100 research outputs found
Adaptive Test of Conditional Moment Inequalities
In this paper, I construct a new test of conditional moment inequalities,
which is based on studentized kernel estimates of moment functions with many
different values of the bandwidth parameter. The test automatically adapts to
the unknown smoothness of moment functions and has uniformly correct asymptotic
size. The test has high power in a large class of models with conditional
moment inequalities. Some existing tests have nontrivial power against
n^{-1/2}-local alternatives in a certain class of these models whereas my
method only allows for nontrivial testing against (n/\log n)^{-1/2}-local
alternatives in this class. There exist, however, other classes of models with
conditional moment inequalities where the mentioned tests have much lower power
in comparison with the test developed in this paper
Comparison and anti-concentration bounds for maxima of Gaussian random vectors
Slepian and Sudakov-Fernique type inequalities, which compare expectations of
maxima of Gaussian random vectors under certain restrictions on the covariance
matrices, play an important role in probability theory, especially in empirical
process and extreme value theories. Here we give explicit comparisons of
expectations of smooth functions and distribution functions of maxima of
Gaussian random vectors without any restriction on the covariance matrices. We
also establish an anti-concentration inequality for the maximum of a Gaussian
random vector, which derives a useful upper bound on the L\'{e}vy concentration
function for the Gaussian maximum. The bound is dimension-free and applies to
vectors with arbitrary covariance matrices. This anti-concentration inequality
plays a crucial role in establishing bounds on the Kolmogorov distance between
maxima of Gaussian random vectors. These results have immediate applications in
mathematical statistics. As an example of application, we establish a
conditional multiplier central limit theorem for maxima of sums of independent
random vectors where the dimension of the vectors is possibly much larger than
the sample size.Comment: 22 pages; discussions and references update
Inference on causal and structural parameters using many moment inequalities
This paper considers the problem of testing many moment inequalities where
the number of moment inequalities, denoted by , is possibly much larger than
the sample size . There is a variety of economic applications where solving
this problem allows to carry out inference on causal and structural parameters,
a notable example is the market structure model of Ciliberto and Tamer (2009)
where with being the number of firms that could possibly enter
the market. We consider the test statistic given by the maximum of
Studentized (or -type) inequality-specific statistics, and analyze various
ways to compute critical values for the test statistic. Specifically, we
consider critical values based upon (i) the union bound combined with a
moderate deviation inequality for self-normalized sums, (ii) the multiplier and
empirical bootstraps, and (iii) two-step and three-step variants of (i) and
(ii) by incorporating the selection of uninformative inequalities that are far
from being binding and a novel selection of weakly informative inequalities
that are potentially binding but do not provide first order information. We
prove validity of these methods, showing that under mild conditions, they lead
to tests with the error in size decreasing polynomially in while allowing
for being much larger than , indeed can be of order
for some . Importantly, all these results hold without any restriction
on the correlation structure between Studentized statistics, and also hold
uniformly with respect to suitably large classes of underlying distributions.
Moreover, in the online supplement, we show validity of a test based on the
block multiplier bootstrap in the case of dependent data under some general
mixing conditions.Comment: This paper was previously circulated under the title "Testing many
moment inequalities
- …