
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. Robust
Generalized Empirical Likelihood for Heavy Tailed Autoregressions with
Conditionally Heteroscedastic Errors (2013)
: revised and resubmitted to Journal of
Multivariate Analysis.
We present a robust Generalized Empirical
Likelihood estimator and confidence region for the parameters of an
autoregression that may have a heavy tailed heteroscedastic error. The
estimator exploits two transformations for heavy tail robustness: a
redescending transformation of the error that robustifies against innovation
outliers, and weighted least squares instruments that ensure robustness
against heavy tailed regressors. Our estimator is consistent for the true
parameter and asymptotically normally distributed irrespective of heavy tails. Parameter
Estimation Robust to LowFrequency Contamination (2014: with A. McCloskey) : submitted.
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. Testing for Granger Causality with Mixed
Frequency Data (2014: with E. Ghysels and K. Motegi): submitted.
It is well known that temporal aggregation has adverse effects
on Granger causality tests. Time series are often sampled at different
frequencies. This is typically ignored, and data are merely aggregated to the
common lowest frequency. We develop a set of Granger causality tests that
explicitly take advantage of 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 mixed frequency causality tests have higher
local asymptotic power as well as more power in finite samples compared to
conventional tests. An Empirical Process PValue Test when a
Nuisance Parameter is Present under Either or Both Hypotheses (2013): submitted.
We present an empirical process method for transforming
a test statistic pvalue in the presence of a nuisance parameter under either
hypothesis. The pvalue transform represents the uniform measure of the
parameter space, or occupation time [OT], on which the null hypothesis is
rejected. We reject at significance level a when the OT is greater than a, and the
asymptotic probability of a Type I error is bounded by α. Thus,
conveniently the OT both operates like a test statistic because large values
indicate rejection of the null, and like a pvalue compliment because its
values are bounded between 0 and 1 and rejection of the null occurs when the
OT is above a. Further, power in the original test naturally
translates to the OT test, while the OT test achieves a nontrivial power
improvement over the original test: even if the original test is not
consistent, as long as it has power on a dense subset of the nuisance
parameter space with Lebesgue measure greater than α then the OT test is
consistent. Finally, computation time is dramatically shorter than a popular
bootstrapsimulation method. Examples and numerical experiments are given
involving tests of functional form, GARCH effects and white noise robust to
heavy tails. Uniform Interval Estimation for an AR(1) Process with AR Errors (2014: with Deyuan Li and Liang Peng): submitted.
An empirical likelihood method
was recently proposed 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.
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. 

Econometrics Workshops 



Software 



Econometrics Links 



Data Sources 



Research Resources 



Journals 



Statistics Links 



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 



Photos 









































































































































































































































































































































































































