Linear probability model: Difference between revisions

Revision as of 15:13, 19 July 2010

Revision as of 15:14, 19 July 2010

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==The model==

The model assumes that, for a binary outcome ([[Bernoulli trial]]), ''Y'', and its associated vector of explanatory variables, ''X'',]]<ref name=Cox>Cox, D.R. (1970) ''Analysis of Binary Data'', Methuen. ISBN 0416-10400-2(Section 2.2)</ref>

: <math> \Pr(Y=1 | X=x) = x'\beta . </math>

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and hence the vector of parameters &beta; can be estimated using [[least squares]]. This method of fitting would be [[Efficiency (statistics)|inefficient]]<ref name=Cox>Cox, D.R. (1970) ''Analysis of Binary Data'', Methuen. ISBN 0416-10400-2(Section 2.2)</ref> This method of fitting can be improved<ref name=Cox/> by adopting an iterative scheme based on [[weighted least squares]], in which the model from the previous iteration is used to supply estimates of the condotional variances, var(''Y''|''X=x''), which would vary between observations. This approach can be related to fitting the model by [[maximum likelihood]].<ref name=Cox/>

A drawback of this model for the parameter of the [[Bernoulli distribution]] is that, unless restrictions are placed on <math> \beta </math>, the estimated coefficients can imply probabilities outside the [[unit interval]] <math> [0,1] </math> . For this reason, models such as the [[logit model]] or the [[probit model]] are more commonly used.

==References==