Nonparametric Discrete Choice Models With Unobserved Heterogeneity.docx

《Nonparametric Discrete Choice Models With Unobserved Heterogeneity.docx》由会员分享，可在线阅读，更多相关《Nonparametric Discrete Choice Models With Unobserved Heterogeneity.docx（59页珍藏版）》请在得力文库 - 分享文档赚钱的网站上搜索。

1、 SMU Cox School of Business Research Paper Series Working Paper No. 07-005 August 2007 Nonparametric Discrete Choice Models with Unobserved Heterogeneity Richard A. Briesch Cox School of Business Southern Methodist University Pradeep K. Chintagunta Graduate School of Business University of Chicago R

2、osa L. Matzkin Department of Economics Northwestern University This paper can be downloaded without charge from the Social Science Research Network electronic library at: http:/ Nonparametric Discrete Choice Models with Unobserved Heterogeneity1 Richard A. Briesch Cox School of Business Southern Met

3、hodist University Pradeep K. Chintagunta Graduate School of Business University of Chicago Rosa L. Matzkin Department of Economics Northwestern University August 2007 1 Corresponding author. Richard Briesch; Cox School of Business; Southern Methodist University; PO Box 750333; Dallas, Tx 75275-0333.

4、 Email: rbrieschmail.cox.smu.edu, phone: 214-768-3180, fax: 214-768-4099. 2 Abstract In this research, we provide a new method to estimate discrete choice models with unobserved heterogeneity that can be used with either cross-sectional or panel data. The method imposes nonparametric assumptions on

5、the systematic subutility functions and on the distributions of the unobservable random vectors and the heterogeneity parameter. The estimators are computationally feasible and strongly consistent. We provide an empirical application of the estimator to a model of store format choice. The key insigh

6、ts from the empirical application are: 1) consumer response to cost and distance contains interactions and non-linear effects which implies that a model without these effects tends to bias the estimated elasticities and heterogeneity distribution and 2) the increase in likelihood for adding non-line

7、arities is similar to the increase in likelihood for adding heterogeneity, and this increase persists as heterogeneity is included in the model. JEL Classification Code: C14 ; C23 ; C33 ; C35. Keywords: Random Effects; Heterogeneity; Discrete Choice; Nonparametric 3 1. INTRODUCTION Since the early w

8、ork of McFadden (1974) on the development of the Conditional Logit Model for the econometric analysis of choices among a finite number of alternatives, a large number of extensions of the model have been developed. These extensions have spawned streams of literature of their own. One such stream has

9、 focused on relaxing the strict parametric structure imposed in the original model. Another stream has concentrated on relaxing the parameter homogeneity assumption across individuals. This paper contributes to both these areas of research. We introduce methods to estimate discrete choice models whe

10、re all functions and distributions are nonparametric, individuals are allowed to be heterogeneous in their preferences over observable attributes, and the distribution of these preferences is also nonparametric. As is well known in discrete choice models, each individual possesses a utility for each

11、 available alternative, and chooses the one that provides the highest utility. The utility of each alternative is the sum of a subutility of observed attributes - the systematic subutility - and an unobservable random term - the random subutility. For example, in a model of a commuter choosing betwe

12、en various means of transportation, the alternatives may be car and bus, and the attributes observable to the researcher may be the cost and time associated with each alternative. The utility to the commuter for a means of transportation is the sum of a function (the systematic subutility) of the ti

13、me and cost of that means of transportation and the effects of other attributes or factors that are not observed by the researcher. These latter effects are represented by an unobservable random term that represents the value of a subutility of unobserved attributes of the means of transportation, s

14、uch as comfort. Manski (1975) developed an econometric model of discrete choice that did not require specification of a parametric structure for the distribution of the unobservable random subutilities. This semiparametric, distribution-free method was followed by other semiparametric distribution-f

15、ree methods, developed by Cosslett (1983), Manski (1985), Han (1987), Ichimura (1989), Powell, Stock and Stoker (1989), Horowitz (1992), and Klein and Spady (1993), Moon (2004), among others. More recently, Geweke and Keane (1997) and Hirano (2002), have applied mixing techniques that allow nonparam

16、etric estimation of the error term in Bayesian models. Similarly, Klein and Sherman (2002) propose a method that allows non-parametric estimation of the density as well as the parameters for ordered response models. These methods are termed semiparametric because they require a parametric structure

