1,005 research outputs found
Nonparametric identification of positive eigenfunctions
Important features of certain economic models may be revealed by studying
positive eigenfunctions of appropriately chosen linear operators. Examples
include long-run risk-return relationships in dynamic asset pricing models and
components of marginal utility in external habit formation models. This paper
provides identification conditions for positive eigenfunctions in nonparametric
models. Identification is achieved if the operator satisfies two mild
positivity conditions and a power compactness condition. Both existence and
identification are achieved under a further non-degeneracy condition. The
general results are applied to obtain new identification conditions for
external habit formation models and for positive eigenfunctions of pricing
operators in dynamic asset pricing models
Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression
We study the problem of nonparametric regression when the regressor is
endogenous, which is an important nonparametric instrumental variables (NPIV)
regression in econometrics and a difficult ill-posed inverse problem with
unknown operator in statistics. We first establish a general upper bound on the
sup-norm (uniform) convergence rate of a sieve estimator, allowing for
endogenous regressors and weakly dependent data. This result leads to the
optimal sup-norm convergence rates for spline and wavelet least squares
regression estimators under weakly dependent data and heavy-tailed error terms.
This upper bound also yields the sup-norm convergence rates for sieve NPIV
estimators under i.i.d. data: the rates coincide with the known optimal
-norm rates for severely ill-posed problems, and are power of
slower than the optimal -norm rates for mildly ill-posed problems. We then
establish the minimax risk lower bound in sup-norm loss, which coincides with
our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV
estimators. This sup-norm rate optimality provides another justification for
the wide application of sieve NPIV estimators. Useful results on
weakly-dependent random matrices are also provided
Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions
We show that spline and wavelet series regression estimators for weakly
dependent regressors attain the optimal uniform (i.e. sup-norm) convergence
rate of Stone (1982), where is the number of
regressors and is the smoothness of the regression function. The optimal
rate is achieved even for heavy-tailed martingale difference errors with finite
th absolute moment for . We also establish the asymptotic
normality of t statistics for possibly nonlinear, irregular functionals of the
conditional mean function under weak conditions. The results are proved by
deriving a new exponential inequality for sums of weakly dependent random
matrices, which is of independent interest.Comment: forthcoming in Journal of Econometric
Counterfactual Sensitivity and Robustness
Researchers frequently make parametric assumptions about the distribution of
unobservables when formulating structural models. Such assumptions are
typically motived by computational convenience rather than economic theory and
are often untestable. Counterfactuals can be particularly sensitive to such
assumptions, threatening the credibility of structural modeling exercises. To
address this issue, we leverage insights from the literature on ambiguity and
model uncertainty to propose a tractable econometric framework for
characterizing the sensitivity of counterfactuals with respect to a
researcher's assumptions about the distribution of unobservables in a class of
structural models. In particular, we show how to construct the smallest and
largest values of the counterfactual as the distribution of unobservables spans
nonparametric neighborhoods of the researcher's assumed specification while
other `structural' features of the model, e.g. equilibrium conditions, are
maintained. Our methods are computationally simple to implement, with the
nuisance distribution effectively profiled out via a low-dimensional convex
program. Our procedure delivers sharp bounds for the identified set of
counterfactuals (i.e. without parametric assumptions about the distribution of
unobservables) as the neighborhoods become large. Over small neighborhoods, we
relate our procedure to a measure of local sensitivity which is further
characterized using an influence function representation. We provide a suitable
sampling theory for plug-in estimators and apply our procedure to models of
strategic interaction and dynamic discrete choice
Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression
This paper makes several important contributions to the literature about
nonparametric instrumental variables (NPIV) estimation and inference on a
structural function and its functionals. First, we derive sup-norm
convergence rates for computationally simple sieve NPIV (series 2SLS)
estimators of and its derivatives. Second, we derive a lower bound that
describes the best possible (minimax) sup-norm rates of estimating and
its derivatives, and show that the sieve NPIV estimator can attain the minimax
rates when is approximated via a spline or wavelet sieve. Our optimal
sup-norm rates surprisingly coincide with the optimal root-mean-squared rates
for severely ill-posed problems, and are only a logarithmic factor slower than
the optimal root-mean-squared rates for mildly ill-posed problems. Third, we
use our sup-norm rates to establish the uniform Gaussian process strong
approximations and the score bootstrap uniform confidence bands (UCBs) for
collections of nonlinear functionals of under primitive conditions,
allowing for mildly and severely ill-posed problems. Fourth, as applications,
