ADVANCED ECONOMETRICS I Theory (3/3) Instructor: Joaquim J. S. - PowerPoint PPT Presentation

ADVANCED ECONOMETRICS I Theory (3/3) Instructor: Joaquim J. S. Ramalho E.mail: jjsro@iscte-iul.pt Personal Website: http://home.iscte-iul.pt/~jjsro Office: D5.10 Course Website: https://jjsramalho.wixsite.com/advecoi Fénix: https://fenix.iscte-iul.pt/disciplinas/03089

3. Discrete Choice Models 3.2. Models for Ordered Choices Ordered choices: Values for the dependent variable: 𝑍 ∈ 0,1, … , 𝑁 − 1 Latent model: ∗ = 𝑦 𝑗 ′ 𝛾 + 𝑣 𝑗 𝑍 𝑗 ▪ 𝑦 𝑗 cannot include an intercept Individual behaviour observed only by intervals: ∗ ≤ 𝛿 0 0 if 𝑍 𝑗 ∗ ≤ 𝛿 𝑛 , 𝑛 if 𝛿 𝑛−1 < 𝑍 1 ≤ 𝑛 ≤ 𝑁 − 2 𝑍 𝑗 = ൞ 𝑗 ∗ > 𝛿 𝑁−2 𝑁 − 1 if 𝑍 𝑗 ▪ Example: ∗ is a latent measure of the health status – 𝑍 𝑗 – 𝑍 𝑗 is an observed health indicator: poor, satisfactory, good, excellent Assumption: the 𝛿 𝑘 ’s are not known 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 2

3. Discrete Choice Models 3.2. Models for Ordered Choices Probabilities: Aim: ∗ in a given interval ▪ Modelling the probability of observing 𝑍 𝑗 Each probability is based on the same 𝐻 ∙ functions used with binary choices, being given by: ∗ ≤ 𝛿 𝑛 |𝑦 𝑗 𝑄𝑠 𝑍 𝑗 = 𝑛|𝑦 𝑗 = 𝑄𝑠 𝛿 𝑛−1 < 𝑍 𝑗 ∗ ≤ 𝛿 𝑛 |𝑦 𝑗 − 𝑄𝑠 𝑍 ∗ < 𝛿 𝑛−1 |𝑦 𝑗 = 𝑄𝑠 𝑍 𝑗 𝑗 ′ 𝛾 + 𝑣 𝑗 ≤ 𝛿 𝑛 |𝑦 𝑗 − 𝑄𝑠 𝑦 𝑗 ′ 𝛾 + 𝑣 𝑗 < 𝛿 𝑛−1 |𝑦 𝑗 = 𝑄𝑠 𝑦 𝑗 ′ 𝛾|𝑦 𝑗 − 𝑄𝑠 𝑣 𝑗 < 𝛿 𝑛−1 − 𝑦 𝑗 ′ 𝛾|𝑦 𝑗 = 𝑄𝑠 𝑣 𝑗 ≤ 𝛿 𝑛 − 𝑦 𝑗 ′ 𝛾 − 𝐻 𝛿 𝑛−1 − 𝑦 𝑗 ′ 𝛾 = 𝐻 𝛿 𝑛 − 𝑦 𝑗 Hence, the general case is: ′ 𝛾 𝐻 𝛿 0 − 𝑦 𝑗 if 𝑛 = 0 ′ 𝛾 − 𝐻 𝛿 𝑛−1 − 𝑦 𝑗 ′ 𝛾 if 1 ≤ 𝑛 ≤ 𝑁 − 2 𝐻 𝛿 𝑛 − 𝑦 𝑗 𝑄𝑠 𝑍 𝑗 = 𝑛|𝑦 𝑗 = ൞ ′ 𝛾 1 − 𝐻 𝛿 𝑁−2 − 𝑦 𝑗 if 𝑛 = 𝑁 − 1 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 3

3. Discrete Choice Models 3.2. Models for Ordered Choices Estimation: Parameters to be estimated: ▪ 𝛾 ▪ 𝛿 0 , … , 𝛿 𝑁−2 Estimation method: ▪ Maximum likelihood Most common models: ▪ Ordered logit ▪ Ordered probit Stata ologit Y 𝑌 1 … 𝑌 𝑙 oprobit Y 𝑌 1 … 𝑌 𝑙 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 4

