
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)
: submitted.
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. Robust
Score and Portmanteau Tests of Volatility Spillover (2012: with M. Aguilar): revised and resubmitted to Journal of Econometrics (2^{nd} round).
This paper presents a variety of tests
of volatility spillover that are robust to heavy tails generated by large
errors or GARCHtype feedback. The tests are couched in a general conditional
heteroskedasticity framework with idiosyncratic shocks that are only required
to have a finite variance if they are independent. We negligibly trim test
equations, or components of the equations, and construct heavy tail robust
score and portmanteau statistics. We develop the tailtrimmed sample
correlation coefficient for robust inference, and prove that its Gaussian
limit under the null hypothesis of no spillover has the same standardization
irrespective of tail thickness. Further, if spillover occurs within a
specified horizon, our test statistics obtain power of one asymptotically. A
Monte Carlo study shows our tests provide significant improvements over
extant GARCHbased tests of spillover, and we apply the tests to financial
returns data. 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 transform the data with a redescending function
that permits identification of the parameter for the candidate model, and
asymptotic normality, while leading to a classic limit theory when the data
have a finite fourth moment. The transform itself can lead to asymptotic bias
in our estimators, hence we correct the bias. 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. 

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(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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