we obtain the first asymptotic pointwise and uniform inference results for
plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss
(DL) welfare functionals under low-level conditions when demand is estimated
via sieve NPIV. Empiricists could read our real data application of UCBs for
exact CS and DL functionals of gasoline demand that reveals interesting
patterns and is applicable to other markets.Comment: This paper is a major extension of Sections 2 and 3 of our Cowles
Foundation Discussion Paper CFDP1923, Cemmap Working Paper CWP56/13 and arXiv
preprint arXiv:1311.0412 [math.ST]. Section 3 of the previous version of this
paper (dealing with data-driven choice of sieve dimension) is currently being
revised as a separate pape
Monte Carlo Confidence Sets for Identified Sets
In complicated/nonlinear parametric models, it is generally hard to know
whether the model parameters are point identified. We provide computationally
attractive procedures to construct confidence sets (CSs) for identified sets of
full parameters and of subvectors in models defined through a likelihood or a
vector of moment equalities or inequalities. These CSs are based on level sets
of optimal sample criterion functions (such as likelihood or optimally-weighted
or continuously-updated GMM criterions). The level sets are constructed using
cutoffs that are computed via Monte Carlo (MC) simulations directly from the
quasi-posterior distributions of the criterions. We establish new Bernstein-von
Mises (or Bayesian Wilks) type theorems for the quasi-posterior distributions
of the quasi-likelihood ratio (QLR) and profile QLR in partially-identified
regular models and some non-regular models. These results imply that our MC CSs
have exact asymptotic frequentist coverage for identified sets of full
parameters and of subvectors in partially-identified regular models, and have
valid but potentially conservative coverage in models with reduced-form
parameters on the boundary. Our MC CSs for identified sets of subvectors are
shown to have exact asymptotic coverage in models with singularities. We also
provide results on uniform validity of our CSs over classes of DGPs that
include point and partially identified models. We demonstrate good
finite-sample coverage properties of our procedures in two simulation
experiments. Finally, our procedures are applied to two non-trivial empirical
examples: an airline entry game and a model of trade flows
Nonparametric Stochastic Discount Factor Decomposition
Stochastic discount factor (SDF) processes in dynamic economies admit a
permanent-transitory decomposition in which the permanent component
characterizes pricing over long investment horizons. This paper introduces an
empirical framework to analyze the permanent-transitory decomposition of SDF
processes. Specifically, we show how to estimate nonparametrically the solution
to the Perron-Frobenius eigenfunction problem of Hansen and Scheinkman (2009).
Our empirical framework allows researchers to (i) recover the time series of
the estimated permanent and transitory components and (ii) estimate the yield
and the change of measure which characterize pricing over long investment
horizons. We also introduce nonparametric estimators of the continuation value
function in a class of models with recursive preferences by reinterpreting the
value function recursion as a nonlinear Perron-Frobenius problem. We establish
consistency and convergence rates of the eigenfunction estimators and
asymptotic normality of the eigenvalue estimator and estimators of related
functionals. As an application, we study an economy where the representative
agent is endowed with recursive preferences, allowing for general (nonlinear)
consumption and earnings growth dynamics
Measurements of Stellar Properties through Asteroseismology: A Tool for Planet Transit Studies
Oscillations occur in stars of most masses and essentially all stages of
evolution. Asteroseismology is the study of the frequencies and other
properties of stellar oscillations, from which we can extract fundamental
parameters such as density, mass, radius, age and rotation period. We present
an overview of asteroseismic analysis methods, focusing on how this technique
may be used as a tool to measure stellar properties relevant to planet transit
studies. We also discuss details of the Kepler Asteroseismic Investigation --
the use of asteroseismology on the Kepler mission in order to measure basic
stellar parameters. We estimate that applying asteroseismology to stars
observed by Kepler will allow the determination of stellar mean densities to an
accuracy of 1%, radii to 2-3%, masses to 5%, and ages to 5-10% of the
main-sequence lifetime. For rotating stars, the angle of inclination can also
be determined.Comment: To appear in the Proceedings of the 253rd IAU Symposium: "Transiting
Planets", May 2008, Cambridge, M
Correcting stellar oscillation frequencies for near-surface effects
In helioseismology, there is a well-known offset between observed and
computed oscillation frequencies. This offset is known to arise from improper
modeling of the near-surface layers of the Sun, and a similar effect must occur
for models of other stars. Such an effect impedes progress in asteroseismology,
which involves comparing observed oscillation frequencies with those calculated
from theoretical models. Here, we use data for the Sun to derive an empirical
correction for the near-surface offset, which we then apply three other stars
(alpha Cen A, alpha Cen B and beta Hyi). The method appears to give good
results, in particular providing an accurate estimate of the mean density of
each star.Comment: accepted for publication in ApJ Letter
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