# Linear probability model

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

${\displaystyle \Pr(Y=1|X=x)=x'\beta .}$

For this model,

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

and hence the vector of parameters β can be estimated using least squares. This method of fitting would be inefficient,[1] and 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, ${\displaystyle \operatorname {Var} (Y|X=x)}$ , which would vary between observations. This approach can be related to fitting the model by maximum likelihood.[1]

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

## References

1. Cox, D. R. (1970). "Simple Regression". Analysis of Binary Data. London: Methuen. pp. 33–42. ISBN 0-416-10400-2.
• Amemiya, Takeshi (1985). "Qualitative Response Models". Advanced Econometrics. Oxford: Basil Blackwell. pp. 267–359. ISBN 0-631-13345-3.
• 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.