Economic Choices

Daniel McFadden

The American Economic Review, Vol. 91, No. 3. (Jun., 2001), pp. 351-378.

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http://www.jstor.org Thu May 31 10:53:46 2007 Economic Choicest

This Nobel lecture discusses the microecono- developments in the economic theory of choice, metric analysis of choice behavior of consumers and modifications to this theory that are being who face discrete economic alternatives. Before forced by experimental evidence from cognitive the 1960's, economists used consumer theory psychology. I will close with a survey of statis- mostly as a logical tool, to explore conceptually tical methods that have developed as part of the the properties of alternative market organiza- research program on economic choice behavior. tions and economic policies. When the theory Science is a cooperative enterprise, and my was applied empirically, it was to market-level work on choice behavior reflects not only my or national-accounts-level data. In these appli- own ideas, but the results of exchange and col- cations, the theory was usually developed in laboration with many other scholar^.^ First, of terms of a representative agent, with market- course, is my co-laureate James Heckman, who level behavior given by the representative among his many contributions pioneered the agent's behavior writ large. When observations important area of dynamic discrete choice anal- deviated from those implied by the representa- ysis. Nine other individuals who played a major tive agent theory, these differences were swept role in channeling microeconometrics and into an additive disturbance and attributed to choice theory toward their modern forms, and data measurement errors, rather than to unob- had a particularly important influence on my served factors within or across individual own work, are Zvi Griliches, L. L. Thurstone, agents. In statistical language, traditional con- Jacob Marschak, Duncan Luce, Amos Tversky, sumer theory placed structural restrictions on Danny Kahneman, Moshe Ben-Akiva, Charles mean behavior, but the distribution of responses Manski, and Kenneth Train. A gallery of about their mean was not tied to the theory. their photographs is shown in Figure 1. I wish In the 1960's, rapidly increasing availability particularly to cite Griliches, Marschak, and of survey data on individual behavior, and the Tversky, robbed by death of their own chances advent of digital computers that could analyze to win Nobel prizes. these data, focused attention on the variations in demand across individuals. It became important I. A Brief History to explain and model these variations as bast of consumer theory, rather than as ad hoc distur- Classical economic theory postulates that bances. This was particularly obvious for dis- consumers seek to maximize their self-interest, Crete choices, such as transportation mode or and that self-interest has broadly defined con- occupation. The solution to this problem has led sistency properties across different decisions. to the tools we have today for microeconometric At one level, the theory is virtually tautological, analysis of choice behavior. I will first give a as in this description from a principles textbook brief history of the development of this subject, by Frank Taussig (1912): and dace mv own contributions in context. Af- ter that, I will discuss in some detail more recent An object can have no value unless it has utility. No one will give anything for an article unless it yield him satisfaction. 'This article is a revised version of the lecture Daniel Doubtless people are sometimes foolish, McFadden delivered in Stockholm, Sweden on December 8, and buy things, as children do, to please a 2000, when he received the Bank of Sweden Prize in Eco- moment's fancy; but at least they think at nornic Sciences in Memory of Alfred Nobel. The article is copyright O The Nobel Foundation 2000 and is published here with the permission of the Nobel Foundation. *: Department of Economics, University of California, ' Any accounting of credit for my contributions to eco- Berkeley, CA 94720. Many of the author's publications nomics has to include Leo Hurwicz, John Chipman, Marc cited in this paper are posted at http://elsa.berkeley.edu/ Nerlove, and Hirofumi Uzawa, who attracted me to the field -1ncfadden. and taught me most of what I know. 351 352 THE AMERICAN ECONOMIC REVIEW JUNE 2001

Zvi Griliches L. L. Thurstone

Duncan Luce Danny Kahneman

Moshe Ben-Akiva Charles Manski Kenneth Train VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES 353

the moment that there is a wish to be on discrete choice behavior, and that seemed a gratified. promising place to start. In a seminal paper on psychophysical dis- The concept of rational consumer behavior was crimination, L. L. Thurstone (1927) introduced given a much more specific meaning in the a Law of Comparative Judgment in which al- perfection of the classical theory by John Hicks ternative i with true stimulus level V, is per- and Paul Samuelson, where self-interest is de- ceived with a normal error as Vi + as,. The fined in terms of stable, innate preferences, and choice probability for a paired comparison then in Herbert A. Simon's (1978) words, "The ra- satisfied P{,,,,(1) = @((V, - V,)/a), a form tional man of economics is a maximizer, who now called the binomial probit model. When the will settle for nothing less than the best." perceived stimuli Vi + s, are interpreted as Theorists considered heterogeneous prefer- levels of satisfaction, or utility, this can be in- ences, but this complication was ignored in em- terpreted as a model for economic choice. pirical studies of market demand that employed Thurstone's work was introduced into econom- the representative consumer device. A con- ics by Jacob Marschak (1960), who explored sumer with preferences represented by a utility the theoretical implications for choice probabil- function U(x) of a vector x of consumption ities of maximization of utilities that contained levels of various goods would maximize this random elements. Marschak called this the Ran- utility subject to a budget constraint p x 5 a, dom Utility Maximization (RUM) model. where p is a vector of prices and a is income, at An influential study of choice behavior by R. a demand function x = d(a, p). This mapping Duncan Luce (1959) introduced an Independence was then assumed to hold at the market level from Irrelevant Alternatives (IIA) axiom that sim- with a disturbance E added to account for dis- plified experimental collection of choice data by crepancies in observed data, x = d(a, p) + allowing multinomial choice probabilities to be E. The disturbance was interpreted as coming inferred from binomial choice experiments. The from measurement error in x, or possibly from IIA axiom states that the ratio of choice probabil- consumer mistakes in optimization. Only rep- ities for alternatives i and j is the same for every resentative demand d(a, p) carried restrictions choice set C that includes both i and j; i.e., Pc(i)l imposed by consumer theory. P,( j) = pjiJt(i)/~{i,j,(j).' Luce showed for The rapidly increasing availability of mi- positive probabilities that IIA implies strict utili- croeconomic data in the 1960's led econo- ties wi such that Pc(i) = wil&,, w,.Marschak metricians to consider more carefully the proved for a finite universe of objects that IIA specification of individual agent behavior. In implies RUM. 1957, Zvi Griliches pointed out that random I proposed for Cottingham's research an econo- elements appearing in the constraints or objec- metric version of the Luce model in which the tives of economic agents would produce distur- strict utilities were specified as functions of ob- bances in observed behavior whose properties served attributes of the alternative freeway routes, depended on their source and whether they were known to the agents (Griliches, 1957; Yair Mundlak, 1963; Griliches and Vidar Ringstad, 1970). I began working on these problems in 1962, in a study of production functions for electricity (Melvyn Fuss et al., 1978; McFadden, In this formula, Vk was a systematic utility that 1978a). I took to be a linear function of measured at- In 1965, a Berkeley graduate student, Phoebe tributes of alternative k, such as construction Cottingham, asked me for suggestions on how cost, route length, and areas of parklands and she might analyze her thesis data on freeway open space taken, with coefficients that re- routing choices by the California Department of flected the tastes of the decision makers, and C Highways. The problem was to devise a com- putationally tractable model of economic deci- sion making that yielded choice probabilities The axiom can also be written as P,(i) = P,(i) . P,(A) Pc(i) for the alternatives i in a finite feasible set for i E A C C, a variant that allows some alternatives to C. I was familiar with the work of psychologists have a zero probability of being chosen. 354 THE AMERICAN ECONOMIC REVIEW JUNE 2001 was a finite set containing the feasible choice pendently during the same decade seems to be alternatives. I called this a conditional logit the direct connection that I provided to con- model since in the case of binomial choice it sumer theory, linking unobserved preference reduced to the logistic model used in biostatis- heterogeneity to a fully consistent description tics, and in the multinomial case it could be of the distribution of demands (McFadden, interpreted as the conditional distribution of de- 1974a). mand given the feasible set of choice alterna- I had an opportunity to develop additional tives C. Today, (I) is more commonly called applications of discrete choice analysis during the multinomial logit (MNL) model, and I will a visit to the Massachusetts Institute of Tech- use this more common terminology. I devel- nology in 1970. At that time, Peter Diamond oped a computer program to estimate the MNL and Robert Hall had developed a separable- model by maximum likelihood, a nontrivial task utility, multistage budgeting, representative in those days, and Cottingham completed consumer model for the complex of consumer her thesis before the program was working transportation decisions, including commute (Cottingham, 1966). However, I was eventually mode choice, and frequency, timing, and des- able to use the model to analyze her data tination of shopping trips. They invited me to (McFadden, 1968, 1976). operationalize their model so that it could be The characterization of alternatives in the estimated from data on individual trip-taking MNL model in terms of their "hedonic" at- behavior. I did so using a nested version of the tributes was natural for this problem, and fol- MNL model, with the nesting levels corre- lowed the psychometric tradition of describing sponding to the separable utility structure and alternatives in terms of physical stimuli. In em- with inclusive values carrying the impact of pirical consumer theory, this was an early im- lower-level decisions into higher levels in the plementation of the hedonic formulation of the same way that sub-budgets are carried consumer problem developed by Griliches through multistage budgeting problems (196 1) and Kevin Lancaster (1966). (McFadden, 1974b; Thomas Domencich and As part of my development of the MNL model, McFadden, 1975). My treatment of inclusive I investigated further its RUM foundations. I values turned out to be approximately right, but showed that the Luce model was consistent with a a superior exact formula for inclusive values, RUM model with independent identically distrib- utilizing what has come to be known as the log uted additive disturbances if and only if these sum formula, was discovered by Moshe Ben- disturbances had a distribution called Extreme Akiva (1972). Value Type I. Earlier and independently, Tony Beginning in 1972, I organized a large re- Marley had established sufficiency (Luce and Pat search project at Berkeley, with support from Suppes, 1965). Ket Richter and I also established the National Science Foundation, for the pur- a general necessary and sufficient condition for pose of developing tools for transportation plan- choice probabilities to be consistent with RUM, ning based on rnicroeconometric analysis of an Axiom of Revealed Stochastic Preference individual travel decisions. Participants in- (ARSP): choice probabilities are RUM-consistent cluded Kenneth Train and Charles Manski. As a if and only if for any finite sequence of events (C,, natural experiment to test and refine nested i,), where C, is a set of feasible alternatives and i, MNL models and other empirical RUM models, is a choice, the sum of the choice probabilities my research group studied the impact of BART, does not exceed the maximum number of a new fixed-rail rapid transit system being built these events consistent with a single prefer- in the San Francisco Bay Area. We collected ence order (McFadden and Marcel K. Richter, data on the travel behavior of a sample of indi- 1970, 1990). viduals in 1972, prior to the introduction of Viewed as a statistical model for discrete BART, and estimated models that were then response, the MNL model was a small and in used to predict the behavior of the same indi- retrospect obvious contribution to microecono- viduals in 1975 after BART began operation. metric analysis, although one that has turned out Table 1 summarizes results for the journey-to- to have many applications. The reason my for- work. In this table, a MNL model estimated mulation of the MNL model has received more using the pre-BART commuter data was evalu- attention than others that were developed inde- ated at the realized attributes of the alternatives, VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES 355