3. Discrete Choice Models 3.2. Models for Ordered Choices Partial effects: Each 𝑌 𝑘 affects 𝑁 probabilities: ∆𝑌 𝑘 = 1 ⟹ ∆𝑄𝑠 𝑍 = 𝑛 𝑌 ′ 𝛾 −𝛾 𝑘 𝑕 𝛿 0 − 𝑦 𝑗 if 𝑛 = 0 ′ 𝛾 − 𝑕 𝛿 𝑛−1 − 𝑦 𝑗 ′ 𝛾 𝛾 𝑘 𝑕 𝛿 𝑛 − 𝑦 𝑗 if 1 ≤ 𝑛 ≤ 𝑁 − 2 = ൞ ′ 𝛾 𝛾 𝑘 𝑕 𝛿 𝑁−2 − 𝑦 𝑗 if 𝑛 = 𝑁 − 1 The sign of 𝛾 𝑘 is informative about the direction of ∆𝑄𝑠 𝑍 = 0 𝑌 and ∆𝑄𝑠 𝑍 = 𝑁 − 1 𝑌 but not of the changes in the remaining probabilities 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 5

3. Discrete Choice Models 3.3. Models for Multinomial Choices Multinomial choices: Values for the dependent variable: 𝑍 ∈ 0,1, … , 𝑁 − 1 Latent model: ▪ Each individual has a given utility associated with each alternative: ′ 𝛾 + 𝑣 𝑗𝑛 𝑉 𝑗𝑛 = 𝑦 𝑗𝑛 ▪ The selected alternative is the one that maximizes utility: 𝑄𝑠 𝑍 𝑗 = 𝑛|𝑦 𝑗 = 𝑄𝑠 𝑉 𝑗𝑛 = 𝑛𝑏𝑦 𝑉 𝑗1 , … 𝑉 𝑗𝑁 |𝑦 𝑗 Main models: ▪ Multinomial Logit: 𝑉 𝑗𝑛 ~ 𝐻𝑣𝑛𝑐𝑓𝑚 and 𝑉 𝑗𝑛 independent ∀𝑛 ▪ Multinomial Probit: 𝑉 𝑗𝑛 ~ 𝑂𝑝𝑠𝑛𝑏𝑚 ▪ Nested Logit ▪ Random Parameters Logit 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 6

3. Discrete Choice Models 3.3. Models for Multinomial Choices Explanatory variables: 𝑦 𝑗𝑛 may include: ▪ 𝑦 𝑗𝑛 : variables that are different across individuals and alternatives ▪ 𝑦 𝑛 : variables that differ across alternatives but not individuals ▪ 𝑦 𝑗 : variables that differ across individuals but not alternatives Example: ▪ 𝑍 𝑗 - selected means of transport to go to work ▪ 𝑦 𝑗𝑛 - time that each individual 𝑗 takes in going to work when using transport 𝑛 ▪ 𝑦 𝑛 - price of transport 𝑛 ▪ 𝑦 𝑗 - age of individual 𝑗 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 7

3. Discrete Choice Models 3.3. Models for Multinomial Choices Multinomial logit: ′ 𝛾+𝑦 𝑗 ′ 𝛾 𝑛 𝑓 𝑦 𝑗𝑛 ′ 𝛾 + 𝑦 𝑗 ′ 𝛾 𝑛 = 𝑄𝑠 𝑍 𝑗 = 𝑛|𝑦 𝑗𝑛 = 𝐻 𝑛 𝑦 𝑗𝑛 ′ 𝛾+𝑦 𝑗 ′ 𝛾 𝑘 𝑁−1 𝑓 𝑦 𝑗𝑘 σ 𝑘=0 𝛾 𝑛 has to be normalized, that is for one alternative (base outcome) its value is set to zero 𝛾 cannot include a constant term Independence of Irrelevant Alternatives (IIA) – the odds ratio between two alternatives does not depend on the remaining alternatives: ′ 𝛾+𝑦 𝑗 ′ 𝛾 𝑛 = 𝑓 𝑦 𝑗𝑛 𝑄𝑠 𝑍 𝑗 = 𝑛|𝑦 𝑗𝑛 ′ 𝛾+𝑦 𝑗 ′ 𝛾 𝑚 𝑄𝑠 𝑍 𝑗 = 𝑚|𝑦 𝑗𝑛 𝑓 𝑦 𝑗𝑚 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 8

