
Home Page of Jonathan B. Hill
Associate Professor of Economics University of North Carolina – Chapel Hill


CV (pdf) 



LINKS 

GEL Estimation for
GARCH Models with Robust Empirical Likelihood Inference (2013: with Artem
Prokhorov): Journal of Econometrics
(conditionally accepted)
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). Journal of Econometrics: forthcoming.
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). Journal of Business and Economic Statistics: forthcoming.
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. Uniform Interval Estimation for an AR(1) Process with AR Errors (2014: with Deyuan Li and Liang Peng). Statistica Sininca: forthcoming.
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. Robust
Generalized Empirical Likelihood for Heavy Tailed Autoregressions with
Conditionally Heteroscedastic Errors (2013).
Journal of Multivariate Analysis
135, 131152.
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 (2015: with M. Aguilar). Journal of
Econometrics 184, 3761.
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. Trimming is either simple based on an indicator
function, or smoothed. In particular, 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. We
discuss the choice of trimming portion, including a smoothed pvalue over a
window of extreme observations. 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. Finally, based on ideas in Patton
(2011), we construct a heavy tail
robust forecast improvement statistic, which allows us to demonstrate that
our spillover test can be used as a model specification pretest to improve
volatility forecasting. Tail
Index Estimation for a Filtered Dependent Time Series (2014). Statistica
Sinica 25, 609630.
We prove Hill's (1975) tail index
estimator is asymptotically normal where the employed data are generated by a
stationary parametric process {x(t)}. We assume x(t)
is an unobservable function of a parameter q that is estimable.
Natural applications include regression residuals and GARCH filters. Our main
result extends Resnick and Stărică's (1997) theory for estimated AR
i.i.d. errors and Ling and Peng's (2004) theory for estimated ARMA i.i.d.
errors to a wide range of filtered time series since we do not require x(t)
to be i.i.d., nor generated by a linear process
with geometric dependence. We assume x(t) is bmixing with possibly hyperbolic dependence, covering ARMAGARCH
filters, ARMA filters with heteroscedastic errors of unknown form, nonlinear
filters like threshold autoregressions, and filters based on misspecified
models, as well as i.i.d.
errors in an ARMA model. Finally, as opposed to existing results we do not
require the plugin for q to be supern^{1/2}convergent
when x(t) has an infinite variance allowing a far
greater variety of plugins including those that are slower than n^{1/2 }, like QMLtype estimators for GARCH models. Robust
Estimation and Inference for Heavy Tailed GARCH (2015). Bernoulli
21, 16291669.
We develop two new estimators for GARCH
models with possibly heavy tailed asymmetrically distributed errors. The
first estimator arises from negligibly trimming QML criterion equations
according to error extremes. The second imbeds negligibly transformed errors
into QML score equations for a Method of Moments estimator. In this case we
exploit a subclass of redescending transforms that includes tailtrimming
and functions popular in the robust estimation literature, and we recenter
the transformed errors to minimize small sample bias. The negligible
transforms allow both identification of the true parameter and asymptotic
normality. We present a consistent estimator of the covariance matrix that
permits classic inference without knowledge of the rate of convergence. A
simulation study shows our Method of
Moments estimator is best overall, and both of our estimators trump existing
ones for sharpness and approximate normality including QML, LogLAD (Peng and
Yao 2003, Linton et al 2010) and QuasiMaximum Weighted Exponential
Likelihood (Zhu and Ling 2012). Finally, we apply the tailtrimmed QML estimator
to financial data. Unified
Interval Estimation for Random Coefficient Autoregressive Models (2014: with L. Peng). Journal of Time Series Analysis 35, 35, 282297.
The quasi maximum likelihood estimation of random
coefficient autoregressive models requires coefficient
randomness if nonstationary cases are allowed. In this paper we propose
empirical likelihood methods based on a weighted score equation to construct
a confidence interval for the coefficient. We do not need to distinguish
whether the coefficient is random or deterministic and whether the process is
stationary or nonstationary. A simulation study confirms the good finite
sample behavior of the proposed methods, and we apply our methods to study
U.S. macroeconomic data. Expected
Shortfall Estimation and Gaussian Inference for Infinite Variance Time Series (2015). Journal
of Financial Econometrics 13, 144.
