scholarly journals ESTIMATION FOR A NONSTATIONARY SEMI-STRONG GARCH(1,1) MODEL WITH HEAVY-TAILED ERRORS

2009 ◽  
Vol 26 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Oliver Linton ◽  
Jiazhu Pan ◽  
Hui Wang
Keyword(s):  
Asymptotic Distribution ◽  
Stationary Solution ◽  
Bootstrap Method ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Local Minimizer ◽  
Stable Law ◽  
Moment Conditions ◽  
Least Absolute Deviations ◽  
Heavy Tailed

This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate $\root \of n $ to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponent κ ∈ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponent κ ∈ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.

scholarly journals Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wensheng Wang ◽  
Anwei Zhu
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Random Scenery ◽  
Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi  (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)  (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

A simple proof of a random stable limit theorem

1970 ◽  
Vol 7 (2) ◽  
pp. 502-504 ◽  
Author(s):  
Stephen R. Kimbleton
Keyword(s):  
Limit Theorem ◽  
Elementary Proof ◽  
General Theorem ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Limit ◽  
Stable Law ◽  
Sufficient Conditions For Convergence ◽  
Necessary And Sufficient

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.

On the First-Order Autoregressive Process with Infinite Variance

1989 ◽  
Vol 5 (3) ◽  
pp. 354-362 ◽  
Author(s):  
Ngai Hang Chan ◽  
Lanh Tat Tran
Keyword(s):  
Strong Consistency ◽  
Domain Of Attraction ◽  
Unit Roots ◽  
Autoregressive Process ◽  
Ordinary Least Squares ◽  
Seasonal Difference ◽  
Infinite Variance ◽  
Stable Law ◽  
First Order ◽  
Heavy Tailed

For a first-order autoregressive process Yt = βYt−1 + ∈t where the ∈t'S are i.i.d. and belong to the domain of attraction of a stable law, the strong consistency of the ordinary least-squares estimator bn of β is obtained for β = 1, and the limiting distribution of bn is established as a functional of a Lévy process. Generalizations to seasonal difference models are also considered. These results are useful in testing for the presence of unit roots when the ∈t'S are heavy-tailed.

A simple proof of a random stable limit theorem

1970 ◽  
Vol 7 (02) ◽  
pp. 502-504
Author(s):  
Stephen R. Kimbleton
Keyword(s):  
Limit Theorem ◽  
Elementary Proof ◽  
General Theorem ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Limit ◽  
Stable Law ◽  
Sufficient Conditions For Convergence ◽  
Necessary And Sufficient

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.

scholarly journals The effect of random scale changes on limits of infinitesimal systems

1978 ◽  
Vol 1 (3) ◽  
pp. 339-372
Author(s):  
Patrick L. Brockett
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Stable Law ◽  
Scale Parameters ◽  
Exact Relationship ◽  
Random Scale

SupposeS={{Xnj,   j=1,2,…,kn}}is an infinitesimal system of random variables whose centered sums converge in law to a (necessarily infinitely divisible) distribution with Levy representation determined by the triple(γ,σ2,M). If{Yj,   j=1,2,…}are independent indentically distributed random variables independent ofS, then the systemS′={{YjXnj,j=1,2,…,kn}}is obtained by randomizing the scale parameters inSaccording to the distribution ofY1. We give sufficient conditions on the distribution ofYin terms of an index of convergence ofS, to insure that centered sums fromS′be convergent. If such sums converge to a distribution determined by(γ′,(σ′)2,Λ), then the exact relationship between(γ,σ2,M)and(γ′,(σ′)2,Λ)is established. Also investigated is when limit distributions fromSandS′are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.

scholarly journals On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

2006 ◽  
Vol 43 (02) ◽  
pp. 421-440
Author(s):  
Remigijus Leipus ◽  
Vygantas Paulauskas ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Domain Of Attraction ◽  
Stationary Solutions ◽  
Partial Sums ◽  
Infinite Variance ◽  
Finite Variance ◽  
Stable Law ◽  
Stable Lévy Process ◽  
Heavy Tailed ◽  
Independent Identically Distributed

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X t = a t X t−1 + ε t with random (renewal-reward) coefficient, a t , taking independent, identically distributed values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, ε t , belonging to the domain of attraction of an α-stable law (0 &lt; α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of A j near the unit root a = 1, we show that the partial sums process of X t converges to a λ-stable Lévy process with index λ &lt; α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance X t to that of infinite-variance X t .

