Applied Econometrics

1. Applied Econometrics William Greene Department of Economics Stern School of Business

2. Applied Econometrics 21. Discrete Choice Modeling

3. A Microeconomics Platform Consumers Maximize Utility (!!!) Fundamental Choice Problem: Maximize U(x1,x2,…) subject to prices and budget constraints A Crucial Result for the Classical Problem: Indirect Utility Function: V = V(p,I) Demand System of Continuous Choices The Integrability Problem: Utility is not revealed by demands

4. Theory for Discrete Choice Theory is silent about discrete choices Translation to discrete choice Existence of well defined utility indexes: Completeness of rankings Rationality: Utility maximization Axioms of revealed preferences Choice sets and consideration sets – consumers simplify choice situations Implication for choice among a set of discrete alternatives Commonalities and uniqueness Does this allow us to build “models?” What common elements can be assumed? How can we account for heterogeneity? Revealed choices do not reveal utility, only rankings which are scale invariant

5. Modeling the Binary Choice Ui,suv = ?suv + ?Psuv + ?suvIncome + ?i,suv Ui,sed = ?sed + ?Psed + ?sedIncome + ?i,sed Chooses SUV: Ui,suv > Ui,sed Ui,suv - Ui,sed > 0 (?SUV-?SED) + ?(PSUV-PSED) + (?SUV-?sed)Income + ?i,suv - ?i,sed > 0 ?i > -[? + ?(PSUV-PSED) + ?Income]

6. What Can Be Learned from the Data? (A Sample of Consumers, i = 1,…,N)

7. Application 210 Commuters Between Sydney and Melbourne Available modes = Air, Train, Bus, Car Observed: Choice Attributes: Cost, terminal time, other Characteristics: Household income First application: Fly or Other

8. Binary Choice Data

9. An Econometric Model Choose to fly iff UFLY > 0 Ufly = ?+?1Cost + ?2Time + ?Income + ? Ufly > 0 ? ? > -(?+?1Cost + ?2Time + ?Income) Probability model: For any person observed by the analyst, Prob(fly) = Prob[? > -(?+?1Cost + ?2Time + ?Income)] Note the relationship between the unobserved ? and the outcome

11. Modeling Approaches Nonparametric – “relationship” Minimal Assumptions Minimal Conclusions Semiparametric – “index function” Stronger assumptions Robust to model misspecification (heteroscedasticity) Still weak conclusions Parametric – “Probability function and index” Strongest assumptions – complete specification Strongest conclusions Possibly less robust. (Not necessarily)

12. Nonparametric

13. Semiparametric MSCORE: Find b’x so that sign(b’x) * sign(y) is maximized. Klein and Spady: Find b to maximize a semiparametric likelihood of G(b’x)

14. MSCORE

15. Klein and Spady Semiparametric

16. Parametric: Logit Model

17. Logit vs. MScore

18. Parametric Model Estimation How to estimate ?, ?1, ?2, ?? It’s not regression The technique of maximum likelihood Prob[y=1] = Prob[? > -(?+?1Cost + ?2Time + ?Income)] Prob[y=0] = 1 - Prob[y=1] Requires a model for the probability

19. Completing the Model: F(?) The distribution Normal: PROBIT, natural for behavior Logistic: LOGIT, allows “thicker tails” Gompertz: EXTREME VALUE, asymmetric, underlies the basic logit model for multiple choice Does it matter? Yes, large difference in estimates Not much, quantities of interest are more stable.

21. Estimated Binary Choice (Probit) Model

22. Estimated Binary Choice Models

24. Marginal Effects in Probability Models Prob[Outcome] = some F(?+?1Cost…) “Partial effect” = ? F(?+?1Cost…) / ?”x” (derivative) Partial effects are derivatives Result varies with model Logit: ? F(?+?1Cost…) / ?x = Prob * (1-Prob) * ? Probit: ? F(?+?1Cost…) / ?x = Normal density ? Scaling usually erases model differences

25. The Delta Method

26. Marginal Effects for Binary Choice Logit Probit

27. Estimated Marginal Effects

28. Marginal Effect for a Dummy Variable Prob[yi = 1|xi,di] = F(?’xi+?di) =conditional mean Marginal effect of d Prob[yi = 1|xi,di=1]=Prob[yi= 1|xi,di=0] Logit:

29. (Marginal) Effect – Dummy Variable HighIncm = 1(Income > 50)

30. Computing Effects Compute at the data means? Simple Inference is well defined Average the individual effects More appropriate? Asymptotic standard errors. (Not done correctly in the literature – terms are correlated!) Is testing about marginal effects meaningful?

31. Average Partial Effects

32. Elasticities Elasticity = How to compute standard errors? Delta method Bootstrap Bootstrap the individual elasticities? (Will neglect variation in parameter estimates.) Bootstrap model estimation?

