Induced order statistics (IOS) arise when sample units are reordered according to the
value of an auxiliary variable, and the associated responses are analyzed in that induced
order. We derive sharp convergence rates for IOS under weak assumptions that accommodate
both interior and boundary conditioning points — addressing limitations in existing methods
used for regression discontinuity designs and k-nearest-neighbor approaches. The paper
establishes sharp marginal rates for approximating target conditional distributions in
Hellinger and total variation distances, uncovering a clear trade-off between smoothness
and speed of convergence, with explicit conditions governing how the number of nearest
neighbors can grow with sample size.
This paper introduces a test for conditional stochastic dominance between two distributions at prespecified values of a conditioning covariate, referred to as target points. The test uses a one-sided Kolmogorov--Smirnov statistic computed from induced order statistics, the outcomes attached to the conditioning observations closest to the target point, and compares it to a critical value that, given the number of neighbors, requires no resampling, kernel smoothing, or parametric assumptions. The same procedure applies whether the outcomes are continuous, discrete, or mixed, and requires only continuity of the conditional distributions in the conditioning variable. We establish asymptotic validity under two frameworks: one in which the number of neighbors is held fixed, where the induced order statistics converge to independent draws from the conditional distributions at the target point; and one in which it grows with the sample size, where we obtain an explicit rate that accommodates both an estimated target point and a data-dependent choice of the number of neighbors. We connect the test to permutation-based inference, provide a refined critical value for discrete outcomes, propose a rule for selecting the tuning parameters, and illustrate the procedure in two empirical applications whose recorded outcomes exhibit mass points. Monte Carlo simulations confirm its strong finite-sample performance.
Partial Identification & Moment Inequalities
This strand studies inference in models where the data restrict parameters to a set rather
than a point. Work covers confidence regions, tests, subvector inference, specification
testing, and computational guidance for moment inequality models.
Models defined by moment inequalities have become a standard modeling framework for
empirical economists across a wide range of fields. We present a guide to empirical
practice intended to help applied researchers navigate all the decisions required to
frame a model as a moment inequality model and construct confidence intervals for the
parameters of interest. The template has four steps: (a) a behavioral decision model,
(b) moving from the decision model to a moment inequality model, (c) choosing a test
statistic and critical value, and (d) accounting for computational challenges. Each
step is illustrated in an empirical application studying identification of expected
sunk costs of offering a product in a market.
This paper surveys recent literature on inference in partially identified models.
After reviewing basic concepts — including the definition of a partially identified
model and the identified set — we turn to the construction of confidence regions in
partially identified settings, emphasizing the importance of requiring uniform
consistency in level over relevant classes of distributions. The survey is mainly
limited to models in which the identified set is characterized by a finite number of
moment inequalities, or the closely related class in which the identified set is a
function of such a set. We conclude with some thoughts on fruitful directions for
future research.
This paper introduces a bootstrap-based inference method for functions of the
parameter vector in a moment (in)equality model. The procedure controls asymptotic
size uniformly over a large class of data distributions and improves upon the two
existing methods with uniform size control — projection-based and subsampling
inference. Relative to projection-based methods, ours weakly dominates in terms of
finite-sample power, strictly dominates in terms of asymptotic power, and is
typically less computationally demanding. Relative to subsampling, it strictly
dominates in asymptotic power and appears less sensitive to the choice of tuning
parameter.
This paper studies specification testing in partially identified models defined by
moment (in)equalities. The literature has suggested a test based on checking whether
confidence sets are empty, referred to as Test BP. We propose two new specification
tests — Test RS and Test RC — that achieve uniform asymptotic size control and
dominate Test BP in terms of power in any finite sample and in the asymptotic limit.
This paper studies the behavior, under local misspecification, of several confidence
sets commonly used for inference in moment (in)equality models. We propose the amount
of asymptotic confidence size distortion as a criterion to choose among competing
inference methods. Under weak assumptions we find two main results: (i) confidence
sets based on subsampling and generalized moment selection suffer from the same degree
of asymptotic confidence size distortion, despite the latter yielding smaller expected
volume under correct specification; and (ii) the asymptotic confidence size of
confidence sets based on the quasi-likelihood ratio statistic can be an arbitrarily
small fraction of that based on the modified method of moments statistic.