17、for the systematic subutility of the observable characteristics. 4 A second stream of literature has focused on relaxing the parametric assumption about the systematic subutility. Matzkin (1991)s semiparametric method freed the systematic subutility of a parametric structure, while maintaining a par

18、ametric structure for the distribution of the unobservable random subutilities. Matzkin (1992, 1993) also proposed fully nonparametric methods where neither the systematic subutility nor the distribution of the unobservable random subutility are required to possess parametric structures. Finally, a

19、third stream of literature has focused on incorporating consumer heterogeneity into choice models. Wansbeek, et al (2001) noted the importance of including heterogeneity in choice models to avoid finding weak relationships between explanatory variables and choice. However, they also note the difficu

20、lty of incorporating heterogeneity into nonparametric and semiparametric models. Further, Allenby and Rossi (1999) noted the importance of allowing heterogeneity in choice models to extend into the slope coefficients. This extension is required because “optimal marketing decisions must account for t

21、he substantial uncertainty on decision criteria which often involve non-linear functions of model parameters” (p. 58). Specifications that have allowed for heterogeneous systematic subutilities include those of Heckman and Willis (1977), Albright, Lerman and Manski (1977), McFadden (1978), and Hausm

22、an and Wise (1978). These papers use a particular parametric specification, i.e., a specific continuous distribution, to account for the distribution of systematic subutilities across consumers. Heckman and Singer (1984) propose estimating the parameters of the model without imposing a specific cont

23、inuous distribution for this heterogeneity distribution. Ichimura and Thompson (1993) have developed an econometric model of discrete choice where the coefficients of the linear subutility have a distribution of unknown form, which can be estimated. More recently, much empirical work has been done t

24、hat allows some relaxation of the heterogeneity distributions. Lancaster (1997) allows for non-parametric identification of the distribution of the heterogeneity in Bayesian models. Taber (2000) and Park et al (2007) apply semiparametric techniques to dynamic models of choice. Briesch, Chintagunta a

25、nd Matzkin (2002) allow consumer heterogeneity in the parametric part of the choice model while restricting the non-parametric function to be homogeneous. Dahl (2002) applies non-parametric techniques to transition probabilities and dynamic models. Finally, Pinkse, et al (2002) allow for heterogenei

26、ty in semiparametric models of aggregate-level choice. The method that we develop here combines the fully nonparametric methods for estimating discrete choice models (Matzkin (1992, 1993) with a method that allows us to estimate the distribution of 5 unobserved heterogeneity nonparametrically (Heckm

27、an and Singer (1984) as well. The unobserved heterogeneity variable is included in the systematic subutility in a nonadditive way (Matzkin (1999, 2003). We provide conditions under which the systematic subutility, the distribution of the nonadditive unobserved heterogeneity variable, and the distrib

28、ution of the additive unobserved random subutility can all be nonparametrically identified and consistently estimated from individual choice data. The method can be used with either cross sectional or panel data. These results update Briesch, Chintagunta, and Matzkin (2002). We apply the proposed me

29、thodology to study the drivers of grocery store-format choice for a panel of households. There are two main types of formats that supermarkets (i.e., grocery stores or chains) classify themselves into everyday low price (EDLP) stores or high-low price (Hi-Lo) stores. The former offer fewer promotion

30、s of lower “depth” (i.e., magnitude of discounts) than the latter. The main tradeoff facing consumers is that EDLP stores are typically located farther away (longer driving distances) than Hi- Lo stores although their prices, on average, are lower than those at Hi-Lo stores leading to a lower total

31、cost of shopping “basket” for the consumer. Since there is strong evidence that the value of time (associated with the driving distance) is heterogeneous across households, and there is no consensus across the previous empirical results about the particular shape of the utility for driving distance

32、and expenditure, we think that the proposed method is ideally suited to understanding the nature of the tradeoff between distance and expenditure facing the consumer. To decrease the well-known dimensionality problems associated with relaxing parametric structures, we use a semiparametric version of

33、 our model. In particular, only the subutility of distance to the store and cost of refilling inventory at the store are nonparametric. We allow this subutility to be heterogeneous across consumers, and provide an estimator for both, the subutilities of the different types, the distribution of types

