
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) : revised and resubmitted
to 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. 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 _t 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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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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