Content deleted Content added
ThingsLittle (talk | contribs) No edit summary |
|||
Line 1:
In [[statistics]], a '''linear probability model''' (LPM) is a special case of a [[binary regression]] model. Here the [[dependent and independent variables|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 [[dependent and independent variables|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]]), <math>Y</math>, and its associated vector of explanatory variables, <math>X</math>,<ref name=Cox>{{cite book |last=Cox |first=D. R. |year=1970 |title=Analysis of Binary Data |___location=London |publisher=Methuen |isbn=0-416-10400-2 |chapter=Simple Regression |pages=33–42 }}</ref>
Line 10:
A drawback of this model is that, unless restrictions are placed on <math> \beta </math>, the estimated coefficients can imply probabilities outside the [[unit interval]] <math> [0,1] </math>. For this reason, models such as the [[logit model]] or the [[probit model]] are more commonly used.
==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 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
:<math>{\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) </math>
:<math> = {\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)</math>
:<math>=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}.</math>
But this is the Linear Probability Model,
:<math>P(y =1\mid \mathbf x )= \beta_0 + \mathbf x'\beta</math>
with the mapping
:<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 Logistic, we obtain the [[logit model]], while if we assume that it is the Normal, we obtain the [[probit model]].
== See also ==
| |||