(0), V^{9[)
R'^.
When
9i
is
When
9j
is
in the
on the boundary of the
S 50, the local deviations A should be constrained to the local parameter
space V^{9j) of the specified form. This situation
case, as characterized in
=
9j.
Andrews
in the partially identified
models
is
similar to the one arising in the point-identified
(1999). Unlike in the point-identified case, the
is
more
an exception.
of a rule rather than
boundary problem
It arises
even in the
simplest leading cases, as will be seen in Section 3.3.
Another important
set
is
the subset of the local parameter space Vn{di) where the local deviations
A are constrained to be towards the interior of the identified
A„(6i/) := {0}
UiXeR"^
-.91
+ X/Vn'
set:
€ int(0;)
for all n'
In addition, the following limit version of An{9i) will play an important
Aoc(^/) := {0}
U{XeR''
Note that since int(0/) C 0,
is
A„(6'/)
-.91
is
+
A/i/n e int(0/)
11
[n,
oo)}.
role:
for all sufficiently large
a subset of Vn(0/), which
also a subset of V^{9j).
G
is
n}.
a subset of V^{9j), and A^{9i)
Given above
=
Cn
^
definitions,
i
shall obtain the following representations of the coverage statistic
Qn)
supe^ge, 4(6'/, 0)
en{9i,0)-Mgj^^/^^Q
\ snpg^^Q^
^
-
sup n (QniOl)
f
we
s^Pe!ede,,\eAn(ei) ^n{Oi, A)
\ ^^'Pe,eaeI^e^r^{e,) ^n(^/. A)
In this representation, the
first
+
-
en{0i,\)
if
g„:=0,
if
Qn .=
mUeQ Qn{e),
Op(l)
if
0,
ifng„74pO.
+ Op(l)
infe,ee,,Aev„{9,) 4(6*/, A)
ng„ -*p
equality follows by definition of the local empirical process £„(^, A),
To guarantee non-
while the second equality emerges from the analysis given in the appendix.
degenerate asymptotics for this
statistic,
we
will require that £n{Oi,X)
converges to a sufficiently
well-behaved random element looi^i, A), in the finite-dimensional sense coupled with some uniformity,
which we
will call the
This form of convergence
quasi-uniform convergence.
will
be
sufficient to
establish that
^
_^
^ .^
\ s^Pe,ede,,xeAocie,) ^oo{di,X)
The
{9j
1
—
-
inig,^Q,^xev^{e,) ^oo(6'7,A)
if
ng„ y^p
0.
following conditions ensure that this convergence takes place.
Assumption C.l
0/
ifng„^pO,
s'^Pe,ede,,\eAooie!) ^oo{0i,\),
\
^
dQi, for any
G int(0/)
e.
:
When
(Degenerate Asymptotics on the Interior).
e
inigi
>
there exists large
^qq^
Since in{Si, A)
>
\\dj
—
0, this
0'j\\
>
enough S
>
and
int(07)
not empty, so that
enough n such that for
large
=
5/^/n}, supg^g/^^^j Kn(^/,0)|
is
/„((5)
:—
with probability no less than
implies that
nqn
=
in(Or,X)
inf
=
wp
^
1.
eiee,MVn{ei)
Condition C.l
is
motivated by the fact that in interval regression examples
£n{9i,X)
We
=
wp
->
1 for
any fixed
di
for large
n
G int(0/) and A G W^.
provide examples and further discussion of this interesting phenomenon in Section
3.3.
Condition C.2 puts further convergence conditions by appropriately matching the large sample
behavior of function in{0,\) with that of some function ^oo(^,
A),
which we
limit of in{9i,X). In order to state the condition, define the following
u„(A) :=
sup
ln{di, A)
and
e,edei
/ri(A)
:=
inf
in{9j, A) for
is
the quasi-uniform
key functionals
n < oo and n
=
oo.
e,edej
Assumption C.2 (Quasi-Uniform Convergence Near Boundary).
K
call
any compact subset ofW^. Then
12
Let 9j G
50/ and A G K, where
i.
£n(^/)A)
>
which
continuous in X for each
(a)
ii.
is
is
ifQi
converges weakly in finite-dimensional sense to some function ioo{Si,X)
=
stochastically equi- continuous; ifQj
to
0,
dj.
50/; u„(A) converges weakly
7^
>
Uao{X) in finite- dimensional sense, and (b) Un{\)
to
dQj, then
(a)
(uca{X),looW) in finite-dimensional sense, and
jointly converge weakly
{un{X),lnW)
(un{X) and ln{X) are stochastically
(h)
equi- continuous.
supg^^sQ^^xeAo^ie,) ^oc (Oi,
iii.
<
>^)
oo
a.s.
Condition C.2 extends the standard conditions used to derive the asymptotics of criterion func-
where 0/
tions in the point-identified case,
(1994).
Although asymptotics
near the boundary
is
be non-degenerate.
to
and
for verifying A.S. C.2-(i) requires that
converges in finite-dimensional sense to some limit ^co(^,
inf transformations of £„(^/,A) over
imposes the conditions required
C.2-(i.-ii.)
for obtaining the limit distribution of coverage statistics
•^ra(^i-^)
Newey and McFadden
e.g.
degenerate, as condition C.l states, the asymptotics
in the interior is
assumed
a single point Oj, see
is
90/, denoted Un{X) and
C.2-(ii) requires that
'^)-
sup and
ln{X), converge in finite-dimensional
sense to respective sup and inf transformations of ^oo(^/>-^), denoted as Uoo{X) and looiX).
and
requires that u„(A)
hmit coverage
Note that
Z„(A) are stochastically equicontinuous.
statistic C, since snpg^ ^qq^ xeAa^idi) ^oo (^7) A) will
in C.2-(ii),
it is
K
is
is
be one of the components of
stochastically equicontinuous over
fails
interval-censored outcome), while
it
Assumption C.3
least as large as
uniformly in {Oj,X) such that iy{6j,X)
positive constants Ci
is
and S
of interest (the regression
holds in others (the generalized
(n{di, A)
Condition C.3
some cases
to hold in
(Local Quadratic Bound). For any
such that with probability at
I
—
C.
•
•
of
model with
moments).
depend on
i>(6i,X)
5f
=
inf^'ge^
||#/ -H
Xj \fn
~
&|||,
for set estimates,
and along with C.l and
C.3 extends, to the
standard conditions needed to obtain the rate of convergence of extremum
Theorem
13
for
e.
C.2 allows us to verify the main previous high-level conditions A. 2 and A.S.
timators in point-identified case; see conditions in
K
0, there is a sufficiently large positive
min[;y(6'/, A),
> K/^/n, where
that do not
>
method
C.2-(ii).
for large enough n
e
> Ci n
e
needed to obtain rate of convergence
set-identified case, the
the
90/ x K,
any compact subsets of K'^. This stronger and simpler condition certainly implies
However, this stronger condition
some
C.2-(iii) insures tightness of
possible to require that
(6 J, A) H^ in{9i, A)
where
It also
5.52 in
Van der Vaart
(1998) and
es-
Newey
and McFadden (1994),
Theorem
p. 2185.
3.1 (Limits of C„). Suppose A.l and C.1-C.3 hold. Then A. 2 holds, in particular,
\ supe,eae/,AsAoo(e/) ^oo{Oi,X) -in{e,ee,,\ev„{0,) ioo{Oi,X),
—>p
where nqn
Theorem
may be
is
necessarily occurs if qn '=
3.1 verifies
easily checked in
Assumption A. 2. The theorem
dQi and
inf
in Q.
under a
statistic attains
set of conditions,
which
a limit distribution, which
and sup transformations of the quasi-uniform
limit foo(^/,A)
we would
like to
know
certain properties of the level sets.
3.2 (Rate of Convergence and Sandwich Property). Suppose A.l and C.l
C
Then, for some positive constants
I.
non-empty
the local parameter spaces A^{9j) and Voo(^/)-
In addition to this basic result,
Theorem
is
states that
examples of interest, the coverage
determined by the appropriate
over
or i/int(0/)
ifnqn/*pO
For any k
—p
oo such that k
-
C.3 hold.
and C'
=
Op{n),
Qj C Cn[k) C G^" wp
-^
1,
where En
= C yk/n,
so that
< C \Jk/n wp
dH{Cn{k),Qj)
II.
For any k G
[co,ci] In?!, the
Qj C
and, provided &
—
>
C„(fc)
C
9;" wp -^
^
at
sup Qbi9)
3.2 verifies
Assumption A. 3
vergence of the level sets to the identified
metric,
is
1,
where Cn
polynomial
- sup
=
C
hin/y/n,
rate,
Qb{9))
=
Op(l).
6ee,
eeS;"
Theorem
1;
sandwich condition A. 3 takes place,
oo and b/n
6(
->
for
parametric models and characterizes the rate of con-
set.
The
essentially Xj \fn.
14
rate of convergence, as
measured by Hausdorff
Analysis of Regression with Interval-Censored Outcomes.
3.3.
Xi
=
1
and suppose E\Yi]
<
£[¥2].
Then Qj =
{9
<0<
E[Yi\
:
First consider the case
£[¥2]}, that
is
9/
=
when
[£;[yi],£;[y2]]
Then
Qn{9)
=
s2
[Y^-eY_^
„n2
/,-,
,
+ {Y2-ey_
and
Suppose that
V^iXi - EYi,Y2 - EY2)' ->d {Wi, W2)'
~ A^(0, fi).
Then
Udi, A) =
and the finite-dimensional
^
wp
limit of
{ln{EYr,y), in{EY2,\))'
is
Therefore the finite-dimensional limit of £„(^,A)
^oo(^/, A)
Theorem
in{di,X)
= [Wi -
3.3 verifies C.1-C.3 for this
.
G {EYuEY2),
1 if 0/
\)li{ei
=
is
((W^i-A)2_,(Ty2-A)2_)'.
given by
EYi)
+ {W2 - xtnei =
example so that
-^00(^7,
A)
is
EY2).
also the quasi-uniform limit of
This implies by Theorem 3.1 that
Cn^d C=
sup
= max
£oo{Oi,X)
[{Wi)l
{Wit]
,
.
e/€9e,,A6Aoc(«'/)
One can
use the above distribution to obtain the critical values and corresponding level set with the
required coverage property.
