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).