
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): under revision for
resubmission to Journal of
Econometrics.
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. 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. An Empirical Process PValue Test for Handling
Nuisance and Tuning Parameters (2014): submitted.
We present an empirical process
method for smoothing a pvalue or the related test statistic in the presence of
nuisance and/or tuning parameter l. 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 l 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. 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 e_ects, a heavy tail robust white noise test, and a
consistent (non)identi_cation 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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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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