This paper addresses optimal inference for parameters that are partially identified
in moment inequality models. Using a large-deviations criterion for optimality, I
show that inference based on the empirical likelihood ratio statistic is optimal.
I also introduce a new empirical likelihood bootstrap that provides a valid
resampling method for moment inequality models, overcoming implementation challenges
that arise from non-pivotal limit distributions. Monte Carlo simulations confirm
favorable finite-sample performance of the proposed framework.
Randomization & Permutation Inference
This strand develops randomization-based and permutation tests valid under weak distributional
assumptions. Applications include regression discontinuity designs, small-cluster settings,
and tests for stochastic dominance at target points.
This paper provides a user's guide to the general theory of approximate randomization
tests (ARTs) developed in Canay, Romano, and Shaikh (2017) when specialized to linear
regressions with clustered data. An important feature of the methodology is that it
applies to settings in which the number of clusters is small — even as small as five.
We provide a step-by-step algorithmic description of how to implement the test and
construct confidence intervals for the parameter of interest. We also present three
novel results: the method admits an equivalent implementation based on weighted
scores; the test and confidence intervals are invariant to studentization; and the
confidence intervals are convex for scalar parameters. Companion R and Stata packages
facilitate implementation and replication of the empirical exercises.
In the regression discontinuity design (RDD), it is common practice to assess design
credibility by testing continuity of the density of the running variable at the
cutoff. We propose an approximate sign test for density continuity based on
g-order statistics, studied under two asymptotic frameworks: one where the number
of local observations is fixed as sample size grows, and one where it grows slowly.
Under both, the test has limiting rejection probability under the null not exceeding
the nominal level. The test is easy to implement, asymptotically valid under weaker
conditions than competing methods, and exhibits finite-sample validity under stronger
conditions. Simulations confirm good size control. We apply the test to Lee (2008)'s
well-known RDD application on incumbency advantage.
It is common in the RDD to assess design credibility by testing whether means of
baseline covariates do not change at the cutoff. We propose a permutation test based
on induced ordered statistics for the null hypothesis of continuity of the
distribution of baseline covariates, and introduce a novel asymptotic framework
that keeps the number of local observations fixed as the overall sample size grows.
The new test is easy to implement, asymptotically valid under weak conditions,
exhibits finite-sample validity under stronger conditions, and has favorable power
relative to tests based on means. Simulations show excellent size control across
designs. We apply the test to Lee (2008)'s RDD application on incumbency advantage.
This paper develops a theory of randomization tests under an approximate symmetry
assumption. Randomization tests provide a general means of constructing tests that
control size in finite samples when the data distribution exhibits symmetry under
the null. We provide conditions under which the same construction yields tests that
asymptotically control the probability of false rejection when the limiting
distribution of a function of the data exhibits symmetry under the null. An important
application is to settings where data are grouped into a fixed number of clusters
with many observations each. We show the approximate symmetry requirement holds
under weak assumptions, including cluster heterogeneity and cross-cluster dependence.
This paper studies Hodges–Lehmann optimality of tests in a general setup, comparing
tests by the exponential rates of growth to one of their power functions at a fixed
alternative while keeping asymptotic sizes bounded. We present two sets of sufficient
conditions for Hodges–Lehmann optimality that extend the scope of the analysis to
setups not covered by existing conditions. Applications to testing moment conditions
and overidentifying restrictions show that the empirical likelihood test satisfies
our new conditions, and that the GMM and generalized empirical likelihood tests are
Hodges–Lehmann optimal under mild primitive conditions, supporting the view that
Hodges–Lehmann optimality is a weak asymptotic requirement.
Experiments & Clustered Data
This strand covers inference in randomized experiments with covariate-adaptive assignment
and settings where treatment is assigned at the cluster level. Topics include bootstrap
validity with few clusters, non-ignorable cluster sizes, and decomposition of treatment
effects when outcomes are delayed.
This paper studies settings where the outcome is not immediately realized after
treatment assignment — a feature ubiquitous in empirical settings. The period
between treatment and outcome allows other observed actions to occur and affect
the outcome. We study several regression-based estimands routinely used to capture
the average treatment effect and shed light on their interpretation in terms of
ceteris paribus effects, indirect causal effects, and selection terms. The three
most popular estimands do not generally satisfy strong sign preservation: they may
be negative even when treatment positively affects the outcome conditional on any
combination of other actions. The most popular regression that includes other
actions as controls satisfies strong sign preservation if and only if those actions
are mutually exclusive binary variables.
