Working Papers - JBHill (UNC)

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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)

Robust Generalized Empirical Likelihood for Heavy Tailed Autoregressions with Conditionally Heteroscedastic Errors (2013) : under revision for Scandinavian Journal of Statistics.

Paper: PDF

We present a robust Generalized Empirical Likelihood estimator and confidence region for the parameters of an autoregression that may have a heavy tailed error, and the error may be conditionally heteroscedastic of unknown form. The estimator exploits two transformations for heavy tail robustness: a redescending transformation of the errors 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 normal irrespective of heavy tails.

Expected Shortfall Estimation and Gaussian Inference for Infinite Variance Time Series (2012) : conditionally accepted by Journal of Financial Econometrics

Paper: PDF (this version : May 2013)

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.

Robust Score and Portmanteau Tests of Volatility Spillover (2012: with M. Aguilar): Revised and resubmitted to Journal of Econometrics

Paper: PDF (this version: April 2013)
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.

Unified Interval Estimation for Random Coefficient Autoregressive Models (2012: with L. Peng) : Revised and resubmitted to Journal of Time Series Analysis

Paper: PDF (this version: May 2013)

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.

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.

Tail Index Estimation for a Filtered Dependent Time Series (2012) : submitted

Paper: 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 Stărică'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-n^1/2-convergent when x(t) has an infinite variance allowing a far greater variety of plug-ins including those that are slower than n^1/2, like QML-type estimators for GARCH models.

Robust Estimation and Inference for Heavy Tailed GARCH (2012) : submitted

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.

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

Paper: PDF

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.

GEL Estimation for GARCH Models with Robust Empirical Likelihood Inference (2013: with Artem Prokhorov)

Paper: PDF

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.

OLD WORKING PAPERS

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

Paper: PDF (paper plus supplemental appendix)

(This paper has been split into two papers. See above for “Least Tail-Trimmed Squares for Infinite Variance Autoregressions” and “Robust Estimation and Inference for Heavy Tailed GARCH”.)

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)

Paper: PDF

Appendix C : Omitted Proofs : PDF

Appendix D : Omitted Figures and Tables : PDF

Gauss: code

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)

Paper: PDF (this version: Feb. 2008)

Appendix: PDF

Gauss: code

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.