Before proceeding to a more general case, notice that the inferential strategy developed in this
simple example can be used in other problems where 0/
F\
<9 <
is
is
interest. All
G
such that
is
a set of
one needs to do in this
class
to derive the joint asymptotic distribution of the sample analogs of the endpoints of
the interval of interest and obtain via simulation the critical values for the
variables, as outlined
above
for the case
when Fj = EYi and F2 = EY2.
Getting back to more general settings, suppose that
(3.9)
all
F2 where Fi and F2 are functionals of the distribution of the observed data. This
models that provide interval hounds on the parameters of
of models
defined as the set of
X
£ {xi,...,xj}
15
P-a.s.,
maximum
of two
random
where the
0/
is
first
component
determined by the
It is
X
Condition (3.9) assumes that
is 1.
eM."^
>
x'^e
:
assumed that 0/ C 0, where
X
n
and
[xj)
x'^d
<
t^ {xj)
a compact subset of
is
(3.11)
<
:= {xj,j
for all j
90/ =
and int(0/)
is
empty
Define ti{x)
=
The
identified set
{Oi e
0/
0/
in R'^, so that
E{Yi\x) and T2{x)
"
(3.13)
=
n{xj)
=n
x'^ej
:
=
if
J}.
J) has rank d,
{xj) or x'jOj
= 50/
<
and that the d x J matrix
IR"^,
which rules out redundant parameterization. Then the boundary of the
(3.12)
discrete.
is
set of inequalities:
0/ := {e
(3.10)
X
of
=
and only
ti (xj)
if
some
for
,
=
ti{xj)
identified set
j
<
T2{xj) for
dQj
is
J},
some
j.
E{Y2\x) and consider the objective function
'=1
E
^^'
=
fc
i>2,
j
=
l,...,J.
i:Xj=Xj
In this case for
:= ^ "121=1 ^-^i
rij
—
^j]>
J
^riJ2^
"
ln{9i,X)
{{n{xj)
-
- x'j\/V^)l +
x'^9j
ihixj)
-
x)e,
- x'^X/V^t)
i=i
Define Wij := y/n{Ti[xj)
-
ti{xj))
and W2J := \/n{T2{xj) -T2{xj))
for j
=
1,
...,
J.
Assume
also that
a central limit theorem and a law of large numbers apply so that
{mi,W2i),...,iWij,W2j))' -^d {{Wn,W2i),...,{Wij,W2j))'
—»p
Tij/n
Theorem
for
pj
each j
=
1,
^ J^{0,n),
and
J.
...,
3.3 (Interval Regression). Let the basic conditions specified in Section 3.2 hold and also
Then C.1-C.3 and A.1-A.3
(3.9)- (3.14) hold.
are satisfied. Moreover,
J
ioo{9i, A)
^
^
f
= J2pj [(^U -
x'^X)ll{x'j9j
=
T,{xj))
+
{W2j
-
x'^Xf_l{x'^9j
=
wp
This theorem
orems
—
>
1
occurs
verifies
2.1, 2.2, 3.1,
and
when int(0/)
T2ix^))]
i/ng„ -+p
supe,gaQ,;^gA<^(g^)^oo(6'/, A)
\ s^Pe,ede,MAc^{9,)^ooi9i,X) -migj^Q,^XeV^{e,) ^oo(^/,A)
where nqn
=
is
non-empty or
if
qn
ifnqny^pO
'.= 0.
Assumptions C.1-C.3 and A.1-A.3, which implies that the
3.2
results of
The-
apply to the interval regression case. Therefore the inference procedure
16
proposed
Theorems
properties given in
The
Further, the confidence regions have the stochastic
in Section 2 is valid in this case.
and
3.1
3.2.
distribution of the limit variable
but this does not create a problem
simplified
much
C
is
not pivotal, and has no
for the inferential
further, unlike in the trivial case with
known
method proposed
Xi
=
One can
I.
closed analytical form,
C can not
in Section 2.
simplify the leading term as
J
sup
=
^oc(^/, A)
Vp, \{Wi,)ll{x]ei = TiiXj)) +
sup
Analysis of the Structural
ple,
and then generalize
it.
=
T2{xj))]
,
-"
^-^^
but other terms do not appear to simplify
3.4.
{W2j)U{x'.ei
l
S/eae,
e,€ae,,A6Aoo(fl/)
above expressions.
in the
Moment
Equations Model.
We
first
work out a simple exam-
Consider the usual two-stage-least-squares model
which leads to the following objective function
Assume
<
that
r
=
rank
EXZ' <
dim(^), so that the identified set
linear subspace of R'^ intersected with 0.
Q,
Note that 0/ has empty
when
there
is
weak
=
{e
=
eo
0/
+
S
is
Qn{9)
is
of an r-dimensional
defined as follows: for any ^o such that Ee{9o)Z
ee
such that
5'
EXZ' =
pivotal
and can be inverted
Suppose
is
for simplicity that g„
is
AniOl)
=
(a„(0/)
= -^
+
no longer pivotal.
:=
(which
is
also true
when
q-a
:= inf9ge(5n(&) but dim(Z)
A'-
Y, ^^^0 (-
E ^'^0
i=l
i=l
Y.{Yi - XleI)X^
= 4=
E
'
[^r.{Bi)
+
A'-
Y. XiZl)
,
i=l
^ii^I)Z^
Under standard sampling conditions
n
-
Y
1=1
^
ZiZ- -^p EZZ',
-
In
the empirical
dim(A')), so that
ir,{eu A)
0,
in the case
for confidence intervals.
the partially identified case, the statistic most relevant to making inference on 0/
process {Qn{Si),di £ ©/) which
=
0}.
Under the usual circumstances, even
interior relative to R'^.
identification,
Qr consists
n
Y
2
^i^'i -*P
EZX', and
=1
17
{e{ej)Z, Oi e
0/}
is
Donsker.
<
Hence the finite-dimensional and quasi-uniform
^00(^7, A)
where {A{9i),9i e 6/)
under
is
iid
a zero
We
is
=
weak
given by
is
+ X'EXzJ (^EZZ'Y^ (^A{9:) + X'EXZ'^'
(^A{ei)
the
limit of £„(0/,A)
limit of the empirical process {An{Oj),d]
€ 0/). For instance,
sampling, provided that {e{6i)Z,9i € 6/} has a square-integrable envelope, the Umit A(-)
mean Gaussian
process with covariance kernel given by EA{9j)A{9'j)'
"
T
1
= An{9iy(^"rt
i=l
Notice that the limit
is
Ee{9j)e{9'j)ZZ'.
and thus
therefore conclude that C.1-C.3 are satisfied
C„
=
-1
An{9j) -^d C
ZrZ'^
=
sup A{9 1)' [E Z Z]'' A{9 j)
e,ee,
^
not pivotal and depends on knowing Q/}^ Also, compactness of
necessary condition for the limit variable C to be
0/
is
the
finite.
Next, we generalize our method to the general nonlinear method of moments. Let the following
partial identification condition hold:
This
is
> C
\\Emi{9)f
(^l^-.
C
There are positive constants
min[ ini
and
such that
S
uniformly for 9 € 0.
\\9-9'j\\,Sf,
both a partial identification condition and a smoothness assumption. This condition implies
that Emi{9)
=
if
an only
ii
9 £ Qj,
and
also
imposes smoothness on the behavior of Emi{9)
for
points 9 near 0/.
In the point-identified case, global identification and the
VeEmi{9) near
9] ordinarily
set identified case, the
imply
Jacobian
(3.15); see e.g.
may be
Theorem
is
:
positive eigenvalue of
Theorem
hold and
EZ'Xv =
In
Hence
if
rank
pj — ^|| >
EZ'X
is
iEX'Z){EZ'X), we have \\EZ'X{9 -
Pakes and Pollard (1998). In the
0,
the vector {9
non-zero, for
9})\\'^
> Cq
= EZ'X{9 — 9]),
— 9})
is
orthogonal
Cq denoting the minimal
\\9
-
9*j\\'^.
3.4 (Generalized Method-of Moments). Let the basic conditions specified in Section 3.1
let
i.
0/ = 90/,
continuous Jacobian G{9)
(A(e/)'
0}.
3.3 in
IV model we have that Emi{9)
the closest point to 9 in 0/. Provided that
to the hyperplane {v
rank and continuity of the Jacobian
degenerate, which necessitates a statement of a more careful
condition (3.15). For example, in the previous linear
where 9]
full
the
case
+ ySXZ')
ii.
=
6e o
S/eEmi{9),
Wn{9)
>
dim(z)
[EZZ']-^ (A{e,)
Cn
Donsker
{mi{9),9 € 0}
v.
dim(2;),
and
class, Hi.
= W{6) +
qn
=
Emi{9)
Op{l) uniformly in 9, where
infgge Qn(S),
+ X'EXZ') and
^d C=
,
sup e^{ei,0)B,eB,
18
satisfy (3.15)
inf
ioo{9i,\)o,eei.>.€Vocie:)
then
and have
W{9)
im{9i,\)
is
positive definite
and continuous for
t^iOj, A)
=
(A(^/)
Then, C.1-C.3 and A.1-A.3 hold, and
all 9.
+ X'G{e,))' W{ej)
{A{9i)
+
X'G{9i))
A{9j)
i—>
weak
the
is
—>p
nq-n
necessarily occurs if qn :=
The theorem
orems
Section 2
in
is
and
3.2
=
3.1
and
3.2.
4.
We
We
illustrate
-£'[n~"'
= inf^ge Qn{9)
X^"_j ?nj(^/)
^"_^
but dim{mi{9))
<
mi(^/)'].
Above,
dim(^).
GMM case.
Therefore the inference procedure proposed in
Further, the confidence regions have the stochastic properties given
Similarly to the interval regression case,
distribution of the limit variable
does not create a problem
qn
if
lim„_oo
C.1-C.3 and A.1-A.3, which imphes that the results of The-
apply to the
valid in this case.
