The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes The Probit & Logit Models Econometrics II Ricardo Mora Department of Economics Universidad Carlos III de Madrid Máster Universitario en Desarrollo y Crecimiento Económico Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Outline 1 The Random Utility Model 2 The Probit & Logit Models 3 Estimation & Inference 4 Probit & Logit Estimation in Stata Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choosing Among a few Alternatives Set up: An agent chooses among several alternatives: labor economics: participation, union membership, ... demographics: marriage, divorce, # of children,... industrial organization: plant building, new product,... regional economics: means of transport,.... We are going to model a choice of two alternatives (not dicult to generalize...) The value of each alternative depends on many factors U0 = b0x0 + e0 U1 = b1x1 + e1 e0;e1 are eects on utility on factors UNOBSERVED TO ECONOMETRICIAN Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choosing Among a few Alternatives Set up: An agent chooses among several alternatives: labor economics: participation, union membership, ... demographics: marriage, divorce, # of children,... industrial organization: plant building, new product,... regional economics: means of transport,.... We are going to model a choice of two alternatives (not dicult to generalize...) The value of each alternative depends on many factors U0 = b0x0 + e0 U1 = b1x1 + e1 e0;e1 are eects on utility on factors UNOBSERVED TO ECONOMETRICIAN Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choosing Among a few Alternatives Set up: An agent chooses among several alternatives: labor economics: participation, union membership, ... demographics: marriage, divorce, # of children,... industrial organization: plant building, new product,... regional economics: means of transport,.... We are going to model a choice of two alternatives (not dicult to generalize...) The value of each alternative depends on many factors U0 = b0x0 + e0 U1 = b1x1 + e1 e0;e1 are eects on utility on factors UNOBSERVED TO ECONOMETRICIAN Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choosing Among a few Alternatives Set up: An agent chooses among several alternatives: labor economics: participation, union membership, ... demographics: marriage, divorce, # of children,... industrial organization: plant building, new product,... regional economics: means of transport,.... We are going to model a choice of two alternatives (not dicult to generalize...) The value of each alternative depends on many factors U0 = b0x0 + e0 U1 = b1x1 + e1 e0;e1 are eects on utility on factors UNOBSERVED TO ECONOMETRICIAN Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choice Under the RUM If b1x1 − b0x0 ≥ e0 − e1 then choice = 1 If b1x1 − b0x0 < e0 − e1 then choice = 0 agent chooses 1 if observed advantages of 1 outweight the unobserved net advantage of 0 note that e = e0 − e1 is dened by the data collection process, not by the decision process Fundamental Assumption: e = e0 − e1 ∼ F Pr (choice = 1) = PrF (e ≤ b1x1 − b0x0) Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choice Under the RUM If b1x1 − b0x0 ≥ e0 − e1 then choice = 1 If b1x1 − b0x0 < e0 − e1 then choice = 0 agent chooses 1 if observed advantages of 1 outweight the unobserved net advantage of 0 note that e = e0 − e1 is dened by the data collection process, not by the decision process Fundamental Assumption: e = e0 − e1 ∼ F Pr (choice = 1) = PrF (e ≤ b1x1 − b0x0) Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choice Under the RUM If b1x1 − b0x0 ≥ e0 − e1 then choice = 1 If b1x1 − b0x0 < e0 − e1 then choice = 0 agent chooses 1 if observed advantages of 1 outweight the unobserved net advantage of 0 note that e = e0 − e1 is dened by the data collection process, not by the decision process Fundamental Assumption: e = e0 − e1 ∼ F