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Goodness of Fit Measure McFaddens pseudo-R 2 L 0 denotes the log likelihood function when only a constant is included in the model L denotes the unconstrained log likelihood function where P denotes the percentage of observations in the sample with y i =1.

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Merits of Pseudo R 2 Lies in the unit interval If the model provides no predictive power, then As the models fit improves, then and However, no interpretation to scale between 0 and 1

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Cross Tabulation of Hits and Misses Let Predicted Actual

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Problems with Success Table The prediction rule is arbitrary. –No linkage made to the costs of the individual errors made. It may be more costly to make an error to classify a yes as a no than to misclassify a no as a yes. –Some loss function would be more helpful in this case. There is no way to judge departures from a diagonal table.

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Extensions of Binary Choice Models Bivariate and Multivariate Choices Expanding the Number of Alternatives –Ordered Choices –Unordered Choice Logit (multinomial/conditional) Nested Logit Mixed Logit Multivariate Probit

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Bivariate Choice An obvious extension of binary choice models arises when two or more such choices are made Examples : –Labor literature: job market participation decisions – may be a combinations of single and multiple decisions –Technology adoption (e.g., conservations tillage and soil testing) –Medical Decisions: Treatment combinations (e.g., psychiatric help and drug treatments) –Smoking: Decision to smoke and to educate one-self regarding risks –Education: homeownership and decision of children to stay in school.

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Bivariate Choice An obvious extension of binary choice models arises when two or more such choices are made. One approach to modeling these choices is to assume that a system of latent variables exists; e.g., with two choices: but we observe the dummy variables

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Bivariate Probit Bivariate probit emerges if the residuals are assumed to be joint standard normal So the corresponding log-likelihood function is simply where

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Expanding the Choice Set Beyond Two Defining the Choice Set Deriving Choice Probabilities Identification in Choice Models

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Source Train, K., (2002), Discrete Choice Methods with Simulation, Cambridge, MA: Cambridge University Press, Ch. 2.

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Discrete Choice Models Discrete choice models characterize decision making in which the available alternatives are Mutually exclusive Exhaustive Finite Defining the relevant choice set is a crucial step in the analysis

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Mutually Exclusive Not very restrictive - one can usually redefine the choice set to satisfy this restriction Example #1 (Train): Home Heating Fuel Choice –Electric –Natural Gas –Oil –Wood –Other Issue: Some households have duel-fuel or multiple systems

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Possible Solutions Explicitly include combinations Electric only Natural Gas only Oil only Wood only Natural gas with electric room heaters Etc. Focus on primary heating system Potential problems: –It may be difficult to identify primary system from available data –Secondary heaters may be important users of energy (e.g., electric space heaters)

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Example #2: Recreation Demand Consider the following choice set for a summer vacation: 1.Yellowstone National Park (Idaho, Wyoming, South Dakota) 2.Badlands National Park (South Dakota) 3.Grand Tetons National Park (Wyoming) 4.Mount Rushmore National Memorial (South Dakota) 5.Devils Tower National Monument (Wyoming) For many vacationers, the alternatives in this choice set are not mutually exclusive

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1 3 2 4 5

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Devils Tower

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Alternative Solutions Redefine alternatives in terms of portfolios Yellowstone only Yellowstone and Grand Tetons only Grand Tetons and Mount Rushmore only Etc. Sequence may matter as well Focus on Primary Destination Potential Problems: Identifying primary destination – even in minds of recreationists Constructing corresponding explanatory variables (e.g. travel costs)

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Exhaustive Criteria Readily satisfied by including a none of the above alternative Heating: no heating Recreation demand: stay at home or all other sites

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Relevance Choice set should be defined not only to be mutually exclusive and exhaustive, but also relevant Choices must be ones from which agent actually chooses, not just the universe of alternatives

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Example: Recreation Demand Should the choice set consist of –All feasible sites? Nothing is revealed about not visiting an unknown site Computationally intractable –Only sites visited in past five years? Useful information is revealed about sites that are not visited. –Sites within a given distance? Limits applicability of results to, say, day trips. Similar issues emerge in job and career search literature Ideally, one would model both choices and accumulation of information about choice set.

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Finite or Countable Choice Sets This is a restrictive characteristic –Excludes the how much decision –Focuses on the which or how many decision These decisions may be tied –Electric power: Choice of rate structure and quantity of electricity consumed –Labor: Choice of participation decision and reservation wage or how much to work –Recreation: Choice of where to visit and how many trips to take or trip duration

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Ordered Choice Models Ordered responses arise in many empirical settings: Opinion surveys: asking if you strongly agree, slightly agree, slightly disagree, or strongly disagree with a statement. Educational data: level of schooling: grade school graduate, high school graduate, some college education, college graduate, some advanced degree, etc. Employment data: unemployed, part-time, full time. Bond ratings Post-release performance of prisoners etc.

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Ordered Choice Models (contd) Typically motivated by assuming an underlying latent variable Typically: Observe:

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Ordered Choice Models (contd) The resulting choice probabilities become:

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Graphically – Binary Choice

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Graphically

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Ordered Probit Most applications assume that the unobserved component of the latent variable is normally distributed; i.e., so that where Equivalently: Normalization required: Either no constant or

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Multinomial Logit Obviously, a second generalization of the binary choice set-up arises when more than two unordered alternatives are presented to decision makers. –Much of the early research arose from studies of travel mode choice in the transportation literature, with work by McFadden (1974a,b) and Domencich and McFadden (1975). –More recently applied to Modeling recreation site selection: Hausman, Leonard, and McFadden (1995) and Morey, Rowe, and Watson (1993) Telecommunications service selection: Train, McFadden, and Ben- Akiva (1987). Occupational choice: Schmidt and Strauss (1975).

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ML – RUM Specification The utility from individual i choosing alternative j is given by: where i.e.,

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ML Choice Probabilities Given the distributional assumptions and representative agent specification, then defining we have that:

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Merits of ML Specification The log-likelihood model is globally concave in its parameters (McFadden, 1973) Choice probabilities lie strictly within the unit interval and sum to one The log-likelihood function has a relatively simple form

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The IIA Assumption In the logit model for any two alternatives, j and k, the ratio of choice probabilities is So that Relative choice are independent of the number and characteristics of all remaining choice probabilities

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Blue Bus/Red Bus Example Suppose initially J=2, with individuals choosing between riding a blue bus (B) or a train (T) to work, with Consider introducing a red bus (R) alternative. One might reasonably assume that together with IIA restrictions, this implies that One would more reasonably expect that

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Nested Logit The standard logit model imposes considerable structure on the distribution of preferences, primarily through the iid assumption The Generalized extreme value (GEV) distribution provides a generalization that allows for a richer pattern of correlations Nested logit is one member of the class of GEV models and the most commonly used

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Nested Logit Nested Logit is appropriate in application where the set of alternatives can be segmented into groups (or nests) satisfying the conditions: –Choices from among alternatives within a group satisfy the IIA assumption –Choices from among alternatives in two different nests are independent alternatives in any other nests (IIN – independence from irrelevant nests)

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Discrete Choice Modeling William Greene Stern School of Business New York University.

Discrete Choice Modeling William Greene Stern School of Business New York University.

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