Theorems
or
verifies the conditions
2.1, 2.2, 3.1,
a zero-mean Gaussian process
limit of the process 9j i—> An(0/),
with the covariance function £'[A(^/)A(^/)']
ifnqnT^pO
^oo{^/,A)
suPe;ee,,AGAo„(fl,)^oo(i9/,A) -infe,ee;,AeK»(e/)
where 9j
,
C
not pivotal, and has no
is
known
method proposed
for the inferential
C depends on 0/. Hence the
closed analytical form, but this
in Section 2.
Computation and Empirical Monte Carlo
our methods above with an empirical Monte Carlo that uses data from the CPS.
also provide details
on the computational method that can be used to construct the confidence
sets.
4.1.
Computation. An
subsampUng method
issue in the
is
being able to use an
algorithm to construct level sets of an objective function. Ideally, one would
of grid points
and evaluate the function on those
efficient
like to
use a rich set
points. However, as the dimension of 9 increases,
constructing a simple uniform grid becomes computationally infeasible.
The Metropolis-Hastings
algorithm provides a computationally attractive method for generating adaptive grid
details of the numerical
approach we use
=
is
1.
Generate a grid of points Q
2.
Given a starting critical value
objective function as:
3.
C„(k)
=
At each subsampling stage
j
:
j
Cj_b,n
=
4.
Compute the Q-quEintile c(q) of
5.
Then compute C„(e(a))
''This algorithm
is
{^g
and
Cq
G B
:
=
k=Colnn, we can compute the level set
— ming ec„(i[) Qn(^g)) < k}.
l,...,Bn,
(Cj^b.m j
=
-
1,
in
'^
of the
compute Qb(^g) for all ^gGCn(k), and the
Qb(6'g)
-
mine^gc.{k) Qb(Sg))
•
••iBn}
mine^gc„(k) Qbl^^g))
<
c(q')}.
a valuable method of generating grid points adaptively, so that they are placed in relevant
regions only. See Robert and Casella (1998) for a detailed description; and Chernozhukov and
examples
The
in the following algorithm:
b(max9^gc.(k)
n(Q„(6'g)
sets.
using the Metropolis-Hastings algorithm.
n(Qn(&g)
where
subsample coverage statistic, e.g.
=
summarized
{9i, ...,9^^)
{^g G
numerical
non-likelihood settings.
19
Hong
(2003) for related
An
important practical issue
consistency of estimate Cq
is
is
the choice of the initial point
not affected by the choice of cq as long as cq
Hence a reasonable procedure
(2.2).
For example, in
GMM
for selecting cq
how
2.2 suggests the
bounded
is
in the sense of
can be based on pointwise testing procedures.
a pointwise critical value for testing Q{9)
of a chi-squared variable with degrees of freedom equal to the
section for
Theorem
cq.
to choose cq in the interval regression case.
of the asymptotic distribution of the coverage statistic
More
=
at ^
number
is
given by the a-quantile
of equations.
See the next
generally, one can use the a-quantile
computed under the assumption that
9j
is
a
singleton.
We examine the actual finite-
Empirical Monte Carlo: Returns to Schooling in the CPS.
4.2.
sample performance of the
inferential procedures
Population Survey. Our "population"
wave
of the
proposed in this paper using data from Current
a sample of white
is
CPS. The wages and salaries
men
series are not top
ages 20 to 50 from the
March 2000
coded or censored and so we are able
to use this population to construct
1.
confidence regions for the returns to schooling parameters in the point identified case, and
2.
confidence regions covering the identified set
Our Monte Carlo
summary
when we
artificially
based on random sampling from the original data
is
statistics for
bracket the data.
set.
Table
1
below provides
our data (population). The "true" returns to schooling coefficient
Table
1.
Variable
Wages and
Salaries
Education
Population
Summary
is
.0533
Statistics
Obs
Mean
Std Dev
Min
Max
13290
66667.6
51968.41
1
513472
13290
11.77
1.89
1
16
with a constant term of 3.91 obtained from a least squares regression of log of wages on education
in the "population"
4.3.
0/
.
We
by describing the
start first
details of the
computational procedure.
Starting values and Implementation Details: The algorithm used to obtain estimates of
is
the same as the one described on page 20 above.
1.
We
draw a sample
of size
The
steps of the procedure are as follows:
n from the above population
(this
population will be
artificially
bracketed below).
2.
We
build an initial estimate C„(co) at the starting cutoff level cq (choosing cq
is
described
next).
3.
In the subsampling step,
we obtain
the consistent estimate, Ca of the cutoff level by subsam-
pling the coverage statistic.
20
At
this point
One may
In
The
all
we
levels that
we
For example, in the method of moments case, the
initial level set of
critical value
the objective
is
=
the following way. Let Yf^
-^{Yu
+
function Qn{d) applied to the data {Yii,Y2i,Xi,i
point-identified at
some value
6^,
and
for
<
Yu =
and
Y2i)
is
problem
specific.
from a x^ with an appropriate degrees
of freedom can be used as the starting value. In the interval regression example,
value Co
get are very close
use the numerical procedures as described in Section 4.1.
used to construct the
is
step 3.^^
Calnn and then repeat
iterations.
of the above steps
starting value cq that
=
cq
but we find that the cutoff
iterate several times,
no or at most two
after
4.
one can go back to step 2 above and set
Y2i
=
Vj",
we choose an
initial
then we use the objective
This generates an auxihary model, which
n).
which we can compute the
is
limit distribution of
n(Q„(0g) - Q„(eS)),
and take
which
its
quantiles as the starting value
a consistent
is
method
cq.
In what follows, we used subsampling to estimate
of estimating cq by
Theorem
2.6.1 in Politis,
Romano, and Wolf
cq,
(1999).
Properties of the Set Inference Procedure in the Point-identified model. Here, the
4.4.
data on v/ages
is
not interval measured (the model
point identified), but
is
we
nonetheless apply
our procedure to obtain the confidence region. The objective function that we use
is
minimum
the
distance objective function
J=16
Qn{e)
(4.1)
= Y.
i=l
^(fi(x,) "
x'^e)l
+
(f2(:r,)
Since wages are not interval-measured, we have that f\{xj)
obtained are compared to the the joint confidence
approach. In Figure
we provide graphs
1,
subsampling procedure
In Table
sample
2,
sizes:
n = 1000 and n
b
=
4.5.
for the case
sizes.
where n
x'^e)l.
T2{xj),j
=
I,..., J.
obtained using the usual
n = 400 and n = 2000
The
Wald
results
inference
observations and see that the
close to the eUipse obtained using the usual chi-squared approximation.
we examine the coverage
sequence of subsample
and
is
for
ellipse
=
-
=
=
2000, using
As we can
1000,
it
We provide the coverage for two
simulations. We report the coverage for a
of our inferential procedure
see,
R =
600
.
coverage seems monotonic initially in the subsample size
peaks at 6
=
200 while
for the case
where n
=
2000,
it
peaks at
500 and comes close to 95%.
Properties of the Set Inference Procedure in the Set-Identified Model. To examine
the identified set of parameters in the case of censoring of the dependent variable, we bracket the
income data into 15
different categories.
These brackets are
In the interval regression example and similar situations, In
21
(in
thousands)
n may be dropped.
Figure
Point Identified Case: Subsampling vs x
1.
C
g
Subsampling
0,065
Ellipses
Ellipse
0.06
S
- Chi Squared Ellipse
UJ
,
0.055
3.4
3.6
Constant
Table
2.
Finite-Sample Coverage Property in the Point Identified Model
Subsample
b=50
b=80
b=120
b=200
b=300
85.1
88
87.2
93.1
91.2
b=200
b=300
b=400
b=500
b=600
86.1
88.2
90.5
95.01
93.3
n=1000
Coverage (^95%;
n=2000
Coverage
[0,5]
[5,7.5]
[30,35]
,
,
[7.5,10]
[35,40]
,
,
[40,50]
(<dh7o)
[10,12,5]
[50,60]
,
Notice here that the topcoding
obtain 718 observations in the
,
,
[12.5,15]
[60,75]
is artificially
first
random without replacement from
artificially
[15,20]
set at
,
[20,25]
,
[25,30]
[100,150]
,
[150,100000]
,
$100 million.
To give a
flavor of the data,
we
bracket, 1211 belong to [50,60], 1598 belong to [100, 150] while
through our 95% confidence region C„(c. 95)
Using the
,
[75,100]
,
734 are making above 150. The objective function
at
Size
for
is
the same as the one used
sample
sizes
=
n
earlier.
In Figure 2
600,2000,4000, and 10000 drawn
the original "population".
bracketed population, we can obtain the identified set by collecting the
parameters that satisfy the following
set of
J
inequalities corresponding to the set of
the level of education takes:
E[y\\xj\
<
60
+
biXj
<
E[y2\xj],
22
j
=
1,
...,
J.
J
values that
We
then graph the identified set 0/ along with the
as the sample
n=
95%
size increases, the set estimates shrink
600, our set estimate of the intercept
Figure
C„(c.95):
2.
9/
is
confidence region Cn(c. 95).
towards the identified
[3.2,4.71] while the true range
is
Figure
vs
n=600, b=60.
Cn(c.95).
Notice that
For example, at
set.
[3.6,4.4].
9/
3.
vs
n=2000, b=200.
0,18
0.16
0.14
Ik.
V
^^k-
— Subsampljng ConfiOence Set
c
0-12
D
UJ
(A
E
E
5 0.06
|oos
Subsamplinq Confidence Set
^^.
So.OB
-^ 0.0a
—
^^
"^^^j^—
0.04
0.02
Sel
Constant
Figure
Figure
Cn(c.95).
4.
9/
vs
9/
5.
Cn(c.95).
n=4000, b=400.
vs
n=10000,
b=2000.
3 0.06
a:
Conridencfl Set
3
3.5
4
4.5
Esbmale
5
Constant
To
calculate coverage probabilities,
we draw
R=
600 random samples from the population and
record whether our set estimates using these samples cover the identified
set.
