Estimator for the Heavy Tailed Index with the Stable Distribution

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

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Regression with Stable Errors Based on Order Statistics

Fluctuation and Noise Letters ◽

10.1142/s0219477522500146 ◽

2021 ◽

Author(s):

Reza Alizadeh Noughabi ◽

Adel Mohammadpour

Keyword(s):

Order Statistics ◽

Stable Distribution ◽

Optimal Number ◽

Least Square ◽

Regression Coefficients ◽

Regression Parameters ◽

The Mean ◽

Heavy Tailed ◽

Ordered Data ◽

Error Order

Classical regression approaches are not robust when errors are heavy-tailed or asymmetric. That may be due to the non-existence of the mean or variance of the error distribution. Estimation based on trimmed data, which ignored outlier or leverage points, has an old history and frequently used. This procedure chooses fixed cut-off points. In this work, we use this idea recently applied for initial estimates of regression coefficients with heavy-tailed stable errors. We propose an effective procedure to calculate the cut-off points based on the tail index and skewness parameters of errors. We use the property of the existence of some moments of stable distribution order statistics. Data are trimmed based on ordered residuals of a least square regression. However, the trimmed data’s optimal number is determined based on the number of error order statistics whose variance exists. Then, we use the rest of the ordered data to estimate the regression coefficients. Based on these order statistics’ joint distribution, we analytically compute the bias and variance of the introduced estimator of regression parameters that was impossible for regression with stable errors.

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DEA Model Considering Outputs with Stochastic Noise and a Heavy-Tailed (Stable) Distribution

Advances in Data Science and Adaptive Analysis ◽

10.1142/s2424922x18500055 ◽

2018 ◽

Author(s):

Hassan Naseri ◽

S. Esmaeil Najafi ◽

Abbas Saghaei

Keyword(s):

Stable Distribution ◽

Stochastic Noise ◽

Dea Model ◽

Heavy Tailed

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DEA model considering outputs with stochastic noise and a heavy-tailed (stable) distribution

INFOR Information Systems and Operational Research ◽

10.1080/03155986.2019.1624476 ◽

2019 ◽

Vol 58 (1) ◽

pp. 87-108

Author(s):

Hassan Naseri ◽

S. Esmaeil Najafi ◽

Abbas Saghaei

Keyword(s):

Stable Distribution ◽

Stochastic Noise ◽

Dea Model ◽

Heavy Tailed

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Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations

Mathematics ◽

10.3390/math9080816 ◽

2021 ◽

Vol 9 (8) ◽

pp. 816

Author(s):

Eunju Hwang

Keyword(s):

Unit Root ◽

Stable Distribution ◽

Asymptotic Properties ◽

Tail Index ◽

Asymptotic Distributions ◽

Uniform Limit ◽

Limit Theory ◽

Heavy Tailed ◽

Heteroscedastic Variance ◽

Ar Models

This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient ρ=ρn is derived uniformly over stationary values in [0,1), focusing on ρn→1 as sample size n tends to infinity. For tail index α∈(0,4) of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on 1−ρn2, but no condition on the rate of ρn is required. It is shown that, for the tail index α∈(0,2), the LSE is inconsistent, for α=2, logn/(1−ρn2)-consistent, and for α∈(2,4), n1−2/α/(1−ρn2)-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index α∈(0,4); and no restriction on the rate of ρn is necessary.

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An Alternative to Cohen's κ

European Psychologist ◽

10.1027/1016-9040.11.1.12 ◽

2006 ◽

Vol 11 (1) ◽

pp. 12-24 ◽

Cited By ~ 19

Author(s):

Alexander von Eye

Keyword(s):

Simulation Study ◽

Null Hypothesis ◽

Categorical Variables ◽

Alternative Measure ◽

Rater Agreement ◽

Verbal Processing ◽

Heavy Tailed ◽

Applicant Selection

At the level of manifest categorical variables, a large number of coefficients and models for the examination of rater agreement has been proposed and used. The most popular of these is Cohen's κ. In this article, a new coefficient, κ s , is proposed as an alternative measure of rater agreement. Both κ and κ s allow researchers to determine whether agreement in groups of two or more raters is significantly beyond chance. Stouffer's z is used to test the null hypothesis that κ s = 0. The coefficient κ s allows one, in addition to evaluating rater agreement in a fashion parallel to κ, to (1) examine subsets of cells in agreement tables, (2) examine cells that indicate disagreement, (3) consider alternative chance models, (4) take covariates into account, and (5) compare independent samples. Results from a simulation study are reported, which suggest that (a) the four measures of rater agreement, Cohen's κ, Brennan and Prediger's κ n , raw agreement, and κ s are sensitive to the same data characteristics when evaluating rater agreement and (b) both the z-statistic for Cohen's κ and Stouffer's z for κ s are unimodally and symmetrically distributed, but slightly heavy-tailed. Examples use data from verbal processing and applicant selection.

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