TY - JOUR

T1 - Some properties of inferences in misspecified linear models

AU - Severini, Thomas A.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1998/9/15

Y1 - 1998/9/15

N2 - Let Y denote an n × 1 vector of observations such that Y = μ + σε where μ is an unknown n × 1 vector, σ > 0 is an unknown parameter, and ε is an n × 1 vector of independent standard normal random variables. A linear regression analysis is often based on a model for μ such as μ = Xβ where X is a known n × p matrix of independent variables and β is a p × 1 vector of unknown parameters. When the assumption that μ = Xβ for some β holds, the results of the analysis can be interpreted as applying to μ, the mean of Y. In this paper, the properties of inferences based on the model μ = Xβ are considered without assuming that the model holds. It is shown that many of the usual properties continue to hold, although with respect to μ*, the vector of form Xβ closest to μ, rather than with respect to μ. Hence, the results of a linear regression analysis have a certain type of validity that applies whether or not the model is correctly specified.

AB - Let Y denote an n × 1 vector of observations such that Y = μ + σε where μ is an unknown n × 1 vector, σ > 0 is an unknown parameter, and ε is an n × 1 vector of independent standard normal random variables. A linear regression analysis is often based on a model for μ such as μ = Xβ where X is a known n × p matrix of independent variables and β is a p × 1 vector of unknown parameters. When the assumption that μ = Xβ for some β holds, the results of the analysis can be interpreted as applying to μ, the mean of Y. In this paper, the properties of inferences based on the model μ = Xβ are considered without assuming that the model holds. It is shown that many of the usual properties continue to hold, although with respect to μ*, the vector of form Xβ closest to μ, rather than with respect to μ. Hence, the results of a linear regression analysis have a certain type of validity that applies whether or not the model is correctly specified.

UR - http://www.scopus.com/inward/record.url?scp=0032530036&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032530036&partnerID=8YFLogxK

U2 - 10.1016/S0167-7152(98)00082-0

DO - 10.1016/S0167-7152(98)00082-0

M3 - Article

AN - SCOPUS:0032530036

SN - 0167-7152

VL - 40

SP - 149

EP - 153

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

IS - 2

ER -