scholarly journals Spectrum of large random Markov chains: Heavy-tailed weights on the oriented complete graph

2017 ◽  
Vol 06 (02) ◽  
pp. 1750006 ◽  
Author(s):  
Charles Bordenave ◽  
Pietro Caputo ◽  
Djalil Chafaï ◽  
Daniele Piras
Keyword(s):  
Probability Measure ◽  
Domain Of Attraction ◽  
Oriented Graph ◽  
Singular Values ◽  
Empirical Measure ◽  
Stable Law ◽  
Smoothness Assumption ◽  
Infinite Tree ◽  
Heavy Tailed ◽  
First Moment

We consider the random Markov matrix obtained by assigning i.i.d. non-negative weights to each edge of the complete oriented graph. In this study, the weights have unbounded first moment and belong to the domain of attraction of an alpha-stable law. We prove that as the dimension tends to infinity, the empirical measure of the singular values tends to a probability measure which depends only on alpha, characterized as the expected value of the spectral measure at the root of a weighted random tree. The latter is a generalized two-stage version of the Poisson weighted infinite tree (PWIT) introduced by David Aldous. Under an additional smoothness assumption, we show that the empirical measure of the eigenvalues tends to a non-degenerate isotropic probability measure depending only on alpha and supported on the unit disk of the complex plane. We conjecture that the limiting support is actually formed by a strictly smaller disk.

scholarly journals On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

2006 ◽  
Vol 43 (2) ◽  
pp. 421-440
Author(s):  
Remigijus Leipus ◽  
Vygantas Paulauskas ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Domain Of Attraction ◽  
Stationary Solutions ◽  
Partial Sums ◽  
Infinite Variance ◽  
Finite Variance ◽  
Stable Law ◽  
Stable Lévy Process ◽  
Heavy Tailed ◽  
Independent Identically Distributed

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation Xt = atXt−1 + εt with random (renewal-reward) coefficient, at, taking independent, identically distributed values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, εt, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of Aj near the unit root a = 1, we show that the partial sums process of Xt converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance Xt to that of infinite-variance Xt.

scholarly journals MIXED CAUSAL-NONCAUSAL AR PROCESSES AND THE MODELLING OF EXPLOSIVE BUBBLES

2019 ◽  
Vol 35 (6) ◽  
pp. 1234-1270 ◽  
Author(s):  
Sébastien Fries ◽  
Jean-Michel Zakoian
Keyword(s):  
Conditional Distribution ◽  
Stable Distribution ◽  
Financial Time Series ◽  
Domain Of Attraction ◽  
Real Data ◽  
Autoregressive Processes ◽  
Financial Time ◽  
Causal Representation ◽  
Heavy Tailed ◽  
Non Gaussian

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.

REDUNDANCY OF MOMENT CONDITIONS AND THE EFFICIENCY OF OLS IN SUR MODELS

2008 ◽  
Vol 24 (5) ◽  
pp. 1456-1460 ◽  
Author(s):  
Hailong Qian
Keyword(s):  
Least Squares ◽  
Asymptotic Efficiency ◽  
Generalized Method Of Moments ◽  
Sufficient Conditions ◽  
Generalized Least Squares ◽  
Ordinary Least Squares ◽  
American Statistical Association ◽  
Orthogonality Condition ◽  
Moment Conditions ◽  
Necessary And Sufficient

In this note, based on the generalized method of moments (GMM) interpretation of the usual ordinary least squares (OLS) and feasible generalized least squares (FGLS) estimators of seemingly unrelated regressions (SUR) models, we show that the OLS estimator is asymptotically as efficient as the FGLS estimator if and only if the cross-equation orthogonality condition is redundant given the within-equation orthogonality condition. Using the condition for redundancy of moment conditions of Breusch, Qian, Schmidt, and Wyhowski (1999, Journal of Econometrics 99, 89–111), we then derive the necessary and sufficient condition for the equal asymptotic efficiency of the OLS and FGLS estimators of SUR models. We also provide several useful sufficient conditions for the equal asymptotic efficiency of OLS and FGLS estimators that can be interpreted as various mixings of the two famous sufficient conditions of Zellner (1962, Journal of the American Statistical Association 57, 348–368).

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