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 < 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 < 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 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 Time Series Regression With a Unit Root and Infinite-Variance Errors

1990 ◽  
Vol 6 (1) ◽  
pp. 44-62 ◽  
Author(s):  
P.C.B. Phillips
Keyword(s):  
Unit Root ◽  
Stable Process ◽  
Nuisance Parameter ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Finite Variance ◽  
Quadratic Functional ◽  
Time Series Regression ◽  
Limit Laws ◽  
Stable Law

In [4] Chan and Tran give the limit theory for the least-squares coefficient in a random walk with i.i.d. (identically and independently distributed) errors that are in the domain of attraction of a stable law. This paper discusses their results and provides generalizations to the case of I(1) processes with weakly dependent errors whose distributions are in the domain of attraction of a stable law. General unit root tests are also studied. It is shown that the semiparametric corrections suggested by the author in other work [22] for the finite-variance case continue to work when the errors have infinite variance. Surprisingly, no modifications to the formulas given in [22] are required. The limit laws are expressed in terms of ratios of quadratic functional of a stable process rather than Brownian motion. The correction terms that eliminate nuisance parameter dependencies are random in the limit and involve multiple stochastic integrals that may be written in terms of the quadratic variation of the limiting stable process. Some extensions of these results to models with drifts and time trends are also indicated.

scholarly journals Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

2010 ◽  
Vol 42 (2) ◽  
pp. 509-527 ◽  
Author(s):  
Donata Puplinskaitė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Moving Average ◽  
Domain Of Attraction ◽  
Random Coefficient ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Stable Law ◽  
Slope Coefficient ◽  
Finite Dimensional

Contemporaneous aggregation ofNindependent copies of a random-coefficient AR(1) process with random coefficienta∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalizationN1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficientahas probability density vanishing regularly ata= 1 with exponentb∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

scholarly journals Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

2010 ◽  
Vol 42 (02) ◽  
pp. 509-527 ◽  
Author(s):  
Donata Puplinskaitė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Moving Average ◽  
Domain Of Attraction ◽  
Random Coefficient ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Stable Law ◽  
Slope Coefficient ◽  
Finite Dimensional

Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 &lt; α &lt; 2) is discussed. We show that, under the normalization N 1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅ t } exhibits long memory. In particular, for {X̅ t }, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

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.

Regression and autoregression with infinite variance

1974 ◽  
Vol 6 (4) ◽  
pp. 768-783 ◽  
Author(s):  
Marek Kanter ◽  
W. L. Steiger
Keyword(s):  
Least Squares ◽  
Unit Circle ◽  
Unbiased Estimate ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Linear Combinations ◽  
Stable Law ◽  
Least Squares Estimate ◽  
Least Squares Estimates ◽  
Independent Identically Distributed

The theory of the linear model is incomplete in that it fails to deal with variables possessing infinite variance. To fill an important part of this gap, we give an unbiased estimate, the “screened ratio estimate”, for λ in the regression E(X|Z) = λX; X and Z are linear combinations of independent, identically distributed symmetric random variables that are either stable or asymptotically Pareto distributed of index α ≤ 2. By way of comparison, the usual least squares estimate of λ is shown not to converge in general to any constant when α < 2. However, in the autoregression Xn = a1Xn-1 + … + akXn-k + Un, the least squares estimates are shown to be consistent as long as the roots of 1 - a1x2 - a2x2 - … - akxk = 0 are outside the complex unit circle, Xn is independent of Un+j,j ≥ 1, and the Un are independent and identically distributed and in the domain of attraction of a stable law of index a ≤ 2. Finally, the consistency of least squares estimates for finite moving averages is established.

Asymptotic Theory of LAD Estimation in a Unit Root Process with Finite Variance Errors

1996 ◽  
Vol 12 (1) ◽  
pp. 129-153 ◽  
Author(s):  
Miguel A. Herce
Keyword(s):  
Unit Root ◽  
Infinite Variance ◽  
Finite Variance ◽  
Test Statistics ◽  
Finite Sample ◽  
Least Absolute Deviations ◽  
Autoregressive Parameter ◽  
Finite Sample Properties ◽  
Heavy Tailed ◽  
Lad Estimation

In this paper we derive the asymptotic distribution of the least absolute deviations (LAD) estimator of the autoregressive parameter under the unit root hypothesis, when the errors are assumed to have finite variances, and present LAD-based unit root tests, which, under heavy-tailed errors, are expected to be more powerful than tests based on least squares. The limiting distribution of the LAD estimator is that of a functional of a bivariate Brownian motion, similar to those encountered in cointegrating regressions. By appropriately correcting for serial correlation and other distributional parameters, the test statistics introduced here are found to have either conditional or unconditional normal limiting distributions. The results of the paper complement similar ones obtained by Knight (1991, Canadian Journal of Statistics 17, 261-278) for infinite variance errors. A simulation study is conducted to investigate the finite sample properties of our tests.

Regression and autoregression with infinite variance

1974 ◽  
Vol 6 (04) ◽  
pp. 768-783 ◽  
Author(s):  
Marek Kanter ◽  
W. L. Steiger
Keyword(s):  
Least Squares ◽  
Unit Circle ◽  
Unbiased Estimate ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Linear Combinations ◽  
Stable Law ◽  
Least Squares Estimate ◽  
Least Squares Estimates ◽  
Independent Identically Distributed

The theory of the linear model is incomplete in that it fails to deal with variables possessing infinite variance. To fill an important part of this gap, we give an unbiased estimate, the “screened ratio estimate”, for λ in the regression E(X|Z) = λX; X and Z are linear combinations of independent, identically distributed symmetric random variables that are either stable or asymptotically Pareto distributed of index α ≤ 2. By way of comparison, the usual least squares estimate of λ is shown not to converge in general to any constant when α &lt; 2. However, in the autoregression X n = a 1 X n-1 + … + a k X n-k + U n , the least squares estimates are shown to be consistent as long as the roots of 1 - a 1 x 2 - a 2 x 2 - … - a k x k = 0 are outside the complex unit circle, X n is independent of U n+j ,j ≥ 1, and the U n are independent and identically distributed and in the domain of attraction of a stable law of index a ≤ 2. Finally, the consistency of least squares estimates for finite moving averages is established.

M-ESTIMATION FOR A SPATIAL UNILATERAL AUTOREGRESSIVE MODEL WITH INFINITE VARIANCE INNOVATIONS

2010 ◽  
Vol 26 (6) ◽  
pp. 1663-1682 ◽  
Author(s):  
S.M. Roknossadati ◽  
M. Zarepour
Keyword(s):  
Simulation Study ◽  
Unit Root ◽  
Autoregressive Model ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Stable Law ◽  
Limiting Behavior ◽  
Asymptotically Normal ◽  
M Estimation ◽  
Unit Root Models

We study the limiting behavior of the M-estimators of parameters for a spatial unilateral autoregressive model with independent and identically distributed innovations in the domain of attraction of a stable law with index α ∈ (0, 2]. Both stationary and unit root models and some extensions are considered. It is also shown that self-normalized M-estimators are asymptotically normal. A numerical example and a simulation study are also given.

Weak convergence and first passage times

1975 ◽  
Vol 12 (02) ◽  
pp. 324-332
Author(s):  
Allan Gut
Keyword(s):  
First Passage Time ◽  
Domain Of Attraction ◽  
Passage Time ◽  
Partial Sums ◽  
First Passage ◽  
Stable Law ◽  
Common Distribution Function ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Passage Times

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk &gt; c‥kp ], c ≥ 0, 0 ≤ p &lt; 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 &lt; α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

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