IntroductionMixed logit and probit models allow random coefficients whose distribution in the population is estimated. Consider, for example, the model in Chapter 6, of anglers' choice among fishing ...sites. The sites are differentiated on the basis of whether campgrounds are available at the site. Some anglers like having campgrounds at the fishing sites, since they can use the grounds for overnight stays. Other anglers dislike the crowds and noise that are associated with campgrounds and prefer fishing at more isolated spots. To capture these differences in tastes, a mixed logit model was specified that included random coefficients for the campground variable and other site attributes. The distribution of coefficients in the population was estimated. Figure 11.1 gives the estimated distribution of the campground coefficient. The distribution was specified to be normal. The mean was estimated as 0.116, and the standard deviation was estimated as 1.655. This distribution provides useful information about the population. For example, the estimates imply that 47 percent of the population dislike having campgrounds at their fishing sites, while the other 53 percent like having them.The question arises: where in the distribution of tastes does a particular angler lie? Is there a way to determine whether a given person tends to like or dislike having campgrounds at fishing sites?A person's choices reveal something about his tastes, which the researcher can, in principle, discover.
Logit Train, Kenneth E.
Discrete Choice Methods with Simulation,
01/2003
Book Chapter
Choice ProbabilitiesBy far the easiest and most widely used discrete choice model is logit. Its popularity is due to the fact that the formula for the choice probabilities takes a closed form and is ...readily interpretable. Originally, the logit formula was derived by Luce (1959) from assumptions about the characteristics of choice probabilities, namely the independence from irrelevant alternatives (IIA) property discussed in Section 3.3.2. Marschak (1960) showed that these axioms implied that the model is consistent with utility maximization. The relation of the logit formula to the distribution of unobserved utility (as opposed to the characteristics of choice probabilities) was developed by Marley, as cited by Luce and Suppes (1965), who showed that the extreme value distribution leads to the logit formula. McFadden (1974) completed the analysis by showing the converse: that the logit formula for the choice probabilities necessarily implies that unobserved utility is distributed extreme value. In his Nobel lecture, McFadden (2001) provides a fascinating history of the development of this path-breaking model.To derive the logit model, we use the general notation from Chapter 2 and add a specific distribution for unobserved utility. A decision maker, labeled n, faces J alternatives. The utility that the decision maker obtains from alternative j is decomposed into (1) a part labeled Vnj that is known by the researcher up to some parameters, and (2) an unknown part εnj that is treated by the researcher as random: Unj = Vnj + εnj ∀ j.
Bayesian Procedures Train, Kenneth E.
Discrete Choice Methods with Simulation,
01/2003
Book Chapter
IntroductionA powerful set of procedures for estimating discrete choice models has been developed within the Bayesian tradition. The breakthough concepts were introduced by Albert and Chib (1993) and ...McCulloch and Rossi (1994) in the context of probit, and by Allenby and Lenk (1994) and Allenby (1997) for mixed logits with normally distributed coefficients. These authors showed how the parameters of the model can be estimated without needing to calculate the choice probabilities. Their procedures provide an alternative to the classical estimation methods described in Chapter 10. Rossi et al. (1996), Allenby (1997), and Allenby and Rossi (1999) showed how the procedures can also be used to obtain information on individual-level parameters within a model with random taste variation. By this means, they provide a Bayesian analog to the classical procedures that we describe in Chapter 11. Variations of these procedures to accommodate other aspects of behavior have been numerous. For example, Arora et al. (1998) generalized the mixed logit procedure to take account of the quantity of purchases as well as brand choice in each purchase occasion. Bradlow and Fader (2001) showed how similar methods can be used to examine rankings data at an aggregate level rather than choice data at the individual level. Chib and Greenberg (1998) and Wang et al. (2001) developed methods for interrelated discrete responses. Chiang et al. (1999) examined situations where the choice set that the decision maker considers is unknown to the researcher.
MotivationSo far we have examined how to simulate choice probabilities but have not investigated the properties of the parameter estimators that are based on these simulated probabilities. In the ...applications we have presented, we simply inserted the simulated probabilities into the log-likelihood function and maximized this function, the same as if the probabilities were exact. This procedure seems intuitively reasonable. However, we have not actually shown, at least so far, that the resulting estimator has any desirable properties, such as consistency, asymptotic normality, or efficiency. We have also not explored the possibility that other forms of estimation might perhaps be preferable when simulation is used rather than exact probabilities.The purpose of this chapter is to examine various methods of estimation in the context of simulation. We derive the properties of these estimators and show the conditions under which each estimator is consistent and asymptotically equivalent to the estimator that would arise with exact values rather than simulation. These conditions provide guidance to the researcher on how the simulation needs to be performed to obtain desirable properties of the resultant estimator. The analysis also illuminates the advantages and limitations of each form of estimation, thereby facilitating the researcher's choice among methods.
