Explosive AR(1) process with independent but not identically distributed errors
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Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
Explosive AR(1) process with independent but not identically distributed errors Tae Yoon Kim1 · Sun Young Hwang2 · Haejune Oh3 Received: 18 February 2019 / Accepted: 30 September 2019 © Korean Statistical Society 2020
Abstract Anderson (The Annals of Mathematical Statistics 30(3):676–687, 1959) studied the limiting distribution of the least square estimator for explosive AR(1) process under the independent and identically distributed (iid) condition on error i.e., X t = ρ X t−1 + et where ρ > 1 and et is iid error with Ee = 0 and Ee2 < ∞. This paper is mainly concerned about the limiting distribution of the least square estimator of ρ, that is ρ, ˆ when errors are not identically distributed. In addition, we provide an approximate −j description of the limiting distribution of n−1 j=0 ρ en− j when ρ > 1 as n → ∞. Keywords Explosive AR(1) process · Non-identical distribution · Least square estimator Mathematics Subject Classification 62M10
1 Introduction and main results Consider the following non-stationary AR(1) process defined by X t = ρ X t−1 + et , t = 1, 2, . . .
B
(1)
Haejune Oh [email protected] Tae Yoon Kim [email protected] Sun Young Hwang [email protected]
1
Department of Statistics, Keimyung University, Daegu, Korea
2
Department of Statistics, Sookmyung Womens University, Seoul, Korea
3
Department of Information and Statistics and Research Institute of Natural Science, Gyeongsang National University, Jinju, Korea
123
Journal of the Korean Statistical Society
where the coefficient ρ ≥ 1, the initial value X 0 = 0 and the error process {et , t ≥ 1} is iid with mean zero and finite variance. Observe that Xt =
t−1
ρ j et− j .
j=0
The random walk (RW) refers to ρ = 1. For ρ > 1, {X t , t ≥ 0} is referred to as an ‘explosive’ process. See, for instance, Hwang and Basawa (2005) and Hwang (2013). The two cases, viz., ρ = 1 (RW) and ρ > 1 (explosive), are well investigated and documented in the literatures for the case of iid errors. In this paper, we are mainly concerned about the explosive case (ρ > 1) which depends on the initial condition X 0 or e0 critically. In the literatures it is employed to study economic bubble and often referred to as random chaotic system in the sense that it is highly sensitive to the initial condition. See, e.g., Kim and Hwang (2019) or Lee (2018). When X 1 , X 2 , . . . , X n denote a sample of size n, the least squares estimator of ρ, ρˆ =
n−1
X t X t+1 /
t=1
n−1
X t2 ,
t=1
is known to be n-consistent and ρ n -consistent according to ρ = 1 and ρ > 1, respectively (see Fuller 1996, Ch.10). Now the following expression is useful to derive limit distributions of ρ, ˆ n−1 t−1
n−1 ρˆ − ρ =
t=1 X t et+1 n−1 2 t=1 X t
=
t=1
j j=0 ρ et− j et+1 n−1 2 t=1 X t
=
Un Vn
where Un =
i−1 n
ρ i− j−1 ei e j
i=2 j=1
and
Vn =
n−1 t=1
⎛ ⎞2 t−1 n−1 t−1 n−1 t−1 ⎝ X t2 = ρ j et− j ⎠ = ρ j+k et− j et−k . t=1
j=0
t=1 j=0 k=0
For ρ > 1, Anderson (1959) showed that und
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