For every sample, we
construct the set estimate, and calculate coverage in the following manner. Numerically,
23
we
store the
5.5
6
Table
3.
Finite-Sample Coverage Property in the Set Identified Model
Subsample Size
N=1000
Coverage ('95%;
N=2000
Coverage (<d^%)
identified set
Cn{Ca)-
0/ and Cni&a)
As Table
3
as arrays,
and Figures
b=100
b=150
b=200
b=250
b=300
84.1
90.2
91.3
88.5
82.8
b=200
b=300
b=400
b=500
b=600
86.34
90.1
92.2
94.3
91.2
and
tlien clieck wlietlier all the points in
2-5 show, the coverage
set estimates converge to the identified set as the
of our inferential
methods supports the
samples
This paper provided confidence regions
are contained in
similar to the point identified case and the
size increases.
The numerical performance
large sample theory developed in
5.
The proposed
is
0/
Theorems
3.1-3.3.
Conclusion
for identified sets in
inference procedures are criterion function based,
models with partial
identification.
and our confidence regions are certain
level sets of the criterion function in finite samples. In the case
when
the model
is
point-identified,
our confidence sets reduce to the conventional confidence regions based on inverting the likelihood
or other criterion functions.
The proposed procedure was shown
to
be valid under general yet
simple conditions. Along with inferential procedures, we have developed methods of analyzing the
asymptotic behavior of econometric criterion functions under
set identification
the rates of convergence of the confidence regions to the identified set.
regressions with interval data
and
set identified
method
of
We
and
also characterized
applied our methods to
moments problems.
We
also assessed the
performance of the methods in Monte Carlo experiments based on the Current Population Survey
data.
We
found that the methods perform well and in accordance with the asymptotic theory
developed.
24
Appendix
We
A.
Appendix
use the following notation for empirical processes in the sequel: for
1=1
E
E
denotes expectation and
wp —
,
(-B/(l^.))/=/-
P* and P, and corresponding notions
probabilities,
under P*
X)
denotes expectation evaluated at estimated functions /:
van der Vaart and Wellner (1996).
in distribution
{Y,
1=1
*
Ef{W,) =
Outer and inner
W=
»
—>p
means "with the inner
1
of
weak converegence are defined
—j
denotes convergence in outer probability, and
"with the inner probability no smaller than
1
—
e for
probability approaching 1,"
sufficiently large
n"
A
.
as in
means convergence
and "wp >
table of notation
is
1
—
e"
means
given at the
end of the Appendix.
Appendix
Lemma
Proofs of Lemma
B.
B.l.
Proof of
B.2.
Proof of Theorem
2.1. Proof
B.3.
Proof of Theorem
2.2. Step
2.1. Proof
(B.l)
2.1,
Theorem
2.1,
and Theorem
2.2.
D
given in the main text.
is
is
I.
D
given in the main text.
We
Cj,6,n
have that
=
sup
ai,{Qj,b{d)
-
Qj.b)
,
esc„(c)
where
qji,
:=
if
Qn
=
and
Qj^i,
:= infgge Qj,b{S) otherwise; Qj,b{&) denotes the criterion function defined
using the j-th subset of the data only. Define
B„
- B-' J2 nCi,6,n
Gb,n{x)
(B.2)
s„
<X} = B'' Y,
<X' {Cj,b,n " Cj,b.n)},
HCj,b,n
j=l
i=l
where
(B.3)
Cj,b,n
=
sup at {Qj,b{0)
-
qj,b)
ese,
In what follows, the main step
is
to
show that
Cj^b,n
can be replaced by
This
Cj^b,n-
will
be possible due to
the sandwich assumption, and despite that the constant k used in construction of the preliminary set estimate
Cn{k)
is
By
data-dependent.
A. 3
wp —
>
1,
for
some deterministic
0^" we have
e,cCnCk)Qe]\
(B.4)
where 6^" := {0i +t:
(B.5)
set
\\t\\
<
en,9i G Qi}- Hence
sup ab{Qj,b{d) -
qj,b)
see,
<
sup
wp
-
1
for all
ab{Qj,b{d)
-
< B„
i
qj.b)
eec„(c)
V
'^'''''
'
>
< sup
ab{Qj,b{0)
-
gj,b)-
see',"
^
C,,t,„
25
'
>
^
Cj,i,,„
'
Hence wp
—
>
1
B„
a, Js) = B-'
(B.6)
By
A. 2 C6
^d
^
B„
<x}<
l{C,-,b,„
<
G6,„(i)
Gi,,„(a:)
s B"'
^
l{C„b,n
<
x}.
C and by A.
=
C6
(B.7)
+ 0p(l)^d
Ct
C,
so that by Step II
Gi,„(a:)-^p
(B.8)
if
X
is
G(2;) :=
P{C <
G6,„(j;) ->p
i},
G(a;)
= P{C <
k},
a continuity point of G{x), which proves that:
G6,nW--p G{x)
(B.9)
at the continuity points
x of G(x). Furthermore,
(B.IO)
if
G
is
continuous at c^
= G-l{a)^j,
c^
=
c^
=
G'''(a),
G-'{a),
since convergence of distribution function at continuity points implies the convergence of quantile functions
at continuity points. Finally the claim
I,
that
P(e/eG„(c,))-»a,
(B.ll)
follows
by Theorem
Step
II.
2.1.
This part shows (B.8). Write
Br,
_
= P{Cb<x} + o^{\)
= P{C <x} +
at the continuity points
x of G{x)
= P{C <
(B.13)
where
i}, as long as
>
n
(a) follows
and the
6
—
»
n—
CO,
Y^rl^Y. 1(^^.^-
by the variance bound
(B.15)
0,
>
oo;
from
(B.14)
in Politis,
Op{l)
for
bounded
Romano, and Wolf
<
(b) follows
Cb^d
and a-mixing
by
C, as
definition of convergence in distribution on M.
26
o (1)
j
[/-statistics for i.i.d. series
(1999); and
=
:r}
b^
oo,
series given
on pages 45 and 72
Likewise, conclude
B„
_
= B-'Y,l{Cj,b,n <
Gb,nix)
X}
'"'
(B.16)
=
x of G{x)
at the continuity points
Step
Wp ^
III.
= P{C <
= P{C <x} +
Op{l)
x}.
e,cCn(k)^h{e,,CnCk))=0.
wp
-^
1
=^
Cn{k)ce'-
(B.18)
Hence
Op{l)
1,
(B.17)
Then,
P{Cb < x} +
wp —
h{Cnik),e,)<hie''',e,)<en.
1
>
d//(e;,C„(fc))<e„.
(B.19)
Thus Claim
II is
D
proven.
Appendix
Proof of Theorem
C.
3.1
First recall that
Cn := sup n{Qn{di) -
q-n)
e,se,
^^^^
^
We
f
supe,ge, 4(5/, 0)
if
g„:=0,
\
supe^ge^ £n(6i7,0)-infe^+;^/y;;ge 4(6li,A)
if
gn := infege
seek to establish that Cn -^d C, where
/Q^N
^
_
ifng„->pO
supe,g3e^;^gA„(e,) ^oo(6'/,A),
I
\ supe,gge^,;^gA«(9,). ^oo(6'/,A) -infs,ge,,Agv^(9,) ^oc(6'/,A)
AooC^*/) := {0}
(C.3)
U
{A e
Ko((9;) := {A
(C.4)
Step
I:
Case when ng„ -^p
0.
G
R"^
K"^
61;
:
:
Since nqn —*p
+ A/^n
+
5/
in
what
follows
Then we would
we
e int(e/)
A/v/n G
9
if
ng„
for all sufficiently large
>p
0,
n},
for all sufficiently large n}.
we have that
0,
c„=
(C.5)
Hence
Qn (5)-
sup e„{e,,o)
+
op{i).
ignore the Op(l) term.
like to
show the following simple, but important representation:
C„=
(C.6)
sup 4(e;,0)=
sup
9iee,
4(^7, A)
e,£ee,,>ieA„(«;)
where
An{ei) := {0}
(C.7)
To prove
6'i
+
(C.6),
X* /y/n
=
we need
9i. If 9/
to
U {A e
show that
R''
for
:
61/
+ A/Vn'
£ int(e/)
for all n'
€
[n,
oo)}.
each 5 G ©/, there exists 9} G 50/ and A* G An(&J) such that
G 96/, then simply take 6)
=
dj
27
and A*
=
0.
If 9:
G int(0/) take a sufficiently small
ball centered at 6i of radius 5
>
,
denoted Bs{Oi), such that Bs{9i)
C
mt(0/). Then start expanding the
radius of the ball 5 so that to find the minimal value of S such that the boundary of the ball includes a point
G d&j. Such radius
9}
Oj all belong to int(6/)
construction
In
what
exists
by compactness of 0/. Then the points on the
G (96/ and A* G An(^J). Hence (C.6)
9*j
follows,
we
is
\* /y/n/, n' £
=
1.
G/ has an empty
2.
0/ has a nonempty interior relative to 0, so that 0/
By
Cn=
interior relative to ©, so that
sup 4(6'/,
0)=
is
sup
(9/
—
O}
9*j)y/n.
and
By
j^
90/.
sup
^00(^7,0).
B,eae,
A^{9,)
=
{0},
also rewrite (C.8) using general notation
Cn=
sup
sup en{9i,0)=
Biee,
2.
=
90/,
C=
en{0i,O)-^d
=
An{ei)
Case
segment between
where A'
empty,
(C.9)
(CIO)
0/
e,ede,
9,eei
we can
line
C.2-(i)
{C.8)
Since int(0/)
oo),
distinguish two cases:
Case
1.
[n,
true.
Case
Case
so that
+
and can be parameterized as 9}
For any 5
C
(^ [9, X) -^d
,
=
sup
9;ese,,AgA„(9,)
>
^co(9/,A).
e,eaei,xeA,^{e,)
decompose
Cn=meix[C:{5),Cn{5)],
(C.ll)
where
sup
C;((5):=
fCl2l
'
^
and In(S)
=
{9i
G int(0/)
By Assumption
:
infg'
C.l for any
gge;
e
>
11^/
Cn((5)
:=
-
^/ll
>
i5/\/n}, as defined in
=
liminfP.(c;(<5)
also that Cn{5)
>
sup
4(^7,0),
8;ee,\/„(i)
Assumption
C.l.
there exists 5 sufficiently large such that
(C.13)
Observe
and
in{Oi,0)
o,eui6)
by construction. Hence
for
0J
any
e
> 1-e.
