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Home Page of Jonathan B. Hill

Associate Professor of Economics

University of North Carolina – Chapel Hill

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WORKING PAPERS

(Under Submission or Invited Revision for Publication)

 

GEL Estimation for GARCH Models with Robust Empirical Likelihood Inference (2013: with Artem Prokhorov): under revision for Journal of Econometrics (1st round)

 

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: expbul1a  Paper: PDF

 

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We construct a Generalized Empirical Likelihood estimator for a GARCH(1,1) model with possibly heavy tailed errors. The estimator imbeds tail-trimmed estimating equations allowing for over-identifying conditions, asymptotic normality and efficiency for very heavy-tailed data due to feedback or idiosyncratic noise. We show the empirical probabilities from the tail-trimmed Continuously Updated Estimator or CUE-GMM elevate weight for usable large values, assign large but not maximum weight to extreme observations, and give the lowest weight to non-leverage points. Finally, we present robust versions of Generalized Empirical Likelihood Ratio, Wald, and Lagrange Multiplier tests, and an efficient and robust moment estimator with an application to expected shortfall estimation.

 

 

 

Robust Generalized Empirical Likelihood for Heavy Tailed Autoregressions with Conditionally Heteroscedastic Errors (2013) : submitted.

 

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Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: expbul1a  Supplemental Material: PDF

 
 

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 (2012: with M. Aguilar): revised and resubmitted to Journal of Econometrics (2nd round).

 

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: expbul1a  Paper: PDF (this version: March 2014)

 

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: expbul1a  Supplemental Appendix: PDF

 
 

This paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by large errors or GARCH-type 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. We develop the tail-trimmed 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. A Monte Carlo study shows our tests provide significant improvements over extant GARCH-based tests of spillover, and we apply the tests to financial returns data.

 

 

Heavy Tail Robust Frequency Domain Estimation (2014: with A. McCloskey) : submitted.

 

*      Paper: PDF (March 2014: typo’s removed)

*      Supplemental Appendix: PDF (March 2014: typo’s removed)

 
 

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 transform the data with a redescending function that permits identification of the parameter for the candidate model, and asymptotic normality, while leading to a classic limit theory when the data have a finite fourth moment. The transform itself can lead to asymptotic bias in our estimators, hence we correct the bias. In the case of symmetrically distributed data we compute the mean-squared-error of our biased estimator and characterize the mean-squared-error 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 for Average Treatment Effects (2013: with S. Chaudhuri) : submitted.

 

*      Paper: PDF

*      Supplemental Appendix: PDF

 
 

We study the probability tail properties of the Inverse Probability Weighting (IPW) estimators of the Average Treatment Effect when there is limited overlap in the covariate distribution. Our main contribution is a new robust estimator that performs substantially better than existing IPW estimators. In the literature either the propensity score is assumed bounded away from 0 and 1, or a fixed or shrinking sample portion of the random variable Z that identifies the average treatment effect by E[Z] = ATE is trimmed when covariate values are large. In a general setting we propose an asymptotically normal estimator that negligibly trims Z adaptively by its large values which sidesteps dimensionality, bias and poor correspondence properties associated with trimming by the covariates, and provides a simple solution to the typically ad hoc choice of trimming threshold. The estimator is asymptotically normal and unbiased whether there is limited overlap or not. In the event there is only one covariate, we also propose an improved robust IPW estimator that trims when the covariate is large. We then work within a latent variable model of the treatment assignment and characterize the probability tail decay of Z. We show when Z exhibits power law tail decay due to limited overlap, and when it has an infinite variance in which case existing estimators do not necessarily have a Gaussian distribution limit. We demonstrate the tail decay property of Z, and study the tail-trimmed estimators by Monte Carlo experiments. We show that our estimator has lower bias and mean-squared-error, and is closer to normal than an existing robust IPW estimator in its suggested form, and in the improved form we propose here.

 

 

Testing for Granger Causality with Mixed Frequency Data (2014: with E. Ghysels and K. Motegi): submitted.

 

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: expbul1a  Paper: PDF (original draft: 2013; this draft: March 2014)

 

 

It is well known that temporal aggregation has adverse effects on Granger causality tests. Time series are often sampled at different frequencies. This is typically ignored, and data are merely aggregated to the common lowest frequency. We develop a set of Granger causality tests that explicitly take advantage of 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 mixed frequency causality tests have higher local asymptotic power as well as more power in finite samples compared to conventional tests.

