In statistics, a linear probability model is a special case of a binary 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 linear regression.
The model assumes that, for a binary outcome (Bernoulli trial), , and its associated vector of explanatory variables, ,[1]
For this model,
![E[Y|X]=\Pr(Y=1|X)=x'\beta ,](http://img.tfd.com/wiki/math/1c/02479ad18df900bce02e137c69c241676e6466bf.svg)
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, , 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 , the estimated coefficients can imply probabilities outside the unit interval . For this reason, models such as the logit model or the probit model are more commonly used.
![[0,1]](http://img.tfd.com/wiki/math/70/738f7d23bb2d9642bab520020873cccbef49768d.svg)
See also
References
Further reading
- Aldrich, John H.; Nelson, Forrest D. (1984). "The Linear Probability Model". Linear Probability, Logit, and Probit Models. Sage. pp. 9–29. ISBN 0-8039-2133-0.
- 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.