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Binary Dependent Variables Chapter 12

Binary Dependent Variables Chapter 12. P ( y = 1| x ) = G ( b 0 + x b ). Binary Dependent Variables. A linear probability model can be written as P( y = 1| x ) = b 0 + x b

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Binary Dependent Variables Chapter 12

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  1. Binary Dependent Variables Chapter 12 P(y = 1|x) = G(b0 + xb) FIN357 Li

  2. Binary Dependent Variables • A linear probability model 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 FIN357 Li

  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(-z2/2) • 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 FIN357 Li

  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 FIN357 Li

  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 FIN357 Li

  6. Interpretation of Probits and Logits • In general we care about the effect of x on P(y = 1|x), that is, we care about ∂p/ ∂x • For the linear case, this is easily computed as the coefficient on x • For the nonlinear probit and logit models, it’s more complicated: • ∂p/ ∂xj = g(b0 +xb)bj, where g(z) is dG/dz FIN357 Li

  7. For logit model • ∂p/ ∂xj =bj exp(b0 +xb ) / (1 + exp(b0 +xb) ) 2 • For Probit model: • ∂p/ ∂xj = bjf (b0 +xb) = bj (2p)-1/2exp(-(b0 +xb)2/2) FIN357 Li

  8. Interpretation (continued) • Can examine the sign and significance (based on a standard t test) of coefficients, FIN357 Li

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