scholarly journals Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance

2020 ◽  
Vol 52 (1) ◽  
pp. 237-265 ◽  
Author(s):  
Vytautė Pilipauskaitė ◽  
Viktor Skorniakov ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Stable Distribution ◽  
Stability Index ◽  
Random Coefficient ◽  
Infinite Variance ◽  
Stable Limit ◽  
Normal Attraction ◽  
Tail Distribution ◽  
Increase Rate ◽  
Domain Of Normal Attraction

AbstractWe discuss the joint temporal and contemporaneous aggregation of N independent copies of random-coefficient AR(1) processes driven by independent and identically distributed innovations in the domain of normal attraction of an $\alpha$ -stable distribution, $0< \alpha \le 2$ , as both N and the time scale n tend to infinity, possibly at different rates. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $\beta > 0$ , we show that, for $\beta < \max (\alpha, 1)$ , the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $\alpha$ , $\beta$ and the mutual increase rate of N and n. The paper extends the results of Pilipauskaitė and Surgailis (2014) from $\alpha =2$ to $0 < \alpha < 2$ .

UNIT ROOT TESTS WITH INFINITE VARIANCE ERRORS

2001 ◽  
Vol 20 (4) ◽  
pp. 461-483 ◽  
Author(s):  
Sung K. Ahn ◽  
Stergios B. Fotopoulos ◽  
Lijian He
Keyword(s):  
Unit Root ◽  
Unit Root Tests ◽  
Infinite Variance

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).

The Bounds Based on the Functions of Observations for Maximum of Stable Law

1972 ◽  
Vol 15 (2) ◽  
pp. 285-287
Author(s):  
A. K. Basu ◽  
M. T. Wasan
Keyword(s):  
Stable Law ◽  
Partial Sum ◽  
Normal Attraction ◽  
Image Position ◽  
Domain Of Normal Attraction

Gnedenko and Kolmogorov [3, pp. 181-182] have shown that if Xn with law F(x) belong to the domain of normal attraction of a stable law of index 0<α<2, i.e. if partial sum Sn/an1/α converges in distribution to some stable law Vα, a>0 then there exist c1 and c2 such thatand

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