
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. Testing for Granger Causality with Mixed
Frequency Data (2014: with E. Ghysels and K. Motegi): revised and resubmitted to Journal of Econometrics (1^{st}
round).
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
invited resubmission to Journal of
Business and Economic Statistics (1^{st} round)
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. Simple Granger Causality Tests for Mixed Frequency Data (2015: with E .Ghysels
and K. Motegi) : submitted.
This paper presents simple
Granger causality tests applicable to any mixed frequency sampling data
setting, and feature remarkable power properties even with relatively small
low frequency data samples and a considerable wedge between sampling
frequencies (for example, quarterly and daily or weekly data). Our tests are
based on a seemingly overlooked, but simple, dimension reduction technique
for regression models. If the number of parameters of interest is large then
in small or even large samples any of the trilogy test statistics may not be
well approximated by their asymptotic distribution. A bootstrap method can be
employed to improve empirical test size, but this generally results in a loss
of power. A shrinkage estimator can be employed, including Lasso, Adaptive
Lasso, or Ridge Regression, but these are valid only under a sparsity assumption which does not apply to Granger
causality tests. The procedure, which is of general interest when testing
potentially large sets of parameter restrictions, involves multiple
parsimonious regression models where each model regresses
a low frequency variable onto only one individual lag or lead of a high
frequency series, where that lag or lead slope parameter is necessarily zero
under the null hypothesis of noncausality. Our test is then based on a max
test statistic that selects the largest squared estimator among all parsimonious
regression models. Parsimony ensures sharper estimates and therefore improved
power in small samples. Inference requires a simple simulationbootstrap step
since the test statistic has a nonstandard limit distribution. We show via
Monte Carlo simulations that the max test is more powerful than existing mixed frequency Granger causality tests in small
samples. An empirical application examines Granger causality over rolling windows
of U.S.\ macroeconomic data from 19622013 using a mixture of high and low
frequency data. 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 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 unconfoundedness of
treatment assignment conditional on covariates, such limited overlap is
manifested in the propensity score for certain units being very close (but
not equal) to 0 or 1. This renders IPW estimators possibly heavy tailed, and
with a slower than root(n) 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 or truncation is ultimately based on the
covariates, ignoring important information about the inverse probability
weighted 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 is robust to
limited overlap more generally. In terms of the propensity score, which is
generally unknown, we plugin its parametric estimator in the infeasible Z,
and then negligibly trim the resulting feasible Z adaptively by its large
values. Trimming can lead to bias in general, hence we estimate and remove
the bias using new theory and methods. Our estimator sidesteps
dimensionality, bias and poor correspondence properties associated with
trimming by the covariates or propensity score. Monte Carlo experiments
demonstrate that trimming by the covariates or the propensity score requires
the removal of a substantial portion of the sample to render a low bias and
close to normal estimator, while our estimator has low bias and meansquared
error, and is close to normal, based on very little trimming. An Empirical Process PValue Test for Handling
Nuisance and Tuning Parameters (2015): under revision for Journal of the American Statistical Association (reject and resubmit)
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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(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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