>
there exists 5 sufficiently large such that
liminfpJc„=C„(5)| > 1-e.
(C.14)
Observe that
Cn{5)=
(C.15)
sup
4(^7,
A).
9yese,,AeA„(e,)n{||A||<5}
By Assumption C.2
(i.-ii.)
any compact
sup
(C16)
^
'
set
i—»
K
e.n{e,,-)^
e,eae,
'
where A
for
sup
i^{9,,-)mL°°{K),
e,ede,
supg^ggg^ ^oo(^/, A) has uniformly continuous paths.
The next claim
Continuous Mapping Theorem that
C„(<5)
fC17)
^
'
The proof
(CIS)
^d
C((5)
:=
sup
eoo{9,,\).
e,eae,,xe\^{e,)n{\\x\\<s}
'
of this claim
is
given below. Hence for any closed set
limsupP'{C„(<5) G
n—OO
F
F} < P{C{5) G F}.
28
is
that (C.16) implies by
Hence by (C.13) and (C.14)
for
any
e
>
there exists
Vimsup P* {Cn e F}
n—*oo
(C.19)
Therefore, taking
e
—
»
and
J^
—
»
oo accordingly,
it
limsupP*{C„
(C.20)
^
enough such that
Sc large
<P{C{5,)e F} +
6.
follows that
gF} <P{CeF},
n—too
where
C := lim C
/p91\
=
(5)
sup
icx>{di,^)
a.s.
The limit C exists in R by the Monotone Convergence Theorem. By construction C > a.s., and by Assumption
C.3(iii) C < oo a.s. Conclude that by the Portmanteau lemma (van der Vaart and Wellner (1996), p. 20) and
(C.20) that
Cn -^d C.
(C.22)
Proof of
Cn((5)
^d
C{5).
Observe that
K„{Oi) C
(C.23)
and An{6i)
/ A„(0/),
in
the sense that An{9i)
uz^, n-
(C.24)
since by definitions given earlier for
„„
all
A^{d,)
no
U^=„ A^^;)
(C.25)
C
A„(6'/)
>
=
A^(ei), for
is
all
n >
no, all no
>
a nested sequence and An(0i)
= n-^1 u-
=
,„ Ar,{9j)
1
—
>
A^(5/),
i.e.
A^{e,),
1
n^
A^i9j) and
„„
A„(e;)
=
A„„(^;).
Therefore,
4
sup
Cn(5):=
(61/,
A)
etede,,xeA„„{e,)n{]\M\<s}
<Cn{6)=
(C.26)
sup
4(6'/, A)
9,ede,,xeA„{e,)n{\\M\<S}
4(^/,A).
sup
<5;(<5):=
e,eae,,xeA„{e,)n{\\x\\<s}
and
sup
C((5):=
eoo{Oi,X)
e,eae,.xeA„^(Bi)n{\\x\\<s]
(C.27)
<C{5):=
sup
^oo(^/,A).
«;eSe,,A6Ac„(9,)n{||A||<,S}
By
(C.16) and the Continuous
C:{5)
(C.28)
and
for
(C.29)
Mapping Theorem
-d
any fixed Hq
Cn{5) -»d C(J)
Let the underlying probability space be denoted as
(C.30)
C{5)
{0,,J^,
sup
P). Observe that for every
^oo(^/,A)
e,ede,.xsA„ie,)n{\\x\\<s}
29
oi
e H,
is
some
necessarily attained at
compact and
A^{6*i{lo))
n
G dQj and A* (a;) e A^{d}{u)) D
d^iuj)
<
{||A||
5}
=
C{5)
(C.31)
Moreover as no
—
(C.32)
»
That
also compact.
is
{||A||
<
5},
because the set 90/
is
is
^oc(6''(w),A*(a;))a.s.
oo,
AnoiOi)
n
/ A„{ei) n
<S}
{||A|i
<
{|iA||
<5}
for
each 8, e
99/
hence
Koi^iH) n
(C.33)
so that by continuity of A h^ ioo{6i, A),
{||Ai|
we have
C(5)
(C.34)
Note that to obtain (C.34) we use that
exists a.s.
<5}y^
A^(9;(u;))
<
5}
a.s.
a.s.
monotonicity increasing in no due to (C.32), so that lim„o_^oo
£(<5)
and by (C.27)
lim
no—oo
<
C(<5)
Moreover, for &/(w) defined above there exists a sequence
(a;) a.s.
{||A||
that
C(5) = C(&)
/ nolim
—oo
C_[S) is
(C.35)
A*
n
Hence by continuity of A
lim
(C.36)
TlQ
Now we
—»00
i—> loai^i, A) at
Mm
C((5)>
C{5)
a.s.
X^^Jyui) in An„(6J(u;))n{||A||
<
S) such that A*^(cij)
—
each 5/ G 99/,
i^{e]{uj),\n,{uj))
=
io.[e'',{u),\'{uj))=C{5)&.s.
Tig—'OO
F
are ready to close the argument. Let
P{C{5) < F}
*=^
be any
number such that P{C{5)
real
lim P*{Cn{5)
= F} =
0,
then
< F}
(2)
< liminfP4C„(5) < F}
(3)
<
lim sup
P*{C„ (5) < F}
"""^
(C.37)
(4)
<
lim sup
P*{C„ (5)
<F}
n—*oo
(5)
< P{C{S} < F}
^
no —'OO
where
(1) is
for
any no
>
1
P{C{6)<F},
by (C.28) and the Portmanteau lemma (van der Vaart and Wellner (1996),
(C.26), (5) by the
such that P{C(5)
Portmanteau lemma, and
= F} =
by (C.34) and the Portmanteau lemma. Thus
0,
liminf P.{C„(<5)
(C.38)
(6) is
n— oo
< F} =
lim sup
n— oo
P*{Cn (5) < F} = P{C{5) < F}.
Thus, by the Portmanteau lemma
Cn{S)
(C.39)
Step
II.
Case when 9/
Case
1.
= 99/
and nq^
9/ has a nonempty
-f*j, 0:
p. 20),
^a
C{5).
Consider two cases:
interior relative to
30
0, so that 9/
^
dQi,
(2)-(4)
for
any
by
F
Case
Case
1.
0/ has an empty
2.
In this case
interior relative to 0, so that
ng„ -^p
Case
2.
I
has taken care
We
0,
of.
have by definition of £n(^/, A) and Cn,
C„=
(C41)
We
90/.
by Assumption C.l
(C.40)
which Step
=
0/
sup ^„{e/,0)-
inf
(.n{e,,\).
need to show that
sup £n(6l/,0),
(C.42)
tn[e,,\)\^A
inf
e,+A/v^se
Ve,ee,
sup
^oc(6l/,A),
inf
^oo(6'/,A)
e/eee,,AeVoo(e,)
\e,eae,A6A„{e,)
/
Define
Mn-=
(C.43)
inf
M:= e,sae;,Aev„(e;)
^ool^^/.A).
and
e„{Oi,\)
inf
fl,+A/7Hee
Mn
Below we show only the marginal convergence
the arguments of the proofs of Step
I
and Step
-*d M..
The joint convergence
(C.42) follows by combining
and applying the Cramer- Wold Device.
II
Write
Mn
(C.44)
=
min[.Mn
{K),Mn[K%
where
Mn{K):=
<'.(«/, A),
inf
9,
+ A/yS'ee, d(9i,A)>K/v^
(0.45)
M^{K):=
fl/+A/vnee,
inf
u(e/,A)<A-/yTr
4 (9/, A),
where
v[Qu\)~
(C.46)
By Assumption
C.3, for
any
>
0,
there
is
some constants Ci >
and
5
>
ll^/
+ A/V^ -
6>;i|.
K large enough so that
liminfP.{A?„(/i:)
n —oo
(C.47)
for
e
inf
> Ci
min[/C2, nJ^]}
that do not depend on
>
1
-
e,
e.
First observe that
Mn{K)=
=
(C.48)
=
^n(^/,A)
inf
^'nf
(-niOiA)
inf
^n(e/,A),
e;€Se,,Aev„(8,)n{||A||<A'}
where
Vn{ei) := {A e M"^
(C.49)
Equality
(1) in (C.48) is trivial. First,
can be represented
trivially satisfies
eis
6J
+
^/nv{9i,X)
:
6>/
+ A/\/^ 6
0,
for all n'
by compactness of 0/ every
X* j \/n where A*
=
yJnv{Qi, A)
< K.
31
< X.
5/
e
[n,
c»)}
+ Xj -Jn
such that v(Qu A)
Second, every Qi
+
Xj \/n with
< Kj \/n
||A|| < if
Equality
(2) in (C.48) is
more
To sliow
subtle.
{{6j
the intersection of two convex sets (by Assumption A.l
{9j
+ X/^,\\\ <
note that {9i
it is
r]Q,9j G 0/}. Note that
+
(2)
+ Bk^^{0))
Bj^^^{0)) n
is
convex) that contain
is
A", 5/
G 0/}
convex and necessarily contains
9[.
Hence
n 6 =
dj, since
for
every
= dj + \l \/n € {9i + 5^/y^(0)) n 0, points on the linear segment between 9' and 9i are also contained
{9i + Bjii^{Q)) n 0. Therefore A = s/n{9' - 9i) e Vn{9]) n {||A|| < K}, and representation (2) follows.
9'
By Assumption C.2
(i.-ii.)
inf
(C.50)
4(^/,-)
^
inf
Equipped with (C.48) and (C.50), we can show through the use
MniK)^^ MiK)~
(C51)
The proof
of this claim
m
ioo{9i,-)
L°°{K).
of the Continuous
inf
Mapping Theorem that
^oo(5/,A).
stated below.
is
Hence by (C.47)-(C.51),
for
any
e
>
there
lim^infP.{A^„
(C.52)
a sufficiently large K{e) such that
is
=
> 1-e.
