
Home Page of Jonathan B. Hill
Associate Professor of Economics University of North Carolina – Chapel Hill


CV (pdf) 



LINKS 

(Under
Submission or Invited Revision for Publication) GEL Estimation for
GARCH Models with Robust Empirical Likelihood Inference (2013: with Artem
Prokhorov): revised and resubmitted to Journal
of Econometrics (1^{st} round)
We construct a Generalized Empirical Likelihood estimator
for a GARCH(1,1) model with a possibly heavy tailed error. The estimator
imbeds tailtrimmed estimating equations allowing for overidentifying
conditions, asymptotic normality, efficiency and empirical likelihood based
confidence regions for very heavytailed random volatility data. We show the
implied probabilities from the tailtrimmed Continuously Updated Estimator
elevate weight for usable large values, assign large but not maximum weight
to extreme observations, and give the lowest weight to nonleverage points.
We derive a higher order expansion for GEL with imbedded tailtrimming
(GELITT), which reveals higher order bias and efficiency properties,
available when the GARCH error has a finite second moment. Higher asymptotics
for GEL without tailtrimming requires the error to have moments of
substantially higher order. We use first order asymptotics and higher order
bias to justify the choice of the number of trimmed observations in any given
sample. We also present robust versions of Generalized Empirical Likelihood
Ratio, Wald, and Lagrange Multiplier tests, and an efficient and heavy tail
robust moment estimator with an application to expected shortfall estimation.
Finally, we present a broad simulation study for GEL and GELITT, and demonstrate
profile weighted expected shortfall for the Russian Ruble  US Dollar
exchange rate. We show that tailtrimmed CUEGMM dominates other estimators
in terms of bias, mse and approximate normality. Uniform Interval Estimation for an AR(1) Process with AR Errors (2014: with Deyuan Li and Liang Peng): revised and resubmitted
to Statistica Sininca.
An empirical likelihood method
was proposed in Hill and Peng (2014) to construct a unified interval estimation
for the coefficient in an AR(1) model, regardless of whether the sequence was
stationary or near integrated. The error term, however, was assumed
independent, and this method fails when the errors are dependent. Testing for
a unit root in an AR(1) model has been studied in the literature for
dependent errors, but existing methods cannot be used to test for a near unit
root. In this paper, assuming the errors are governed by an AR(p) process, we
exploit the efficient empirical likelihood method to give a unified interval
for the coefficient by taking the structure of errors into account.
Furthermore, a jackknife empirical likelihood method is proposed to reduce
the computation of the empirical likelihood method when the order in the AR
errors is not small. A simulation study is conducted to examine the finite
sample behavior of the proposed methods. Testing for Granger Causality with Mixed
Frequency Data (2014: with E. Ghysels and K. Motegi): revised and resubmitted to Journal of Econometrics.
We develop Granger causality
tests that apply directly to data sampled at different frequencies. We show
that taking advantage of mixed frequency data allows us to better recover
causal relationships when compared to the conventional common low frequency
approach. We also show that the new causality tests have higher local
asymptotic power as well as more power in finite samples compared to
conventional tests. In an empirical application involving U.S. macroeconomic
indicators, we show that the mixed frequency approach and the low frequency
approach produce very different causal implications, with the former yielding
more intuitively appealing result. Parameter
Estimation Robust to LowFrequency Contamination (2014: with A. McCloskey) : under revision for Journal of Business and Economic
Statistics.
We provide methods to robustly estimate
the parameters of stationary ergodic shortmemory time series models in the
potential presence of additive lowfrequency contamination. The types of
contamination covered include level shifts (changes in mean) and monotone or
smooth time trends, both of which have been shown to bias parameter estimates
towards regions of persistence in a variety of contexts. The estimators
presented here minimize trimmed frequency domain quasimaximum likelihood
(FDQML) objective functions without requiring specification of the
lowfrequency contaminating component. We provide two approaches, allowing
for either thin or heavytailed data. When proper sample sizedependent
trimmings are used, the FDQML estimators are consistent and asymptotically
normal, asymptotically eliminating the presence of any spurious persistence.
These asymptotic results also hold in the absence of additive lowfrequency
contamination, enabling the practitioner to robustly estimate model
parameters without prior knowledge of whether contamination is present.
Popular time series models that fit into the framework of this article
include ARMA, stochastic volatility, GARCH and ARCH models. We explore the
finite sample properties of the trimmed FDQML estimators of the parameters of
some of these models, providing practical guidance on trimming choice.
Empirical estimation results suggest that a large portion of the apparent
persistence in certain volatility time series may indeed be spurious. Heavy
Tail Robust Frequency Domain Estimation (2014: with A. McCloskey) : submitted.
We develop heavy tail robust frequency
domain estimators for covariance stationary time series with a parametric
spectrum, including ARMA, GARCH and stochastic volatility. We use robust
techniques to reduce the moment requirement down to only a finite variance.
In particular, we negligibly trim the data, permitting both identification of
the parameter for the candidate model, and asymptotically normal frequency
domain estimators, while leading to a classic limit theory when the data have
a finite fourth moment. The transform itself can lead to asymptotic bias in
