Linear probability model: Difference between revisions

Revision as of 15:11, 17 December 2023

Latest revision as of 20:08, 22 May 2025

(One intermediate revision by one other user not shown)

Line 14:

==Latent-variable formulation==

More formally, the LPM can arise from a latent-variable formulation (usually to be found in the [[econometrics]] literature<ref name=Amemiya>{{cite journal |last=Amemiya |first=Takeshi |year=1981 |title=Qualitative Response Models: A Survey|journal=Journal of Economic Literature |volume =19 |number =4 |pages=1483–1536 }}</ref>), as follows: assume the following regression model with a latent (unobservable) dependent variable:

: <math>y^* = b_0+ \mathbf x'\mathbf b + \varepsilon,\;\; \varepsilon\mid \mathbf x\sim U(-a,a).</math>

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

Define the indicator variable <math> y = 1</math> if <math> y^* >0</math>, and zero otherwise, and consider the conditional probability

Line 39:

:<math>\beta_0 = \frac {b_0+a}{2a},\;\; \beta=\frac{\mathbf b}{2a}.</math>

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 Logisticlogistic, we obtain the [[logit model]], while if we assume that it is the Normalnormal, we obtain the [[probit model]] and, if we assume that it is the logarithm of a [[Weibull distribution]], the [[Generalized linear model|complementary log-log model]].

== See also ==