# Linear Probability Functions Family

(Redirected from Linear Statistical Model)

## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/linear_probability_model Retrieved:2015-2-16.
• In statistics, a linear probability model is a special case of a binomial regression model. Here the observed 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 simple linear regression.

The model assumes that, for a binary outcome (Bernoulli trial), $Y$, and its associated vector of explanatory variables, $X$,[1] :$\Pr(Y=1 | X=x) = x'\beta .$

For this model, :$E[Y|X] = \Pr(Y=1|X) =x'\beta,$

and hence the vector of parameters β can be estimated using least squares. This method of fitting would be inefficient.[1] This method of fitting 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, $Var(Y|X=x)$, which would vary between observations. This approach can be related to fitting the model by maximum likelihood.[1]

A drawback of this model for the parameter of the Bernoulli distribution is that, unless restrictions are placed on $\beta$, the estimated coefficients can imply probabilities outside the unit interval $[0,1]$. For this reason, models such as the logit model or the probit model are more commonly used.

1. Cite error: Invalid <ref> tag; no text was provided for refs named Cox