
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) 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. A Smoothed PValue Test When There is a
Nuisance Parameter under the Alternative (2015): under revision for Journal of the American Statistical Association (reject and resubmit)
We present a new test for
cases when there is a nuisance parameter l under the alternative hypothesis. The test exploits
the pvalue occupation time [PVOT], the measure of the subset of l on which a pvalue test based on a test statistic T_{n}(l) rejects the null hypothesis. The PVOT has only
been explored in Hill and Aguilar (2013) and Hill (2012) as a way to smooth
over a trimming parameter for heavy tail robust test statistics. Our key
contributions are: (i) we show that a weighted average local power of a test
based on T_{n}(l) is identically a weighted average mean PVOT, and the PVOT used for
our test is therefore a point estimate of the weighted average probability of
PV test rejection; (ii) the critical value of our test is the significance
level itself, making inference easy (as opposed to supremum and average test
statistic transforms which require a bootstrap method for pvalue
computation); (iii) we only require T_{n}(l) to have a known or bootstrappable limit distribution,
hence we do not require root(n)Gaussian asymptotics as is nearly always
assumed; (iv) the pvalue may reflect local asymptotics, allowing for weakly
identified models; and (v) in a numerical experiment in which local
asymptotic power is computed, the PVOT test is virtually equivalent to a test
with the greatest weighted average power. We give examples of PVOT tests of
omitted nonlinearity, GARCH effects and structural breaks, and a simulation
study demonstrates the merits of PVOT test of omitted nonlinearity and GARCH
effects.
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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Beijing 

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places) 

Montreal 
Quebec City 

Bergen 
Tromso 

Eureka 
Cape Anne 

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Sur 

Toledo Spain 
Connemara 



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