A^„(/i:(e))}
Note that
^=
(C 53)
M
exists a.s. in
<
by C.2,
By
R
inf
^cc (^/, A)
=
lim
by the Monotone Convergence Theorem. Since
M < oo
a.s., i.e
M
(C.51)-(C.52), for any
is
>
e
{9i, A)
>
and
tight.
and each closed
limsupP*{7W„
€F} <
set
F,
lim sup F*
"-"°°
(C.54)
(.^z
M (K)
{Mn{K {e))
e
F} +
e
"-*°°
<P{M{K{€)) eF}+e.
Letting
e
—
>
and K{e)
—
»
oo accordingly,
follows that
limsupP'{A^„
n —-oo
(C.55)
By
it
the Portmanteau
Lemma
conclude that
Mn--d M.
(C.56)
Prooi of
GF} <P{A^ eF}.
MnjK) ^i M{K).
(C.57)
Observe that
Vn,{9i)
C
Vn{9,)
C VU6i),
for all
n > no
so that
M„{K)~
tn{9i,X)
inf
er^de,,\&v„(e,)n{\\\\\<K}
<Mr.{K)=
rC58)
inf
<M;:(A'):=
iniOiA)
inf
in{9i,X)-
s,s5e,,Aev„„(e,)n{||Aii<K}
By
(C.50) and the Continuous
(C.59)
Mapping Theorem
Mn{K)^iM{K):=
for
any fixed no
inf
e,eae,,Aev„„(e;)n{||A||<A-}
32
ioo{9uX),
finite at least for
A
=
in
and
MniK)
(C.60)
-^d
M{K)
-
C^
inf
e,ede,.>~ev^{8,)n{\\x\\<K}
{Oi
,
A)
Moreover, as no "^ oo
(C.61)
Vn„iei)
so that by continuity of A
n
{||A||
<K}/'
i—» iooi^i, ^) ^^
{i|Ai|
< A^
for
each Oj G 90/,
each dj 6 96/
\M
M(K)
(C.62)
The
n
V^iei)
a.s.
details of proving (C.62) are nearly identical to those given in (C.31)-(C.36), so are not repeated.
Let
F
be any
real
number such that
P{M{K) = F} =
P{M{K) < F} =
lim
n—'oo
>
lira
then
0,
P,{M^{K) < F}
<F}
sup P'{X„(i<:)
n—'OD
(3)
>
<F}
liminfP,{7\/(„(A')
"^°=
{C.63)
(4)
>^mmf
P,{Mn{K)<F}
n—*oo
(5)
> P{M{K) < F}
^
no—>oo
where
(6) is
(1) is
P{M{K) <
by the Portmanteau lemma and (C.60),
(2)-(4)
by (C.62) and the Portmanteau lemma. Therefore,
for
for
any no >
F},
..
by (C.58),
any
real
F
(5)
n— oo
n— oo
Hence by the Portmanteau lemma
Mn{K)^d M{K).
(C.65)
Appendix
Steps
Step
I
I:
and
II
prove Claim
Case when nq^
—>p
I
and Step
0.
III
D.
Proof of Theorem
proves Claim
Recall that Q{9i)
=
4(e/, A) := n{Qn(0i
(D.l)
Since ng„
—p
0,
C„=
the Proof of
Theorem
3.1
II.
+ \/V^) -
sup £„(6I/,0)
+
Q{ei)).
Op(l).
we have that
sup 4(e/,0)
e,ee,
33
3.2
and
we have that
(D.2)
(D.3)
by the Portmanteau lemma, and
such that
P{Ai{K) = F} =
limsupP*{X„(A:) < P} = \imMP,{Mn{K) < F} = P{M{K) < F}.
(C.64)
From
1
=
Op.(l).
Hence since k
—»p
oo
<
C„
(D.4)
wp
fc
-^
=>
1
K —^
Condition C.3 implies that for any
wp —+
^
6/ C Cn{k) wp
K/k
oo with
=>
1
-^
h{ei,Cnik))
=0 wp ^
1.
there are positive constants Ci and S such that
1
Ci
(D.5)
uniformly
By
in (0/, A)
n
and since Q(9j)
definition of Cn(fc)
sup
+
nQr^{0)
< 4(e^, a)
min[i.(6i/, A)^, 5^]
K/^,
>
such that u{9[, A)
(D.6)
•
where
A)
i^(5/,
=
infe^^e;
11^/
^/\/n -
+
6'j\\.
=
=
Op{l)
sup
+ Op(l) <
4(6'/,A)
fc.
e/+>./v^ec„(fc)
eec„(fc)
Hence
9/+A/v^€C„{fc)
(D.7)
Then the claim
is
that
wp —
»
C^k) C
where 0^ := {dj
+1
:
||i||
<
Then
4fc
by (D.9) and by
(a) is
wp —
>
(b) is
1,
by (D.5),
fc
Op(l).
inf
<^
Ci
n
—
oo and k/n -^
(c) is
Combining (D.4) and (D.8)
it
•
+ A/v^-^;!
l^z
1
»
»
e2('^/^')"'/v^,
G 6/}. Suppose otherwise, then
wp —
for this pair [Oj, A),
(D.IO)
4fc
c,6[
v(ei,\)=
(D.9)
+
1
(D.8)
where
implies4(6'/,A) <fc
min[i/(6i7,
A)^
5^]
<
for
some
6i
+
X/^Jn e Cn{k),
>2(fc/Ci)'/VVn.
4((9/, A)
<
fc
+
Op(l),
so that Ci n-min[i/(5/, A)^, 5^]
> C\-n-rmn[A{k / C\) / n, 5'^] >
by (D.7). This yields a contradiction. Thus, (D.8)
wp —
follows that
»
Step H: Case when ng„ y^p
0.
Observe the relationship between the "centered" and "un-centered"
dence regions. Let the un-centered region be denoted as before:
(D.12)
Cnic)~{9:niQn{9))<c],
and the centered version be denoted as
(in this
proof only):
Cn{c):= {e:n{Qni9)-qn)<c], where g„ :=
(D.13)
inf
Observe that the following key relationship between the two sets
(D.14)
(D.15)
Cn{k)
fc
—
»
oo but k
=
Op{n).
From
= Cn{k + nqn)
the proof of
Mn
=
nqn
=
for all
Theorem
2.1
Op(l) since
/c>
0.
we know that
Mn
-^d
Hence
(D.16)
true.
dH{ei,Cn{k)) < h(Cn{k),ei) < /i(e'/'^^'>'''/^,e/) < 2ik/c,y^^/v^.
(D.ii)
Let
is
1
k:=Ck +
nqn)
=
34
k-il
+
Opil)).
M.
Q„ (61).
confi-
By
(D,14)
(D.17)
CrrCk)
Hence by (D.ll) and (D.16)
=
wp —
follows that
it
^H (^(fc), 6/) = dnCC^ (fc),
SO
Cn{k),
and the claim now
Step
2{k/C,f'^l^<
(2(fc/Ci)>/VV^)
II,
that for k G
[cq, ci]
Inn,
•
0/ C Cn{k) C ej", where
(D.19)
6(
wp —
»
—
{^!
We
+
*
•
IKII
^n>0l £ ©/}. and En —>
—
have that k 6
[co,ci]
wp —
-Inn
>
1.
sup Qi{e)
-
some
C>
0.
=
Then, using Q{6i)
<
6(
sup Qb{0)
-
sup Qb{e))
a sequence of positive constants.
is
wp
->
1,
I
where
Cn
=C
In
sup QhiOi))
sup
<
+ \/Vb -
biQbiei
Qie,))
- sup HQbie,)) -
sup
=
max[
4(6/, A)
- sup
.
sup
=max[
sup
4(9/, A), sup 4(^/,0)]- sup li,{9[,0)
e,ee,
e,ee,
4(6'/,0)
+
=
max[
sup
in
e;se,
e,eei
sup 4(6'/,0)]
4(6'/, 0),
8;eaej
e,ee/
sup 4(^7,0)
+ Op(l)- sup
etee,
(a) follows
sup ^(e/.O)] - sup 4(^/,0)
Op(l),
9/eae,
=
ib{0i,O)
e,ee,
e,eae,,\x\<Vbe„
21)
Q{e,))
e,ee,
e,ee,,\M<^<^i
assumed
n/^/n
0,
ei+x/Vbse)"
where
Op(l),
e,ee,
Bee]"
(13
=
aee,
Hence, by Claim
9/ C Cn{k) C 0;"
(D.20)
for
+Op(l)),
1
dee)"
©/"
(1
follows.
proves Claim
III
1
»
dH{Cnik),e!) < dH{Cn(k),e,) <
(D,18)
0/).
+ Op(l) -
sup 4(5/, 0)
8;ee,
4(6'/,0)
e,ee,
from the uniform stochastic equicontinuity of the process A
i-->
sup^^ggg^ ib{Si,^) over
K
C.2 and from
V^En^O,
(D.22)
which follows from the assumption that
h/n
(D.23)
(b) follows
(D24)
—0
at
polynomial
rate, so that
vftcn
ex
\/b/n
by the Continuous Mapping Theorem and proof of Theorem
sup 4(6/,
^e,ee,
f
0),
sup
B,ede,
4(6'/,0))^d
'
sup
f
^
eiede,,>.eA^{e,)
3.1
In
n
—
»
0;
which implies that as
^oo(6'/,A),
sup
e,ede,
6
—
oo
t^[e,,0)\.
'
D
35
Appendix
Verification of A.l
Proof of Theorem
E.
3.3
immediate from the stated assumptions. Therefore we
is
will focus
on verifying C.l to
C.3.
Step
x'j6
d
=
>
dim(6»).
and
<
x'^O
96/
<
Denote Tk := {Tk{xj),j
T={tGR'^:71<f<
is
9/
identified set
T2 {xj) for all j
<
J}.