Cell counts Predicted choices Actual choices Auto alone Carpool Bus BART Total Auto alone Carpool Bus BART

Total 352.4 144.5 94.0 40.0 63 1

Predicted share (percent) 55.8 22.9 14.9 6.3 (Standard error) (percent) (1 1.4) (10.7) (3.7) (2.5) Actual share (percent) 59.9 21.7 12.2 6.2

including the new BART alternative, that were travel demand problem and applications such as available to each of the 631 subjects who were education and occupation choices, demand for surveyed after BART began operation. The cell consumer goods, and location choices have led counts are the sums of the predicted probabili- to adoption of these methods in a variety of ties for the sample individuals classified by their studies of choice behavior of both consumers actual post-BART choice. The standard errors and firms. in the predicted shares are calculated taking into account the precision of model parameter 11. Refinements of Economic Choice Analysis estimates. There were some systematic errors in our At a choice conference in Paris in 1998, a predictions. We overestimated willingness to working group (Ben-Akiva et al., 1999) laid out walk to BART, and underestimated willingness the elements in a contemporary view of the to drive alone. In retrospect, the methods we theory of choice; an adaptation is shown in used to assign an alternative-specific effect for Figure 2. The figure describes one decision- the new BART mode, and to account for sub- making task in a lifelong sequence, with ear- stitution between modes, were much inferior to lier information and choices operating through the market research and modeling methods that experience and memory to provide context for are used today. However, our overall forecasts the current decision problem, and the results of for BART were quite accurate, particularly in this choice feeding forward to influence future comparison to the official 1973 forecast, ob- decision problems. The heavy arrows in this tained from aggregate gravity models, that figure coincide with the economists' standard BART would carry 15 percent of commute model of the choice process, a theory of rational trips. We were lucky to be so accurate, given the choice in which individuals collect information standard errors of our forecasts, but even dis- on alternatives, use the rules of probability to counting luck, our study provided strong evi- convert this information into perceived at- dence that disaggregate RUM-based models tributes, and then go through a cognitive pro- could outperform conventional methods. Our cess that can be represented as aggregating the procedures were also more sensitive to the op- perceived attribute levels into a stable one- erational policy decisions facing transportation dimensional utility index which is then maxi- planners. On the basis of our research, and other mized. The lighter arrows in the diagram studies of the effectiveness of RUM-based correspond to psychological factors that enter travel demand analysis, these methods have decision-making; these I will discuss later. The been widely adopted for transportation planning concepts of perception, preference, and process around the world. Details of our research are appear in both economic and psychological found in McFadden et al. (1977) and McFadden views of decision-making, but with different (1978b). The obvious similarities between the views on how they work. 356 THE AMERICAN ECONOMIC REVIEW JUNE 2001

Stated Perception$ Experience Inforniation

Time & Dollar Budgets, Choice Set Constraints

Preferenccr)

Stated Preferentec

A. Fundamentals ioral hypothesis started from the standard model, with randomness attributed to unobserved hetero- The heart of the standard or rational model of geneity in tastes, experience, and information on economics is the idea that consumers seek to the attributes of alternatives. Parameterizing the maximize innate, stable preferences over the utility function and the distribution of the random quantities and attributes of the commodities factors yielded parametric models for the choice they consume. This holds even if there are in- probabilities, conditioned on observed attributes termediate steps in which raw goods are trans- of alternatives and characte~isticsof the decision formed by the individual to produce maker. The MNL model is a tractable example. It satisfactions that are the proximate source of is useful to review this derivation of the RUM utility, e.g., travel is an input to employment, explanation of choice behavior, talung a careful and shopping activities are inputs to household look at the meaning of its fundamental elements, production. An important feature of the theory and the scope and limitations of the models that is the consumer sovereignty property that pref- come out. I believe this is particularly true for erences are predetermined in any choice situa- analysts who want to try to combine economic tion, and do not depend on what alternatives are market data with experimental data on prefer- available. Succinctly, desirability precedes ences, or who want to bring in cognitive and availability. psychometric effects that are ignored in the stan- The standard model has a vaguely biological dard model. flavor. Preferences are determined from a ge- In the standard model, consumers have pref- netically coded taste template. The model al- erences over levels of consumption of goods lows experience to influence how preferences and leisure. When goods have hedonic at- consistent with the template are expressed. tributes, preferences are defined to incorporate However, most applications of the standard the consumer's subjective perceptions of these model leave out dependence on experience, and attributes. The expressed preferences of the much of the power of this model lies in its consumers are functions of their taste template, ability to explain most patterns of economic experience, and personal characteristics, includ- behavior without having to account for experi- ing both observed and unobserved components. ence or perceptions. Mild regularity conditions allow us to represent The original formulation of RUM as a behav- preferences by a continuous real-valued utility VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES 357 function of the characteristics of the consumer, come net of the cost of the alternative, the wage and consumption levels and attributes of goods. rate, and goods prices, the consumer will choose Consumers are heterogeneous in unobserved leisure and consumption levels of remaining characteristics such as their taste templates and goods to maximize utility subject to budget and the mechanisms they use to form perceptions. I time constraints. The level of utility attained is will assume that the unobserved characteristics then a function of the attributes of the discrete vary continuously with the observed character- alternative. observed consumer characteristics. istics of a consumer. For example, the tastes and a uniformly distributed random vector charac: perceptions of an individual change smoothly terizing unobserved consumer characteristics, with age as long as there are no major shifts in and the economic variables that determine the observed characteristics. Technically, this is an budget constraint: net nonwage income, the assumption that unobserved characteristics are a wage rate, and goods prices. The theory of continzlous random field indexed by the ob- optimization implies that this is a classical in- served characteristics. An implication of this direct utility function, with the properties that it assumption is that the conditional distribution has a closed graph and is quasi-convex and of the unobserved characteristics will depend homogeneous of degree zero in the economic continuously on the observed characteristics. variables, and increasing in net nonwage in- This assumption is not very restrictive, and can come. Under fairly mild conditions, it is possi- essentially be made true by construction. ble to require that the indirect utility function be One important restriction that consumer sov- convex, rather than quasi-convex, in the eco- ereignty places on the conditional distribution nomic variables. The last step in applying the of unobserved consumer characteristics is that it standard model to discrete choice is to require cannot depend on current economic variables the consumer's choice among the feasible alter- such as nonwage income, the wage rate, and natives to maximize conditional indirect utility. goods prices, which determine feasibility The functional form of the canonical indirect through the consumer's budget, but are ex- utility function will depend on the structure of cluded from influencing tastes. The conditional preferences, including the trade-off between distribution can however depend on the individ- goods and leisure as nonwage income or the wage ual's history of economic status and choices, rate change, the role of household production in through the operation of experience on the ex- determining how goods combine to satisfy needs, pression of preferences. Under mild regularity and separability properties of preferences. The conditions, the random field of unobserved con- original 1970 formulation of the RUM model for sumer characteristics can be written as a con- travel demand applications fit into this framework, tinuous transformation of a uniform continuous in some variantof the form random field; this is an extension of an elemen- tary result from probability theory that a uni- U= V+ 7 and variate random variable Y with distribution F can be written almost surely as Y = Fp'(v) with v a uniform (0,1) random variable. This transformation can then be absorbed into the In this formula, a is nonwage income, c is the cost definition of the utility function, so that the of the alternative, w is the wage rate, with (a,c, w) dependence of the utility function on unob- all expressed in real terms with other goods prices served consumer characteristics can be repre- implicit, t is the time required by the alternative, x sented canonically as a continuous function of a is a vector of other observed attributes of the uniformly distributed random vector. alternative, s is a vector of observed characteristics I consider discrete choice from feasible sets of the consumer, and z(x,s) is a vector of pre- containing finite numbers of mutually exclusive specified functions of the arguments. The (a,p, y) and exhaustive alternatives that are character- are parameters, and 0 determines the elasticity of ized by their observed attributes, with other the demand for leisure and is commonly assumed aspects of consumer behavior taking place in to be either zero or one, but can be a parameter in the background. Suppose for the moment that (0,1) corresponding to a Stone-Geary specifica- the consumer is assigned a specific discrete tion for systematic utility (McFadden and Kenneth alternative. Given this alternative, nonwage in- Train, 1978). The 7 is an additive disturbance 358 THE AMERICAN ECONOMIC REVIEW JUNE 2001 summarizing the effects of unobserved consumer uous mixed partial derivatives that alternate in characteristics. When q = -log(-log(&)) and the sign, with nonnegative odd mixed derivatives. E are uniformly distributed and independent Then F(ql, ... ,7,) = exp(-H(eCv', ... ,eCvJ))is a across alternatives, the disturbances are indepen- joint distribution function whose one-dimensional dently identically extreme value distributed and marginals are extreme value distributions. Con- produce a MNL model (1) in which the systematic sider a RUM model ui = + qi for a set of utility has the form (2) for each k E C. alternatives C = (1, ... , J), where the 7's have A natural question to ask in retrospect is how this distribution. Then E maxi ui = log(H(eV1,... , restrictive this specification is, and to what degree eVf))+ [, where [ = 0.57721 ... is Euler's con- it can be modified to accommodate more general stant. The RUM choice probabilities are given by RUM-consistent behavior. The answer is that both the derivatives of this expectation, with the closed the linear dependence of systematic utility on eco- form nomic variables and the distributional assumption yielding the IIA property are quite special. While the model works well as an empirical approxima- tion in surprisingly many applications, it implies a uniform pattern of substitution between alterna- tives that may not be behaviorally plausible. A number of more flexible and more or less tractable One example of a GEV generating function is families of models have been developed with the linear function H = w, + ... + w,; this more general dependence on explanatory vari- yields the MNL model. More complex GEV ables and/or distributions of unobservables that models are obtained by repeated application of permit more general patterns of substitution be- the following result: If sets A, B satisfy A U tween alternatives. B = C, and wA, w,, and w, are the corre- sponding subvectors of (w,, ... , w,), if B. Models for RUM-Consistent H~(w,) and H,(w,) are GEV generating func- Clzoice Probabilities tions in wA and w,, respectively, and if s 2 1, then ~'(w,) = HA(w~)"" + HB(wB) is a The MNL model has proven to have wide GEV generating function in w,. The parameter empirical applicability, but as a theoretical 11s is called an inclz~sive value coejjicient. model of choice behavior its IIA property is Nested MNL models are defined by applying unsatisfactorily restrictive. Examples due to this recursion repeatedly to nonoverlapping sets John Chipman (1960) and Gerard Debreu A and B, and the argument shows they are (1960), later elaborated as the "red bus, blue RUM-consistent. bus" problem in transportation applications, Mixtures of RUM-consistent choice models are show that we can sometimes expect this model again RUM-consistent. For example, if H(w,, ... , to fail. Nested MNL models, generalized ex- w,, a) is a family of GEV generating functions treme value (GEV) models, and multinomial indexed by parameters a that determine nesting probit (MNP) models have been developed to structure, weights, and inclusive values, and one relax the restrictive properties of the simple has a distribution over a that does not depend on MNL model. These are often very useful, but economic variables, then the RUM model ui = remain restrictive in the sense that tractable Vi + qi with F(ql, ... ,7,) = E, exp(-H(eCvl, ... , versions fall short of being able to represent all eCW;a)) has E maxi ui = E, log(H(eV1,... , eVJ; RUM-consistent behavior. One family of RUM- a))+ [ and choice probabilities satisfying consistent discrete choice models that is very P,(i) = a(E, maxi ui)ldVi = ~,e'~,(e"l, ... , flexible is the random parameters or mixed eVf;a)/H(eV1, ... ,eVf;a). Useful specializations of multinomial logit (MMNL) model. the GEV family can be found in McFadden, GEV models were introduced and their RUM (1981), Kenneth Small (1987), and C. R. Bhat consistency established in McFadden (1978b). (1998). Define a GEV generating function H(wl, ... , w,) A different approach that established the RUM- to be a nonnegative linear homogeneous function consistency of an important family of nested of w r 0 with the property that H goes to +w MNL models was taken by H. W. C. L. Williams when any argument goes to +a, and with contin- (1977) and Andrew Daly and Stan Zachary VOL. 91 NO. 3 McFADDEN: EC 'ONOMIC CHOICES 359

