Choice Models. Session 1. K. Sudhir Yale School of Management. Spring

Transcription

1 Choice Models Session 1 K. Sudhir Yale School of Management Spring

2 Outline The Basics Logit Properties Model setup Matlab Code Heterogeneity State dependence Endogeneity Model Setup Bayesian Learning Forward Looking Consumers

3 The Basics Logit Properties The Logit Specification Utility Components and Assumptions Denote indirect utility for alternative j as U j Two components: determininstic Vj, random ɛ j U j = V j + ε j, j = 0, 1, 2,... J,V 0 = 0 Random ɛj observed by consumer, not by researcher Assumptions about ɛ j are i.i.d and extreme value distributed i.e., F (εj ) = e e b(ε j a), j = 0, 1, 2,... J Usually set scale parameter b = 1; and location parameter a = 0 e V j Logit Probability: P j = J 1 + e V k k=1

4 The Basics Logit Properties Properties of the Logit Four Issues 1. Distribution of Maximum Utilities 2. Mean and Variance of the Extreme Value Distribution 3. Independence of Irrelevant Alternatives 4. Cross Elasticities 5. Compensating Variation

5 The Basics Logit Properties 1. Inclusive Value Expected utility from a set of choices given utility maximization The distribution of the maximum of a set of EV random variables is also extreme value Expected utility from a set of j choices given EV error terms with is called the inclusive value J Inclusive Value = ln(1 + e V k ) k=1

6 The Basics Logit Properties 2. Mean and Variance The First Two Moments Mean: µ = a + bγ, where γ is the Euler s constant When a = 0, b = 1 for normalization, then µ = When we work with differences in utilities, not relevant But, we will use this in dynamic models Variance: σ 2 = b2 π 2 6 When a = 0, b = 1 for normalization, then σ 2 = π2 6 = Suppose you estimate U = 3 + 4p in one sample and U = 6 + 8p in a second sample What does it mean for the variance in the unobserved utility?

7 The Basics Logit Properties 3. Independence of Irrelevant Alternatives (IIA) What is it and why it may be a problem? IIA is normally a sensible requirement If A is preferred to B out of the choice set A, B, then expanding the choice set to A, B, X, must not make B preferable to A. The alternative X should be irrelevant to the choice between B and A. For the logit model, IIA implies The relative odds of A & B, P(A)/P(B) should not be affected by addition of X But problem when you add similar alternatives to what is in the set

8 The Basics Logit Properties 3. Independence of Irrelevant Alternatives (IIA) Problem with IIA: The Red Bus-Blue Bus Problem Suppose V B and V C are equal Logit share of Bus and Car: 0.5, 0.5 What should share of Red Bus, Blue Bus and Car be? Common sense: 0.25, 0.25, 0.5 Logit: 0.33,0.33,0.33. Why? EV Errors are independent across choices

9 The Basics Logit Properties 4. Own and Cross Elasticities of Logit Suppose U j = α j βp j + ε j, j = 0, 1, 2,... J Own Elasticity Change in share of brand j w.r.t. own price s j p j = V j p j s j (1 s j ) = βs j (1 s j ) Elasticity ( ) s j sj p j = (βs j (1 s j )) / p j = βp j (1 s j ) Cross Elasticity Change in share of brand i w.r.t. competitor price s j p k = V j p j s j s k = βs j s k Elasticity ( s j sj p k = (βs j s k ) / p k )= βp k s k

10 The Basics Logit Properties 5. Compensating Variation How much does a change in price or product utility affect utility? Compensating Variation is the additional money needed to reach original utility after a change in prices, or a change in product quality (Hicks, 1939) Without Income Effects E(cv) = 1 β [E(U(p 0, X 0 ) E(U(p 1, X 1 )] = 1 J ln(1 + e V j (p 0,X 0 ) ) ln(1 + β j=1 J e V j (p 1,X 1 ) ) j=1

11 Model setup Matlab Code Model Without Heterogeneity Notation U ijt = Q j + X jt β αp jt + ε ijt i: individual/household, j: brand/alternative, t: time U: utility Q: intrinsic preference or quality, X : covariates (except price), p: price β: effect of covariates, α: effect of price ɛ: unobserved (to researcher) utility All parameters are known to individual

