Binary choice models

Summary

I want to redo this. “Binary choice models are not regression models. There is no model for \(y_i\) and there are no error terms \(ε_i\) (see also the latent variable representation of the binary choice model).” is problematic. Sure you can define \(ε_i=y_i-E\left( x_i \right)\) as before and \(y_i=E\left( x_i \right)+ε_i\) is a regression model which we estimate using ML, not NLS.

Emphasize: in the LRM, we model \(p_i=ζ_i\) . Better, model such that \(p_i\) depends positively on \(ζ_i\) without being equal to \(ζ_i\) . If \(ζ_i→∞\) we want \(p_i→1\) . If \(ζ_i→-∞\) we want \(p_i→0\) . This will be the case if we model

\[p_i= \frac{1}{1+e^{-ζ_i}}\]

Setup

• Given: a random sample where the dependent variable \(y_i\) is a dummy variable.
• Define

\[ζ_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}\]

• \(ζ_i\) is important in binary choice models but it is no longer assumed that \(ζ_i=P\left( y_i=1|x_i \right)\) as in the linear probability model.

Binary choice models

• In a binary choice model, we assume that

\[P\left( y_i=1|x_i \right)=F\left( ζ_i \right) , i=1,…,n\]

• where \(F\) is any strictly increasing function with domain \(R\) and range \(\left( 0,1 \right)\) .
• The model is given a name based on the choice of \(F\) .
• If

\[F\left( x \right)= \frac{1}{1+e^{-x}}\]

• then the binary choice model is called a logit model.
• If \(F\left( x \right)\) is the cumulative density function of a standard normal, then the binary choice model is called a probit model.

Binary choice estimation

• We can find consistent estimates of \(β_1 ,…,β_k\) in binary choice models using a general method called maximum likelihood .
• We can find consistent estimates of standard errors of the estimates allowing us to do inference.
• Binary choice models are not regression models. There is no model for \(y_i\) and there are no error terms \(ε_i\) (see also the latent variable representation of the binary choice model).