(1979). The Williams-Daly-Zachary formulation formly by a Bernstein-Weierstrauss polynomial established two results that are useful more gen- in the observed arguments and the uniformly erally. First, they showed that an extreme value distributed vector of unobserved characteris- distributed random variable X can be written as tic~.~This can again be done so that the prob- the sum of two independent random variables Y ability of the approximation changing the and Z, with Z also extreme value distributed, if and preference order is very small. Condition on the only if the scale factor for X is at least as large as uniform random vector that enters the utility the scale factor for Z. Second, they effectively function, and then integrate this vector out to showed that in the family of RUM models with an obtain the MMNL model. additive linear nonwage income term, expected maximum utility behaves like a "representative consumer" indirect utility function with the prop- erty that its price derivatives are proportional to the choice probabilities. A nested MNL model with no income effects has the property that its In this formula, a is a vector of polynomial func- choice probabilities are given by derivatives of its tions of the uniform random vector E, and the Zj top-level inclusive value. Then, one can establish are vectors of polynomial functions of observed that a nested MNL model is consistent with RUM characteristics of the consumer and observed at- by showing, for suitable range restrictions on in- tributes of alternative j. It is immediate from its clusive value coefficients, that its top-level inclu- derivation that every MMNL model of the form sive value meets the necessary and sufficient (4) is RUM-consistent, provided the functions curvature conditions for an indirect utility func- Zj a(&) are indirect utility functions for tion. Proofs of these results are given in McFadden each E. The model (4) has the interpretation of a (1981) and McFadden and Train (2000). MNL model of the usual linear-in-parameters Generalized extreme value families of choice form in which we allow the to va~y models avoid some LIA restrictions, but cannot randomly, and in which we allow a flexible defi- represent all RUM-consistent behavior. The MNP nition of the systematic utility of an alternative by model, obtained from a RUM model with additive introducing a series approximation in the observed normal disturbances that have a general covari- attributes of the alternative, interacted with ob- ance structure, is quite flexible, but its choice served characteristics of the decision maker. In probabilities must usually be written in open form principle, the approximation errors in this formu- as multivariate integrals that require numerical lation-can be bounded and the order of the volv- A . integration. Special restrictions such as factor- nomial required to achieve a desired level of analytic covariance structures are needed to make accuracy can be determined in advance. However, these models tractable (McFadden, 1981, 1984). the quantities this calculation requires are often However, simulation-based estimation methods, unavailable in applications, and itis better to use discussed later, have improved our ability to im- an adaptive or cross-validation method to deter- plement fairly general forms of these models in mine a stopping point for the approximation. The applications. shape restrictions required on Z . a(&) are most Recently, McFadden and Train (2000) have easily imposed component-by-component, with established a somewhat surprising and conve- sign restrictions on the corresponding components nient mixed MNL (MMNL) approximate repre- of a.Theoretically, it is possible to select a basis sentation of any regular RUM-consistent choice so that this can be done without losing the uni- probabilities. Start from the canonical represen- form approximation property, but this-has been tation of the standard model described earlier. done constructively only for one and two di- Make the fairly mild assumption that the class mensions (George Anastassiou and Xiang Yu, of all feasible sets is compact. Perturb the ca- 1992; Lubomir Dechevsky and Spridon Penev, nonical indirect utility functions by adding in- 1997). Alternately, one can proceed without dependent Extreme Value Type I disturbances, scaled so that the probability is very small that the original and perturbed indirect utility func- Other Hamel bases for the approximation can also be tions order alternatives differently. Further, used, and may have advantages in terms of parsimony and approximate the canonical indirect utility uni- the imposition of shape restrictions. 360 THE AMERICAN ECONOMIC REVIEW JUNE 2001 imposing the shape restrictions, and test for Table 3 takes a and y to have independently them in the range of the observations (Donald distributed components that are either normal or Brown and Rosa Matzkin, 1998). lognormal. The table gives percentiles of the One can approximate the distribution of the a estimated parameter distributions. Notable in coefficients in (4) by a distribution concentrated this model is the spread in the distribution of on a finite set of points, with the probability tastes for number of trout, which determines weights at these points treated as parameters. This catch rates, and the division of anglers be- is called a latent class model. It is possible to use tween positive and negative tastes for camp- latent class models to obtain nonparametric esti- grounds and number of access points, which mates of any family of RUM-consistent choice provide convenience but also produce crowd- probabilities by the method of sieves. The latent ing. The elasticity is the percentage increase class model is a single hidden-layer feedfowd in the probability for a site resulting from a neural network (with MNL activation functions), 1-percent increase in the explanatory variable and the asymptotic approximation theory that has for that alternative, calculated at sample been developed for neural networks can be ap- average values for the variables and the plied to establish convergence rates and stopping probabilities. rules (Hal White, 1989, 1992; Bing Cheng and D. Michael Titterington, 1994; Xiaohong Chen and C. Estimating Willingness-to-Pay in Discrete White, 1998; Chunrong Ai and Chen, 1999). It is Choice Models possible to develop other RUM-consistent ap- proximations to families of choice probabilities Applications of discrete choice models to eco- that are useful in some applications (John Dagsvk, nomic policy problems often call for estimation of 1994). Willingness-to-Pay (WTP) for policy changes. Summarizing, I have outlined a result which For example, the Montana trout-fishing study says that any well-behaved RUM model can be sought to determine WTP for the increase in fish approximated by a MMNL model, or alternately stocks that would come from restoration of natural by a latent class model, provided the transfor- riparian conditions. For the MMNL model in Ta- mations of observed variables and the random ble 3, which is independent of nonwage income, distributions that enter these forms are suffi- mean WTP has a convenient exact expected log ciently flexible. The MMNL model was intro- sum form in the systematic utilities before (V!) duced by Scott Cardell and Fred Dunbar (1980). and after (V:.') the change, With the development of convenient simulation methods for estimation (David Revelt and Train, 1998), it has become widely used. (5) WTP To illustrate application of the MMNL model, I will describe a study of trout-fishing destination choice conducted as part of an as- sessment of damage that copper mining caused to recreational fishing in the Clark Fork River This is a case where Hicksian and Marshallian Basin in Montana. A sample of 962 fishing trips measures of consumer surplus coincide, and also to 59 sites on Montana rivers, made by 238 where preferences can be aggregated into repre- anglers, was collected in a household survey sentative "community" preferences (Chipman and conducted by William Desvousges and associ- James Moore, 1990; McFadden, 1999b). ates at Triangle Economic Research (1996). The When the indirect utility function is not linear variables in the study are described in Table and additive in nonwage income, computation 2. These data have been used by Train (1998) to of exact Hicksian compensating variation is estimate MMNL models of the form (4) for much more burdensome. McFadden gives fishing site choice. This study assumes an indi- bounds that will sometimes suffice for policy rect utility model U = a(a - c) - pwt + analysis, and develops Monte Carlo Markov z(x, s)y, where the notation is the same as in (2), Chain methods for numerical calculation of and the parameters (a, p, y) vary randomly over exact WTP. Recently, Anders Karlstrom the population, with a specification that fixes the (2000) has developed numerical methods that ratio pla and for the estimates described in simplify these calculations. VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES 361

