Linear probability model

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In statistics, a linear probability model is a special case of a binomial regression model. Here the 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 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), Y, and its associated vector of explanatory variables, X,[1]

 \Pr(Y=1 | X=x) = x'\beta .

For this model,

 E[Y|X] = \Pr(Y=1|X) =x'\beta,

and hence the vector of parameters β can be estimated using least squares. This method of fitting would be inefficient.[1] This method of fitting can be improved by adopting an iterative scheme based on weighted least squares,[1] in which the model from the previous iteration is used to supply estimates of the conditional variances, Var(Y|X=x), which would vary between observations. This approach can be related to fitting the model by maximum likelihood.[1]

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

References[edit]

  1. ^ a b c d Cox, D. R. (1970). "Simple Regression". Analysis of Binary Data. London: Methuen. pp. 33–42. ISBN 0-416-10400-2. 

Further reading[edit]

  • Wooldridge, Jeffrey M. (2013). "A Binary Dependent Variable: The Linear Probability Model". Introductory Econometrics: A Modern Approach (5th international ed.). Mason, OH: South-Western. pp. 238–243. ISBN 978-1-111-53439-4.