34、, and the additional parameters of the model. Further, we assume that the unobserved component of utility in this application is distributed according to a type-I extreme value distribution. In the next section we describe the model. In Section 3 we state conditions under which the model is identifi

35、ed. In Section 4 we present strongly consistent estimators for the functions and distributions in the model. Section 5 provides computational details. Section 6 presents the empirical application. Section 7 concludes. 2. THE MODEL As is usual in discrete choice models, we assume that a typical consu

36、mer must choose one of a finite number, J, of alternatives, and he/she chooses the one that maximizes the value of a utility function, 6 F 1 which depends on the characteristics of the alternatives and the consumer. Each alternative j is characterized by a vector, zj, of the observable attributes of

37、 the alternatives. We will assume that zj x j, rj , where rj R and x j RK K 1 . Each consumer is characterized by a vector,s RL , of observable socioeconomic characteristics for the consumer. The utility of a consumer with observable socioeconomic characteristics s, for an alternative, j, is given b

38、y V j, s, zj, j where j and denote the values of unobservable random variables. For any given value of , and any j, V j, , is a real valued, but otherwise unknown, function. The dependence of V on allows us to incorporate into the model the possibility that this systematic subutility be different fo

39、r different consumers, even if the observable exogenous variables posses the same values for these consumers. We will denote the distribution of by G and we will denote the normalized distribution of the random vector 1 , . . . , J by F . The probability that a consumer with socioeconomic characteri

40、stics s will choose alternative j when the vector of observable attributes of the alternatives is z z1 , . . . , zJ x 1 , r1 , . . . , x J , rJ will be denoted by p j|s, z;V , F , G . Hence, p j|s, z; V, F, G Pr j|s, z; , V, F dG where Pr j|s, z; , V, F denotes the probability that a consumer with s

41、ystematic subutility V ; will choose alternative j, when the distribution of is F. By the utility maximization hypothesis, Pr j|s, z; , V, F Pr V j, s, x j , rj , j V k, s, xk , rk , k for all k j Pr k j V j, s, x j, rj, V k, s, xk , rk , for all k j which depends on the distribution F. In particula

42、r, if we let * 2 1 , . . . , J 1 , then denote the distribution of the vector 7 F 1 1 j 1 Pr(1 | s, z;,V , F * ) F V (1, s, x1 , r1 ,) V (2, s, x2 , r2 ,),.,V (1, s, x1 , r1 ,) V (J , s, x2 , r2 ,)1 and the probability that the consumer will choose alternative 1 is then p(1 | s, z;V , F * , G) Pr(1

43、| s, z;,V , F * ) dG() F *V (1, s, x1 , r1 ,) V (2, s, x1 , r1 ,),.,V (1, s, x1 , r1 ,) V (J , s, x1 , r1 ,) dG() 1 For any j , Pr( j | s, z;,V , F * ) can be obtained in an analogous way, letting * denote the distribution of 1 j , . . . , J j . A particular case of the polychotomous choice model is

44、 the Binary Threshold Crossing Model, which has been used in a wide range of applications. This model can be obtained from the Polychotomous Choice Model by letting J 2, 2 1 , and s, x 2 , r2 , V 2, s, x 2 , r2 , 0. In other words, the model can be described by: y V x, r, 1 if y 0 y 0 otherwise wher

45、e y is unobservable. In this model, * denotes the distribution of . Hence, for all x, r, Pr(1 | x, r;,V , F ) F * V ( x, r,)and for all x, r the probability that the consumer will choose alternative 1 is p(1 | s, z;V , F , G ) Pr(1 | s, z;,V , F ) dG () F * V (x, r,) dG () 3. NONPARAMETRIC IDENTIFIC

46、ATION Our objective is to develop estimators for the function V and the distributions F and G , without requiring that these functions and distributions belong to parametric families. It follows from the definition of the model that we can only hope to identify the distributions of the vectors F * 8

47、 1 F 1 1 J 1 j 1 j , . . . , J j for j 1, . . . , J. Let F * denote the distribution of 1 . Since from * we can obtain the distribution of j j 2, . . . , J , we will deal only with the identification of F * F * . F * . We let DEFINITION: The function V and the distribution F and G are identified in a set W F G such that V , F , G W F G if V, F, G W F G Pr j|s, z; V ; , F dG Pr j|s, z;V ; , F dG for j 1, . . . , J, a. s. implies that That is, (V , F , G )