On the Strong Mixing Property for Linear Sequences

Городецкий, В. В.

doi:10.1137/1122049

Cited by 153 publications

(58 citation statements)

References 4 publications

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“…In…nite order distributed lags, for example, need not be mixing due to density smoothness requirements, including ARFIMA, nonlinear ARMA-GARCH and some long memory processes. See Gorodetskii (1977), Andrews (1984), Guegan and Ladoucette (2001), Carrasco and Chen (2002) and Wu (2005).…”

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confidence: 99%

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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confidence: 99%

On Tail Index Estimation for Dependent, Heterogeneous Data

Hill

2010

Econom. Theory

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“…For example, Chanda (1974) first presented that linear stochastic process is strong mixing. Gorodeskii (1977) showed that linear process is strong mixing under certain conditions and also provided the convergent rate for the strong mixing coefficient. Withers (1981) further gave an alternative set of conditions for linear processes to be strong mixing, and proved the strong mixing coefficient is polynomial decay under some conditions.…”

Section: Introductionmentioning

confidence: 95%

Bahadur representation of linear kernel quantile estimator of VaR under assumptions

Xiang-lan

Yang

et al. 2010

Journal of Statistical Planning and Inference

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Bahadur representation and its applications have attracted a large number of publications and presentations on a wide variety of problems. Mixing dependency is weak enough to describe the dependent structure of random variables, including observations in time series and longitudinal studies. This note proves the Bahadur representation of sample quantiles for strongly mixing random variables (including ρ-mixing and φ-mixing) under very weak mixing coefficients. As application, the asymptotic normality is derived. These results greatly improves those recently reported in literature.

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“…Assumption 1 includes a wide variety of possible data-generating mechanisms. For example, both the ARMA process and the MA(∞) process can become strongly mixing under some regularity conditions (see, e.g., Gorodetskii [1977] and Withers [1981]). The mixing condition in this assumption essentially controls the extent of permissible temporal dependence in the process in the relation to the probability of outlier occurrences.…”

Section: A(l)x Tmentioning

confidence: 99%

Spurious Regressions of Stationary AR(p) Processes with Structural Breaks

Chu¹,

Kozhan²

2010

Studies in Nonlinear Dynamics &Amp; Econometrics

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When a pair of independent series is highly persistent, there is a spurious regression bias in a regression between these series, closely related to the classic studies of Granger and Newbold (1974). Although this is well known to occur with independent I(1) processes, this paper provides theoretical and numerical evidence that the phenomenon of spurious regression also arises in regressions between stationary AR(p) processes with structural breaks, which occur at different points in time, in the means and the trends. The intuition behind this is that structural breaks can increase the persistence levels in the processes (e.g., Granger and Hyung (2004)), which then leads to spurious regressions. These phenomena occur for general distributions and serial dependence of the innovation terms.

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On the Strong Mixing Property for Linear Sequences

Cited by 153 publications

References 4 publications

On Tail Index Estimation for Dependent, Heterogeneous Data

On Tail Index Estimation for Dependent, Heterogeneous Data

Bahadur representation of linear kernel quantile estimator of VaR under assumptions

Spurious Regressions of Stationary AR(p) Processes with Structural Breaks

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