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.

MULTIVARIATE AUTOREGRESSION OF ORDER ONE WITH INFINITE VARIANCE INNOVATIONS

2008 ◽  
Vol 24 (3) ◽  
pp. 677-695 ◽  
Author(s):  
M. Zarepour ◽  
S.M. Roknossadati
Keyword(s):  
Autoregressive Model ◽  
Domain Of Attraction ◽  
Ordinary Least Squares ◽  
Infinite Variance ◽  
Regularity Conditions ◽  
Vector Autoregressive Model ◽  
Stable Law ◽  
Limiting Behavior ◽  
Vector Autoregressive ◽  
Multivariate Autoregression

We consider the limiting behavior of a vector autoregressive model of order one (VAR(1)) with independent and identically distributed (i.i.d.) innovations vector with dependent components in the domain of attraction of a multivariate stable law with possibly different indices of stability. It is shown that in some cases the ordinary least squares (OLS) estimates are inconsistent. This inconsistency basically originates from the fact that each coordinate of the partial sum processes of dependent i.i.d. vectors of innovations in the domain of attraction of stable laws needs a different normalizer to converge to a limiting process. It is also revealed that certain M-estimates, with some regularity conditions, as an appropriate alternative, not only resolve inconsistency of the OLS estimates but also give higher consistency rates in all cases.

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 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 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.

Autoregressive processes with infinite variance

1977 ◽  
Vol 14 (02) ◽  
pp. 411-415 ◽  
Author(s):  
E. J. Hannan ◽  
Marek Kanter
Keyword(s):  
Least Squares ◽  
Autoregressive Model ◽  
Infinite Variance ◽  
Autoregressive Processes ◽  
Stable Law ◽  
Least Squares Estimators ◽  
Data Points

The least squares estimators β i(N), j = 1, …, p, from N data points, of the autoregressive constants for a stationary autoregressive model are considered when the disturbances have a distribution attracted to a stable law of index α &lt; 2. It is shown that N1/δ(β i(N) – β) converges almost surely to zero for any δ &gt; α. Some comments are made on alternative definitions of the βi (N).

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 EXPLOITING INFINITE VARIANCE THROUGH DUMMY VARIABLES IN NONSTATIONARY AUTOREGRESSIONS

2013 ◽  
Vol 29 (6) ◽  
pp. 1162-1195 ◽  
Author(s):  
Giuseppe Cavaliere ◽  
Iliyan Georgiev
Keyword(s):  
Unit Root ◽  
Asymptotic Properties ◽  
Ordinary Least Squares ◽  
Infinite Variance ◽  
Local Power ◽  
Test Statistic ◽  
Asymptotically Normal ◽  
Normal Test ◽  
Order Of Magnitude ◽  
Near Unit Root

We consider estimation and testing in finite-order autoregressive models with a (near) unit root and infinite-variance innovations. We study the asymptotic properties of estimators obtained by dummying out “large” innovations, i.e., those exceeding a given threshold. These estimators reflect the common practice of dealing with large residuals by including impulse dummies in the estimated regression. Iterative versions of the dummy-variable estimator are also discussed. We provide conditions on the preliminary parameter estimator and on the threshold that ensure that (i) the dummy-based estimator is consistent at higher rates than the ordinary least squares estimator, (ii) an asymptotically normal test statistic for the unit root hypothesis can be derived, and (iii) order of magnitude gains of local power are obtained.

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 distribution tails and expectations of maxima in critical branching processes

1996 ◽  
Vol 33 (3) ◽  
pp. 614-622 ◽  
Author(s):  
K. A. Borovkov ◽  
V A. Vatutin
Keyword(s):  
Related Problem ◽  
Branching Processes ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Global Maximum ◽  
Stable Law ◽  
Distribution Tail ◽  
Limit Behaviour ◽  
Watson Process

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.

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.

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