the limit distribution of our estimators when the fourth moment does not
exist, hence we correct the bias using extreme value theory that applies
whether tails decay according to a power law or not. In the case of
symmetrically distributed data, we compute the meansquarederror of our
biased estimator and characterize the meansquarederror minimization number
of sample extremes. A simulation experiment shows our QML estimator works
well and in general has lower bias than the standard estimator, even when the
process is Gaussian, suggesting robust methods have merit even for thin
tailed processes. Robust
Estimation and Inference for Average Treatment Effects (2014: with S. Chaudhuri) : submitted.
We study the probability tail properties
of the Inverse Probability Weighting (IPW) estimators of the Average
Treatment Effect (ATE) when there is limited overlap between the covariate
distributions of the treatment and control groups. Under strong ignorability,
such limited overlap is manifested in the propensity score for certain units
being very close (but not equal) to 0 or 1, rendering IPW estimators possibly
heavy tailed, and with a slow rate of convergence. Most existing estimators
are either based on the assumption of strict overlap, i.e. the propensity
score is bounded away from 0 and 1; or they truncate the propensity score; or
trim observations based on a variety of techniques based on covariate or
propensity score values. Trimming and truncation is ultimately based on the covariates,
ignoring important information about the random variable Z that identifies
ATE by E[Z]= ATE. Using a threshold crossing model for treatment assignment,
we explain the possibility of irregular identification by showing Z can have
a power law tail under limited overlap, with heavier or thinner tails based
on the relative distribution tails of the treatment assignment covariate and
error. We then propose a new tailtrimmed IPW estimator whose performance,
unlike that of the existing supposedly robust IPW estimators, is robust to
limited overlap more generally. This new estimator negligibly trims Z
adaptively by its large values and thus sidesteps dimensionality, bias and
poor correspondence properties associated with trimming by the covariates.
The estimator is asymptotically normal and unbiased whether there is limited
or strict overlap, and we use higher order asymptotics to determine a
reasonable trimming policy. Monte Carlo experiments demonstrate that our
estimator has potentially significantly lower bias and low meansquared
error, and is closer to normal, than existing IPW estimators. It also shows
that trimming by the covariates can require the removal of a substantial
portion of the sample to render a low bias and close to normal estimator. An Empirical Process PValue Test for Handling
Nuisance and Tuning Parameters (2015): submitted.
We
present an empirical process method for smoothing a pvalue, or the related
test statistic, in the presence of nuisance and/or tuning parameters. We do
not require root(n) Gaussian asymptotics, and our test can work in
conjunction with Andrews and Cheng (2012, 2013, 2014)'s methods of robust
inference when a subset of parameters are possibly weakly identified. Our
test is particularly relevant when Andrews and Cheng (2012, 2013, 2014)'s
methods are not appropriate: when the parameter is not logically estimated
nor necessarily part of the data generating process, or when root(n) Gaussian
asymptotics does not hold, including tests of omitted nonlinearity and GARCH
effects, and heavy tail robust tests, or distribution tail inference. Power
in the original test naturally translates to power in our test, while our
test can achieve a nontrivial power improvement over the original test.
Examples and numerical experiments are given involving tests of functional
form, GARCH effects, a heavy tail robust white noise test, and a consistent
(non)identification robust test of Smooth Transition Autoregression.
Robust
MEstimation for Heavy Tailed Nonlinear ARGARCH (2011).
We
develop new tailtrimmed Mestimation methods for heavy tailed Nonlinear
ARGARCH models. Tailtrimming allows both identification of the true
parameter and asymptotic normality for nonlinear models with asymmetric
errors. In heavy tailed cases the rate of convergence is infinitesimally
close to the highest possible amongst Mestimators for a particular loss
function, hence super root(n)convergence can be achieved in nonlinear AR
models with infinite variance errors, and arbitrarily near
root(n)convergence for GARCH with errors that have an infinite fourth
moment. We present a consistent estimator of the covariance matrix that
permits classic inference without knowledge of the rate of convergence, and
explore asymptotic covariance and bootstrap meansquarederror methods for
selecting trimming parameters. A simulation study shows the estimator trumps
existing ones for AR and GARCH models based on sharpness, approximate
normality, rate of convergence, and test accuracy. We then use the estimator
to study asset returns data. 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 seminonparametric and nonparametric measures, estimators and tests
of bivariate tail dependence for noniid data based on tail exceedances and
events. The measures and estimators capture extremal dependence decay over
time and can be rescaled 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 nonextremal properties and joint distribution
specifications. Finally, we study the extremal associations within and
between equity returns in the 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 chisquared under a null of "extremal white
noise", as long as extremes are NearEpochDependent, 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. 

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Miscellaneous Links 

Academic 



Personal
(places I’ve lived) 

Beijing 

San Fran. 

San Diego 
Miami 

Seattle 



Personal (favorite
places) 

Montreal 
Quebec City 

Bergen 
Tromso 

Eureka 
Cape Anne 

Edinburg 

Big
Sur 

Toledo Spain 
Connemara 



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