Linear probability model

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, a linear probability model (LPM) is a special case of a <a href="/facts/Binary_regression/YERKnRcr">binary regression</a> model. Here the <a href="/facts/Dependent_and_independent_variables/dINfzIF2">dependent variable</a> 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 <a href="/facts/Dependent_and_independent_variables/dINfzIF2">explanatory variables</a>. For the "linear probability model", this relationship is a particularly simple one, and allows the model to be fitted by <a href="/facts/Linear_regression/5n998IhK">linear regression</a>.
The model assumes that, for a binary outcome (<a href="/facts/Bernoulli_trial/7fm8snCf">Bernoulli trial</a>), 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, and its associated vector of explanatory variables, 
 
 
 
 X
 
 
 {\displaystyle X}
 
,

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

For this model,

E
        [
        Y
        
          |
        
        X
        ]
        =
        0
        ⋅
        Pr
        (
        Y
        =
        0
        
          |
        
        X
        )
        +
        1
        ⋅
        Pr
        (
        Y
        =
        1
        
          |
        
        X
        )
        =
        Pr
        (
        Y
        =
        1
        
          |
        
        X
        )
        =
        
          x
          ′
        
        β
        ,
      
    
    {\displaystyle E[Y|X]=0\cdot \Pr(Y=0|X)+1\cdot \Pr(Y=1|X)=\Pr(Y=1|X)=x'\beta ,}

and hence the vector of parameters β can be estimated using <a href="/facts/Least_squares/50jIwbxC">least squares</a>. This method of fitting would be inefficient, and can be improved by adopting an iterative scheme based on <a href="/facts/Weighted_least_squares/BpCgNF5m">weighted least squares</a>, in which the model from the previous iteration is used to supply estimates of the conditional variances, 
 
 
 
 Var
 ⁡
 (
 Y
 
 |
 
 X
 =
 x
 )
 
 
 {\displaystyle \operatorname {Var} (Y|X=x)}
 
, which would vary between observations. This approach can be related to fitting the model by <a href="/facts/Maximum_likelihood/0Yq2dpQD">maximum likelihood</a>.
A drawback of this model is that, unless restrictions are placed on 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
, the estimated coefficients can imply probabilities outside the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> 
 
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle [0,1]}
 
. For this reason, models such as the <a href="/facts/Logit_model/mGj6mc8y">logit model</a> or the <a href="/facts/Probit_model/k7hCvtYj">probit model</a> are more commonly used.

Linear probability model open-in-new

Linear probability model