TABLE2-EXPLANATORY VARIABLES FOR TROUT-FISHINGDESTINATION CHOICE IN MONTANA

Variable Description Mean Standard deviation Trout stock Hundreds of fish per 1,000' of stream Trip cost Travel cost to the site, including the variable cost of driving and the value of time spent driving (calculated at % the angler's wage, or P = a/3) Access Number of state-designated access areas per USGS block Aesthetics Rating 0 to 3 (Montana River Information System) Campgrounds Number of campgrounds per USGS block Major Major fishing area (Angler's Guide to Montana) Restricted Number of restricted species at the site (e.g., mandated catcwrelease) during some of year Logsize Log of number of USGS blocks that contain the site

Source: Adapted from Kenneth E. Train, "Recreation Demand Models with Taste Differences over People," Land Economics, Vol. 74, No. 2. O 1998. Reprinted by permission of the University of Wisconsin Press.

Distribution of coefficient Parameter Proportion Elasticity Variable distribution 10 percentile Median 90 percentile positive (at median coefficient) Trout stock Lognormal 0.015 0.056 0.207" 1.O 0.098 Standard error 0.014 0.034 0.070 Trip cost Lognormal -0.253" -0.091'1' -0.032" 0.0 -7.945 Standard error 0.030 0.006 0.004 Access Normal -3.369" -0.950" 1.470" 0.31 -0.161 Standard error 0.715 0.361 0.392 Aesthetics Lognormal 0.152* 0.452" 1.342* 1.O 0.616 Standard error 0.060 0.103 0.159 Campgrounds Normal -2.005" 0.116 2.237" 0.53 0.022 Standard error 0.693 0.323 0.591

Major Normal - 1.795" 1.018* 3.831'" 0.68 0.56 Standard error 0.401 0.289 0.642 Restricted Normal -1.651* -0.499" 0.653* 0.29 -0.166 Standard error 0.305 0.131 0.171 Logsize Fixed 0.9835" 0.9835" 0.9835* 1.O 0.967 Standard error 0.108 0.108 0.108

Source: Adapted froin Kenneth E. Train, "Recreation Demand Models with Taste Differences over People," Land Econotnics, Vol. 74, No. 2. O 1998. Reprinted by permission of the University of Wisconsin Press. * Significant at the 1-percent level.

D. Dynamic Models study RUM theory as an intra-consumer, as well as an inter-consumer, model of taste variation, A major opening up of the study of economic providing a link to psychological models of choice behavior occurs when one turns to data decision-making. Analysis of the dynamics of on repeated choices by the same individuals, discrete choice has been pioneered by James and the dynamics of discrete choice. It is in this Heckman (1981a, b), who recognized the criti- panel framework that the operation of experi- cal roles of initial values and recursive structure ence on the evolution of perceptions and tastes, in well-specified dynamic models and is respon- postulated in Figure 2, can be examined empir- sible for the fundamental development of ically. Repeated decisions also allow one to appropriate econometric methods. Dynamic 362 THE AMERICAN ECONOMIC REVIEW JUNE 2001 models have important applications to issues of minants of attitudes; they also influence the labor supply and job status, and also to a variety perceptions that feed into the choice process of subjects ranging from the timing of purchases (see the light arrows in Figure 2). In these of new goods to life-cycle decisions like theories, the economists' calculus of utility as- retirement. sessment and maximization is reduced to one of An important element in analysis of the dynam- many factors in the decision-making environ- ics of discrete choice has been the integration of ment, with an influence that is often overridden expectations into choice decisions, through dy- by context effects, emotion, and errors in per- namic optimization (John Rust, 1994) and through ception and judgment (see Ola Svenson, 1979; interactions between agents (Jean-Jacques Laffont Tommy Garling, 1992; George Lowenstein, and Quang Vuong, 1996). The last topic is related 1996). Experimental evidence and self-reported to a more general issue in discrete choice analysis. decision protocols support the view that heuris- In many microeconomic data sets, the explanatory tic rules are the proximate drivers of most hu- variables behind an economic choice can be man behavior. The psychologist Drazen Prelec treated as predetermined, because the feedback (1991) distinguishes this view of decision- from decisions of individual consumers to market- making from utility-maximization models by level economic variables is weak. However, in the cognitive processes involved: dynamic models where current unobservables are not necessarily independent of past history, or in Decision analysis, which codifies the ra- "thin" markets where there is strategic interaction tional model, views choice as a funda- between agents, feedbacks become strong enough mentally technical problem of choosing so that it is necessary to deal with endogeneity in the course of action that maximizes a uni- explanatory variables. dimensional criterion, utility. The primary mental activity is the reduction of multi- E. Discrete/Continuous Choice ple attributes or dimensions to a single one, through specification of value trade- offs. For rule-governed action, the funda- Discrete and continuous components of mental decision problem is the quasi-legal economic decisions are fully integrated in eco- one of constructing a satisfying interpre- nomic choice theory, through common prefer- tation of the choice situation. The primary ences and time and budget constraints. mental activity involved in this process is However, this integration has rarely been car- the exploration of analogies and distinc- ried through in empirical study of consumer tions between the current situation and behavior. Jeffrey Dubin and McFadden (1984) other canonical choice situations in which develop a consistent model of discrete and con- a single rule or principle unambiguously tinuous decisions for application to choice and applies. ... The purpose of rules must be derived from some weakness of our nat- use of consumer products, but the cost of com- ural cost-benefit accounting system, and putational tractability is a highly restrictive pa- one might expect to find rules proliferat- rameterization. Further development of this ing in exactly those choice domains topic, perhaps using semiparametric estimation where a natural utilitarianism does not to relax model restrictions, is needed. produce satisfactory results.

111. The Psychology of Choice Behavior Human behavior may be governed by rules, but it is possible that these rules simply encode In psychological theories of the choice pro- preferences. The evolutionary and behavioral cess, the individual is less organized, and more arguments used to explain the reinforcement of adaptive and imitative, than in the economists' self-protective rules systems also suggest that standard model. Psychological descriptions of selection will favor rules systems that consis- decision-making are both colorful and intuitive. tently advance self-interest. Many psychologists Attitudes play a major role in determining how argue that behavior is far too sensitive to con- consumers define the decision-making task. In text and affect to be usefully related to stable the words of Daniel Kahneman (1997), "Econ- preferences. However, if there are underlying omists have preferences; psychologists have preferences, then even if the link from prefer- attitudes." Affect and motivation are key deter- ences to rules is quite noisy it may be possible VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES 363 to recover these preferences and use them to you are reminded of this, St. Emilion looks correctly evaluate economic policies, at least as closer. Could this illusion affect behavior? It an approximation that is good enough for gov- may be significant that Figure 3 was adapted ernment policy work. from a brochure published by the commune of The existence of underlying preferences is St. Emilion. And in fact St. Emilion is more a vital scientific question for economists. If crowded than Margaux, perhaps as a result of the answer is affirmative, then the evidence on enophiles' illusions. However, I doubt that this decision-making from cognitive psychology is due to mass misreading of maps by travelers implies only that economists must look through to Bordeaux. We learn to be suspicious of our the smoke screen of rules to discern the deeper perceptions. We may see things cockeyed, but preferences that are needed to value economic we adopt conservative behavioral strategies, policies. This is a difficult task, but not an such as measuring map distances, that prevent impossible one. If the answer is negative, then us from deviating too far from our self-interest. economists need to seek a foundation for policy In light of this example, how sliould a scien- analysis than does not require that the concept tist go about predicting travel decisions of map of "greatest good for the greatest number" be readers? One place to start is the library of meaningful. I am guardedly optimistic that the optical illusions. These certainly help to reveal question has an affirmative answer. The first the cognitive processes involved in vision. reason is that many behavioral deviations from However, it is very difficult to synthesize this the economists' standard model are explained library into a forecasting system that is broadly by perceptual illusions and information process- predictive. Another starting point is a crude ing errors, rather than a more fundamental "you see what a camera sees" model of vision. breakdown in the definition of self-interest. The We know from the very existence of optical second is that many of the rules we do use are illusions that this model is not universally true. essentially defensive, protecting us from bad Despite this, the crude model is broadly predic- choices. To illustrate, consider the simplified tive, and even more so if it is relaxed to accom- road map of the wine-producing region around modate some systematic illusions. I consider Bordeaux shown in Figure 3. this a good analogy for economists deciding Bordeaux appears to be closer to St. Emilion how to predict economic choice behavior. Until than to Margaux. However, you will immedi- the day comes when brain science understands ately recognize that this is a version of the how the cognitive mechanisms operate in Fig- classical Muller-Lyer optical illusion in which ure 2 for a broad spectrum of economic deci- the distances are actually the same. Even after sions, I suspect that the standard model, enhanced to account for the most systematic perceptual illusions, will prove to be the best platform for evaluating most economic policies.