33. Estimated Income Elasticity for Air Choice Model

34. Odds Ratio – Logit Model Only Effect Measure? “Effect of a unit change in the odds ratio.”

35. Ordered Outcomes E.g.: Taste test, credit rating, course grade Underlying random preferences: Mapping to observed choices Strength of preferences Censoring and discrete measurement The nature of ordered data

36. Modeling Ordered Choices Random Utility Uit = ? + ?’xit + ?i’zit + ?it = ait + ?it Observe outcome j if utility is in region j Probability of outcome = probability of cell Pr[Yit=j] = F(?j – ait) - F(?j-1 – ait)

37. Movie Madness

38. Health Care Satisfaction (HSAT)

39. Ordered Probability Model

40. Ordered Probabilities

41. Five Ordered Probabilities

42. Coefficients

43. Effects in the Ordered Probability Model

44. Ordered Probability Model for Health Satisfaction

45. Ordered Probability Effects

46. Ordered Probit Marginal Effects

47. Multinomial Choice Among J Alternatives • Random Utility Basis Uitj = ?ij + ?i ’xitj + ?i’zit + ?ijt i = 1,…,N; j = 1,…,J(i); t = 1,…,T(i) • Maximum Utility Assumption Individual i will Choose alternative j in choice setting t iff Uitj > Uitk for all k ? j. • Underlying assumptions Smoothness of utilities Axioms: Transitive, Complete, Monotonic

48. Utility Functions The linearity assumption and curvature The choice set Deterministic and random components: The “model” Generic vs. alternative specific components Attributes and characteristics Coefficients Part worths = preference weights = coefficients Alternative specific constants Scaling

49. The Multinomial Logit (MNL) Model Independent extreme value (Gumbel): F(?itj) = 1 – Exp(-Exp(?itj)) (random part of each utility) Independence across utility functions Identical variances (means absorbed in constants) Same parameters for all individuals (temporary) Implied probabilities for observed outcomes

50. Specifying Probabilities

51. Observed Data Types of Data Individual choice Market shares Frequencies Ranks Attributes and Characteristics Choice Settings Cross section Repeated measurement (panel data)

52. Data on Discrete Choices Line MODE TRAVEL INVC INVT TTME GC HINC 1 AIR .00000 59.000 100.00 69.000 70.000 35.000 2 TRAIN .00000 31.000 372.00 34.000 71.000 35.000 3 BUS .00000 25.000 417.00 35.000 70.000 35.000 4 CAR 1.0000 10.000 180.00 .00000 30.000 35.000 5 AIR .00000 58.000 68.000 64.000 68.000 30.000 6 TRAIN .00000 31.000 354.00 44.000 84.000 30.000 7 BUS .00000 25.000 399.00 53.000 85.000 30.000 8 CAR 1.0000 11.000 255.00 .00000 50.000 30.000 321 AIR .00000 127.00 193.00 69.000 148.00 60.000 322 TRAIN .00000 109.00 888.00 34.000 205.00 60.000 323 BUS 1.0000 52.000 1025.0 60.000 163.00 60.000 324 CAR .00000 50.000 892.00 .00000 147.00 60.000 325 AIR .00000 44.000 100.00 64.000 59.000 70.000 326 TRAIN .00000 25.000 351.00 44.000 78.000 70.000 327 BUS .00000 20.000 361.00 53.000 75.000 70.000 328 CAR 1.0000 5.0000 180.00 .00000 32.000 70.000

53. Estimated MNL Model

54. Estimated MNL Model

55. Estimated MNL Model

56. Estimated MNL Model

57. Estimated MNL Model

58. Model Fit Based on Log Likelihood Three sets of predicted probabilities No model: Pij = 1/J (.25) Constants only: Pij = (1/N)?i dij [(58,63,30,59)/210=.286,.300,.143,.281) Estimated model: Logit probabilities Compute log likelihood Measure improvement in log likelihood with R-squared = 1 – LogL/LogL0 (“Adjusted” for number of parameters in the model.) NOT A MEASURE OF “FIT!”

59. Fit the Model with Only ASCs

60. CLOGIT Fit Measures Based on the log likelihood

61. Effects of Changes in Attributes on Probabilities Partial Effects: Effect of a change in attribute “k” of alternative “m” on the probability that choice “j” will be made is Proportional changes: Elasticities

62. Elasticities for CLOGIT Request: ;Effects: attribute (choices where changes occur ) ; Effects: GC(*) (INVT changes in all choices)

63. Choice Based Sampling Over/Underrepresenting alternatives in the data set Biases in parameter estimates? (Constants only?) Biases in estimated variances Weighted log likelihood, weight = ?j / Fj for all i. Fixup of covariance matrix ; Choices = list of names / list of true proportions $

64. Choice Based Sampling Estimators

65. Changes in Estimated Elasticities

66. The I.I.D Assumption Uitj = ?ij + ?i ’xitj + ?i’zit + ?ijt F(?itj) = 1 – Exp(-Exp(?itj)) (random part of each utility) Independence across utility functions Identical variances (means absorbed in constants) Restriction on scaling Correlation across alternatives? Implication for cross elasticities (we saw earlier) Behavioral assumption, independence from irrelevant alternatives (IIA)