Limited Dependent Variables Po=7x)=G(o+xB) ◆y*=B0+x月+t,y=max(O,y*) Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Limited Dependent Variables P(y = 1|x) = G(b0 + xb) y* = b0 + xb + u, y = max(0,y*)
Binary dependent variables e Recall the linear probability model, which can be written as P(=1x)=o+xB va drawback to the linear probability model is that predicted values are not constrained to be between 0 and 1 e An alternative is to model the probability as a function, G(Bo +xB), where 0<G(k<1 Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Binary Dependent Variables Recall the linear probability model, which can be written as P(y = 1|x) = b0 + xb A drawback to the linear probability model is that predicted values are not constrained to be between 0 and 1 An alternative is to model the probability as a function, G(b0 + xb), where 0<G(z)<1
The Probit model o One choice for G(z) is the standard normal cumulative distribution function(cdf) ◆G()=中()三Jdv)dv, where) is the standard normal, so d=)=(2)- exp(-z2/2) o This case is referred to as a probit model o Since it is a nonlinear model, it cannot be estimated by our usual methods Use maximum likelihood estimation Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 The Probit Model One choice for G(z) is the standard normal cumulative distribution function (cdf) G(z) = F(z) ≡ ∫f(v)dv, where f(z) is the standard normal, so f(z) = (2p) -1/2exp(-z 2 /2) This case is referred to as a probit model Since it is a nonlinear model, it cannot be estimated by our usual methods Use maximum likelihood estimation
The logit model o Another common choice for G(z)is the logistic function, which is the cdf for a standard logistic random variable ◆G(z) =exp(z/1+ exp(2J=A() e This case is referred to as a logit model, or sometimes as a logistic regression Both functions have similar shapes -they are increasing in z, most quickly around 0 Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 The Logit Model Another common choice for G(z) is the logistic function, which is the cdf for a standard logistic random variable G(z) = exp(z)/[1 + exp(z)] = L(z) This case is referred to as a logit model, or sometimes as a logistic regression Both functions have similar shapes – they are increasing in z, most quickly around 0
Probits and logits Both the probit and logit are nonlinear and require maximum likelihood estimation e No real reason to prefer one over the other e Traditionally saw more of the logit, mainly because the logistic function leads to a more easily computed model o Today, probit is easy to compute with standard packages, so more popular Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 Probits and Logits Both the probit and logit are nonlinear and require maximum likelihood estimation No real reason to prefer one over the other Traditionally saw more of the logit, mainly because the logistic function leads to a more easily computed model Today, probit is easy to compute with standard packages, so more popular