Linear probability model: Difference between revisions

Revision as of 19:38, 12 December 2013

Revision as of 19:42, 12 December 2013

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In [[statistics]], a '''linear probability model''' is a special case of a [[binomial regression]] model. Here the [[dependent and independent variables|observed variable]] for each observation takes values which are either 0 or 1. The probability of observing a 0 or 1 in any one case is treated as depending on one or more [[dependent and independent variables|explanatory variables]]. For the "linear probability model", this relationship is a particularly simple one, and allows the model to be fitted by [[simple linear regression]].

The model assumes that, for a binary outcome ([[Bernoulli trial]]), ''<math>Y''</math>, and its associated vector of explanatory variables, <math>X</math>,<ref name=Cox>{{cite book |last=Cox |first=D. R. |year=1970 |title=Analysis of Binary Data |___location=London |publisher=Methuen |isbn=0-416-10400-2 |chapter=Simple Regression |pages=33–42 }}</ref>

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

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For this model,

:<math> E[Y|X] = \Pr(Y=1|X) =x'\beta,</math>

and hence the vector of parameters β can be estimated using [[least squares]]. This method of fitting would be [[Efficiency (statistics)|inefficient]].<ref name=Cox /> This method of fitting can be improved by adopting an iterative scheme based on [[weighted least squares]],<ref name=Cox/> in which the model from the previous iteration is used to supply estimates of the conditional variances, <math>\Var(Y|X=x)</math>, 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.