A. Cognitive Zllusioizs

The preceding discussion has treated the psy- chological view of decision-making as a theoret- ical alternative to the standard model, but there is now also substantial evidence that in a laboratory setting individuals will sometimes make decisions that deviate strikingly and systematically from the predictions of the standard model. The exper- imental results of Kahneman and Amos Tver- sky (e.g., Tversky and Kahneman, 1974, 1981; Kahneman and Tversky, 1979, 1984) have been particularly influential in forcing economists to ST. EMILION rethink the standard model. Table 4, from FIGURE3. ROADSIN THE WINE-PRODUCINGREGION McFadden (1999a), lists some of the (over- NEAR BORDEAUX lapping) cognitive phenomena identified by 364 THE AMERICAN ECONOMIC REVIEW JUNE 2001

cognitive psychologists and behavioral econo- sponse bias that if unrecognized might seri- mists that appear to influence behavior. ously distort economic policy analysis. One important cognitive phenomenon is an- clzoriizg, in which responses are pulled toward C. Bias in Stated Willingness-to-Pay numerical prompts, even when they are unin- formative (Tversky and Kahneman, 1974). A The second study, by Donald Green et al. psychological explanation for anchoring is that (1998), asks subjects recruited from visitors to a a prompt creates in the subject's mind, at least science museum to state their willingness-to- temporarily, the possibility that the uncertain pay to save off-shore seabirds from small oil quantity could be either above or below the spills. Subjects were assigned randomly to con- prompt. This could result from classical psycho- trol and treatment groups. Both groups were physical discrimination errors, or from a cogni- given the following preamble: tive process in which the subject treats the question as a problem-solving task and seeks an There is a population of several million appropriate framework for "constructing" a cor- seabirds living off the Pacific coast, from rect solution. Evidence suggests that individuals San Diego to Seattle. The birds spend are poor natural statisticians, placing too much most of their time many miles away from weight on readily available information and ex- shore and few people see them. It is esti- emplars, and too little on background informa- mated that small oil spills kill more than tion that is more difficult to retrieve. Education 50,000 seabirds per year, far from shore. Scientists have discussed methods to pre- trains individuals to use problem-solving proto- vent seabird deaths from oil, but the so- cols in which responses to questions are based lutions are expensive and extra funds will not only on substantive knowledge, but also on be required to implement them. It is usu- contextual cues as to what a correct response ally not possible to identify the tankers might be. Consequently, it is no surprise if that cause small spills and to force the subjects apply these protocols and use numeri- companies to pay. Until this situation cal prompts in forming responses. changes, public money would have to be spent each year to save the birds. We are B. Bias in Reported Consumption interested in the value your household would place on saving about 50,000 sea- birds each year from the effects of off- I will describe two experiments that show an- shore oil spills. choring is at least a problem for measurement in economic surveys. The first, taken from Michael Hurd et al. (1998), is concerned with response bias The control group was then given this open- when subjects are asked to report on economic ended question: quantities they may not know with certainty. These authors conducted an experiment in the AHEAD panel, a large study of the elderly in the If you could be sure that 50,000 seabirds United States. Subjects were asked about their would be saved each year, what is the monthly consumption, using an unfolding brack- MOST your household would pay in extra ets format that asked for yeslno responses to a federal or state taxes per year to support series of numerical prompts. The pattern of an operation to save the seabirds? The prompts given to each subject was selected by operation will stop when ways are found to prevent oil spills, or to identify the experimental design. For the range of initial tankers that cause them and make their prompts used in the experiment, from $500 per owners pay for the operation. month to $5,000 per month, this led the implied $- per year. median consumption levels to vary from $895 per month to $1,455 per month; see Figure The treatment groups were given the referen- 4. More detailed information on the experi- dum question: mental results is given in Table 5. The distri- butions of responses for the different If you could be sure that 50,000 seabirds treatment groups show convincingly that the would be saved each year, would you agree anchoring phenomenon can introduce re- to pay ($5) in extra federal or state taxes VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES

TABLE4-COGMTIVE EFFECTSIN DECISION-MAKING

Effect Description Anchoring Responses are influenced by cues contained in the question Availability Responses rely too heavily on readily retrieved information, and too little on background information Context Previous questions and interviewer interaction color perception Framinglreference point Question format changes saliency of different aspects of the cognitive task Focal Quantitative information is stored andlor reported categorically Primacylrecency Initial and recently experienced events are the most salient Projection Responses are consonant with the self-image the subject wishes to project Prospect The likelihoods of low probability events are misjudged, and treated either as too likely or as zero Regression Causality and permanence are attached to past fluctuations, and regression to the mean is underestimated Representativeness High conditional probabilities induce overestimates of unconditional probabilities Rule-driven Motivation and self-control induce strategic responses Saliency The most salient aspects of the question are overemphasized Status quo Current status and history are privileged Superstition Elaborate causal structures are attached to coincidences Temporal Temporally inconsistent time discounting

Starting Value = $2,000

Starting Value = $5,000

Starting Value = $500

1,000 x (Dollars)

Source: The above is from McFadden et al. (1998), "Consumption and Saving Balances of the Elderly," in Frontiers in tlze Econonzics of Aging, edited by David A. Wise and published by the University of Chicago Press. O 1998 by the National Bureau of Economic Research. All rights reserved.

per year to support an operation to save the This question was then followed up by an open- seabirds? The operation will stop when ended question: ways are found to prevent oil spills, or to identify the tankers that cause them and What is the MOST that you would be make their owners pay for the operation. willing to pay? Yes No. $- per year. 366 THE AMERICAN ECONOMIC REVIEW JUNE 2001

TABLE5-CONSUMPTION: SAMPLESIZES, MEDIANS, AND MEANS

Percentage Medlans Means Starting Number with of bracket gate Sample open-ended responses Standard Standad Standard Standard Treatment amount size response completed Nonpuametrich errorC pararnetucd en.01' Nonparametricf errorg ~ararnet~ic" errorL

1 2 3 4 5 6 7

2 and 5 3 and 7 1 and 6 OE fi~st (1, 2, 7) Fo~ced (3, 5, 6) Pooled (1, 2, 3. 5, 6. 7)

Open-ended lesponses 1,378 1,000 9 1,253 64 Ovetall 1.855 1,378 84.0 1,163 21 1,170 19 1,696 26 1,531 22

So~irce:The above is from McFadden et al. (1998),"Consumption and Saving Balances of the Elderly," in Frontiers in the Econonlics of Aging, edited by David A. Wise and published by the University of Chicago Press. O 1998 by the National Bureau of Economic Research. All rights reserved. " Subjects were first asked for an open-ended response, with unfolding brackets if there was no response to the open-ended question. Exponential of linearly interpolated CCDF of log consumption, with the CCDF estimated using a "saturated" multinomial nlodel for all respondents. 'The standard error is estimated by rn(a - b)/2p~"~,where (b,a)is the log consun~ptionbracket containing the estimator, in is the median, andp is the bracket probability. This estimator assumes that log consumption is uniformly distributed within the bracket containing the median. dExponential of p, where p is the mean of a lognormal distribution fitted by MLE to bracket frequencies of log consumption. "The standard error is estimated by mo(~r12~)"~,where 111 is the estimated median and o is the estimated standard deviation of log consumption. Y,,O,(L~~+ b,)l2,where (b,, a,) is a log consumption bracket and p, is the bracket probability. Standard enor is estimated by (2;(112 (a, + b,) - ~n)~p,)"21~L'2, where (b,,a,)is a log consumption bracket, pi is the bracket probability, and nz is the median. Exponential of p + 02/2, where p and a are estimates of the mean and standard deviation of log consumption. ' The standard error is estimated by po(1 + ~/2)"21~1'2,where o is the estimated standard deviation of log consumption.

The numerical prompt of $5 in the referendum Comparing open-ended and follow-up re- question was varied across several levels set by sponses, the median WTP in the control group experimental design, with the treatments se- is $25, and the median WTP in the treatment lected to coi~espondto specified quantiles of the group varied from $10 when the prompt was $5 control group's distribution of responses. If sub- to $50 when the prompt was $400. The refer- jects conform to the economists' standard endum responses also show an anchoring effect, model, their preferences are innate and will not with higher pluralities for "yes" at higher be anchored to the numerical prompts contained prompts than in the control group. These pro- in the referendum questions. In fact, the re- duce a nonparametric estimate of $167 for mean sponse patterns suggest the prompt creates an WTP in the treatment group, compared with a "anchor" for the construction of preferences. mean of $64 in the control group, again statis- Figure 5 shows the differences in response fre- tically significant. Put another way, the effect of quencies for the control and treatment groups; a one-dollar increase in the prompt is to increase details are given in Table 6. The anchoring mean response by 28 cents. This experiment effects are large, and statistically significant. also showed that anchoring in response to the VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES

Treatment Group

10 100 1000 x (Dollars)

FIGURE5. WTP FOR SEABIRDS Source: Reprinted from Resource and Energy Economics, Vol. 20, Donald Green, Karen Jacowitz, Daniel Kahneman, and Daniel McFadden, "Referendum Contingent Valuation, Anchoring, and Willingness to Pay for Public Goods," pp. 85-116, copyright 1998, with permission from Elsevier Science.