 

 

An Empirical Process P-Value Test when a Nuisance Parameter is Present under Either or Both Hypotheses (2013): submitted.

 

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We present an empirical process method for transforming a test statistic p-value in the presence of a nuisance parameter under either hypothesis. The p-value transform represents the uniform measure of the parameter space, or occupation time [OT], on which the null hypothesis is rejected. We reject at significance level a when the OT is greater than a, and the asymptotic probability of a Type I error is bounded by α. Thus, conveniently the OT both operates like a test statistic because large values indicate rejection of the null, and like a p-value compliment because its values are bounded between 0 and 1 and rejection of the null occurs when the OT is above a. Further, power in the original test naturally translates to the OT test, while the OT test achieves a non-trivial power improvement over the original test: even if the original test is not consistent, as long as it has power on a dense subset of the nuisance parameter space with Lebesgue measure greater than α then the OT test is consistent. Finally, computation time is dramatically shorter than a popular bootstrap-simulation method. Examples and numerical experiments are given involving tests of functional form, GARCH effects and white noise robust to heavy tails.

 

 

Central Limit Theory for Tail-Trimmed Sums of Heavy-Tailed Dependent, Heterogeneous Data (2010): under revision for Stochastic Processes and their Applications.

 

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We present Gaussian central limit theorems for tail-trimmed sums of a heavy tailed weakly dependent process in the Feller class. We show how the results imply asymptotic normality for sample tail-trimmed variances and covariances, and a super-root(n)-convergent least squares estimator for infinite variance autoregressions.

 

OLD WORKING PAPERS

 

Robust M-Estimation for Heavy Tailed Nonlinear AR-GARCH (2011).

 

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: expbul1a  Paper: PDF (paper plus supplemental appendix)

 

 

We develop new tail-trimmed M-estimation methods for heavy tailed Nonlinear AR-GARCH models. Tail-trimming 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 M-estimators 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 mean-squared-error 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)

 

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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 semi-nonparametric and nonparametric measures, estimators and tests of bivariate tail dependence for non-iid data based on tail exceedances and events. The measures and estimators capture extremal dependence decay over time and can be re-scaled 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 non-extremal properties and joint distribution specifications. Finally, we study the extremal associations within and between equity returns in the U.S., U.K. and Japan.

 

 

Gaussian Tests of 'Extremal White Noise' for Dependent, Heterogeneous, Heavy Tailed Time Series with an Application (2008) 

 

Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: Description: expbul1a  Paper: PDF (this version: Feb. 2008)

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We develop a portmanteau test of extremal serial dependence. The test statistic is asymptotically chi-squared under a null of "extremal white noise", as long as extremes are Near-Epoch-Dependent, 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.

 

 

Econometrics Workshops

UNC/Triangle Workshops

 

Software

Gauss Code List

Gauss Links

 

Econometrics Links

Econometrics Links

Econometrics Books

Resources for Students

Resources on the Net

Texts and Notes

Resources E'metrics and Fin.

Econometrics Texts

Econometricians

 

Data Sources

Data Links

 

Research Resources

Web of Science

JSTOR

EconLit, MathSciNet

NBER Working Papers

CEPR Discussion Papers

EconWPA, EconPapers

Econbase, Authors

CiteSeer, RePEc, IDEAS

Eco5.com

 

Journals

J. Amer. Stat. Assoc.

J. Royal Stat. Soc. B

Annals of Statistics

Annals of Probability

Bernoulli

Econometric Theory

Econometrica

 

Statistics Links

General Links

Liens Statisitique

Journals

American Statistical Association

Joint Statistical Meetings

Statistical Society of Canada

 

Miscellaneous Links

Academic

Economics Dept.'s

American Universities

Canadian Universities

European Universities

 

Personal (places I’ve lived)

Nicaragua

Madrid

Beijing

Tilburg

Boulder

San Fran.

San Diego

Miami

Seattle

Nürmberg

 

Personal (favorite places)

Montreal

Quebec City

Bergen

Tromso

Eureka

Cape Anne

The Giddings

Heidelberg

Delft

Cat Ba

Edinburg

Amsterdam

Point Reyes

Big Sur

Toledo Spain

Connemara

Boulder

Telluride

 

Photos

Boulder in Snow

Point Reyes

Telluride Bridge

Craig na Managh

North Carolina Fall

Colorado Rockies

Cibola National Forest

Rain in Rockies

Telluride Aspen

Woods near NC home

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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