12 Model setup Matlab Code Model Without Heterogeneity Model for U ijt = Q j + X jt β αp jt + ε ijt, j = 0, 1,..., J Normalize: Q 0 + X 0t β αp 0t = 0 P ijt = 1 + e Q j +X jt β αp jt J k=1 e Q k+x kt β αp kt Parameters: θ = {Q j, j = 0, 1,..., J, β, α}

13 Model setup Matlab Code Model Without Heterogeneity Writing out the likelihood (Guadagni and Little Model) Individual Likelihood y ijt = 1 if i buys j at time t; 0 otherwise Overall Likelihood L i = T J P y ijt ijt t=1 j=1 L = ln(l) = N T J P y ijt ijt i=1 t=1 j=1 N T J i=1 t=1 j=1 by Maximum Likelihood P y ijt ijt

14 Model setup Matlab Code Programming Model Without Heterogeneity Normalizing the Likelihood

15 Model setup Matlab Code Programming Model Without Heterogeneity The Likelihood Function

16 Model setup Matlab Code Programming Model Without Heterogeneity The Likelihood Function

17 Heterogeneity State dependence Endogeneity Latent Class Heterogeneity (Kamakura and Russell, 1989) Writing out the likelihood Individual i belongs to one of S discrete segments s, with own parameters θ s Pijt s = 1 + e Qs j +X jtβ s α s p jt J k=1 e Qs k +X ktβ s α s p kt

18 Heterogeneity State dependence Endogeneity Latent Class Heterogeneity (Kamakura and Russell, 1989) Writing out the likelihood Individual Likelihood yijt = 1 if i buys j at time t; 0 otherwise Overall Likelihood L = = N L i s = S i=1 s=1 N ln(l) = i=1 s=1 π s T i J t=1 j=1 T S π s L i s P y ijt ijt s J P y ijt ijt s t=1 j=1 ( N S ) ln π s L i s K. Sudhir i=1 MGTs=1 756: Empirical Methods in Marketing

19 Heterogeneity State dependence Endogeneity Continuous Heterogeneity (Gonul and Srinivasan, 1993) Writing out the likelihood is done using simulated maximum likelihood (SML) Individual preferences θ are a drawn from multivariate normal distribution For each draw d of θfrom the normal distribution, compute the probability of choice Pijt d = 1 + e Qd j +X jtβ d α d p jt J k=1 e Qd k +X ktβ d α d p kt

20 Heterogeneity State dependence Endogeneity Continuous Heterogeneity (Gonul and Srinivasan, 1993) Writing out the likelihood Individual Likelihood yijt = 1 if i buys j at time t; 0 otherwise Overall Likelihood L = = N L i d = i=1 d=1 N ln(l) = i=1 d=1 T i J t=1 j=1 D T D P y ijt ijt d J P y ijt ijt d t=1 j=1 L i d ( N D ) ln L i d K. Sudhiri=1 MGT 756: d=1empirical Methods in Marketing

21 Heterogeneity State dependence Endogeneity Discrete & Continuous Heterogeneity Comparing the likelihoods Discrete Heterogeneity L = = N S i=1 s=1 N ln(l) = i=1 s=1 π s T S π s L i s J t=1 j=1 P y ijt ijt ( N S ) ln π s L i s i=1 s=1 Continuous Heterogeneity L = N D T J P y ijt ijt d i=1 d=1 t=1 j=1 N D = i=1 d=1 L i d ( N D ) ln(l) = ln L i d i=1 d=1

22 Heterogeneity State dependence Endogeneity Incorporating State Dependence Does past choice affect future choice or outcomes? U ijt = Q j + X jt β αp jt + γy jt 1 +ε ijt, j = 0, 1,..., J When γ > 0, inertia, when γ < 0, variety seeking First order inertia How did G&L model state dependence? start with equal S jt across brands S jt = λy jt 1 + (1 λ)s jt 1

23 Heterogeneity State dependence Endogeneity Separating State Dependence from Heterogeneity How do experience or initial endowment or preference affect choice and outcomes? Identifying the Hand of Past: Distinguishing State Dependence from Heterogeneity, Heckman (1991, AER) Examples 1. Does early education (nurture) or intrinsic ability (nature) lead to better performance? 2. Are certain persons prone to criminality, or does crime breed crime? 3. Does unemployment affect future unemployment because of loss of work experience or market stigma? 4. Does early entry confer an advantage, or does it proxy ability of early entrants?

