This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 oddsratio 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....
View Full
Document
 Fall '08
 Staff

Click to edit the document details