Asymptotic Inference for Nearly Nonstationary AR(1) Processes

N. H. CHAN; C. Z. WEI

doi:10.1214/aos/1176350492

Asymptotic Inference for Nearly Nonstationary AR(1) Processes

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

A first-order autoregressive process, Y_t= βY_t₋₁+ ε_t, is said to be nearly nonstationary when β is close to one. The limiting distribution of the least-squares estimate bn for β is studied when Y_t is nearly nonstationary. By reparameterizing β to be 1 − γ/n, γ being a fixed constant, it is shown that the limiting distribution of τ_n= (∑ⁿ_t=1Y²_t−1)^1/2(b_n− β) converges to ℒ(γ) which is a quotient of stochastic integrals of standard Brownian motion. This provides a reasonable alternative to the approximation of the distribution of τ_n proposed by Ahtola and Tiao (1984). © 1987 Institute of Mathematical Statistics

Original language	English
Pages (from-to)	1050-1063
Journal	Annals of Statistics
Volume	15
Issue number	3
DOIs	https://doi.org/10.1214/aos/1176350492
Publication status	Published - Sept 1987
Externally published	Yes

Research Keywords

Autoregressive process
Least squares
Nearly nonstationary
Stochastic integral

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title = "Asymptotic Inference for Nearly Nonstationary AR(1) Processes",

abstract = "A first-order autoregressive process, Yt = βYt−1 + εt, is said to be nearly nonstationary when β is close to one. The limiting distribution of the least-squares estimate bn for β is studied when Yt is nearly nonstationary. By reparameterizing β to be 1 − γ/n, γ being a fixed constant, it is shown that the limiting distribution of τn = (∑nt=1Y2t−1)1/2(bn − β) converges to ℒ(γ) which is a quotient of stochastic integrals of standard Brownian motion. This provides a reasonable alternative to the approximation of the distribution of τn proposed by Ahtola and Tiao (1984). {\textcopyright} 1987 Institute of Mathematical Statistics",

keywords = "Autoregressive process, Least squares, Nearly nonstationary, Stochastic integral",

author = "CHAN, {N. H.} and WEI, {C. Z.}",

year = "1987",

month = sep,

doi = "10.1214/aos/1176350492",

language = "English",

volume = "15",

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journal = "Annals of Statistics",

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publisher = "Institute of Mathematical Statistics",

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TY - JOUR

T1 - Asymptotic Inference for Nearly Nonstationary AR(1) Processes

AU - CHAN, N. H.

AU - WEI, C. Z.

PY - 1987/9

Y1 - 1987/9

N2 - A first-order autoregressive process, Yt = βYt−1 + εt, is said to be nearly nonstationary when β is close to one. The limiting distribution of the least-squares estimate bn for β is studied when Yt is nearly nonstationary. By reparameterizing β to be 1 − γ/n, γ being a fixed constant, it is shown that the limiting distribution of τn = (∑nt=1Y2t−1)1/2(bn − β) converges to ℒ(γ) which is a quotient of stochastic integrals of standard Brownian motion. This provides a reasonable alternative to the approximation of the distribution of τn proposed by Ahtola and Tiao (1984). © 1987 Institute of Mathematical Statistics

AB - A first-order autoregressive process, Yt = βYt−1 + εt, is said to be nearly nonstationary when β is close to one. The limiting distribution of the least-squares estimate bn for β is studied when Yt is nearly nonstationary. By reparameterizing β to be 1 − γ/n, γ being a fixed constant, it is shown that the limiting distribution of τn = (∑nt=1Y2t−1)1/2(bn − β) converges to ℒ(γ) which is a quotient of stochastic integrals of standard Brownian motion. This provides a reasonable alternative to the approximation of the distribution of τn proposed by Ahtola and Tiao (1984). © 1987 Institute of Mathematical Statistics

KW - Autoregressive process

KW - Least squares

KW - Nearly nonstationary

KW - Stochastic integral

UR - https://www.jstor.org/stable/2241815

U2 - 10.1214/aos/1176350492

DO - 10.1214/aos/1176350492

M3 - RGC 21 - Publication in refereed journal

SN - 0090-5364

VL - 15

SP - 1050

EP - 1063

JO - Annals of Statistics

JF - Annals of Statistics

IS - 3

ER -

Asymptotic Inference for Nearly Nonstationary AR(1) Processes

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Research Keywords

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