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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): revised and resubmitted to Journal of Econometrics (1st round)

 

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We construct a Generalized Empirical Likelihood estimator for a GARCH(1,1) model with a possibly heavy tailed error. The estimator imbeds tail-trimmed estimating equations allowing for over-identifying conditions, asymptotic normality, efficiency and empirical likelihood based confidence regions for very heavy-tailed random volatility data. We show the implied probabilities from the tail-trimmed Continuously Updated Estimator elevate weight for usable large values, assign large but not maximum weight to extreme observations, and give the lowest weight to non-leverage points. We derive a higher order expansion for GEL with imbedded tail-trimming (GELITT), which reveals higher order bias and efficiency properties, available when the GARCH error has a finite second moment. Higher asymptotics for GEL without tail-trimming 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 tail-trimmed CUE-GMM 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): revised and resubmitted to Journal of Econometrics (1st round).

 

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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 Low-Frequency Contamination (2014: with A. McCloskey) : under revision for invited resubmission to Journal of Business and Economic Statistics (1st round)

 

*      Paper: PDF (Oct. 2014)

*      Supplemental Appendix: PDF (Oct. 2014)

 
 

We provide methods to robustly estimate the parameters of stationary ergodic short-memory time series models in the potential presence of additive low-frequency 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 quasi-maximum likelihood (FDQML) objective functions without requiring specification of the low-frequency contaminating component. We provide two approaches, allowing for either thin or heavy-tailed data. When proper sample size-dependent 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 low-frequency contamination, enabling the practitioner to robustly estimate model parameters without prior knowledge of whether contamination is present. Popular time series models that fit 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.

 

 

Simple Granger Causality Tests for Mixed Frequency Data (2015: with E .Ghysels and K. Motegi) : submitted.

 

*      Paper: PDF (June 2015)

*      Supplemental Appendix: PDF (June 2015)

 
 

This paper presents simple Granger causality tests applicable to any mixed frequency sampling data setting, and feature remarkable power properties even with relatively small low frequency data samples and a considerable wedge between sampling frequencies (for example, quarterly and daily or weekly data). Our tests are based on a seemingly overlooked, but simple, dimension reduction technique for regression models. If the number of parameters of interest is large then in small or even large samples any of the trilogy test statistics may not be well approximated by their asymptotic distribution. A bootstrap method can be employed to improve empirical test size, but this generally results in a loss of power. A shrinkage estimator can be employed, including Lasso, Adaptive Lasso, or Ridge Regression, but these are valid only under a sparsity assumption which does not apply to Granger causality tests. The procedure, which is of general interest when testing potentially large sets of parameter restrictions, involves multiple parsimonious regression models where each model regresses a low frequency variable onto only one individual lag or lead of a high frequency series, where that lag or lead slope parameter is necessarily zero under the null hypothesis of non-causality. Our test is then based on a max test statistic that selects the largest squared estimator among all parsimonious regression models. Parsimony ensures sharper estimates and therefore improved power in small samples. Inference requires a simple simulation-bootstrap step since the test statistic has a non-standard limit distribution. We show via Monte Carlo simulations that the max test is more powerful than existing mixed frequency Granger causality tests in small samples. An empirical application examines Granger causality over rolling windows of U.S.\ macroeconomic data from 1962-2013 using a mixture of high and low frequency data.

 

 

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

 

*      Paper: PDF (Sept. 2014)

*      Supplemental Appendix: PDF (Sept. 2014)

 
 

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 trim the data, permitting both identification of the parameter for the candidate model, and asymptotically normal frequency domain estimators, while leading to a classic limit theory when the data have a finite fourth moment. The transform itself can lead to asymptotic bias in the limit distribution of our estimators when the fourth moment does not exist, hence we correct the bias using extreme value theory that applies whether tails decay according to a power law or not. 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 and Inference for Average Treatment Effects (2014: with S. Chaudhuri) : submitted.

 

*      Paper: PDF (March 2015)

*      Supplemental Appendix: PDF (March 2015)

 
 

We study the probability tail properties of Inverse Probability Weighting (IPW) estimators of the Average Treatment Effect (ATE) when there is limited overlap between the covariate distributions of the treatment and control groups. Under unconfoundedness of treatment assignment conditional on covariates, such limited overlap is manifested in the propensity score for certain units being very close (but not equal) to 0 or 1. This renders IPW estimators possibly heavy tailed, and with a slower than root(n) rate of convergence. Most existing estimators are either based on the assumption of strict overlap, i.e. the propensity score is bounded away from 0 and 1; or they truncate the propensity score; or trim observations based on a variety of techniques based on covariate or propensity score values. Trimming or truncation is ultimately based on the covariates, ignoring important information about the inverse probability weighted random variable Z that identifies ATE by E[Z]= ATE. Using a threshold crossing model for treatment assignment, we explain the possibility of irregular identification by showing Z can have a power law tail under limited overlap, with heavier or thinner tails based on the relative distribution tails of the treatment assignment covariate and error. We then propose a new tail-trimmed IPW estimator whose performance is robust to limited overlap more generally. In terms of the propensity score, which is generally unknown, we plug-in its parametric estimator in the infeasible Z, and then negligibly trim the resulting feasible Z adaptively by its large values. Trimming can lead to bias in general, hence we estimate and remove the bias using new theory and methods. Our estimator sidesteps dimensionality, bias and poor correspondence properties associated with trimming by the covariates or propensity score. Monte Carlo experiments demonstrate that trimming by the covariates or the propensity score requires the removal of a substantial portion of the sample to render a low bias and close to normal estimator, while our estimator has low bias and mean-squared error, and is close to normal, based on very little trimming.

 

 

An Empirical Process P-Value Test for Handling Nuisance and Tuning Parameters (2015): under revision for Journal of the American Statistical Association (reject and resubmit)

 

*      Paper: PDF

*      Supplemental Appendix: PDF

 

 

We present an empirical process method for smoothing a p-value, or the related test statistic, in the presence of nuisance and/or tuning parameters. We do not require root(n) Gaussian asymptotics, and our test can work in conjunction with Andrews and Cheng (2012, 2013, 2014)'s methods of robust inference when a subset of parameters are possibly weakly identified. Our test is particularly relevant when Andrews and Cheng (2012, 2013, 2014)'s methods are not appropriate: when the parameter is not logically estimated nor necessarily part of the data generating process, or when root(n) Gaussian asymptotics does not hold, including tests of omitted nonlinearity and GARCH effects, and heavy tail robust tests, or distribution tail inference. Power in the original test naturally translates to power in our test, while our test can achieve a non-trivial power improvement over the original test. Examples and numerical experiments are given involving tests of functional form, GARCH effects, a heavy tail robust white noise test, and a consistent (non)identification robust test of Smooth Transition Autoregression.

 

 

OLD WORKING PAPERS

 

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

 

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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) 

 

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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

 Duke/NCSU

 

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Journals

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J. Royal Stat. Soc. B

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Statistics Links

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Miscellaneous Links

Academic

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Personal (places I’ve lived)

Nicaragua

Madrid

Beijing

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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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