J) for
=
A;
determined by a set of inequalities: 0/ := [9 ^
is
assumed that the d x J matrix
It is
=
{eeR'^ .X'8
=
T2}
is
compact and
X
X
<
:= {xj,j
J)
is
R''
some
for
t
has
t
:
rank,
full
<
Ti
6/
<
t
T2}.
compact.
is
assumed that ©/ C O.
It is
determined as
59/ =
(E.2)
That
from any
Of
=
G In{S)
{61
G 9/
G int(9/)
[Oi
:
:
=
x'^O
infgjgae/
j5pr3^^«>o
Suppose otherwise that k
=
\\Tk{xj)-x^9,\\
(E.4)
^
_
^^1^,
=
T2 (xj)
^ ^/V^}
^'jW
some
for
,
j}.
implies that x'^Oj
=
=
x'.O'j
Tk{xj) for
>
such that
v{j,k).
some j and k and some
9'j
G 99/. This
Hence
0.
99/
^^^ ^ dQjy9'j G
^^11
must be bounded away
observe that there exists k
||xj||5/y'n. Indeed,
^ei^dej,Wjede!:x'/j=Mxj),
^ dQj. Hence k >
poses a contradiction to 9;
~
11^/
This implies Xj9j
0.
or x'j9
ti {xj)
by the distance proportional to
Tk (xj)
(E-3)
:
x'/j
=
Tk{xj),
\/{j, k).
\\Xj\\
Recall that
>
||xj||
for all j since the first
1
component
\\Tk{xj)-x'Jj\\>K
(E.5)
of Xj is
1.
V5/^96/,
||e^-^/||||xj||
inf
Hence
y{j,k).
Thus,
(E.6)
9i
G In{5)
=>
Ti [xj)
-
x'^9i
< —^\\xj\\
or T2 (xj)
-
x'^Oi
>
Vj.
-^||a;j||,
Recall
(E.7)
4 (e/, A)
=
^ ^ fv^
j=i
Iv
(fi {x^)
-
J
12
x'^9,)
-
x'^x]
L
J
-j-
+ j] !^ [^ (f2
[xi)
-
x'^9j)
-
x'^X\
L
11,
J
j=i
Hence
2
(E.8)
4(e/,0)
= ^^[w'i,-v/^(a:;e/-ri(a;,))l
+
j=i
^
2
!^
[l^^^
- ^^
(x^^z
-
T2 (x,))
1
.
j=i
Hence
sup
(E.9)
14
(0/, 0)
I
<
e,ei^(5)
Choose any
(E.IO)
:
of rank
Thus
1, 2.
Bi
(E.l)
Since
The
Verification of C.l:
I.
Ti [xj)
e
>
0.
Since
||xj||
limsupmaxP
n—'oo
^ ^"
\W,^
-
k5\\x^
>
1,
(Vl^ij
Wij
>
=
Op
K(5||xj||)
111
'
-'+
Jit
and W2J
(1)
<
e
and
= Op
36
E-
Jit
(1) for
"
n—>c»
[^2,-
each j
maxlimsupP
J<J
iSJ
+
(1^2^
+
^5\\x^
<
J, for 5 sufficiently large,
111
'
< — '«<5||a:j||) <
e-
C.l
is
now
verified, since
it
follows that for any
sup
(E.ll)
e
>
there exists 5 sufficiently large such that
0,
|4{6'/,0)|
=
wp > 1-e.
e,ei„(6)
Step
II.
Verification of part
ft
first
that for fixed 61 G
L
90/ and
J 4-
Mapping Theorem the
J
e^
(E,14)
C 0/
,
^
2
= Y.Pi Ki> - ^^^]
1
(^J^^
W'
.
C.2-ii(a) requires us to
=
"^1
(^j))
re
<
such that
x'jfii
(xji,
Z
=
1, ...,
rank d and
d) has
^
90/, that
=
{kt,ji)
is
_
1
[x'^di
=
the interior of
T2 (Xj))
0/
is
.
nonempty
among
The
the finite subset of points
systems of d-equations of the form:
all
Tki{xj,),
a^^A]
limit theory of supg^gge^ £„ {6j, ).
00 amounts to choosing Oj
that are defined as a collection of solutions to
(E.15)
0/
given by
is
2
+ Ep> K^ -
examine the finite-dimensional
of finding supg^ggg^ in (^/, A) for
I,
finite-dimensional limit of Cn {di A)
Verification of part ii(a) of C.2: First, suppose that
III.
relative to
Vj
(01, X)
Step
in
= 0wp ^1.
42'(9,,A)
Therefore, by the Continuous
problem
by an argument similar to that
fixed A,
(E.13)
Step
where
,
-;
(E.12)
Note
4 (^/, A) = d^' [61, \) + d^' (5;, A)
of C.2: Write
i
l
€ {1,2} x
=
l,...,d,
{1, 2,
...,
J}
for
each
/.
There
is
only a
finite set of
such systems of equations. Then,
sup
(E.16)
e,ede,
= max
4 (0/, A)
^'^v,
4
(61/,
A)
,
for
n <
Hence by the Continuous Mapping Theorem the finite-dimensional weak
by maxejgv,
supe.eae,
^00 {^/i Oi
s-nd therefore the finite-dimensional
weak
00.
maxe,gv,
limit of
4 {^I^ )
limit of supg^gge^ £„ (0/,
)
is
is
given
given by
^oo(6'/,-)-
0/ = 90/,
Second, suppose that
the interior of 0/ empty relative to
V
=V
1
(n
[x,]
=
T2 (.T,))
^ (w,, -
K'^.
Then wp
—
»
1
'
x; a)
'--^
(E.17)
inf
'^
Therefore, by Continuous
infgjge, ^00 (^/i
is
4 (^/, A) =
inf
^
that
)
The
i^
'
(Or, A)
1 (ri
(x,)
=
r^ {xj))pj
[W.j
- x'^xf
.
j<J
Mapping Theorem the
finite-dimensional limit of infg^ge;
joint finite-dimensional convergence of (infe^ge,
37
4 (di,
•)
given by
4 (^7,
4 {di< )) follows
iSupg^gQ,
•)
is
by the Continuous Mapping Theorem.
Step
III.
Verification of part ii(b) of C.2: Let
fixed 6{ e 90/, in {^ii A)
is
K be any compact subset of R''.
€ K, where Vj
tinuity for maxe^gv; ^n {di, A) over X
a
is
finite subset of
To
dQj,
it
verify the stochastic equiconsuffices to
show that
for
any
€ K. This immediately follows from convexity of
stochastically equicontinuous in X
^n (^/! ) in A and finite-dimensional convergence oi £„ {9j, ) to a convex continuous function i^o (^/, ) over R"*,
proven in Step
I.
Likewise, stochastic equicontinuity of infe,ge, in (^/, A) in A G /f also follows from convexity
and finite-dimensional convergence of infe^ge; in
to a convex continuous function infgjge/ ^oo (^/,
•)
(^/i
•)'
cf.
(E,17).
Step IV. Verification of part
whenever
ti [xj]
=
Xjdj,
iii
and
Any A e
of C.2:
<0
x'jX
whenever
56/
A<„ {d[) for 9j e
=
t2 [xj)
satisfies the
property that x'^X
>
Hence
x'^Oj.
J
i^
sup
(E.18)
Vpj
<
{9j, A)
+
{[Wyj]l
=
Step V. Verification of C.3: For each 9n{di,X) such that u(9i,X)
<
[W2,]'i)
infgjge,
oo
a.s.
\\0i
+
X/^/n -
> K/y/n,
9',\\
decompose
(E.19)
Any
61/ -I-
A/x/n
solution ^J
is
=
e;
-I-
subject to
A7v/n, where
1
<p<
depends on
'
{xj^,l
=
l,...,p)
arg
inf
8^dQ
xr^9}
=
has rank p and
{9;, A)) is necessarily
=
one of the
Tk,{xj,),
(fc;,j;)
finitely
r=
(Tfc,
are
md
9\\
WX']]
=
y/nv {e,,X)
>
K.
=
l,...,p,
{1,...,
J}
for
each
sub-matrices of
I.
Matrix X*'X* (which
X*'X* that have
full
rank
zero. Let also
=
({ji,k,),l
=
l,...,p).
zero,
ci||A'||<||^*'A-|U.
Moreover, given any index set JfC* for
,
x^,A'<0
(E.23)
(E.22) at least for one {]' ,k') €
(E.24)
-
X/y/n
e {1,2} x
bounded away from
(E.22)
By
l
{Xj,),l=l,...,p) and JIC*
Since the eigenvalues of (^X*'X*)
+
many p x p
and whose eigenvalues are bounded above away from
(E.21)
\\9i
I
dim(5) binding constraints of the form:
(E.20)
where X* :=
6i;
ci||A*||
<
li^.A'l
all [ji, ki)
if
fc,
=
1
G JK.*:
or
x'^^X*
>Q
if
fc;
=
2.
JK*:
sothati^.A* <
-Ci||A*||
38
if
fc*
=
1
or
x^. A'
>
ci||A*||
if
k*
=
2.
p,
=
4 (01, A)
Decompose
+ d^'
[9,, A)
d^'
(9/, A)
where
,
(E.25)
^^
=
4^) (0;, A)
>—
-
[l^ij-
>p,.(l
+
(x^A*) 1'
- 7^
[u/i,
%/"
(a^j-
A*) ]'
+
o,{l))[Wrj.
1
=n
(r;e;
1 (/=*
=
c,\\y\\]\
+
5
[vi/2,
1)
+
[n/^ (x;.A*)
[k'
=
1)
>mmpj{l + 0p{l))\mm[Wij,-W2jJ <
j<j
ix,))
+
-
J] +Ci||A*||l
J
1^2,.]
' 1 (fc*
- W^j.]\
[cillA'll
I
- v^ {xry)
[k'
=
1
' i (x^.e;
=
2)
=
2)
r^ (x,))
,
+
and
'-'
(E.26)
J
J
Hence, recalling that
>
constants c
|H£|
=
wp >
and
Op{l), for any
1
—
e
>
0,
0,
[m^2j
- V^(x^r -
=
^/nv{9!,\), one has that
where W_ := min
we can
K
select
- X;A*]_l(x;e; <
T2(X,))
>
0.
