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## Limited Dependent Variables and Discrete Choice Modelling

Limited dependent variables considers regression models where the dependent variable takes limited values like zero and one for binary choice mowedels, or a multinomial model where there is a few choices like modes of transportation, for example, bus, train, or a car. Binary choice examples in economics include a woman’s decision to participate in the labor force, or a worker’s decision to join a union. Other examples include whether a consumer defaults on a loan or a credit card, or whether they purchase a house or a car. This qualitative variable is recoded as one if the female participates in the labor force (or the consumer defaults on a loan) and zero if she does not participate (or the consumer does not default on the loan). Least squares using a binary choice model is inferior to logit or probit regressions. When the dependent variable is a fraction or proportion, inverse logit regressions are appropriate as well as fractional logit quasi-maximum likelihood. An example of the inverse logit regression is the effect of beer tax on reducing motor vehicle fatality rates from drunken driving. The fractional logit quasi-maximum likelihood is illustrated using an equation explaining the proportion of participants in a pension plan using firm data. The probit regression is illustrated with a fertility empirical example, showing that parental preferences for a mixed sibling-sex composition in developed countries has a significant and positive effect on the probability of having an additional child. Multinomial choice models where the number of choices is more than 2, like, bond ratings in Finance, may have a natural ordering. Another example is the response to an opinion survey which could vary from strongly agree to strongly disagree. Alternatively, this choice may not have a natural ordering like the choice of occupation or modes of transportation. The Censored regression model is motivated with estimating the expenditures on cars or estimating the amount of mortgage lending. In this case, the observations are censored because we observe the expenditures on a car (or the mortgage amount) only if the car is bought or the mortgage approved. In studying poverty, we exclude the rich from our sample. In this case, the sample is not random. Applying least squares to the truncated sample leads to biased and inconsistent results. This differs from censoring. In the latter case, no data is excluded. In fact, we observe the characteristics of all mortgage applicants even those that do not actually get their mortgage approved. Selection bias occurs when the sample is not randomly drawn. This is illustrated with a labor participating equation (the selection equation) and an earnings equation, where earnings are observed only if the worker participates in the labor force, otherwise it is zero. Extensions to panel data limited dependent variable models are also discussed and empirical examples given.

## Design of Discrete Choice Experiments

Discrete choice experiments are a popular stated preference tool in health economics and have been used to address policy questions, establish consumer preferences for health and healthcare, and value health states, among other applications. They are particularly useful when revealed preference data are not available. Most commonly in choice experiments respondents are presented with a situation in which a choice must be made and with a a set of possible options. The options are described by a number of attributes, each of which takes a particular level for each option. The set of possible options is called a “choice set,” and a set of choice sets comprises the choice experiment. The attributes and levels are chosen by the analyst to allow modeling of the underlying preferences of respondents. Respondents are assumed to make utility-maximizing decisions, and the goal of the choice experiment is to estimate how the attribute levels affect the utility of the individual. Utility is assumed to have a systematic component (related to the attributes and levels) and a random component (which may relate to unobserved determinants of utility, individual characteristics or random variation in choices), and an assumption must be made about the distribution of the random component. The structure of the set of choice sets, from the universe of possible choice sets represented by the attributes and levels, that is shown to respondents determines which models can be fitted to the observed choice data and how accurately the effect of the attribute levels can be estimated. Important structural issues include the number of options in each choice set and whether or not options in the same choice set have common attribute levels. Two broad approaches to constructing the set of choice sets that make up a DCE exist—theoretical and algorithmic—and no consensus exists about which approach consistently delivers better designs, although simulation studies and in-field comparisons of designs constructed by both approaches exist.