We develop robust methods of
nonparametric estimation and inference for the Expected Shortfall of heavy
tailed asset returns. We use a tailtrimming indicator to dampen extremes
negligibly, ensuring standard Gaussian inference, and a higher rate of
convergence than without trimming when the variance is infinite. Trimming,
however, causes bias in small samples and possibly asymptotically when the
variance is infinite, we exploit a rarely used remedy to estimate and utilize
the tail mean that is removed by trimming. Since estimating the tail mean
involves estimation of tail parameters and therefore an added arbitrary
choice of the number of included extreme values, we present weak limit theory
for an ES estimator that optimally selects the number of tail observations by
making our estimator arbitrarily close to the untrimmed estimator, yet still
asymptotically normal. Finally, we apply the new estimators to financial
returns data. Are
There Common Values in FirstPrice Auctions? A TailIndex Nonparametric Test (2013, with A. Shneyerov). Journal
of Econometrics 174, 144164.
We develop a consistent nonparametric test of common
values in firstprice auctions and apply it to British Columbia Timber Sales
data. The test is based on the behavior of the CDF of bids near the reserve
price. We show that the curvature of the CDF is drastically different under
private values (PV) and common values (CV). We then show that the problem of
discriminating between PV and CV is equivalent to estimating the lower tail
index of the bid distribution. Our approach admits unobserved auction
heterogeneity of an arbitrary form. We develop a Hill (1975)type tail index
estimator and find presence of common values BC Timber Sales. Least
TailTrimmed Squares for Infinite Variance Autoregressions (2013). Journal
of Time Series Analysis 34, 168186
We develop a robust least squares
estimator for autoregressions with possibly heavy tailed errors. Robustness
to heavy tails is ensured by negligibly trimming the squared error according
to extreme values of the error and regressors. Tailtrimming ensures
asymptotic normality and superroot(n)convergence with a rate comparable to
the highest achieved amongst Mestimators for stationary data. Moreover,
tailtrimming ensures robustness to heavy tails in both small and large
samples. By comparison, existing robust estimators are not as robust in small
samples, have a slower rate of convergence when the variance is infinite, or
are not asymptotically normal. We present a consistent estimator of the
covariance matrix and treat classic inference without knowledge of the rate
of convergence. A simulation study demonstrates the sharpness and approximate
normality of the estimator, and we apply the estimator to financial returns
data. Finally, tailtrimming can be easily extended beyond least squares
estimation for a linear stationary AR model. We discuss extensions to
QuasiMaximum Likelihood for GARCH, Weighted Least Squares for a possibly
nonstationary Random Coefficient Autoregression, and Empirical Likelihood
for robust confidence region estimation, in each case for models with
possibly heavy tailed errors. Heavy
Tail and PlugIn Robust Consistent Conditional Moment Tests of Functional
Form (2011). In X. Chen and N. Swanson (ed.'s), Recent Advances and Future
Directions in Causality, Prediction, and Specification Analysis: Essays in
Honor of Halbert L. White Jr., pp. 241274. Springer: New York
We
present asymptotic powerone test statistics for heavy tailed time series.
Under the null the regression errors must have a finite mean, and otherwise they
may have arbitrarily heavy tails. If the errors have an infinite variance
then in principle any consistent plugin is allowed, depending on the model,
including those with nonGaussian limits or a subroot(n)convergence rate.
One statistic exploits an orthogonalized test equation that promotes plugin
robustness irrespective of tails. We derive chisquared weak limits,
characterize an empirical process method for selecting the trimming fractile,
and study the finite sample properties. Stochastically Weighted Average
Conditional Moment Tests of Functional Form (2012). Studies in Nonlinear Dynamics and
Econometrics 17, 121141
We
develop a new consistent conditional moment test of functional form based on
nuisance parameter indexed sample moments first presented in Bierens (1982, 1990).
We reduce the nuisance parameter space to known countable sets, which leads
to a weighted average conditional moment test in the spirit of Bierens and
Ploberger's (1997) Integrated Conditional Moment test. The weights are
possibly stochastic in an arbitrary way, integerindexed and flexible enough
to cover a range of tests from average to higher quantile to maximum tests,
the latter of which is impossible in the existing ICM framework.
Nevertheless, the limit distribution under the null and local alternative
belong to the same class as the ICM statistic, hence our test is admissible
if the errors are Gaussian, and a flat weight leads to the greatest weighted
average local power. Moment Condition Tests for
Heavy Tailed Time series (2013, with M. Aguilar). Journal of Econometrics 172, 255274.
We develop an asymptotically chisquared statistic
for testing moment conditions E[m(b)] = 0, where m(b) may be weakly
dependent, scalar components of m(b) may have an infinite variance, and
E[m(b)] need not exist under the alternative.
