Jonathan B. Hill
________________________________________________
Working Papers
(under submission or invited
revision for publication)
Consistent GMM Residuals-Based
Tests of Functional Form (2008): Revised and resubmitted to Econometric
Reviews
|
|
|
This
paper presents a consistent GMM residuals-based test of functional form for
time series models. By relating two moment conditions we deliver a vector
moment condition in which at least one element must be non-zero under the
alternative of model mis-specification: the test will never fail to detect
model mis-specification of any form for large samples, and is asymptotically
chi-squared under the null, allowing for fast and simple inference. A
simulation study reveals the superiority of a randomized test: randomly
selecting the nuisance parameter leads to more power than supremum-tests, and
obtains empirical power nearly equivalent to most powerful tests in most cases
for even relatively small n.
Tail and Non-Tail Memory with Applications to Extreme Value and Robust
Statistics (2008): Revised
and resubmitted to Econometric Theory
|
|
|
New
notions of tail and non-tail dependence are used to characterize separately
extremal and non-extremal information, including tail log-exceedances and
events, and tail-trimmed levels. We prove Near Epoch Dependence (McLeish 1975,
Gallant and White 1988) and L0-Approximability (Pötscher and Prucha 1991) are
equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central
limit theory for important extreme value and robust (tail-trimmed) statistics
under general conditions of memory and heterogeneity. We apply the theory to
characterize the extremal and non-extremal memory properties of possibly very
heavy-tailed GARCH processes and distributed lags, including Asymmetric GARCH
and Nonlinear Autoregressions. Finally, we prove asymptotic normality of tail index
and tail dependence estimators, and a tail-trimmed sum of these persistent,
heterogeneous heavy-tailed data resulting in some of the most general limit
theory available in the extreme value and robust estimation literatures.
Gaussian Tests of 'Extremal White Noise' for Dependent, Heterogeneous, Heavy Tailed Time Series with an Application (2008)
|
|
|
We develop a
portmanteau test of extremal serial dependence. The test statistic is
asymptotically chi-squared under a null of "extremal white noise", as
long as extremes are Near-Epoch-Dependent, covering linear and nonlinear
distributed lags, stochastic volatility, and GARCH processes with possibly unit
or explosive roots. We apply tail specific tests to equity market and exchange
rate returns.
Extremal Memory of Stochastic Volatility with Applications to Tail Shape and Tail Dependence Inference (2008)
|
|
|
We derive the exact joint tail shape of
multivariate stochastic volatility processes with shocks that have a regularly varying
distribution tail. Tail dependence is a function of linear volatility memory
parametrically represented by tail scales, while tail power indices do not
provide any relevant dependence information. Further, a linear function of tail
events and exceedances is itself linearly independent, implying tail index
inference is identical to the iid case. The results are also applied to
non-parametric tail dependence estimation. Both applications hold for a large
array of linear and nonlinear volatility data generating processes.
Robust Estimation and Inference for Extremal Dependence in Time Series (2009)
|
|
|
Dependence between extreme values is
predominantly measured by first assuming a parametric joint distribution
function, and almost always for otherwise marginally iid processes. We develop
semi-nonparametric and nonparametric measures, estimators and tests of
bivariate tail dependence for non-iid data based on tail exceedances and
events. The measures and estimators capture extremal dependence decay over time
and can be re-scaled to provide robust estimators of canonical conditional tail
probability and tail copula notions of tail dependence. Unlike extant
offerings, the tests obtain asymptotic power of one against infinitessimal
deviations from tail independence. Further, the estimators apply to dependent,
heterogeneous processes with or without extremal dependence and irrespective of
non-extremal properties and joint distribution specifications. Finally, we
study the extremal associations within and between equity returns in the U.S.,
U.K. and Japan.
Are There Common Values on BC
Timber Sales? A Tail-Index Nonparametric Test (2009, with A. Shneyerov)
|
|
|
We develop a new nonparametric test of common
values in first-price auctions with a binding reserve price. The test is based
on the behavior of the CDF of bids near the reserve price. We show that this
behavior is drastically different under private values (PV) and common values
(CV). Next, we show that the problem of discriminating between PV and CV is
equivalent to the problem of estimating the lower tail index of the bids
distribution. Our approach allows for unobserved auction heterogeneity of an
arbitrary form, and in particular doesn't require the number of potential
bidders to be observable. Drawing on the existing and recent literature on tail
index estimation, we characterize the B. Hill (1975) tail index estimator for
panels with stochastic dimension and develop semi- and nonparametric estimators
of the asymptotic variance for robust inference. We implement the test on a
sample of British Columbia timber auctions and find strong support for CV.
Stochastically Weighted Average Conditional Moment Tests of Functional
Form (2008)
|
|
|
We develop a new consistent conditional moment
test of functional form based on nuisance parameter indexed sample moments. We
reduce the nuisance parameter space to known countable sets, provide a new
vantage into why existing parametric moment condition tests work, and uncover a
new class of revealing weights. These results are exploited to construct a
weighted average conditional moment test, where the weights are possibly
stochastic in an arbitrary way, integer-indexed and flexible enough to cover a
range of tests from Crámer-von Mises to Kolmogorov-Smirnov. Using a variety of
weights the test statistic obtains power that nearly matches most powerful
tests against a variety of alternatives.
Limit Theory for Kernel-Self Normalized Tail-Trimmed Sums of Dependent, Heterogeneous Data with Applications (2009) |
|
|
Although robust estimation methods were
formalized by the late 1800's, data trimming and truncation for non-iid data
has received little attention. We establish sufficient conditions for weak laws
of large numbers and Gaussian central limit theorems for tail-trimmed sums of
dependent heterogeneous data, where the trimmed process itself satisfies a
mixingale condition. The sum is self-normalized with a consistent kernel
variance estimator for the central limit, so the rate of convergence, tail
thickness and memory persistence do not need to be specified beyond fairly
minimal regulatory conditions, hence robust inference is available. The theory
applies to martingale differences, mixing, geometrically ergodic, and mixingale
processes, including linear and nonlinear distributed lags, and linear and
nonlinear random volatility. We show how the results imply asymptotic normality
for a Tail Trimmed Least Squares estimator for models of infinite variance
data.