Tail and Nontail Memory With Applications to Extreme Value and Robust Statistics

Hill, Jonathan B.

doi:10.1017/s0266466610000514

Cited by 24 publications

(4 citation statements)

References 93 publications

(197 reference statements)

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“…Remark 5: A process fX t g is L p -E-NED with size if and only if it is L s -E-NED with size p= maxfp; sg for any s R p since jI(X t > b mn e u ) P (X t > b mn e u j= t+qn n;t qn )j 1 a:s. See Hill (2008c). But this suggests p is irrelevant since L p -E-NED is equivalent to L s -E-NED.…”

Section: Remarkmentioning

confidence: 99%

“…By independence f t g is trivially F-strong mixing with arbitrary size. If the GARCH process has a unit root, and in many cases an explosive root, then fX t g is still geometrically L 2 -E-NED on f= n;t g with arbitrary E-NED and F-mixing base sizes (Hill 2008c), although fX t g itself need not be mixing nor population …”

Section: Example 4 (Nonlinear Distributed Lag)mentioning

confidence: 99%

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On Tail Index Estimation for Dependent, Heterogeneous Data

Hill

2010

Econom. Theory

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In this paper we analyze the asymptotic properties of the popular distribution tail index estimator by B. Hill (1975) for dependent, heterogeneous processes. We develop new extremal dependence measures that characterize a massive array of linear, nonlinear, and conditional volatility processes with long or short memory. We prove the Hill-estimator is weakly and uniformly weakly consistent for processes with extremes that form mixingale sequences, and asymptotically normal for processes with extremes that are Near Epoch Dependent on the mixing extremes of some arbitrary process. The extremal persistence assumptions in this paper are known to hold for mixing, Lp-NED and some non-Lp-NED processes, including ARFIMA, FIGARCH, explosive GARCH, nonlinear ARMA-GARCH, and bilinear processes, and nonlinear distributed lags like random coe¢ cient and regime switching autoregressions.Finally, we deliver a simple nonparametric estimator of the asymptotic variance of the Hill-estimator, and prove consistency for processes with NED extremes. 1.INTRODUCTION This paper develops an asymptotic theory for the popular distribution tail index estimator due to B. Hill (1975) under general conditions. Many time series in …nance, macroeconomics and meteorology exhibit extreme values that appear to cluster (Leadbetter, Lindgren and Rootzén 1983, Embrechts, Klüppelberg, andMikosch 1997). In order to deliver a Gaussian limit theory that is robust to the nature of persistence and heterogeneity in extremes, we introduce new extremal dependence measures and develop an associated weak and uniform limit theory for dependent, heterogeneous tail arrays.Denote by fX t g = fX t : 1 < t < 1g a stochastic process on some probability measure space, write F t (x) := P (X t x) and assume F t has support on [0; 1). Assume F t (x) := P (X t > x) is regularly varying at 1: for all > 0 and each t (1) limI would like to thank Enno Mammen for insight into the extremal dependence properties developed here, Oliver Linton for discussions concerning the strong-GARCH case, and Holger Drees for comments on an earlier version. In particular, I kindly thank three anonymous referrees, Co-Editor Bruce Hansen and Editor Peter C. B. Phillips for expert commentary that lead to substantial improvements. All errors, of course, are mine alone. y

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Section: Remarkmentioning

confidence: 99%

Section: Example 4 (Nonlinear Distributed Lag)mentioning

confidence: 99%

On Tail Index Estimation for Dependent, Heterogeneous Data

Hill

2010

Econom. Theory

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“…Note that in our context, the data (y t ) undergo a non-Lipschitz transformation (viz., they are squared), and therefore the relationship between moment conditions and memory is not the "standard" one (see, e.g., the IP in Theorem 29.6 in [13]). In principle, moment conditions such as the one in part (ii) could be tested for, for example, using a test based on some tail-index estimator -Hill [19,20] extends the well-known Hill's estimator to the context of dependent data. Other types of dependence could be considered, for example, assuming a linear process for y t -an IP for the sample variance is in [33], Theorem 3.8.…”

Section: Letmentioning

confidence: 99%

Testing for instability in covariance structures

Kao¹,

Trapani²,

Urga³

2018

Bernoulli

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We propose a test for the stability over time of the covariance matrix of multivariate time series. The analysis is extended to the eigensystem to ascertain changes due to instability in the eigenvalues and/or eigenvectors. Using strong Invariance Principles and Law of Large Numbers, we normalise the CUSUMtype statistics to calculate their supremum over the whole sample. The power properties of the test versus alternative hypotheses, including also the case of breaks close to the beginning/end of sample are investigated theoretically and via simulation. We extend our theory to test for the stability of the covariance matrix of a multivariate regression model. The testing procedures are illustrated by studying the stability of the principal components of the term structure of 18 US interest rates.

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“…Although we assume an AR model with i.i.d. error (1), in fact as long as r n →∞ and r n = o ( n ) it is known and , , for a truly vast array of time series, including AR with linear or nonlinear GARCH shocks with geometric or hyperbolic memory decay (see Hill, 2010, 2011b and the citations therein). Further, Hill (2010, Theorem 3) presents a consistent kernel estimator of the asymptotic variance of : where w n , s , t is a kernel function.…”

Section: Empirical Applicationmentioning

confidence: 99%

Least tail‐trimmed squares for infinite variance autoregressions

Hill¹

2012

Journal Time Series Analysis

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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‐‐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.

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Tail and Nontail Memory With Applications to Extreme Value and Robust Statistics

Cited by 24 publications

References 93 publications

On Tail Index Estimation for Dependent, Heterogeneous Data

On Tail Index Estimation for Dependent, Heterogeneous Data

Testing for instability in covariance structures

Least tail‐trimmed squares for infinite variance autoregressions

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