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Unformatted text preview: Multivariate Response Models The response variable is unordered and takes more than two values. The term unordered refers to the fact that response 3 is not more favored than response 2. One choice from a group is selected, the labeling of the choices is arbitrary. Example: Choice of Health Plan Choice of Occupation Transportation Mode for Commuting to Work Y takes the values f ; 1 ;:::;J g J a positive integer X a set of conditioning variables Example: Y occupational choice X contains education, age, gender, race, marital status As in the binary case, we wish to know how ceteris paribus changes in the elements of X a/ect the response probabilities P ( Y = j j X ) for j = 0 ; 1 ;:::;J . Because the probabilities must sum to unity, P ( Y = 0 j X ) is determined once we know the probabilities for j = 1 ;:::;J . Multinomial Logit X a 1 & K vector with &rst element unity P ( Y = j j X ) = exp & X& j 1 + P J h =1 exp ( X& h ) j = 1 ;:::;J Because the probabilities sum to unity P ( Y = 0 j X ) = 1 1 + P J h =1 exp ( X& h ) If J = 1 , let & 1 = & and we have the binary logit model. Note, the model is not derived from an assumption that errors to a latent model are logisitc. Rather, the response probabilities are assumed to be a logistic function. Partial e/ects are complicated. For continuous X k @P ( Y = j j X ) @X k = P ( Y = j j X ) " & jk P J h =1 & hk exp ( X& h ) 1 + P J h =1 exp ( X& h ) # : Even the direction of the e/ect is not determined by & jk . A simpler interpretation of & j is given by the odds-ratio P ( Y = j j X ) P ( Y = 0 j X ) = exp & X& j with change approximately & jk exp & X& j & X k 1 The sign of & jk determines the direction of the e/ect on the odds ratio. To simplify the analysis even further ln & P ( Y = j j X ) P ( Y = 0 j X ) = X& j so that both the sign and the magnitude are deteremined by & jk In general ln & P ( Y = j j X ) P ( Y = h j X ) = X & j & & h Another useful fact. Because P ( Y = j or Y = h j X ) = P ( Y = j j X ) + ( Y = h j X ) it follows that P ( Y = j j Y = j or Y = h;X ) = P ( Y = j j X ) P ( Y = j j X ) + P ( Y = h j X ) which simpli&es as exp X& j exp X& j + exp ( X& h ) = exp X & j & & h 1 + exp X & j & & h which is a logistic function....
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- Fall '08