Nevo on Random-Coefficient Logit

Transcription

1 Nevo on Random-Coefficient Logit March 28, 2003 Eric Rasmusen Abstract Overheads for logit demand estimation for G604. These accompany the Nevo JEMS article. Indiana University Foundation Professor, Department of Business Economics and Public Policy, Kelley School of Business,Indiana University, BU 456, 1309 E. 10th Street, Bloomington, Indiana, Office: (812) Fax: Php.indiana.edu/ erasmuse, Php.indiana.edu/ erasmuse/sdfsdfsdf. I thank Fei Lu for her comments. 1

2 I. The Problem 1 Suppose we are trying to estimate a demand elasticity how quantity demanded responds to price. We might have 20 years of cereal market data on 50 brands of cereal, for a total of 1000 data points. We also have data on 6 characteristics of each brand, and we have demographic data on 4 characteristics of the consumers in each year. Each year each consumer decides which brand of cereal to buy, buying either one or zero units. We do not observe individual decisions, but we model them nonetheless. Since we don t observe individual consumers, I think it is misleading to say how many there are it will not enter the analysis at all. But since Nevo does, let there be I = 400 consumers. The utility of consumer i from buying brand j in year t is 2 (1) u ijt = α i (y i p jt )+x jt β i + ξ jt + ijt i =1,..., 400, j =1,...,50, t =1,..., 20, where y i istheincomeofconsumeri, p jt is the price of product j in year t, x jt is a 6-dimensional vector of observable characteristics of product j in time t, ξ jt is a disturbance scalar (xi) summarizing unobserved characteristics of product j at time t, and ijt is the usual disturbance with mean zero. The parameters to be estimated are consumer i s marginal utility of income, α i, and marginal utility of characteristics, the 6-vector β i. We will assume that jt follows the Type I extreme-value distribution. Nevo inserts this assumption later in his exposition. 1 Start here with the graphs of the problem of estimating supply and demand curves, instrumental variables, and finding instruments. 2 Why not add v it to the equation? Because that would not affect choices between cereals, and we cannot detect utility anyway. 2

3 The Consumer Decision The utility of consumer i from buying brand j in year t is 3 (1) u ijt = α i (y i p jt )+x jt β i + ξ jt + ijt i =1,..., 400, j =1,...,50, t =1,..., 20, where y i istheincomeofconsumeri, p jt is the price of product j in year t, x jt is a 6-dimensional vector of observable characteristics of product j in time t, ξ jt is a disturbance scalar (xi) summarizing unobserved characteristics of product j at time t, and ijt is the usual disturbance with mean zero. The parameters to be estimated are consumer i s marginal utility of income, α i, and marginal utility of characteristics, the 6-vector β i. Consumer i buys good j in time t if it yields him the highest utility of any good, including an outside option, good 0 whose utility we normalize to equal zero. What we observe, though, is not consumer i s decision, but the market share of good j. (Even if we did observe his decision, we would still have to choose between regular logit and BLP logit, though.) 3 Why not add v it to the equation? Because that would not affect choices between cereals, and we cannot detect utility anyway. 3

4 Regular Multinomial Logit One way to proceed would be to assume that all consumers are identical in their taste parameters; i.e., that α i = α and β i = β, and that the ijt disturbances are uncorrelated across i s. Then we have the ordinary multinomial (because there are 51 choices) logit model (at least if we assume that the ijt disturbances are also uncorrelated across t and j s, the problems of serial correlation and of different relationships between products in different years). We would have (5) u ijt = α(y i p jt )+x jt β + ξ jt + ijt i =1,..., 400, j =1,..., 50, t =1,...,20. We still have different incomes for different individuals, but that is OK. We can add them all up, since the coefficient on each person is thesamevalue,α. Then we would have (5 ) u jt = α(y p jt )+x jt β + ξ jt + jt j =1,..., 50, t =1,..., 20. If we assume that jt follows the Type I extreme-value distribution, then this is the multinomial logit model. Somehow market shares in this situation are (6) s jt = e x jt β αp jt +ξ jt k=0 ex kt β αp kt +ξ kt 4

5 Derivatives I (6) s jt = e x jt β αp jt +ξ jt k=0 ex kt β αp kt +ξ kt To figure out the elasticity, we need to calculate s jt. It is helpful to rewrite (6) by defining M j so Then (6.2) s jt = (6.1) s jt = First, suppose k = j. Then (6.4) s jt M j k=0 Mk + M j k=0 Mk. M j ( k=0 M k) 2 0 p jt = k=0 M + M j k ( k=0 M k) 2 = α = αs jt s kt M j k=0 M k M k k=0 M k Mk αm k 5

