Binary choice models, inference
Summary
Binary choice fitted values
- Given estimates \(b_1,…,b_k\) of \(β_1 ,…,β_k\) in the binary choice model we can calculate \(z\) - values
\[z_i=b_1+b_2x_{i,2}+b_3x_{i,3}+…+b_kx_{i,k} , i=1,…,n\]
- \(z_i\) is our estimate of \(ζ_i\) .
- We can find fitted values
\[{\hat{y}}_i=F\left( z_i \right) , i=1,…,n\]
- which are estimates of \(P\left( y_i=1|x_i \right)\) (we can do out of sample predictions as well).
The \(β\) -parameters
- In binary choice models
\[ \frac{∂P\left( y=1|x \right)}{∂x_j}=F'\left( ζ \right)β_j , j=2,…,k\]
- where \(F'\) is the derivative of \(F\) .
- \(∂P\left( y=1|x \right)/∂x_j\) is estimated using
\[F'\left( z \right)b_j\]
Marginal effects, logit and probit
- For the logit model
\[F'\left( x \right)= \frac{e^{-x}}{{\left( 1+e^{-x} \right)}^2}\]
- For the probit model \(F'\) is the probability density function of the standard normal
\[F'\left( x \right)= \frac{1}{\sqrt{2π}}exp \left( - \frac{x^2}{2} \right)\]
The sign of the \(β\) -parameters
\(β_j\) has no immediate interpretation in binary choice models. However, the sign of \(β_j\) can be interpreted:
- If \(β_j=0\) then \(ζ\) does not depend on \(x_j\) and \(P\left( y_i=1|x \right)\) will not depend on \(x_j\) .
- If \(β_j>0\) ( \(β_j<0\) ) then \(ζ\) increases (decreases) with \(x_j\) and \(P\left( y_i=1|x \right)\) will increase (decrease) with \(x_j\) .