8,967 research outputs found
Magnetic fields in the overshoot zone: The great escape
In order that magnetic flux be confined within the solar interior for times comparable to the solar cycle period it has been suggested that the bulk of the solar toroidal field is stored in the convectively stable overshoot region situated beneath the convection zone proper. Such a magnetic field, though, is still buoyant and is therefore subject to Rayleigh-Taylor type instabilities. The model problem of an isolated region of magnetic field embedded in a convectively stable atmosphere is considered. The fully nonlinear evolution of the two dimensional interchange of modes is studied, thereby shedding some light on one of the processes responsible for the escape of flux from the solar interior
Bootstrap-Based Inference for Cube Root Asymptotics
This paper proposes a valid bootstrap-based distributional approximation for
M-estimators exhibiting a Chernoff (1964)-type limiting distribution. For
estimators of this kind, the standard nonparametric bootstrap is inconsistent.
The method proposed herein is based on the nonparametric bootstrap, but
restores consistency by altering the shape of the criterion function defining
the estimator whose distribution we seek to approximate. This modification
leads to a generic and easy-to-implement resampling method for inference that
is conceptually distinct from other available distributional approximations. We
illustrate the applicability of our results with four examples in econometrics
and machine learning
The alpha-effect in rotating convection: a comparison of numerical simulations
Numerical simulations are an important tool in furthering our understanding
of turbulent dynamo action, a process that occurs in a vast range of
astrophysical bodies. It is important in all computational work that
comparisons are made between different codes and, if non-trivial differences
arise, that these are explained. Kapyla et al (2010: MNRAS 402, 1458) describe
an attempt to reproduce the results of Hughes & Proctor (2009: PRL 102, 044501)
and, by employing a different methodology, they arrive at very different
conclusions concerning the mean electromotive force and the generation of
large-scale fields. Here we describe why the simulations of Kapyla et al (2010)
are simply not suitable for a meaningful comparison, since they solve different
equations, at different parameter values and with different boundary
conditions. Furthermore we describe why the interpretation of Kapyla et al
(2010) of the calculation of the alpha-effect is inappropriate and argue that
the generation of large-scale magnetic fields by turbulent convection remains a
problematic issue.Comment: Submitted to MNRAS. 5 pages, 3 figure
Alternative Asymptotics and the Partially Linear Model with Many Regressors
Non-standard distributional approximations have received considerable
attention in recent years. They often provide more accurate approximations in
small samples, and theoretical improvements in some cases. This paper shows
that the seemingly unrelated "many instruments asymptotics" and "small
bandwidth asymptotics" share a common structure, where the object determining
the limiting distribution is a V-statistic with a remainder that is an
asymptotically normal degenerate U-statistic. We illustrate how this general
structure can be used to derive new results by obtaining a new asymptotic
distribution of a series estimator of the partially linear model when the
number of terms in the series approximation possibly grows as fast as the
sample size, which we call "many terms asymptotics"
Inference in Linear Regression Models with Many Covariates and Heteroskedasticity
The linear regression model is widely used in empirical work in Economics,
Statistics, and many other disciplines. Researchers often include many
covariates in their linear model specification in an attempt to control for
confounders. We give inference methods that allow for many covariates and
heteroskedasticity. Our results are obtained using high-dimensional
approximations, where the number of included covariates are allowed to grow as
fast as the sample size. We find that all of the usual versions of Eicker-White
heteroskedasticity consistent standard error estimators for linear models are
inconsistent under this asymptotics. We then propose a new heteroskedasticity
consistent standard error formula that is fully automatic and robust to both
(conditional)\ heteroskedasticity of unknown form and the inclusion of possibly
many covariates. We apply our findings to three settings: parametric linear
models with many covariates, linear panel models with many fixed effects, and
semiparametric semi-linear models with many technical regressors. Simulation
evidence consistent with our theoretical results is also provided. The proposed
methods are also illustrated with an empirical application
Optimal Bandwidth Choice for Robust Bias Corrected Inference in Regression Discontinuity Designs
Modern empirical work in Regression Discontinuity (RD) designs often employs
local polynomial estimation and inference with a mean square error (MSE)