TABLE6-W~LL~NGNESS-TO-PAYTO SAVE 50,000 OFF-SHORESEAB~RDS PER YEAR (NUMBERSARE IN PERCENTAGES)

Starting-point bid Distribution Open-ended $5 $25 $60 $150 $400 12.2 8.5 67.4 25.5 12.2 53.2 8.2 8.5 0.0 2.1 0.0 2.1 Sample size 49 47 P(open-ended response > bid) (percent) 80.2 52.9 (Standard error) (percent) (5.7) (7.1) P(anchored response > bid) (percent) 87.8 66.0 (Standard error) (percent) (4.7) (6.9) Median response $10.00 $25.00 (Standard error) ($2.33) ($1.16) Mean response" $20.30 $45.43 (Standard error) ($3.64) ($12.61) Coefficient Standard error Marginal effect of starting point bid 0.284 0.32 Nonparametric referendum mean" $167.33 $76.90 Parametric referendum mean $265.59 $138.96

Source: Reprinted from Resource and Energy Economics, Vol. 20, [authors] Donald Green, Karen Jacowitz, Daniel Kahneman, and Daniel McFadden, "Referendum Contingent Valuation, Anchoring, and Willingness to Pay for Public Goods," pp. 85-116, copyright 1998, with permission from Elsevier Science. " One observation of $2,000,000 is excluded from the calculation of the open-ended mean. If the open-ended mean WTP of $64.25 is representative of all California adults, then the total state WTP for protecting 50,000 seabirds is $1.49 bil., or $29,800 per bird. The upper bound to the distribution is assumed to equal the largest anchored response, $1,000. The reported standard error is the RMSE at the maximum possible bias, given the upper bound to the distribution. 368 THE AMERICAN ECONOMIC REVIEW JUNE 2001

WTP question paralleled anchoring in re- standard statistical software packages now per- sponses to objective estimation questions, such mit more or less mindless use of these models. as the height of the tallest redwood tree in There is increasing use of nonparametric esti- California. mators, bootstrap methods to refine asymptotic The Green et al. experiment was hypotheti- approximations, Generalized Method of Mo- cal, and subjects were aware that their responses ments procedures for robustness, and simulation would have no direct monetary consequences. methods to overcome problems that are intrac- A natural question for economists to ask is table using conventional computation. There are whether such deviations from the standard a few statistical developments that are specific model continue to appear in market choices to or particularly applicable to discrete choice where real decisions involve real money. The analysis. I will summarize a few of these devel- marketing of consumer goods suggests an affir- opments, concentrating on those in which I have mative answer. Businessmen are taught that had some hand. when selling a targeted product, they can en- hance its appeal by positioning a clearly infe- A. Choice-Based Sampling rior product at nearly the same price (Itamar Simonson and Tversky, 1992). Thus, awareness A choice-based sample is one obtained by illusions appear to be present in real markets, stratification on the basis of response behavior and systematic enough to be exploited by sellers. whose explanation is the target of study. Obser- Economists investigating consumer behavior vations on response and explanatory variables can learn a great deal from careful study of (covariates) are collected within each stratum. market research findings and marketing prac- These are then used for statistical inference on tice. Ultimately, behavioral economists need to the conditional distribution of the response, move beyond stylized descriptions of choice given the covariates. For example, a study of behavior and become involved in market re- occupational choice may draw a sample strati- search experiments that explore directly the na- fied by occupation, so the first stratum is a ture of economic choice processes. There may sample of engineers, the second stratum is a be a further methodological lesson from market sample of educators, and so forth. Data are research. Discovery and exploitation of cogni- collected on covariates such as gender and uti- tive illusions in purchase behavior seems to lization of training subsidies. The observations coexist comfortably with the use of RUM- might then be used to infer the impact of train- consistent discrete response models, adapted to ing subsidies on occupational choice. Choice- use data on perceptions, as a major tool for based samples may be unintentional, the result predicting buyer behavior. of self-selection or stratification in general pur- pose surveys, or may be deliberate, designed to IV. Statistical Methods reduce sampling costs or improve the informa- tiveness or accuracy of responses. The microeconometric analysis of choice be- Statistical methods developed for random havior requires statistical methods for parametric samples will often be inconsistent or inefficient and nonparametric estimation, and diagnostic when applied to choice-based samples. The tools to detect errors in specification and test hy- essential problem is that the analysis is at- potheses. Applications of choice models also re- tempting to infer properties of the conditional quire systems for producing disaggregate and distribution of choices given covariates, using aggregate forecasts and policy scenarios that track observations that are drawn from conditional statistical accuracy. These requirements are ge- distribution of covariates given choices. The neric to applied statistics, but are made more dif- solution to the inference problem is to incor- ficult in this area because natural models derived porate the mapping between the conditional from RUM foundations are usually nonlinear, and distributions in the analysis, either by re- often not particularly tractable. weighting observations so that they behave as Applied RUM analysis, based on the MNL if they were drawn from a random sample, or model and its relatives, has generally relied on by reweighting the probability model for a ran- maximum-likelihood methods and their large dom sample so that it is consistent with the sample properties, and routines available in empirical sampling process. The statistical is- VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES 369

sues in analyzing choice-based samples were Random sampling draws from the table in treated in a seminal paper by Charles Manski and proportion to the cell probabilities. Exogenous Steve Lerman (1977),with further results by Steve stratification draws rows, with probabilities that Cosslett (1981) and Manski and McFadden may differ from the population marginal prob- (1981). The choice-based sampling problem is abilities p(z),and then within a row draws col- closely related to the problem of analysis of self- umns in proportion to their population selected samples. The seminal treatment of selec- conditional probabilities ~(~lz).A simple choice- tion problems was given by Heckrnan (1974, based sampling design draws columns, with prob- 1979), with further contributions by Steve Gold- abilities that may differ from the population feld and Richard Quandt (1973),G. S. Madalla marginal probabilities q(y),then within a column and Forrest Nelson (1975),Jeny Hausman and draws rows in proportion to their conditional David Wise (1976),and Lung-Fei Lee and Robert probabilities ~(zly)= ~(~z)~(z)l~(y). Porter (1984). Papers by David Hsieh et al. More complex endogenous sampling designs (1985),Tony Lancaster and Guido Imbens (1990), are also possible. A general framework that Norman Breslow (1996), and Imbens and Lan- permits a unified analysis of many sampling caster (1996) sharpen the statistical analysis of schemes characterizes the sampling protocol for endogenous samples. Extensions of the basic a stratum s in terms of a probability R(z, y, s) framework for inference in choice-based samples that a member of the population in cell (y, z) have been made for a variety of problems. Imbens will qualify for the stratum. The joint probabil- (1992) has provided methods for combining ity that a member of the population is in cell (y, choice-based survey data with aggregate statistics. z) and qualifies for stratum s is R(z, y, s)~(y lz, McFadden (2001) has studied the problem of P,)p(z) The proportion of the population qual- analysis of endogenously recruited panels. ifying into the stratum, or qual$cation factor, is Table 7 depicts, schematically, the popula- r(s) = C, C,, R(z, y, s)~(ylz)~(z),and the tion probability law for a choice y and a conditional probability of (y,z) given qualifica- vector of covariates z.~The joint probability tion is R(z, y, s)~(ylz)~(z)lr(s).The term of a (y, z) cell can be written as the product R(z, y, s) is sometimes called a propensity of the conditional probability of y given z score. When a fraction of the sample f(s) is times the marginal probability of z, p( y, z) = drawn from stratum s, g(y, z) = C, R(z, y, ~(ylz)~(z).The row sums give the marginal s)P(~Z)~(Z)f(s)lr(s)is the probability for an probability p(z) of z, and the column sums observation from the pooled sample, and give the marginal probability q(y) = C, g(ylz) = P(Ylz)(C, R(z, Y, s)f(s)lr(s))l[C,, P(yz)p(z)of y. Bayes law gives the condi- P( y 'lz)(C, R(z, y ', s)f(s)lr(s))] is the condi- tional probability of z given y, Q(z1y) = tional probability of y given z in this pooled ~(yl~)~(z)l~(y).The target of statistical sample. Note that this conditional probability analysis is the conditional probability ~(ylz), depends on the marginal probability of z only sometimes termed the response probability. through the qualification factors. In applications, P( 1 z) is usually assumed to When the sampling protocol is exogenous be invariant under treatments that alter the [i.e., R(z, y, s) does not depend on y], the marginal probability of z; then knowledge of conditional probability g(ylz) for the pooled ~(ylz)permits the analysis to forecast y in sample equals the population conditional prob- new populations or under policy treatments ability P(y1z). Consequently, any statistical in- where the z distribution is changed.' ference procedure designed to reveal features of the conditional probability ~(ylz)in random samples will apply to an exogenously stratified sample. In particular, if P( lz) is in a paramet- This exposition treats y and z as discrete, but the discussion applies with minor modifications to the case ric family, then maximization of the random where y and/or some components of z are continuous. A conditional probability with this invariance property is sometimes said to define a causal model. It is true that a causal structure will imply this invariance property, but it is for the invariance property, while detection of true causal also possible for the invariance property to hold, making structures is beyond the reach of statistics. For these rea- forecasting possible, without the presence of a deeper causal sons, it is best to avoid the language of causality and structure. Further, there are straightforward statistical tests concentrate instead on invariance properties. 370 THE AMERICAN ECONOMIC REVIEW JUNE 2001