Pr (choice = 1) = PrF (e ≤ b1x1 − b0x0) Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Choice Under the RUM If b1x1 − b0x0 ≥ e0 − e1 then choice = 1 If b1x1 − b0x0 < e0 − e1 then choice = 0 agent chooses 1 if observed advantages of 1 outweight the unobserved net advantage of 0 note that e = e0 − e1 is dened by the data collection process, not by the decision process Fundamental Assumption: e = e0 − e1 ∼ F Pr (choice = 1) = PrF (e ≤ b1x1 − b0x0) Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes The Probit & Logit Models Probit Assumption: e1;e0 ∼ N(0;Σ) so that e ∼ N(0;1) Pr(choice = 1) = Φ(bx) where Φ is the cdf of the standard normal this is called the Probit Model the vector of parameters b can be consistently estimated by ML Logit Assumption: e0 − e1 = e ∼Logistic Pr choice 1 exp(bx) ( = ) = 1+exp(bx) Easy computation! Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes The Probit & Logit Models Probit Assumption: e1;e0 ∼ N(0;Σ) so that e ∼ N(0;1) Pr(choice = 1) = Φ(bx) where Φ is the cdf of the standard normal this is called the Probit Model the vector of parameters b can be consistently estimated by ML Logit Assumption: e0 − e1 = e ∼Logistic Pr choice 1 exp(bx) ( = ) = 1+exp(bx) Easy computation! Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes The Probit & Logit Models Probit Assumption: e1;e0 ∼ N(0;Σ) so that e ∼ N(0;1) Pr(choice = 1) = Φ(bx) where Φ is the cdf of the standard normal this is called the Probit Model the vector of parameters b can be consistently estimated by ML Logit Assumption: e0 − e1 = e ∼Logistic Pr choice 1 exp(bx) ( = ) = 1+exp(bx) Easy computation! Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes The Probit & Logit Models Probit Assumption: e1;e0 ∼ N(0;Σ) so that e ∼ N(0;1) Pr(choice = 1) = Φ(bx) where Φ is the cdf of the standard normal this is called the Probit Model the vector of parameters b can be consistently estimated by ML Logit Assumption: e0 − e1 = e ∼Logistic Pr choice 1 exp(bx) ( = ) = 1+exp(bx) Easy computation! Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes The Probit & Logit Models Probit Assumption: e1;e0 ∼ N(0;Σ) so that e ∼ N(0;1) Pr(choice = 1) = Φ(bx) where Φ is the cdf of the standard normal this is called the Probit Model the vector of parameters b can be consistently estimated by ML Logit Assumption: e0 − e1 = e ∼Logistic Pr choice 1 exp(bx) ( = ) = 1+exp(bx) Easy computation! Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes The Probit & Logit Models Probit Assumption: e1;e0 ∼ N(0;Σ) so that e ∼ N(0;1) Pr(choice = 1) = Φ(bx) where Φ is the cdf of the standard normal this is called the Probit Model the vector of parameters b can be consistently estimated by ML Logit Assumption: e0 − e1 = e ∼Logistic Pr choice 1 exp(bx) ( = ) = 1+exp(bx) Easy computation! Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Example: Marriage Decision Consider a sample of women who have a relation The econometrician only observes 1 marry = if married , = 0 otherwise xm: factors aecting utility of being married xs : factors aecting utility of being single note that something that aects the utility of being married will also aect the utility of being single, but not in the same way (for example, pregnant status) Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Controls in the Marriage Decision 0 age preg Um = bm + bm age + bm pregnant + em 0 age preg Us = bs + bs age + bs pregnant + es Probit Assumption: es − em jx ∼ N(0;1) Pr(marry = 1) = Φ(b0 + bage age + bpreg pregnant) 0 0 b0 = bm − bs age age bage = bm − bs preg preg bpreg = bm − bs 2 2 var(es − em) = ss + sm − 2ss;m = 1 Ricardo Mora The Probit Model The Random Utility Model The Probit & Logit Models Estimation & Inference Probit & Logit Estimation in Stata Summary Notes Controls in the Marriage Decision 0 age preg Um = bm + bm age + bm pregnant + em 0 age preg Us = bs + bs age + bs pregnant + es Probit Assumption: es − em