Score tests are a natural application, and in general a variety of tests can
be heavytail robustified by our method, including white noise, GARCH
affects, omitted variables, distribution, functional form, causation,
volatility spillover and overidentification. The test statistic is derived
from a tailtrimmed sample version of the moments evaluated at a consistent
plugin for b. Depending on the test in question and heaviness
of tails, the plugin may be any consistent estimator including subroot(T)convergent and/or asymptotically
nonGaussian ones, since b can be assured not to affect the test statistic
asymptotically. We adapt bootstrap, pvalue occupation time, and covariance
determinant methods for selecting the trimming fractile in any sample, and
apply our statistic to tests of white noise, omitted variables and volatility
spillover. We find it obtains sharp empirical size and strong power, while
conventional tests exhibit size distortions. Consistent GMM ResidualsBased
Tests of Functional Form (2013). Econometric Reviews 32, 361383.
This
paper presents a consistent GMM residualsbased test of functional form for
time series models. By relating two moment conditions we deliver a vector
moment condition in which at least one element must be nonzero if the model
is misspecified: the test will never fail to detect misspecification of any
form for large samples, and is asymptotically chisquared under the null,
allowing for fast and simple inference. A simulation study reveals randomly
selecting the nuisance parameter leads to more power than supremumtests, and
can obtain empirical power nearly equivalent to the most powerful test for
even relatively small n. Extremal Memory of
Stochastic Volatility with an Application to Tail Shape Inference
(2011). Journal of Statistical Planning
and Inference 141, 663676.
In this paper we characterize joint tails and tail
dependence for a class of stochastic volatility processes. We derive the
exact joint tail shape of multivariate stochastic volatility processes with
innovations that have a regularly varying distribution tail. This is used to
give four new characterizations of tail dependence. In three cases tail
dependence is a function of linear volatility memory parametrically
represented by tail scales, while tail power indices do not provide any
relevant dependence information. In the fourth case a linear function of tail
events and exceedances is itself linearly independent, implying tail index
inference based on the Hill (1975) estimator is identical to the iid
case. Tail
and NonTail Memory with Applications to Extreme Value and Robust Statistics (2011). Econometric Theory 27, 844884.
New
notions of tail and nontail dependence are used to characterize separately extremal
and nonextremal information, including tail logexceedances and events, and
tailtrimmed levels. We prove Near Epoch Dependence (McLeish 1975, Gallant
and White 1988) and L0Approximability
(Pötscher and Prucha 1991) are equivalent for tail events and tailtrimmed
levels, ensuring a Gaussian central limit theory for
important extreme value and robust statistics under general conditions. We
apply the theory to characterize the extremal and nonextremal memory
properties of possibly very heavy tailed GARCH processes and distributed
lags. This in turn is used to verify Gaussian limits for tail index, tail
dependence and tail trimmed sums of these data, allowing for Gaussian
asymptotics for a new TailTrimmed Least Squares estimator for heavy tailed processes. Institutions and Growth Volatility (2011: with N. Anbarci and H. Kirmanoglu). Economic Papers 30, 233–252. Recently some studies provided evidence that democratic political institutions generate less volatile growth. These studies, however, do not provide any link between democracy and investment volatility. Here, we focus on the specific channel that links individualistic societies and low growth volatility. We test whether investment volatility and consequently growth volatility are lower in individualistic societies.We construct a twoequation system of investment and income growth volatility, allowing various measures of individualism to influence growth volatility both directly and indirectly. We find that individualism significantly directly and indirectly influences growth volatility negatively. On Tail Index Estimation
for Dependent, Heterogeneous Data (2010). Econometric Theory 26, 13981436.
In this
paper we analyze the asymptotic properties of the popular distribution tail
index estimator by B. Hill (1975) for possibly heavytailed, heterogenous,
dependent processes. We prove the Hill estimator is weakly consistent for
processes with extremes that form mixingale sequences, and asymptotically
normal for processes with extremes that are nearepochdependent on the
extremes of a mixing process. Our limit theory covers infinitely many ARFIMA
and FIGARCH processes, stochastic recurrence equations, and bilinear
processes. Moreover, we develop a simple nonparametric kernel estimator of
the asymptotic variance of the Hill estimator, and prove consistency for
extremalNED processes. On
Functional Central Limit Theorems for Dependent, Heterogeneous Arrays with
Applications to Tail Index and Tail Dependence Estimation (2009). Journal of Statistical Planning and
Inference: 139, 20912110.
