Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion
Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion
The asymptotic distribution of the least squares estimator in threshold regression is expressed in terms of a compound Poisson process when the threshold effect is fixed and as a functional of two-sided Brownian motion when the threshold effect shrinks to zero. This paper explains the relationship between this dual limit theory by showing how the asymptotic forms are linked in terms of joint and sequential limits. In one case, joint asymptotics apply when both the sample size diverges and the threshold effect shrinks to zero, whereas sequential asymptotics operate in the other case in which the sample size diverges first and the threshold effect shrinks subsequently. The two operations lead to the same limit distribution, thereby linking the two different cases. The proofs make use of ideas involving limit theory for sums of a random number of summands.
Brownian motion, Compound Poisson process, Doob's martingale inequality, Sequential asymptotics, Threshold regression
123-126
Yu, Ping
12919dd2-b91f-4996-bbf5-d6f69799f641
Phillips, Peter C.B.
f67573a4-fc30-484c-ad74-4bbc797d7243
1 November 2018
Yu, Ping
12919dd2-b91f-4996-bbf5-d6f69799f641
Phillips, Peter C.B.
f67573a4-fc30-484c-ad74-4bbc797d7243
Yu, Ping and Phillips, Peter C.B.
(2018)
Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion.
Economics Letters, 172, .
(doi:10.1016/j.econlet.2018.08.039).
Abstract
The asymptotic distribution of the least squares estimator in threshold regression is expressed in terms of a compound Poisson process when the threshold effect is fixed and as a functional of two-sided Brownian motion when the threshold effect shrinks to zero. This paper explains the relationship between this dual limit theory by showing how the asymptotic forms are linked in terms of joint and sequential limits. In one case, joint asymptotics apply when both the sample size diverges and the threshold effect shrinks to zero, whereas sequential asymptotics operate in the other case in which the sample size diverges first and the threshold effect shrinks subsequently. The two operations lead to the same limit distribution, thereby linking the two different cases. The proofs make use of ideas involving limit theory for sums of a random number of summands.
Text
From Compound Poisson Process to Brownian Motion
- Accepted Manuscript
More information
Accepted/In Press date: 23 August 2018
e-pub ahead of print date: 5 September 2018
Published date: 1 November 2018
Keywords:
Brownian motion, Compound Poisson process, Doob's martingale inequality, Sequential asymptotics, Threshold regression
Identifiers
Local EPrints ID: 423711
URI: http://eprints.soton.ac.uk/id/eprint/423711
ISSN: 0165-1765
PURE UUID: 03cd8e78-77db-431b-a0bd-9d9dfe822b5c
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Date deposited: 27 Sep 2018 16:30
Last modified: 16 Mar 2024 07:06
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Author:
Ping Yu
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