6 Derivatives II (6.2) s jt = We just found that (6.4) s jt M j k=0 Mk + M j ( k=0 M k) 2 0 p jt = k=0 M + M j k ( k=0 M k) 2 = α = αs jt s kt M j k=0 M k Second, suppose k = j. Then (6.6) s jt M k k=0 M k p jt = αm j + M k=0 M j k ( k=0 M k) 2 = αs jt + αs 2 jt = αs jt (1 s jt ) Mk αm k ( αmj ) We can now calculate the elasticity of the market share: the percentage change in the market share of good j when the price of good k goes up: (6 ) η jkt % s jt % p kt = s jt pkt s jt = αp jt (1 s jt ) if j = k αp kt s kt otherwise. 6

7 Digression on Generality of Different Specifications Nevo says on page 522 that if we did not assume that ijt was i.i.d. (and Type I extreme value) we would not be constraining the model at all by assuming that α i = α and β i = β. I found that confusing. ThereasonIwasconfusedisbecauseifwemaintaintheusualassumption that ijt is uncorrelated with x jt then if we say that β i = β we are requiring the effect of an increase in x jt to be the same for each consumer, whereas allowing ijt to be correlated across consumers just says that the disturbances can be the same and cannot require an increase in x jt tohavethesameeffect for all consumers. But Nevo does not assume that ijt is uncorrelated with x jt.ifwe do not assume that, then he is quite right, that assuming β i = β does not constrain the model at all. The intuition is as follows. One way to depict consumer 1 as having a bigger effect of x than consumer 2 for good j is to let β j1 > β j2. Another way is to say that 1jt > 2jt and 1jt increases with x jt. If we put no constraints on the correlation of ijt with anything else, in fact, then the model u ijt = ijt is perfectly general. (1) u ijt = α i (y i p jt )+x jt β i + ξ jt + ijt j =1,..., 50, t =1,...,20 (5 ) u jt = α(y p jt )+x jt β + ξ jt + jt j =1,...,50, t =1,..., 20 7

8 Problems with Multinomial Logit A problem with multinomial logit is that the theoretical structure of the elasticities in equation (6 ) is unrealistic in two ways. 1. If market shares are small, as is frequently the case, then α(1 s jt ) is close to α, so that own-price elasticities are close to αp jt. This says that if the price is lower, demand is less elastic, less responsive to price, which in turn implies that the seller will charge a higher markup on goods with low marginal cost. There is no particular reason why we want to assume this, and in reality we often see that markups are higher on goods with higher marginal cost, e.g. luxury cars compared to cheap cars. 2. The cross-price elasticity of good j with respect to the price of good k is αp kt s kt ), which only depends on features of good k. If good k raises its price, its loses customers equally to each other brand. 4 This is a standard defect of multinomial logit. 5 A third problem is endogeneity of the prices. In the model, market shares depend on prices and disturbance terms. But prices will also depend on the disturbance terms; if demand is unusually strong, prices will be higher too. This calls for some kind of instrumental variables estimation. (6) s jt = e x jt β αp jt +ξ jt k=0 ex kt β αp kt +ξ kt (6 ) η jkt % s jt % p kt = s jt pkt s jt = αp jt (1 s jt ) if j = k αp kt s kt otherwise. 4 An apparent third feature, that a high price of good k means it has higher cross-elasticities with every other good, is actually the same thing as problem (1). 5 Give the red bus/blus bus/bicycle example, McFadden. 8

9 The Random Coefficients Logit Model 6 A second way to proceed is to use nested logit. I won t describe that here. A third way to proceed is to assume that the parameters are determined by consumer characteristics. Thus, we might assume that (2) α i β i = = α β α β + ΠD i + Σν i, + Π α Π β D i + Σ α Σ β (ν iα, ν iβ ) where D i is a 4 1 vector of person i s observable characteristics, ν i is a 7 1 vector of the effect of person i s unobservable characteristics on his α i and β i parameters; Π is a 7 4 matrix of how parameters (the α i and the 6 β i s) depend on consumer observables, Σ is a 7 7 matrix of parameters, and (ν iα, ν iβ ), (Π α, Π β )and(σ α, Σ β )just split each vector or matrix in two parts. We will denote the distributions of D and ν by PD(D) andpν (ν). Since we ll be estimating the distribution of the consumer characteristics D, you will see the notation ˆP D(D) showuptoo. We llassume that Pν (ν) is multivariate normal. 7 6 Random coefficients is a bad name for this. It parallels random effects in cross sectional modelling. What it really means is individual coefficients. 7 Nevo does not explain the distributions very well. He variously uses ˆP D (D), P D (D), P ν (ν), and ˆP ν (ν). I ve tried to make the notation consistent, but I may simply have misunderstood what he is doing. 9