Load forecasting models employed in the electric utility industry have become increasingly dependent upon information about the electricity used by individual appliances (i.e., end-uses). Currently, ...information on appliance usage is obtained from two fundamentally different sources: (1) engineering estimates and (2) conditional demand estimates. Bayesian analysis provides the means by which these two sources can be formally combined. Observed usage data (via the conditional demand approach) are used to modify a set of prior beliefs (the engineering approach), transforming them into a posterior distribution that describes appliance usage patterns and reflects the evidence provided by both approaches.
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38.
Endogeneity Train, Kenneth E.
Discrete Choice Methods with Simulation,
01/2001
Book Chapter
OverviewSo far we have assumed that the explanatory variables that enter a discrete choice model are independent of the unobserved factors. In many situations, however, the explanatory variables are ...endogenous, that is, are correlated or otherwise not independent of the unobserved factors. Examples include the following:Unobserved attributes of a product can affect its price.In modeling consumers' choices among products, it might be impossible to measure all of the relevant attributes of the various products. In the case of cars, for example, the researcher can obtain information about the fuel efficiency, length, width, horsepower, weight, and many other attributes of each car that is offered by manufacturers, but attributes such as comfort, beauty of the design, smoothness of the ride, handling in curves, expected resale value, and prestige cannot be measured directly. Yet the price of the product can be expected to reflect these unobserved (i.e., unmeasured) attributes. There are two reasons why price is affected. First, insofar as the unobserved attributes are costly for the manufacturer, the price of the product can be expected to reflect these costs. Second, insofar as the unobserved attributes affect demand for the product, a price that is determined by the interaction of demand and supply can be expected to reflect these differences in demand. The end result is that price is correlated with unobserved attributes, rather than being independent as we have assumed so far in this book. …
OverviewThis chapter describes the features that are common to all discrete choice models. We start by discussing the choice set, which is the set of options that are available to the decision maker. ...We then define choice probabilities and derive them from utility-maximizing behavior. The most prominent types of discrete choice models, namely logit, generalized extreme value (GEV), probit, and mixed logit, are introduced and compared within the context of this general derivation. Utility, as a constructed measure of well-being, has no natural level or scale. This fact has important implications for the specification and normalization of discrete choice models, which we explore. We then show how individual-level models are aggregated to obtain market-level predictions, and how the models are used for forecasting over time.The Choice SetDiscrete choice models describe decision makers' choices among alternatives. The decision makers can be people, households, firms, or any other decision-making unit, and the alternatives might represent competing products, courses of action, or any other options or items over which choices must be made. To fit within a discrete choice framework, the set of alternatives, called the choice set, needs to exhibit three characteristics. First, the alternatives must be mutually exclusive from the decision maker's perspective. Choosing one alternative necessarily implies not choosing any of the other alternatives. The decision maker chooses only one alternative from the choice set. Second, the choice set must be exhaustive, in that all possible alternatives are included.
IntroductionMixed logit and probit models allow random coefficients whose distribution in the population is estimated. Consider, for example, the model in Chapter 6, of anglers' choice among fishing ...sites. The sites are differentiated on the basis of whether campgrounds are available at the site. Some anglers like having campgrounds at the fishing sites, since they can use the grounds for overnight stays. Other anglers dislike the crowds and noise that are associated with campgrounds and prefer fishing at more isolated spots. To capture these differences in tastes, a mixed logit model was specified that included random coefficients for the campground variable and other site attributes. The distribution of coefficients in the population was estimated. Figure 11.1 gives the estimated distribution of the campground coefficient. The distribution was specified to be normal. The mean was estimated as 0.116, and the standard deviation was estimated as 1.655. This distribution provides useful information about the population. For example, the estimates imply that 47 percent of the population dislike having campgrounds at their fishing sites, while the other 53 percent like having them.The question arises: where in the distribution of tastes does a particular angler lie? Is there a way to determine whether a given person tends to like or dislike having campgrounds at fishing sites?