— W2:;,i <
W^ij,
large
(^/,A)
(.„
j\-
>
c
(H£
+ c'||A*||)_^,
Let \/nv{9j,X}
=
some
for
||A*||
enough so that W_ > —c'y/nv(9j,X)/2 wp
>
>
1
positive
ii".
—
e.
Since
Then
e,
el'\9j,\)>^cc'v{9,,\f.
(E.27)
Since Cn{9i,X)
>
4^'
(61/,
A),
C.3
is
I.
Verification of A. 1
is
D
verified.
Appendix
Step
T2(X,))
=l
||A*||
>
c'
2
E^
+
F.
Proof of Theorem
3.4
immediate from the stated assumptions and from {mi{9),9 G 8} being
Donsker (hence Glivenko-Cantelli).
Step
II.
Step
III.
Verification of C.l
is
not needed because 9/
4(^7, A)
(G„mi(0/
X Wn{9i
condition
ii.
(F.2)
condition
(F.3)
that {mi{9),
GnTni{9i
where A(9/)
in 9.
= v^E„mi(e/ + X/V^yWn{9i + X/^)V^Enmi{9, +
=
(F.l)
By
= 90/
Verification of C.2 Write
is
+
+
X^^/n)
+ VnEmi{9i +
X/^/n) {Gnmi{9,
e 9} forms a Donsker
+
X/^/n)
class for
+ A/v^) = Gnmi{9,) +
X/^/n)
X/-Jn))'
+
y/^Em,{9i
any compact
Op(l) => A(0,), in
+
set
X/^/n))
.
K,
L°°(9/ x
/f)
the Gaussian process with the covariance functions given in the statement of
iv.,
Wn{9,
+
A/n/^)
=
W{9,)
+
39
Op(l), in
L°°(9/ x
/C).
Theorem
2.4.
By
By
condition
iii.,
^Emi{ei +
(F.4)
=
\l\fn)
+
G(e,)'X
o(l), in
L'^{Qi x K).
Hence
U9i,
(F.5)
Hence
A)
^ ^oo(0/, A) = (A(0;) + G{e,y\)'
C.2-i. is verified, since finite-dimensional
C.2-ii.(a)
and Un{X)
=
is
is
G(0/)'A)
,
in
and Un(A)
implied by weak convergence of
is
L~(e/ x K).
implied by weak convergence in L°°{Qi x K).
supg^gQ^ ^n(^/i A) are continuous transformations of £n(0,
stochastic equicontinuity of ^^(A)
which
convergence
+
by the Continuous Mapping Theorem since the functionals Zn(A)
verified
is
x W^(e,) x (A(0;)
=
inie,£e, ini^i,^)
C.2-ii.(b) follows as well, since
A).
implied by stochastic equicontinuity of £n(^/,A) over 6/ x K,
(-niSi, A) in
L°°(0/ x K) to the quadratic form of a Gaussian process,
^oc(0/,A).
Step IV. Verification of C.3.
By
condition
ii.
G,m.(e)
(F.6)
where A(0)
mi{9) forms a Donsker
^
A(e)
Hence
class.
L°°(G),
in
the Gaussian process with the covariance functions given in the statement of
is
Theorem
2.4.
Hence
(F.7)
=
sup||G„mi(5')||
0p(l).
see
By condition
iv.,
iv. fn -^p
>
^
uniformly inO €
Hence wp
0.
—
>
Q
Wn{d)
>
i
(
>
(F.8)
inf
in
inf
^,n\\
v(B,,x)>K/V^
~
Note the
By
line
different
is
Define ^n
mineig {Wn{9)).
infflge
By
condition
viB,,x)>K/V^
from the preceding one, as
it
'
VnEmi{ei +
X/^)
Op{l)
II
y/n(v{9i,X)^A6'^)
^n{v{9!,XyA5^)\\
II
y/n{v{9i,xyA5^)
^7i{v{9j,xyAS^)\\oo
uses the sup
norm
instead of the Euclidean norm.
the partial identification condition (3.15)
|[v/^£m.(e;+A/0i)||
^
(F,9)
II
for
=
(^W^Emiid, + \/y^) + Gnm^iOi + A/v^)||2)
USi,
(
;;"::,^! „^
.
/^\n{v{e[,X)
inf
— W{9) + 0p{l).
1
some constant
C>
0.
(F.IO)
For a given
e
>
Hoc
y/n{v{ei,X)^A6^)
0,
II—
K
can be
made
arbitrarily large, so that
=£diL_||
wp >
1
—
e
<C/2.
IVn(i)(6'/,A)2A<52)lloo
Hence wp >
1
—
e,
v{B,,x)>K/y/^\n{v{9i,Xy A5^)J
Hence wp >
1
—
Hence C.3
is
v
/
/
e,
en{9!,X)>C' n-{v{9i,Xf
(F.12)
•>/
AS'^)
for all
i/(6i/,
A)
>
J-C/v^.
D
verified.
40
Appendix
Suppose that one
when
well motivated,
when
analysis or
some parameter
interested in a special parameter 9* inside
is
there
it is
value,
rV
a sense in which 9*
is
0/ The
.
9* in
some
inference about
0/
is
the "truth". This case typically arises in non-structural
is
believed that the models are correct representations of data-generating processes for
there
i.e.
of data P. In this scenario,
0/
9'
is
e
such that the model law Pe agrees with the actual stochastic law
R''
not of interest per
is
problem has been already investigated
similar
to Pointwise Inference
G. Relationship
quantile estimation by Chernozhukov and
se,
but rather 9'
in the context of
Hansen
(2003),
is.
In
method
of
moments
settings, a
dynamic model with censoring by Hu
and
(2002),
GMM weak or complete unidentification by
Kleibergen (2002). Imbens and Manski (2004) investigate the Wald type inference about 9* for the special
case where a real parameter of interest
is
known
to
lie
in
an interval with endpoints that can be consistently
Here we provide further insights concerning the pointwise inference
estimated.
in its relation to regionwise
inference.
Assumption A. 4. Suppose
there exists On
aniQniOl)
where C{9i)
a
is
random
has continuous distribution function on
G.l. Suppose
liminf„_cx>
P{e* e C„(c;)| >
that
>
~
oo such that
Qn) -^d
Moreover, for at
variable.
Theorem
—
(0,
0(9!)
least
for dl
one 9j € 0/, C{9i)
oo); otherwise, C{9j)
Assumptions A.l and A. 4
0/6 0/
=
>
with positive probability and
with probability one.
hold. Let c*
=
supg^ Ca{9j).
Then for any
9* in
Qj
a.
Proof:
liminf P{0* e
CnK)} =
liminf P{a„ (Q„(9')
-
qn)
<
supc„(6')}
(G.l)
(1)
> liminfP{a„(Q„(e*)-g„)
<c„(6'-)}
>
(1
or a)
>
a,
D
The theorem above
is
constructed using the Anderson-Rubin pointwise testing principle. Notice that the
confidence set constructed in the theorem above
Theorem
C„(c„(-))
(G.2)
also has the
necessarily smaller than the regionwise confidence set in
is
Notice also that as a byproduct of our derivation, inequality
2.1.
a pointwise
coverage.
Hu
= {r
e 0/
:
On {Qn{9')
-
Qn)
is
generally not equal
is
generally a special case
Qnid)
(is
let
also use this technique for
this set for the case of interval-identified parameter. ^^
of our construction.
all
The
is
quantiles.
However,
this set
9 € 0/.'"
Ca{9) be the subsampling estimate of Ca{9) for each 9 e Qi.
inequalities.
'''This
IV
set Cn{ca{-))
This can be seen by defining the new objective function
^^A more recent paper than ours, by Ho, Pakes and Porter (2004), has also proposed such sets
moment
set
< cU9')}
smaller) than the level set C„{c*) of the function OnQniS)-
= Q„{9)/m3x[cai9),e] for
Further
shows that the
(2001) proposed the set (G.2) in the context of a partially identified
dynamic censored regression model. Chernozhukov and Hansen (2003)
Manski and Imbens (2004) construct
(1) in (G.3)
The
difference
is
an equi-quantile transformation of the original objective function. In many examples
as objective functions
in
the context of
that they do not directly work with objective functions.
have the equi-quantile property by using optimal weights.
41
this
is
unnecessary,
Theorem
G.2. Suppose that Assumptions A.l and A. 4 hold. Let
liminf„_,oo -P] ^*
Proof:
€
Cr,{c*^)
>
0/ C 9; wp
Since
liminf
>a-
—
>
P{r
(Likewise, a corollary
1, it
e C„(c;))
=
liminf
n—too
9* in
Q;
supc<,(^)}
Qn)
<
F{a„ (Qn(r) -
g„)
< c,(r) +
>
liminf
(1
<
Qn)
-
or q)
for
>
Ccie*)}
Op(l)}
a,
subsampling,
Inequality (1) shows that C„(c*
(
Then for any
q
"^°°
from the standard argument
2.1.
P{a„ {Qn{e*) -
limini PianiQuiO")
>
(2) follows
sup^^ Ca{9[).
that Cn{c*^{-)) also covers 6' with probability a.)
is
>
(G.3)
Equality
=
follows that
n—*oo
proof of Theorem
c*
as the one presented in Step 2 of the
e.g.
also covers 6* with probability a.
(•))
).
D
[CONSISTENCY AND RATES HERE TOO?]
Notation and Terms
—>p
—d
wp -^
wp > 1 —
convergence
in (outer) probability
P*
convergence in distribution under P*
1
with inner probability P, converging to one
e
with inner probability P, larger than
closed ball centered at x of radius 5
Bs{x)
I
>
1
—
e for
sufficiently large n,
Q
identity inatrix
normal random vector with mean
A/'(0, a)
T Donsker class
here this
and variance matrix a
means that empirical process /
>—» -4=
5I)"=i(/(^i)
~
EfiyVi))
is
asymptotically Gaussian in L°°{J^), see Vaart (1999)
metric space of bounded over
L°°{J-)
minimum
mineig(A)
T
eigenvalue of matrix
functions, see Vaart (1999)
A
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Date Due
Lib-26-67