Home Page of Jonathan B. Hill

Associate Professor of Economics

University of North Carolina – Chapel Hill

Dept. of Economics
Gardner Hall 208B
University of North Carolina
Chapel Hill, NC 27599-3305

jbhill at email dot unc dot edu



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GEL Estimation for GARCH Models with Robust Empirical Likelihood Inference (2013: with Artem Prokhorov): Journal of Econometrics 190, 18-45.


*       Paper: PDF (this version: Sept. 2015)


*       Supplemental Material: PDF (this version: June 2015)



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 (2016: with E. Ghysels and K. Motegi). Journal of Econometrics 192, 207-230.


*                 Paper: PDF (Initial version: March 2014; this version : January 2015)



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). Journal of Business and Economic Statistics: forthcoming.


*      Paper: PDF (July 2015)


*      Supplemental Appendix: PDF (July 2015)


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. 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 _t 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.



Uniform Interval Estimation for an AR(1) Process with AR Errors (2014: with Deyuan Li and Liang Peng). Statistica Sininca: forthcoming.


*      Paper: PDF



An empirical likelihood method was proposed in Hill and Peng (2014) to construct a unified interval estimation for the coefficient in an AR(1) model, regardless of whether the sequence was stationary or near integrated. The error term, however, was assumed independent, and this method fails when the errors are dependent. Testing for a unit root in an AR(1) model has been studied in the literature for dependent errors, but existing methods cannot be used to test for a near unit root. In this paper, assuming the errors are governed by an AR(p) process, we exploit the efficient empirical likelihood method to give a unified interval for the coefficient by taking the structure of errors into account. Furthermore, a jackknife empirical likelihood method is proposed to reduce the computation of the empirical likelihood method when the order in the AR errors is not small. A simulation study is conducted to examine the finite sample behavior of the proposed methods.



Robust Generalized Empirical Likelihood for Heavy Tailed Autoregressions with Conditionally Heteroscedastic Errors (2013). Journal of Multivariate Analysis 135, 131-152.


*      Paper: PDF


*      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 (2015: with M. Aguilar). Journal of Econometrics 184, 37-61.


*      Paper: PDF


*      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. Trimming is either simple based on an indicator function, or smoothed. In particular, 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. We discuss the choice of trimming portion, including a smoothed p-value over a window of extreme observations. 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. 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 pre-test to improve volatility forecasting.



Tail Index Estimation for a Filtered Dependent Time Series (2014). Statistica Sinica 25, 609-630.


*      Paper: PDF


*      Supplemental Material: PDF



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 Starica'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 b-mixing with possibly hyperbolic dependence, covering ARMA-GARCH filters, ARMA filters with heteroscedastic errors of unknown form, nonlinear filters like threshold autoregressions, and filters based on mis-specified models, as well as i.i.d. errors in an ARMA model. Finally, as opposed to existing results we do not require the plug-in for q to be super-n1/2-convergent when x(t) has an infinite variance allowing a far greater variety of plug-ins including those that are slower than n1/2 , like QML-type estimators for GARCH models.



Robust Estimation and Inference for Heavy Tailed GARCH (2015). Bernoulli 21, 1629-1669.


*      Paper: PDF


*      Supplemental Material: PDF


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 sub-class of redescending transforms that includes tail-trimming and functions popular in the robust estimation literature, and we re-center 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, Log-LAD (Peng and Yao 2003, Linton et al 2010) and Quasi-Maximum Weighted Exponential Likelihood (Zhu and Ling 2012). Finally, we apply the tail-trimmed QML estimator to financial data.



Unified Interval Estimation for Random Coefficient Autoregressive Models (2014: with L. Peng). Journal of Time Series Analysis 35, 35, 282-297.


*      Paper: PDF



The quasi maximum likelihood estimation of random coefficient autoregressive

models requires coefficient randomness if non-stationary 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 non-stationary. 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, 1-44.


*      Paper: PDF



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 returns data.



Are There Common Values in First-Price Auctions? A Tail-Index Nonparametric Test (2013, with A. Shneyerov). Journal of Econometrics 174, 144-164.


*      Paper: PDF (this version: Feb. 2013)

*      Appendix: PDF


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.



Least Tail-Trimmed Squares for Infinite Variance Autoregressions (2013). Journal of Time Series Analysis 34, 168-186


*      Paper: PDF


*      Gauss Code



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.



Heavy Tail and Plug-In 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. 241-274. Springer: New York


*      Paper: PDF


*      Appendix: PDF



We 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 17, 121-141


*      Paper: PDF



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, 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.


*      Paper: PDF


*      Appendix: PDF



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: PDF


*      Appendix: PDF



This 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.



Extremal Memory of Stochastic Volatility with an Application to Tail Shape Inference (2011). Journal of Statistical Planning and Inference 141, 663-676.


*      Paper: PDF



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 Non-Tail Memory with Applications to Extreme Value and Robust Statistics (2011). Econometric Theory 27, 844-884.


*      Paper: PDF



New 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.



            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 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 bands)



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.


*      Paper: PDF


*      Appendix: PDF



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.


*      Paper: PDF



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 90, 145-179.


*      P aper: PDF


*      Appendix PDF



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.


*      Paper: PDF    Appendix: PDF

*      Gauss: code

*      Data



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.


*      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.



 Royal African Company Share Prices during the South Sea Bubble (2002, with Ann Carlos and Nathalie Moyen). Explorations in Economic History 39, 61-87.


*      Paper: PDF


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.