ch17 - Limited Dependent Variables P(y = 1|x = G(0 x y = 0...

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Economics 20 - Prof. Anderson 1 Limited Dependent Variables P ( y = 1| x ) = G ( β 0 + x β ) y* = β 0 + x β + u, y = max ( 0,y* )
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Economics 20 - Prof. Anderson 2 Binary Dependent Variables Recall the linear probability model, which can be written as P( y = 1| x ) = β 0 + x β 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 ( β 0 + x β ), where 0< G ( z )<1
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Economics 20 - Prof. Anderson 3 The Probit Model One choice for G ( z ) is the standard normal cumulative distribution function (cdf) G ( z ) = Φ ( z ) ≡ ∫ φ ( v )d v , where φ ( z ) is the standard normal, so φ ( z ) = (2 π ) -1/2 exp(- 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
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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 )] = Λ ( 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
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Economics 20 - Prof. Anderson 5 Probits and Logits Both the probit and logit are nonlinear and
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