This paper considers inference in cluster randomized experiments when cluster sizes
are non-ignorable — that is, when treatment effects may depend non-trivially on
cluster sizes. We frame the analysis in a super-population framework in which
cluster sizes are random, departing from earlier work that treats them as fixed. We
distinguish between the equally-weighted and size-weighted cluster-level average
treatment effects, and provide inference methods for each in an asymptotic framework
where the number of clusters grows and treatment is assigned via covariate-adaptive
stratified randomization. We also permit sampling only a subset of units within
each cluster and demonstrate implications for commonly used estimators.
This paper studies the wild bootstrap–based test of Cameron, Gelbach, and Miller
(2008). Existing analyses require a large number of clusters. In an asymptotic
framework with a small number of clusters, we provide conditions under which an
unstudentized version of the test is valid, including homogeneity-like restrictions
on the distribution of covariates. We establish that a studentized version may only
overreject by an amount that decreases exponentially with the number of clusters. A
qualitatively similar result holds for score bootstrap-based tests, which permit
testing in nonlinear models.
This paper studies inference in randomized controlled trials with covariate-adaptive
randomization and multiple treatments. We study inference about the average effect of
one or more treatments relative to other treatments or a control. Tests based on a
fully saturated linear regression using the usual heteroskedasticity-consistent
standard errors are invalid — they may have limiting rejection probability strictly
greater than the nominal level. Tests based on suitable variance estimators that we
provide are instead exact. For the case where treatment proportions are constant
across strata, we additionally consider tests based on a linear regression with
strata fixed effects.
This paper studies inference for the average treatment effect in randomized
controlled trials with covariate-adaptive randomization — schemes that stratify on
baseline covariates and assign treatment to achieve balance within strata. The usual
two-sample t-test is conservative, with limiting rejection probability strictly less
than the nominal level. A simple adjustment to the standard error yields a test that
is exact. Analogous results hold for the t-test in a regression of outcomes on
treatment and strata indicators. We also propose a covariate-adaptive permutation
test that only permutes treatment status within strata and show it is exact for
relevant special cases.
Other Methods
Work on detecting discrimination through outcome tests, testability of nonparametric
identification with endogeneity, quantile regression with panel data, and moment
selection in GMM.
The decisions of judges, lenders, journal editors, and other gatekeepers often
generate significant disparities across affected groups. An important question is
whether these disparities reflect relevant differences in individual characteristics
or biased decision making. Becker (1957, 1993) proposed an outcome test of bias
based on differences in post-decision outcomes across groups, inspiring a large
empirical literature. We offer a methodological blueprint for this approach, showing
that models underpinning outcome tests can be recast as Roy models, since
heterogeneous potential outcomes enter directly into the decision maker's choice
equation. Different members of the Roy model family are distinguished by the
tightness of the link between potential outcomes and decisions, with important
implications for defining bias and deriving logically valid outcome tests.
This paper examines three hypothesis testing problems arising in the identification
of some nonparametric models with endogeneity. The first concerns testing necessary
conditions for identification involving mean independence restrictions (completeness
conditions). The second and third concern testing identification directly in models
involving quantile independence restrictions. For each problem, we provide conditions
under which any test will have power no greater than size against any alternative —
concluding that no nontrivial tests for these hypothesis testing problems exist.
This paper provides sufficient conditions that point identify a quantile regression
model with fixed effects, and proposes a simple transformation of the data that
eliminates the fixed effects when they are location shifters. The resulting two-step
estimator is consistent and asymptotically normal as both n and T grow. The
estimator is straightforward to compute and can be implemented in standard
econometrics packages.
This paper proposes a procedure to estimate linear models when the number of
instruments is large, addressing the tradeoff between asymptotic efficiency (favoring
more instruments) and bias (adversely affected by adding instruments). I propose a
kernel weighted GMM estimator using a trapezoidal kernel. I derive the higher-order
mean squared error and show that the trapezoidal kernel generates lower asymptotic
variance than regular kernels. Monte Carlo simulations show the estimator performs
on par with optimal-instrument estimators and improves upon a GMM estimator that
uses all available instruments.