Jonathan B. Hill
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Publications
and Forthcoming Manuscripts
On Tail Index Estimation for
Dependent, Heterogeneous Data (2010) Econometric
Theory 26, in press.
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In this paper we analyze
the asymptotic properties of the popular distribution tail index estimator by
B. Hill (1975) for possibly heavy-tailed, 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 near-epoch-dependent 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
non-parametric kernel estimator of the asymptotic variance of the Hill
estimator, and prove consistency for extremal-NED 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, 2091-2110.
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We establish invariance principles for a
large class of dependent, heterogeneous arrays. The theory equally covers
conventional non-tail 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 Near-Epoch-Dependence 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.
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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 Non-Degenerate
Model Specification Tests Against Smooth Transition and Neural Networks
Alternatives (2008) Annales D’Economie et de Statistique: in press.
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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 most-powerful 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 Long-Run Causation in
Trivariate VAR Processes with a Rolling Window Study of the Money-Income
Relationship (2007) Journal of Applied Econometrics 22,
747-765.
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This paper develops a simple sequential
multiple horizon non-causation 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 over-rejection, 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, 453-473.
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In this paper we prove Wold-type
decompositions with strong-orthogonal 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 Wold-type
decompositions of discrete time processes; completely characterizes when
nonlinear heavy-tailed processes obtain a strong-orthogonal 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, 61-87.
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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.