10 Utility in the Random Coefficients Logit Model Equation (1) becomes 8 (3 ) u ijt = α i (y i p jt )+x jt β i + ξ jt + ijt = α i y i (α + Π α D i + Σ α ν iα )p jt +x jt (β + Π β D i + Σ β ν iβ )+ξ jt + ijt = α i y i +( αp jt + x jt β + ξ jt )+(Π α D i + Σ α ν iα )p jt +x jt (Π β D i + Σ β ν iβ )+ ijt = α i y i +( αp jt + x jt β + ξ jt )+( p jt, x jt )(ΠD i + Σν i ) + ijt = α i y i + δ jt + µ ijt + ijt j =1,..., 50, t =1,...,20. What I have done here is to reorganize the terms to separate them into four parts. First, there is the utility from income, α i y i. This plays no part in the choice, so it will drop out. Second, there is the mean utility, δ jt, which is the component of utility from this choice that is the same across all consumers. Third, there is a heteroskedastic disturbance, µ ijt, and a homoskedastic i.i.d. disturbance, ijt. 8 Note that there is a bad typo in the Nevo paper on p. 520 in equation (3): u instead of µ in each line. 10

11 Market Shares and Elasticities in the BLP Model SomehowifweusetheTypeIextremevaluedistributionfor ijt, then the market share of good j for individuals of type i is (6 ) s ijt = e δ jt +µ ijt k=0 eδ kt +µ ikt. Recall that we denote the distributions of D and ν by PD(D) and Pν (ν). Since we ll be estimating the distribution of the consumer characteristics D, you will see the notation ˆP D(D) showuptoo. The overall market share of good j is found by integrating this across the individual types, weighting each type by its probability in the population: (6 ) s jt = s ijt d ˆP D(D)dP ν (ν) = e δ jt +µ ijt k=0 eδ kt +µ ikt d ˆP D (D)dP ν (ν) The price elasticity of the market share of good j with respect to the price of good k is (6 ) η jkt s jt pkt s jt = p jt s jt α i s ijt (1 s ijt )d ˆP D(D)dP ν (ν) if j = p kt s kt α i s ijt s ikt d ˆP D(D)dP ν (ν) other This is harder to estimate than the ordinary logit model, because of the integrals. 11

12 The Method of Moments. 9 In a typical model the GMM estimator uses the fact that in the population the M regressor vectors x m (which put together as a matrix are denoted X) are independent of the disturbances: (8 ) Ex m =0,m=1,...,M, and chooses the parameter vector being estimated, β, to minimize the square of the sample analog of equation (8 ): (9 ) ˆβ = argmin β ˆ XΦ 1 X ˆ, where Φ is an M M matrix which is a consistent estimator of EX X, which is related to the variance-covariance matrix (but it hasthex sintheretoo). We minimize the square of the sample analog because we want to minimize the magnitude, rather than generate large negative numbers by our choice, and squares are easier to deal with than absolute values (the same choice in OLS versus minimizing absolute values of errors). We weight by Φ because we want to make heavier use of observations that contain more independent information. This includes the serial correlation and heteroskedasticity corrections. The Method of Moments, like Ordinary Least Squares but unlike Maximum Likelihood, does not require us to know the distribution of the disturbances. In this context, though, we will still have to use the assumption that the ijt follow the extreme value distribution, because we need it to calculate the integrals of market shares aggregated across consumer types. 9 The name Method of Moments is a bad name. The first moment of a distribution is the average and the second moment is the variance. The method of moments uses neither; it relies on covariances (well, I guess that is part of the variance-covariance matrix). 12

13 OLS For OLS, we can use Φ = X X, because we assume that EX X = X Iσ 2 X, and equal weighting by σ 2 will not affect the minimization problem so we can drop the weighting. We also assume that y = Xβ +. Equation (9 ) becomes (9 ) ˆβ = argmin β = argmin β ˆ X(X X) 1 X ˆ (y Xβ) Q(y Xβ), where Q X(X X) 1 X.NotethatX Q = X and QX = X. We can rewrite expression (9 ) as follows: (9 ) ˆβ = argmin β = argmin β y Qy y QXβ β X Qy + β) X QXβ y Qy y Xβ β X y + β X Xβ Differentiate this with respect to β and we get the first order condition, (9 ) 2X y + 2X Xβ =0, so ˆβ =(X X) 1 X y. Instrumental variables can be similarly derived. 13

14 For our present problem, we are not making the OLS assumptions. Instead, we will use a matrix Z of observations on M instruments for X, and our moment equations are that the instruments be independent of a disturbance term, ω(θ ), that is a nonlinear function of the parameters θ that we are trying to estimate. Thus, our assumption on the population is that 10 (8) Ez m ω(θ )=0,m=1,...,M. The GMM estimator is then (9) ˆθ = argmin θ ω(θ) ZΦ 1 Z ω(θ), where Φ is a consistent estimator of EZ Z. The tricky part of the theory is in choosing the function ω(θ )that goes into the moment condition. 10 I think there is a typo in Nevo here, on page 531, and z m should replace Z m in equation (8). 14