TABLE7-POPULAT~ON CELL PROBABILITIES

Y I Y 2 ... YJ Sum

z1 PiY llzl)~(zl) P~Y~IzI)~~(zI) ... pi~JlzI)l)izl) OizI)

z2 PiY Ilz2)Piz2) PiY2z2)1)iz2) ... pi~Jlz2)1~iz2) P~ZZ)

z~ P(~1zK)P(zK) p~~2~~K)~)~~K)... P(~JI~K)~(~K) P~K)

Sum ~(YI) q(~2) ... ~(.vJ) 1

sample likelihood function in an exogenously from the population with response y = 2, with stratified sample will have the same pEoperties R(z, y, 2) equal to one if y = 2, and zero as in a random sample.6 However, in an endog- otherwise. This is called an enriched sample. enous sample in which the qualification proba- The qualification factors are 41) = 1 and r(2) = bility R(z, y, s) does not depend on y, the q(2). If q(2) is known, a consistent estimate of the conditional probability g(ylz) for the pooled slope parameter fl in the model can be obtained by sample is not equal to ~(ylz).Consequently, the CML method with g(1lz) = @(-a - zp)f(l)l statistical inference assuming that the data [@(-a - zp)f(l) + @(a + zfi)m1) + f(2)l generation process is described by ~(ylz)is q(2))l. By contrast, likelihood maximization using generally statistically inconsistent. Also, the ~(ylz)is not consistent for p. distribution of covariates in an endogenous An important simplification of the CML sample will differ from their population distri- method occurs for the MNL model. Suppose that bution, with g(z) = p(z)C, (f(s)lr(s)) C, R(z, the vector of covasiates is partitioned into compo- y, s) P( y 1 z), and a corresponding correction nents z = (v, x) with v discrete, and P(~~V,x) = factor must be applied to the sample empirical exp(a, + y,, + xP,)IC)> exp(a,, + Y,,,?, + xp,o. In distribution of z to estimate population quanti- this model, the p, are slope coefficients for the ties consistently. covasiates x, the ayare response-specific effects, Manski and McFadden (1981) propose that and the y,, are interactions of response-specific statistical inference when ~(ylz)is parametric and v-specific effects. Suppose that the qualifica- be based on the conditional likelihood g(ylz), tion probability R(v, x, y, s) does not depend on x. and term this the conditional maximum likeli- The conditional probability g(ylz) is again of hood (CML) method. When the qualification multinomial logit form, with the same p, param- factors r(s) and sample frequencies f(s) are eters but with the remaining parameters shifted; known or can be estimated consistently from e.g., g(ylv, x) = exp(a? + y;, + xp,.)l external samples, and the forms of ~(ylz)and 2,. exp(a;, + y.;,, + xp ,),'with the transformed R(z, y, s) allow identification of any unknown parameters satisfyind + Y;U = ar ,+ K,,, + parameters in R(z, y, s), this approach is con- log(& R(v, y, s)f(s)lr(s)). Consistent estimation of sistent. In general, the probability g(ylz) is not this model requires the inclusion of all the alter- in the same parametric family as P(y1z). To native specific effects and interactions that are illustrate, suppose a population has a binomial modified by sampling factors. How- probit choice probability, ~(21~)= @(a + ever, if these variables are included, then the zp), and ~(1lz) = @(- a - zp). Suppose the slope parameters p), are estimated consistently sample consists of a randomly sampled stratum without further adjustments for endogenous 1 with R(z, y, 1) = 1, plus a stratum 2 drawn sampling.7

Some statistical procedures use propensity score If the raising factors are estimated rather than known, weights to remove correlation of treatment variables and there is an additional contribution to the asymptotic covari- covariates induced by exogenous self-selection. ance matrix (Hsieh et al., 1985). VOL. 91 NO. 3 McFADDEN: EC 'ONOMIC CHOICES 371

B. Computation and Simulation 1984; David Hendry, 1984; Christian Gouri- eroux and Alain Monfort, 1996). Consider- From an era where estimation of a single able room for improvement in simulation multinomial logit model was a major computa- methods remains. In particular, some of the tional task, we have progressed to the point statistical methods for dealing with measure- where simple multinomial logits are virtually ment error and outliers in real data are also instantaneous, even for large numbers of alter- potentially useful for processing simulated natives and observations. This is nearly true for data. nested multinomial logit models, or logit models A model where simulation methods are usu- containing other nonlinear elements, via general ally needed, and relatively easy to apply, is the purpose maximum-likelihood programs, although MMNL model (4). Under the name kernel logit, acheving and verifying convergence in such it has been employed by McFadden (1989); problems remains an art. However, the evaluation Dennis Bolduc (1992); David Brownstone and of choice probabilities that cannot be expressed in Train (1999); and K. Srinivasian and H. closed form, but require numerical integration of Mahmassani (2000) as a computational approx- moderately high dimension, remains a computa- imation to multinomial probit or as a general tionally hard problem. For example, the multino- flexible RUM approximation. Because the mial probit model with an unrestricted covariance MNL model itself is smooth in its parameters a, structure continues to resist conventional compu- the following procedure gives positive, unbi- tation except for special cases. ased, smooth simulators of the MMNL Use of simulation methods has provided the probabilities, and smooth simulators of their most traction in obtaining practical representa- derivatives: Suppose a is given by a smooth tions and estimates for these computationally parametric inverse mapping a(€, 8), where 8 hard models. A simulated sample drawn from a parameterizes the distribution of a and E is trial data generation process (DGP) is an analog uniformly distributed in a hypercube. This of a real sample drawn from the true DGP. If the works easily for cases where the a are multi- simulation procedure is designed so that the variate normal, or transformations of multivar- simulated sample does not "chatter" as one iate normals (e.g., lognormal, truncated varies the trial parameters, then one can esti- normal), and with somewhat more difficulty for mate the true DGP by making the simulated other common distributions. The simulation and real samples congruent. McFadden procedure is then to draw a simulated sample of (1989) develops and formalizes this approach E'S, of size R, either at random or using some to inference, and generalizes simulators for patterned random numbers such as Halton se- the multinomial probit model first introduced quences, fix this sequence for all subsequent by Manski and Lerman (1981). Research in analysis, and treat the approximation P,(i) = the past decade has expanded the library of ER exp(Z(a - c,, w,ti, x,, $1 ' a(€,, 8)>/Zj simulation methods, including the use of exp(Z(a - c,, x,, s) . a(€,, O)), where ER Gibbs, Metropolis-Hastings, and other Monte denotes an empirical expectation with respect to Carlo Markov Chain samplers, use of pseudo- the simulation sample, as if it were exact. A random and patterned random numbers such modest rate requirement on R, that it rise more as Halton and Sobel sequences, and tools such rapidly than the square root of sample size, is as the Method of Simulated Moments, sufficient to guarantee that either maximum Method of Simulated Scores, and the simu- likelihood or method of moments applied using lated EM algorithm (Vassilis Hajivassiliou this formula will contain a negligible simulation and Paul Ruud, 1994; McFadden and Ruud, error in sufficiently large samples. To avoid 1994; Hajivassiliou et al. 1996; Hajivassiliou misleading estimates of precision when sample and McFadden, 1998; Train, 1999; Bhat, sizes and R are moderate, one should use the 2000). These methods have made it feasible sandwich formula for the covariance matrix in to work with quite flexible models, such as possibly misspecified models (McFadden and multinomial probit and mixed multinomial Train, 2000). In applications where the inverse logit models. Statistical simulation is also a transformation a(&, 0) is not tractable, one can powerful tool for model comparison and pol- instead use importance sampling methods or a icy analysis (Thomas Cowing and McFadden, Metropolis-Hastings sampler. 372 THE AMERICAN ECONOMIC REVIEW JUNE 2001

