PUBLISHED JOURNAL ARTICLES (see my
cv for other publications)
Expected Shortfall Estimation and Inference for Infinite Variance Time Series (2012) : conditionally accepted by Journal of Financial Econometrics
We develop robust methods of
non-parametric estimation and inference for the Expected Shortfall of heavy
tailed asset returns. We use a tail-trimming 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
There Common Values in First-Price Auctions? A Tail-Index Nonparametric Test (2013, with A. Shneyerov): Journal of Econometrics 174, 144-164.
(this version: Feb. 2013)
We develop a consistent nonparametric test of common
values in first-price 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.
Tail-Trimmed Squares for Infinite Variance Autoregressions (2013) : Journal
of Time Series Analysis 34, 168-186
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. Tail-trimming ensures asymptotic normality
and super-root(n)-convergence with a rate comparable
to the highest achieved amongst M-estimators for stationary data. Moreover,
tail-trimming 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, tail-trimming can be easily extended beyond least squares
estimation for a linear stationary AR model. We discuss extensions to
Quasi-Maximum Likelihood for GARCH, Weighted Least Squares for a possibly
non-stationary Random Coefficient Autoregression, and Empirical Likelihood for robust confidence region
estimation, in each case for models with possibly heavy tailed errors.
Tail and Plug-In Robust Consistent Conditional Moment Tests of Functional
Form (2011): to appear
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. 241-274. Springer: New York
present asymptotic power-one 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 plug-in is allowed, depending on
the model, including those with non-Gaussian limits or a sub-root(n)-convergence rate. One statistic exploits an orthogonalized test equation that promotes plug-in
robustness irrespective of tails. We derive chi-squared 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 16 (in press)
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, integer-indexed 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, 255-274.
We develop an asymptotically chi-squared 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 heavy-tail robustified by our method, including
white noise, GARCH affects, omitted variables, distribution, functional form,
causation, volatility spillover and over-identification. The test statistic
is derived from a tail-trimmed sample version of the moments evaluated at a consistent
plug-in for b. Depending on the test in question and heaviness
of tails, the plug-in may be any consistent estimator including sub-root(T)-convergent and/or asymptotically
non-Gaussian ones, since b can be assured not to affect the test statistic
asymptotically. We adapt bootstrap, p-value 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 Residuals-Based
Tests of Functional Form (2013): Econometric Reviews 32, 361-383.
paper presents a consistent GMM residuals-based 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 non-zero if the model
is mis-specified: the test will never fail to
detect mis-specification of any form for large
samples, and is asymptotically chi-squared under the null, allowing for fast
and simple inference. A simulation study reveals randomly selecting the
nuisance parameter leads to more power than supremum-tests,
and can obtain empirical power nearly equivalent to the most powerful test
for even relatively small n.
Memory of Stochastic Volatility with an Application to Tail Shape Inference
(2011) Journal of Statistical Planning
and Inference 141, 663-676.
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.
and Non-Tail Memory with Applications to Extreme Value and Robust Statistics (2011) Econometric Theory 27, 844-884.
notions of tail and non-tail dependence are used to characterize separately extremal and non-extremal
information, including tail log-exceedances and
events, and tail-trimmed levels. We prove Near Epoch Dependence (McLeish
1975, Gallant and White 1988) and L0-Approximability (Pötscher and Prucha 1991) are equivalent for tail events and
tail-trimmed 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
non-extremal 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 Tail-Trimmed Least Squares estimator for heavy tailed processes.
and Growth Volatility (2011: with N. Anbarci
and H. Kirmanoglu): Economic Papers 30, 233–252.
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 two-equation 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, 1398-1436.
Paper PDF (working
paper with omitted proofs is here)
Gauss: code (Hill estimator with kernel confidence
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
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.
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
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.
Non-Degenerate Model Specification Tests Against Smooth Transition and Neural
Networks Alternatives (2008) Annales
D’Economie et de Statistique 90, 145-179.
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.
Tests of Long-Run Causation in Trivariate VAR
Processes with a Rolling Window Study of the Money-Income Relationship
(2007) Journal of Applied Econometrics
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.
Orthogonal Decompositions and Nonlinear Impulse Response Functions for
Infinite Variance Processes (2006) Canadian
Journal of Statistics 34, 453-473.
Paper: PDF (working paper with omitted proofs)
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.
African Company Share Prices during the South Sea Bubble
(2002, with Ann Carlos and Nathalie Moyen), Explorations in Economic
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.