# Consistency of the LSE in Linear regression with stationary noise

Guy Cohen; Michael Lin; Arkady Tempelman

Colloquium Mathematicae (2004)

- Volume: 100, Issue: 1, page 29-71
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topGuy Cohen, Michael Lin, and Arkady Tempelman. "Consistency of the LSE in Linear regression with stationary noise." Colloquium Mathematicae 100.1 (2004): 29-71. <http://eudml.org/doc/284040>.

@article{GuyCohen2004,

abstract = {We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also $L_\{p\}$-consistency when the noise is strict sense stationary with continuous spectrum and finite absolute pth moment, p ≥ 1 (even without finite variance).
When the spectral measure of the noise is not continuous, we assume that the non-random regressors are Hartman almost periodic, and obtain a spectral condition for L₂-consistency. An additional assumption on the regressors yields strong consistency for strictly stationary noise sequences.
We also treat the case when the regressors are random sequences, with trends having some good averaging properties and with additive stationary ergodic random fluctuations independent of the noise. When the noise and the fluctuations have disjoint point spectra and the noise is strict sense stationary, we obtain strong consistency of the LSE.
The results are applied to amplitude estimation in sums of harmonic signals with known frequencies.},

author = {Guy Cohen, Michael Lin, Arkady Tempelman},

journal = {Colloquium Mathematicae},

language = {eng},

number = {1},

pages = {29-71},

title = {Consistency of the LSE in Linear regression with stationary noise},

url = {http://eudml.org/doc/284040},

volume = {100},

year = {2004},

}

TY - JOUR

AU - Guy Cohen

AU - Michael Lin

AU - Arkady Tempelman

TI - Consistency of the LSE in Linear regression with stationary noise

JO - Colloquium Mathematicae

PY - 2004

VL - 100

IS - 1

SP - 29

EP - 71

AB - We obtain conditions for L₂ and strong consistency of the least square estimators of the coefficients in a multi-linear regression model with a stationary random noise. For given non-random regressors, we obtain conditions which ensure L₂-consistency for all wide sense stationary noise sequences with spectral measure in a given class. The condition for the class of all noises with continuous (i.e., atomless) spectral measures yields also $L_{p}$-consistency when the noise is strict sense stationary with continuous spectrum and finite absolute pth moment, p ≥ 1 (even without finite variance).
When the spectral measure of the noise is not continuous, we assume that the non-random regressors are Hartman almost periodic, and obtain a spectral condition for L₂-consistency. An additional assumption on the regressors yields strong consistency for strictly stationary noise sequences.
We also treat the case when the regressors are random sequences, with trends having some good averaging properties and with additive stationary ergodic random fluctuations independent of the noise. When the noise and the fluctuations have disjoint point spectra and the noise is strict sense stationary, we obtain strong consistency of the LSE.
The results are applied to amplitude estimation in sums of harmonic signals with known frequencies.

LA - eng

UR - http://eudml.org/doc/284040

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.