15 The BLP Method, I The tricky part of the theory is in choosing the function ω(θ ) that goes into the moment condition. Here is the BLP approach (the numbers of the steps are from Nevo, Appendix, p. 1). (-2) Select arbitrary values for (δ, Π, Σ) as a starting point. Note that you do not need to select α and β separately from δ. (-1) Construct the probability distribution for consumer characteristics, ˆP D(D), using the sample of actual consumer characteristics that you have. 11 (0) Draw random values for (ν i, D i )fori =1,...n s from the distributions Pν (ν) and ˆP D(D) for a sample of size n s, where the bigger you pick n s the more accurate your estimate will be Also figure out Pν (ν); I m not sure how. Nevo slides over this problem on p. 2 of his appendix. Beyond assuming that the distribution is multivariate normal, it seems the modeller also must assume a mean (zero is fine) and variance (which is more arbitrary. 12 On p. 532, Nevo uses ns for the sample size peculiar and bad notation. Pakes (2001, lecture 4) does too. I don t know why P ns is in his equation (11) and later. He has ˆP ν (v) andpd (D), which I think are mistakes. Note that the typo of v for ν shows up here and there in the paper. The Greek letter ν is best avoided in choosing notation because it is so easily confused with v, asthe typesetting of this article shows. 15

16 The BLP Method, II In previous steps, we assembled various starting values and suchlike. Now: (1) Using the starting values and the random values, and using the assumption that the ijt follow the extreme-value distribution, approximate the integral for market share that results from aggregating equation (6 )acrossi by the smooth simulator (text, appendix p. 3). 13 (11) s jt = 1 n s Σ n s i=1s ijt = 1 n s Σ n s i=1 e [δ jt +Σ6 k=1 xk jt (σ k νk i +π k1 D i1 + +π k4 D i4 )]) m=0 e[δ mt +Σ6 k=1 xk mt (σ k νk i +π k1 D i1 + +π k4 D i4 )] where (ν 1 i,...,ν 6 i )and(d i1,...,d i4 )fori =1,...n s are those random draws from the previous step., 13 xxxeverywhere, check to see if it should be m=0 to 50 or 1 to 50. Nevo uses 1 to 50. I think they might come to the same thing becuse the utility of the outside option is normalized to zero. 16

17 The BLP Method, III The previous step ame out with integrals with predicted market shares. Now: (2) Use the following contraction mapping, which, a bit surprisingly, converges. Keeping (Π, Σ) fixed at their starting points, find values of δ by the following iterative process. (12) δ h+1 t = δ h t +(lns t lns t ), where S t is the observed market share. and s t is the predicted market share from step (1) that uses δ t h+1 as its starting point. Start with the arbitrary δ 0 of step (-2). If the observed and predicted market shares are equal, then δ t h+1 = δ h t andth e series has converged. In practice, keep iterating until (lns t lns t ) is small enough for you to be satisifed with its accuracy. 17

18 The BLP Method, IV The previous step ame out with values for δ. Now: (2.5) Pick some starting values for (α, β). (3) Figure out the value of the moment expression using the starting values. First, define (13) ω jt = δ jt (x jt β + αp jt ) Second, figure out the value of the moment expression, (13 ) ω ZΦ 1 Z ω You need the matrix Φ 1 to do this. So, till step (4.5), just use Φ 1 = Z Z as a starting point. 14 (4) Do a minimization search, trying nearby values of (α, βδ, Π, Σ) until the value of the moment expression is close enough to zero for you. (5) Take your converged estimates and use them to compute a new ω, and use that to compute a new value for Φ, Z ωω Z.Thenusethis and go back to steps (3) and (4). Nevo notes that you could then iterate between estimating parameters and estimating the weighting matrix. Both methods are consistent, and neither has more attractive theoretical properties. 14 Nevo s text slides over this problem, but he discusses it on p. 5 of the appendix. 18

19 References Berry, Steven (1994) Estimating Discrete-Choice Models of Product Differentiation, RAND Journal of Economics, 25: Steven Berry, James Levinsohn and Ariel Pakes, Automobile Prices in Market Equilibrium, Econometrica, 63,4: (July 1995). Aviv Nevo, A Practitioner s Guide to Estimation of Random- Coefficients Logit Models of Demand, Journal of Economic and Management Strategy, 9,4: (Winter 2000). Pakes, Ariel (2001) Lecture 4: Graduate Industrial Organization. Demand in Characteristic Space, October 1, 2001, Harvard Economics, http// ec2610/lectures from Pakes/L pdf. 19