Jonathan B. Hill

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Current Projects - Abstracts

Central Limit Theory for Tail-Trimmed Sums of Dependent, Heterogeneous Data, with an Application to Robust Least Squares

Although robust estimation methods were formalized by the late 1800's, data trimming and truncation for non-iid data has received little attention. We establish sufficient conditions for asymptotic normality of a tail-trimmed sum of dependent, heterogeneous data. The sum is self-standardized with a kernel variance estimator so the rate of convergence, tail thickness and memory persistence do not need to be specified, and hypothesis testing is available. The resulting central limit theory applies to martingale differences, mixing, geometrically ergodic, and mixingale processes in general, including linear and nonlinear distributed lags, linear and nonlinear GARCH and AR-GARCH, and stochastic volatility. The theory is applied to asymptotically Gaussian nonlinear least squares estimation of a nonlinear infinite variance difference equation.

Tail Trimmed Generalized Method of Moments (with Eric Renault)

We develop a GMM estimator robust to heavy-tails by trimming an asymptotically vanishing portion of the moment condition process. As long as the moment process is a martingale difference the tail-trimmed GMM estimator is consistent and asymptotically normal, in particular for processes that are heavy-tailed for any reason (e.g. IGARCH with finite kurtosis shocks, or GARCH with infinite kurtosis shocks). The rate of convergence in general depends on tail thickness, where √n is recovered when the moment process has finite variance components.

Minimum Distance Estimation under Non-Standard Conditions with Applications to Tail Trimmed Least Squares and Quasi-Maximum Trimmed Estimation

We analyze the asymptotic properties of Minimum Distance Estimators where the criterion function need not be differentiable for small or large samples. The small samples problem arises from criterion discontinuities due to model nonlinearity and/or trimming or truncation (e.g. Threshold GARCH, Least Trimmed Squares). The large sample problem arises from moment condition failure due to heavy tails, in which case the criterion Jacobian is unbounded (e.g. GMM for Threshold IGARCH). We establish sufficient conditions for consistency and asymptotic normality for a general class of MDE's that covers Method of Moments and M-estimators, including GMM, QML, LAD and NLLS when the criterion is differentiable (e.g. GMM for ARMA), non-differentiable with a smooth limit (e.g. QML for Threshold GARCH), or never differentiable (e.g. GMM for Threshold GARCH with infinite kurtosis). The results are applied to generalized versions of GMM and M-estimation, and specifically to Tail Trimmed Least Squares and its rates of convergence.

Tail Dependence for Time Series : Non-Parametric Characterization, Estimation and Inference

We develop new representations of tail dependence for time series that provide significant details on what tail index and tail copula notions of tail dependence actually represent. We reveal significant shortcomings in these standard measures including mis-classification of tail dependence, inabilities to detect tail dependence, and the inability to model tail dependence decay between x_{t-h} and y_{t} as the lag h increases for all distribution classes repeatedly exploited in this literature. We deliver a complete non-parametric methodology for measuring, estimating and testing for tail dependence, covering a multitude of time series processes, and easily capturing persistence decay where extant methods fail. On the theory side we prove joint weak convergence for tail dependence estimates at multiple lags where non-extremal properties are irrelevant and we do not require a model of the bivariate tail probability. Finally, we analyze daily returns in international equity markets.

Robust Semi-Nonparametric Tail Inference for Asymmetric Time Series

We prove the B. Hill (1975) tail index estimator is asymptotically normal for a large class of stationary dependent processes {x_t}. We assume {x_t } is Lp-Weakly Dependent for some p > 0 (cf. Hannan 1973, Wu and Min 2005), covering mixing and Near Epoch Dependent processes, including Fractionally Integrated ARMA, Nonlinear Autoregressions, IGARCH, Nonlinear GARCH, nonlinear ARMA-GARCH, and stochastic volatility. The results easily lead to a joint limit theory for left and right tail index and tail scale estimators. We deliver asymptotically most powerful semi-nonparametric tests of tail index, and joint tail index and tail scale, symmetry. A simulation study reveals the tests work well for a variety of persistent, heterogeneous, symmetric and asymmetric time series. Finally, we estimate the tail index for equity returns in several international stock markets and test for tail symmetry.

Trimmed Least Square for Dynamic Linear Regressions Models with Heterogeneous Errors

We develop robust least squares estimators for the slope parameter in a stationary dynamic linear regression model. We trim a fixed or vanishing tail quantile of the sample normal equations implied by the first order condition and which govern asymptotics, and deliver trimmed least squares estimators by method of trimmed moments. The resulting Trimmed and Tail Trimmed Least Squares estimators are asymptotically normal for arbitrarily heavy tailed data, we only require the error term to be a martingale difference and impose trivial moment conditions, thus allowing random volatility errors (e.g. GARCH). The results easily extend to a vector regression setting and to nonlinear functional forms. We deliver a simple consistent estimator of the asymptotic covariance matrix. Finally, we demonstrate tail trimming leads to sub-√n or super-√n consistency depending on the relative tail thickness of the errors and regressors, super-√n consistency is always exhibited for infinite variance autoregressions, and fixed quantile trimming always implies √n-consistency. A simulation study reveals TTLS leads to a potentially large improvement in efficiency over TLS for heavy tailed autoregressions.

Adaptive Tail Trimmed QML Estimation for Nonlinear Semi-Strong ARMAX-GARCH Models

We develop a robust Quasi-Maximum Likelihood estimator for semi-strong Nonlinear ARMAX-Nonlinear GARCH processes. The estimator is based on tail trimming nonlinear estimating equations within a method of moments framework, and is asymptotically normal for possibly very heavy-tailed data due to underlying shocks and/or model parameter values. In particular, we only impose trivial moment conditions on the GARCH errors covering non-stationary cases, and allow for asymmetric data generating processes. We propose a unique adaptive method for selecting the keft- and right-tail trimming proportions that exploits a penalized untrimmed criterion and weak limit theory for minimum distance estimator processes on a cadlag space.