C. Specijication Testing: IIA Tests Suppose A is a specified subset of alternatives. Create a new variable zi that is zero for i @ The MNL model, is a powerful tool for anal- A, and for i E A equals log(PA(i)) - ysis of economic choice behavior when its IIA CjEA PAU) log(P,(j)), where P,G) is calculated property is satisfied by an application, since it is from the basic model. A numerically equivalent easily estimated, allows drastic reduction of form is obtained by replacing log(P,(j)) by V, = data collection and computation by samp- xiP. Estimate an expanded MNL model that con- ling subsets of alternatives (McFadden, 1981; t&ns the basic model variables plus one or more of Atherton et al., 1990), and gives an easy for- the new variables zi constructed for different A. mula for forecasting demand for new alterna- The A's can be disjoint, overlapping, andlor tives. On the other hand, as the "red bus, blue nested. Then carrv out a Likelihood Ratio test for bus" example illustrates, the model can produce significance of tie z's, with degrees of freedom seriously misleading forecasts if IIA fails. For equal to the number of added variables after elim- this reason, there was an early interest in devel- inating any that are linearly dependent. If there is oping specification tests that could be used to a single z, then the test can use the t-statistic for the detect failures of IIA. The first proposed test significance of this variable. This test is asymptot- (McFadden et al., 1978; Hausman and McFad- ically equivalent to a score or Lagrange Multiplier den, 1984) required estimating the MNL model test of the basic MM, model against a nested twice, once on a full set of alternatives C, and MNL model in which consume~sdiscriminate second on a specified subset of alternatives A, more sharply between alternatives within A than using the subsample with choices from this sub- they do between alternatives that are not both in A. set. If IIA holds, the two estimates should not be One minus the coefficient of a z variable can be statistically different. If IIA fails and A corre- interpreted as a preliminary estimate of the inclu- sponds to a nest of similar alternatives, then sive value coefficient for the nest A. there will be sharper discrimination within the The test above for a single set A is asymp- subset A, so that the estimates from the second totically equivalent to a one-degree-of-freedom setup will be larger in magnitude than the esti- Hausman-McFadden test focused in the direction mates from the full set of alternatives. Let PA determined by the parameters P; conversely, the denote the estimates obtained from the second test above with the variable Vi replaced by the setup, and a, denote their estimated covariance vector xj for j E A is asymptoti~all~equivalent to matrix. Let p, denote the estimates of the same the original Hausman-McFadden test for A. One parameters obtained from the full choice set, may get a rejection of the null hypothesis that IIA and a, denote their estimated covariance ma- holds either if IIA is in fact false, or if there is tri~.~Hausman and McFadden showed that some other problem with the model specification, the quadratic form (0, - PA)'(% - a,)-' such as omitted variables or a failure of the logit (PC - PA) has an asymptotic chi-square distri- form due to asymmetry or fat tails in the distur- bution when IIA is true. In calculating this test, bances. Rejection of the test will often occur when one must be careful to restrict the comparison of IIA is false even if the set A does not corres~ond parameters, dropping components as necessary, to the true nesting pattern. However, the test will to get a nonsingular array 4 - a,. When this typically have greatest power when A is a nest for is done, the degrees of freedom of the chi- which an IIA failure occurs. square test equals the rank of % - n,. The simple form of the covariance matrix for the D. Specijication Testing: Mixing parameter difference arises because PC is the in MNL Models efficient estimator for the problem. Another test which is particularly easy to com- In light of the theoretical result that any well- pute was proposed by McFadden (1987). Estimate behaved RUM model can be approximated by a the basic MNL model, using all the observations. MMNL model, satisfaction of the IIA property can be recast as a condition that there be no unobserved heterogeneity in the MNL model pa- Some parameters that can be estimated from the full choice set may not be identified in the second setup, in rameters. This suggests that a test for the validity which case PC refers to estimates of the subvector of of the IIA property, and specification test for the parameters that are identified in both setups. explanatory power to be added by introducing VOL. 91 NO. 3 McFADDEN: ECONOMIC CHOICES 373 mixing, can be constructed using a Lagrange Mul- sions along which RP data provides no vari- tiplier approach. The advantage of this method is ation. Large quantities of relevant data can be that the test procedure requires only estimation of collected at moderate cost. There will always base MNL models, so that simulation estimators be questions about how closely cognitive are not needed, and that it can test against a battery tasks in a hypothetical setting can match those of alternatives at the same time. To perform the in a real decision-making environment. Good test, first construct artificial variables zti = (xti - experimental technique can remove the most ~,~)~/2with xtC = EjECx,jPc( j) for selected com- obvious sources of incongruity, but calibra- ponents t of xi, where Pc(j) are the estimated base tion and validation using RP data is usually MNL probabilities. Then reestimate the model needed. with these added variables and use a Wald or Both marketing and economic policy appli- Likelihood Ratio test for the significance of the cations need an analytic framework for com- artificial variables. This test is asymptotically bining RP and SP data, and linking equivalent to a Lagrange Multiplier test of the experience and information to SP responses. hypothesis of no mixing against the alternative In 1984, Ben-Akiva and I specialized the of a MMNL model with mixing in the selected multiple-indicator, multiple-cause (MIMC) components t of the logit model. The degrees of model for this purpose, following the path freedom equals the number of artificial vari- diagram in Figure 2, and adding a hidden ables zti that are linearly independent of x. (latent) layer to handle mappings into discrete McFadden and Train (2000) also generalize the responses (Karl Joreskog and Dag Sorbom, preceding test so that an estimated MMNL 1979; McFadden, 1986; Train et al., 1987; model with some mixing components can be Takayuki Morikawa, 1989; Ben-Akiva and tested against the alternative that additional Morikawa, 1990). Applications have shown mixing components are needed. this to be useful framework for integrating marketing data into forecasting problems E. Market Research Data and Models (Morikawa et al., 1991; Brownstone and Train, 1999; David Hensher et al., 1999; Jor- An important interdisciplinary interaction has dan Louviere et al., 1999). developed between economic choice analysis and market research. The experimental methods V. Conclusions used in market research permit elucidation and measurement of the workings of the decision- Looking back at the development of discrete making process described in Figure 2. In par- choice analysis based on the RUM hypothesis, I ticular, it is possible to elicit stated perceptions, believe that it has been successful because it stated preferences, and attitude scales; we call emphasized empirical tractability and could ad- these stated preference (SP) data in contrast to dress a broad array of policy questions within a the revealedpreference (RP) data obtained from framework that allowed results to be linked observed choices. Most of these variables and back to the economic theory of consumer be- the methods used to measure them come from havior. Some possibilities for development of applied psychology. In particular, conjoint the approach have not yet been realized. The analysis, a method for eliciting stated prefer- RUM foundation for applied choice models has ences within a classical experimental design, been only lightly exploited. Models have gen- provides data that with proper consumer train- erally conformed to the few basic qualitative ing and allowance for cognitive illusions can constraints that RUM imposes, but have not add considerably to the ability to predict con- gone beyond this to explore the structure of sumer decisions. consumer preferences or the connections be- The use of experiments rather than field tween economic decisions along different di- surveys to collect data on consumer decisions mensions and in different areas. The potentially has several major advantages. The environ- important role of perceptions, ranging from ment of hypothetical choice can be precisely classical psychophysical perception of attributes, specified, with a design which allows straight- through psychological shaping of perceptions to forward identification of effects. Innovations reduce dissonance, to mental accounting for times in services can be studied, including dimen- and costs, remains largely unexplored in empirical 374 THE AMERICAN ECONOMIC REVIEW JUNE 2001 research on economic choice. Finally, the vealed Preferences and Stated Intentions." feedback from the empirical study of choice Transportation Research A, 1990, 24(6), pp. behavior to the economic theory of the con- 485-95. sumer has begun, through behavioral and ex- Bhat, C. R. "Accommodating Flexible Substitu- perimental economics, but is still in its tion Patterns in Multidimensional Choice adolescence. Modeling: Formulation and Application to What lies ahead? 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7 Estimation of Response Probabilities From Augmented Retrospective Observations David A. Hsieh; Charles F. Manski; Daniel McFadden Journal of the American Statistical Association, Vol. 80, No. 391. (Sep., 1985), pp. 651-662. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28198509%2980%3A391%3C651%3AEORPFA%3E2.0.CO%3B2-2

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A Method of Simulated Moments for Estimation of Discrete Response Models Without Numerical Integration Daniel McFadden Econometrica, Vol. 57, No. 5. (Sep., 1989), pp. 995-1026. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198909%2957%3A5%3C995%3AAMOSMF%3E2.0.CO%3B2-Z

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Rationality as Process and as Product of Thought Herbert A. Simon The American Economic Review, Vol. 68, No. 2, Papers and Proceedings of the Ninetieth Annual Meeting of the American Economic Association. (May, 1978), pp. 1-16. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197805%2968%3A2%3C1%3ARAPAAP%3E2.0.CO%3B2-4 NOTE: The reference numbering from the original has been maintained in this citation list. http://www.jstor.org

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Choice in Context: Tradeoff Contrast and Extremeness Aversion Itamar Simonson; Amos Tversky Journal of Marketing Research, Vol. 29, No. 3. (Aug., 1992), pp. 281-295. Stable URL: http://links.jstor.org/sici?sici=0022-2437%28199208%2929%3A3%3C281%3ACICTCA%3E2.0.CO%3B2-9

A Discrete Choice Model for Ordered Alternatives Kenneth A. Small Econometrica, Vol. 55, No. 2. (Mar., 1987), pp. 409-424. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28198703%2955%3A2%3C409%3AADCMFO%3E2.0.CO%3B2-I

Recreation Demand Models with Taste Differences over People Kenneth E. Train Land Economics, Vol. 74, No. 2. (May, 1998), pp. 230-239. Stable URL: http://links.jstor.org/sici?sici=0023-7639%28199805%2974%3A2%3C230%3ARDMWTD%3E2.0.CO%3B2-E

Consumer Attitudes and Voluntary Rate Schedules for Public Utilities Kenneth E. Train; Daniel L. McFadden; Andrew A. Goett The Review of Economics and Statistics, Vol. 69, No. 3. (Aug., 1987), pp. 383-391. Stable URL: http://links.jstor.org/sici?sici=0034-6535%28198708%2969%3A3%3C383%3ACAAVRS%3E2.0.CO%3B2-O

Judgment under Uncertainty: Heuristics and Biases Amos Tversky; Daniel Kahneman Science, New Series, Vol. 185, No. 4157. (Sep. 27, 1974), pp. 1124-1131. Stable URL: http://links.jstor.org/sici?sici=0036-8075%2819740927%293%3A185%3A4157%3C1124%3AJUUHAB%3E2.0.CO%3B2-M

The Framing of Decisions and the Psychology of Choice Amos Tversky; Daniel Kahneman Science, New Series, Vol. 211, No. 4481. (Jan. 30, 1981), pp. 453-458. Stable URL: http://links.jstor.org/sici?sici=0036-8075%2819810130%293%3A211%3A4481%3C453%3ATFODAT%3E2.0.CO%3B2-3

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Some Asymptotic Results for Learning in Single Hidden-Layer Feedforward Network Models Halbert White Journal of the American Statistical Association, Vol. 84, No. 408. (Dec., 1989), pp. 1003-1013. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28198912%2984%3A408%3C1003%3ASARFLI%3E2.0.CO%3B2-E

NOTE: The reference numbering from the original has been maintained in this citation list.