24 Heterogeneity State dependence Endogeneity Incorporating Endogeneity Unobserved Attributes and Common Demand Shocks in Utility U ijt = Q j + X jt β αp jt + γy jt 1 + ξ jt + ε ijt, j = 0, 1,..., J Consumers and firms know ξ jt, hence it will be correlated with price Solution Methods LIML (Villas-Boas and Winer 1999 Mgt Sci) p jt = ωw jt + η jt, j = 0, 1,..., J Jointly estimate demand and supply Control Function (Petrin and Train, 2010 Mkt Sci; Pancras and Sudhir, 2007 JMR) Substitute ξ jt with the residuals from the control function for p jt and then estimate demand model

25 Model Setup Bayesian Learning Forward Looking Consumers Developing a The learning environment (1) The Updating Model U jt = Q j + X jt β + ε jt Consumers uncertain about true quality or benefits Q j They have initial priors of quality Q j0 Consumers update prior beliefs after use and information Purchase product based on current belief about quality Q jt

26 Model Setup Bayesian Learning Forward Looking Consumers Developing a Purchase Probability and Likelihood For simplicity, ignore X jt, let U jt = Q j + ε jt, where ε jt is i.i.d. EV They have initial priors regarding quality Q j0 Based on usage or marketing activities, consumers update prior beliefs Purchase product based on current belief about quality Q jt Probability of purchasing j at time t is P jt = 1 + e E( Q jt ) J k=1 e E( Q kt ) = 1 + e E( Q jt ) J k=1 e E( Q jt )

27 Model Setup Bayesian Learning Forward Looking Consumers Bayesian Learning Evolution of consumer beliefs with usage; Learning is a special form of state dependence At time 0, consumer s prior is Q j0 N(Q j0, σ 2 0 ) From usage of j (y j1 = 1), I get a signal d j1, a draw fromd j1 N(Q j, σd 2) Then period 1 posterior is Q j1 N Q j0 σj0 2 1 σ 2 j0 + y j1d j1 σ 2 d + y j1 σ 2 d, 1 1 σ 2 j0 + y j1 σ 2 d

28 Model Setup Bayesian Learning Forward Looking Consumers Purchase Probability in s Moving from Period 1 to Period 2 Probability of purchasing j in period 2 P j2 = 1 + e y j1 d j1 + Q j0 σj0 2 σd 2 1 y j1 σj0 2 + σd 2 J e k=1 Q k0 σk0 2 1 y k1 d k1 + σk0 2 + σ 2 d y k1 σ 2 d

29 Model Setup Bayesian Learning Forward Looking Consumers Purchase Probability in s Likelihood of Purchase at time t and Probability of purchasing j in period t P jt = 1 + e Q j0 σj J e k=1 σj0 2 + τ 1 τ=1 y jτ d jτ Q k0 σk0 2 + t=1 σ2 d τ 1 τ=1 y jτ 1 σk0 2 + σ 2 d τ 1 τ=1 y kτ d kτ t=1 σ2 d τ 1 τ=1 y kτ σ 2 d is straightforward Just use updated Quality formulas for the quality Prior variance has K. to Sudhir be normalized MGT 756: Empirical to 1 Methods in Marketing

30 Model Setup Bayesian Learning Forward Looking Consumers Forward Looking Learning Consumers Consumer anticipate learning and change behavior Consumers anticipate learning and behave in a forward looking manner Erdem and Keane (1995). Consumers will solve dynamic programs The static Bayesian Learning model is incorporated into such dynamic models