optimal bandwidth choice. This bandwidth yields an MSE-optimal RD treatment
effect estimator, but is by construction invalid for inference. Robust bias
corrected (RBC) inference methods are valid when using the MSE-optimal
bandwidth, but we show they yield suboptimal confidence intervals in terms of
coverage error. We establish valid coverage error expansions for RBC confidence
interval estimators and use these results to propose new inference-optimal
bandwidth choices for forming these intervals. We find that the standard
MSE-optimal bandwidth for the RD point estimator is too large when the goal is
to construct RBC confidence intervals with the smallest coverage error. We
further optimize the constant terms behind the coverage error to derive new
optimal choices for the auxiliary bandwidth required for RBC inference. Our
expansions also establish that RBC inference yields higher-order refinements
(relative to traditional undersmoothing) in the context of RD designs. Our main
results cover sharp and sharp kink RD designs under conditional
heteroskedasticity, and we discuss extensions to fuzzy and other RD designs,
clustered sampling, and pre-intervention covariates adjustments. The
theoretical findings are illustrated with a Monte Carlo experiment and an
empirical application, and the main methodological results are available in
\texttt{R} and \texttt{Stata} packages
On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference
Nonparametric methods play a central role in modern empirical work. While
they provide inference procedures that are more robust to parametric
misspecification bias, they may be quite sensitive to tuning parameter choices.
We study the effects of bias correction on confidence interval coverage in the
context of kernel density and local polynomial regression estimation, and prove
that bias correction can be preferred to undersmoothing for minimizing coverage
error and increasing robustness to tuning parameter choice. This is achieved
using a novel, yet simple, Studentization, which leads to a new way of
constructing kernel-based bias-corrected confidence intervals. In addition, for
practical cases, we derive coverage error optimal bandwidths and discuss
easy-to-implement bandwidth selectors. For interior points, we show that the
MSE-optimal bandwidth for the original point estimator (before bias correction)
delivers the fastest coverage error decay rate after bias correction when
second-order (equivalent) kernels are employed, but is otherwise suboptimal
because it is too "large". Finally, for odd-degree local polynomial regression,
we show that, as with point estimation, coverage error adapts to boundary
points automatically when appropriate Studentization is used; however, the
MSE-optimal bandwidth for the original point estimator is suboptimal. All the
results are established using valid Edgeworth expansions and illustrated with
simulated data. Our findings have important consequences for empirical work as
they indicate that bias-corrected confidence intervals, coupled with
appropriate standard errors, have smaller coverage error and are less sensitive
to tuning parameter choices in practically relevant cases where additional
smoothness is available
The Relative Space: Space Measurements on a Rotating Platform
We introduce here the concept of relative space, an extended 3-space which is
recognized as the only space having an operational meaning in the study of the
space geometry of a rotating disk. Accordingly, we illustrate how space
measurements are performed in the relative space, and we show that an old-aged
puzzling problem, that is the Ehrenfest's paradox, is explained in this purely
relativistic context. Furthermore, we illustrate the kinematical origin of the
tangential dilation which is responsible for the solution of the Ehrenfest's
paradox.Comment: 14 pages, 2 EPS figures, LaTeX, to appear in the European Journal of
Physic
A Random Attention Model
This paper illustrates how one can deduce preference from observed choices
when attention is not only limited but also random. In contrast to earlier
approaches, we introduce a Random Attention Model (RAM) where we abstain from
any particular attention formation, and instead consider a large class of
nonparametric random attention rules. Our model imposes one intuitive
condition, termed Monotonic Attention, which captures the idea that each
consideration set competes for the decision-maker's attention. We then develop
revealed preference theory within RAM and obtain precise testable implications
for observable choice probabilities. Based on these theoretical findings, we
propose econometric methods for identification, estimation, and inference of
the decision maker's preferences. To illustrate the applicability of our
results and their concrete empirical content in specific settings, we also
develop revealed preference theory and accompanying econometric methods under
additional nonparametric assumptions on the consideration set for binary choice
problems. Finally, we provide general purpose software implementation of our
estimation and inference results, and showcase their performance using
simulations
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