3. Discrete Choice Models 3.3. Models for Multinomial Choices When all explanatory variables are of the type 𝑦 𝑗𝑛 and 𝑦 𝑛 , the choice between alternatives 𝑛 and 𝑚 is fully explained by diferences in the alternative characteristics: ′ 𝛾 = 𝑓 𝑦 𝑗𝑛 𝑄𝑠 𝑍 𝑗 = 𝑛|𝑦 𝑗𝑛 ′ −𝑦 𝑗𝑚 ′ 𝛾 ′ 𝛾 = 𝑓 𝑦 𝑗𝑛 𝑄𝑠 𝑍 𝑗 = 𝑚|𝑦 𝑗𝑚 𝑓 𝑦 𝑗𝑚 ▪ Is this case, the model is often called ‘ conditional logit ’ Stata asclogit Y 𝑌 1𝑛 …, case( id ) alternatives( varname ) casevars( 𝑌 𝑗 …) basealternative( name ) When all explanatory variables are of the type 𝑦 𝑗 , the choice between alternatives 𝑛 e 𝑚 is fully explained by diferences between 𝛾 𝑛 e 𝛾 𝑚 : ′ 𝛾 𝑛 = 𝑓 𝑦 𝑗 𝑄𝑠 𝑍 𝑗 = 𝑛|𝑦 𝑗 ′ 𝛾 𝑛 −𝛾 𝑚 ′ 𝛾 𝑚 = 𝑓 𝑦 𝑗 𝑄𝑠 𝑍 𝑗 = 𝑚|𝑦 𝑗 𝑓 𝑦 𝑗 Stata mlogit Y 𝑌 1 … 𝑌 𝑙 , baseoutcome( 0 ) 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 9

ҧ 3. Discrete Choice Models 3.3. Models for Multinomial Choices Estimation: ▪ Maximum likelihood based on the following log-likelihood function: 𝑂 ′ 𝛾 + 𝑦 𝑗 ′ 𝛾 𝑛 𝑀𝑀 = ෍ 𝑒 𝑗𝑛 𝑚𝑝𝑕 𝐻 𝑛 𝑦 𝑗𝑛 𝑗=1 ▪ 𝑒 𝑗𝑛 = 1 if individual 𝑗 chooses alternative 𝑛 Partial effects: ▪ ∆𝑌 𝑗𝑘 = 1 ⟹ – ∆𝑄𝑠 𝑍 𝑗 = 𝑛 𝑌 = 𝛾 𝑘 𝐻 𝑛 ∙ 𝑒 𝑗𝑛 − 𝐻 𝑛 ∙ – 𝛾 𝑘 gives the sign of the partial effect ▪ ∆𝑌 𝑗 = 1 ⟹ 𝑁−1 𝛾 𝑛 𝐻 𝑛 ∙ – ∆𝑄𝑠 𝑍 𝛾 = σ 𝑛=1 𝑗 = 𝑛 𝑌 = 𝐻 𝑛 ∙ 𝛾 𝑘 − 𝛾 , onde ҧ – 𝛾 𝑘 gives the sign of the partial effect relative to the base alternative, not the sign of the overall effect 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 10

3. Discrete Choice Models 3.3. Models for Multinomial Choices Testing IIA ▪ Hausman test comparing: – Full multinomial logit model – Multinomial logit model excluding one or more alternatives ▪ If multinomial logit is the correct model, then both models produce consistent estimators (null hypothesis) ▪ If multinomial logit is not the correct model, then the results generated by both models will be different (alternative hypothesis) Stata mlogit Y 𝑌 1 … 𝑌 𝑙 , baseoutcome( 0 ) (ou asclogit …) estimates store Mod1 mlogit Y 𝑌 1 … 𝑌 𝑙 if Y != 3 , baseoutcome( 0 ) (ou asclogit …) estimates store Mod2 hausman Mod1 Mod2 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 11

3. Discrete Choice Models 3.3. Models for Multinomial Choices Multinomial probit: Not affected by the IIA property Very complex, requiring the computation of 𝑁 − 1 integrals The version implemented in Stata assumes independent errors, which eliminates the only advantage of multinomial probit over multinomial logit Stata mprobit Y 𝑌 1 … 𝑌 𝑙 , baseoutcome( 0 ) 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 12

3. Discrete Choice Models 3.3. Models for Multinomial Choices Nested logit: Not affected by the IIA property, grouping the choices in several sets in such a way that: ▪ Within each group, alternatives may be correlated ▪ Between groups, alternatives are independent Results from a sequential decision process – example for a two-level process: Financing Bank debt Own funds Stock market Bank 1 Bank N Lisbon Frankfurt Stata ▪ Level 1 – defining J groups, nlogit … ▪ Level 2 – defining 𝑁 𝑘 choices in each group 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 13

3. Discrete Choice Models 3.3. Models for Multinomial Choices Random parameters logit: Latent model: ′ 𝛾 𝑗 + 𝑣 𝑗𝑛 𝑉 𝑗𝑛 = 𝑦 𝑗𝑛 Most common assumption: 𝛾 𝑗 ~ 𝑂 𝛾, Σ 𝛾 Not affected by the IIA property If Σ 𝛾 = 0 , it reduces to the Multinomial Logit model; hence, comparing the two models allows the IIA property to be tested 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 14