Linear probability model

More formally, the LPM can arise from a latent-variable formulation (usually to be found in the econometrics literature [2]), as follows: assume the following regression model with a latent (unobservable) dependent variable:

${\displaystyle y^{*}=b_{0}+\mathbf {x} '\mathbf {b} +\varepsilon ,\;\;\varepsilon \mid \mathbf {x} \sim U(-a,a).}$

The critical assumption here is that the error term of this regression is a symmetric around zero uniform random variable, and hence, of mean zero. The cumulative distribution function of ${\displaystyle \varepsilon }$ here is ${\displaystyle F_{\varepsilon |\mathbf {x} }(\varepsilon \mid \mathbf {x} )={\frac {\varepsilon +a}{2a}}.}$

Define the indicator variable ${\displaystyle y=1}$ if ${\displaystyle y^{*}>0}$, and zero otherwise, and consider the conditional probability

${\displaystyle {\rm {Pr}}(y=1\mid \mathbf {x} )={\rm {Pr}}(y^{*}>0\mid \mathbf {x} )={\rm {Pr}}(b_{0}+\mathbf {x} '\mathbf {b} +\varepsilon >0\mid \mathbf {x} )}$

${\displaystyle ={\rm {Pr}}(\varepsilon >-b_{0}-\mathbf {x} '\mathbf {b} \mid \mathbf {x} )=1-{\rm {Pr}}(\varepsilon \leq -b_{0}-\mathbf {x} '\mathbf {b} \mid \mathbf {x} )}$

${\displaystyle =1-F_{\varepsilon |\mathbf {x} }(-b_{0}-\mathbf {x} '\mathbf {b} \mid \mathbf {x} )=1-{\frac {-b_{0}-\mathbf {x} '\mathbf {b} +a}{2a}}={\frac {b_{0}+a}{2a}}+{\frac {\mathbf {x} '\mathbf {b} }{2a}}.}$

But this is the Linear Probability Model,

${\displaystyle P(y=1\mid \mathbf {x} )=\beta _{0}+\mathbf {x} '\beta }$

with the mapping

${\displaystyle \beta _{0}={\frac {b_{0}+a}{2a}},\;\;\beta ={\frac {\mathbf {b} }{2a}}.}$

This method is a general device to obtain a conditional probability model of a binary variable: if we assume that the distribution of the error term is Logistic, we obtain the logit model, while if we assume that it is the Normal, we obtain the probit model and, if we assume that it is the logarithm of a Weibull distribution, the complementary log-log model.

• Linear approximation

• 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.
• Horrace, William C., and Ronald L. Oaxaca. "Results on the Bias and Inconsistency of Ordinary Least Squares for the Linear Probability Model." Economics Letters, 2006: Vol. 90, P. 321–327