We
establish invariance principles for a large class of dependent, heterogeneous
arrays. The theory equally covers conventional nontail arrays, and
inherently degenerate tail arrays popularly encountered in the extreme value
literature including sample means and covariances of extreme events and
exceedances. For tail arrays we trim dependence assumptions down to a minimum
by constructing extremal versions of mixing and NearEpochDependence
properties, covering mixing, ARFIMA, FIGARCH, stochastic volatility,
bilinear, random coefficient autoregressive, nonlinear distributed lag and
Extremal Threshold processes, and stochastic recurrence equations. Of practical importance our theory can be used to
characterize the functional limit distributions of B. Hill's (1975) tail
index estimator, the tail quantile process, and multivariate extremal
dependence measures under substantially general conditions. Heavy Tails and Mixed
Distribution Hypothesis (2008). Encyclopedia of Quantitative
Finance, Wiley 2009 : forthcoming.
We outline the Mixed Distribution Hypothesis as a
means to explain heavy tails in financial time series. We discuss the
hypothesis' historical roots, and fully present the most popular, and
original, form of the hypothesis and its implications for modeling asset
returns. Original contributions and modern extensions are cited. Consistent and NonDegenerate
Model Specification Tests Against Smooth Transition and Neural Networks
Alternatives (2008). Annales D’Economie et de
Statistique 90, 145179.
We
develop a regression model specification test that directs maximal power toward
smooth transition functional forms, and is consistent against any deviation
from the null specification. We provide new details regarding whether
consistent parametric tests of functional form are asymptotically degenerate:
a test of linear autoregression against STAR alternatives is never
degenerate. Moreover, a test of Exponential STAR has power attributes
entirely associated with the choice of threshold. In a simulation experiment
in which all parameters are randomly selected the proposed test has power
nearly identical to a mostpowerful test for true STAR, neural network and
SETAR processes, and dominates popular tests. We apply the test to U.S.
output, money, prices and interest rates. Efficient
Tests of LongRun Causation in Trivariate VAR Processes with a Rolling Window
Study of the MoneyIncome Relationship (2007). Journal of Applied Econometrics 22,
747765.
This paper develops a simple sequential
multiple horizon noncausation test strategy for trivariate VAR models (with
one auxiliary variable). We apply the test strategy to a rolling window study
of money supply and real income, with the price of oil, the unemployment rate
and the spread between the Treasury bill and commercial paper rates as
auxiliary processes. Ours is the first study to control simultaneously for
common stochastic trends, sensitivity of test statistics to the chosen sample
period, null hypothesis overrejection, sequential test size bounds, and the
possibility of causal delays. Evidence suggests highly significant direct or
indirect causality from M1 to real income, in particular through the unemployment
rate and M2 once we control for cointegration. Strong
Orthogonal Decompositions and Nonlinear Impulse Response Functions for
Infinite Variance Processes (2006). Canadian
Journal of Statistics 34, 453473.
In this paper we prove Woldtype decompositions with
strongorthogonal prediction innovations exist in smooth, reflexive Banach
spaces of discrete time processes if and only if the projection operator
generating the innovations satisfies the property of iterations. Our theory
includes as special cases all previous Woldtype decompositions of discrete
time processes; completely characterizes when nonlinear heavytailed
processes obtain a strongorthogonal moving average representation; and
easily promotes a theory of nonlinear impulse response functions for infinite
variance processes. We exemplify our theory by developing a nonlinear impulse
response function for smooth transition threshold processes, we discuss how
to test decomposition innovations for strong orthogonality and whether the
proposed model represents the best predictor, and we apply the methodology to
currency exchange rates. Royal
African Company Share Prices during the South Sea Bubble
(2002, with Ann Carlos and Nathalie Moyen). Explorations
in Economic History
39, 6187.
Price bubbles provide a unique opportunity to
test whether investors act rationally and have sufficient knowledge of the
economic environment in which they trade. We focus our attention on the 1720
South Sea bubble episode as experienced by a company not involved in
governmental debt financing—the Royal African Company. Following the example
of the South Sea Company, the Royal African Company lent its funds to
equityholders at a preferential rate. Recognizing this benefit along with the
announced dividends explains a large portion of the bubble. Furthermore, the
unexplained residual does not behave like an exploding bubble, casting doubt
that speculative excess motivated market participants in 1720. Our findings
are indeed consistent with investor rationality, and the unexplained residual
suggests that we are missing information that was available to the British
financial market in 1720. 

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