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


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LINKS 

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: forthcoming.
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 (2014). Bernoulli:
forthcoming.
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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