Multinomial Probit and Logit Models

Multinomial Probit and Logit Models Conditional Logit Model Mixed Logit Model Ani Katchova © 2013 by Ani Katchova. All rights reserved. 2 Multinomial, Conditional, and Mixed Models Overview  Multinomial outcome dependent variable (in wide and long form of data sets)  Indepen...

Multinomial Probit and Logit Models Conditional Logit Model Mixed Logit Model Ani Katchova © 2013 by Ani Katchova. All rights reserved. 2 Multinomial, Conditional, and Mixed Models Overview  Multinomial outcome dependent variable (in wide and long form of data sets)  Independent variables (alternative-invariant or alternative-variant)  Multinomial logit model (coefficients, marginal effects, IIA) and multinomial probit model  Conditional logit model (coefficients, marginal effects)  Mixed logit model 3 Multinomial, Conditional and Mixed Models Multinomial outcome examples  The type of insurance contract that an individual selects.  The product that an individual selects (say type of cereal).  Occupational choice by an individual (business, academic, non-profit organization).  The choice of fishing mode (beach, pier, private boat, charter boat). Multinomial outcome dependent variable  The dependent variable y is a categorical, unordered variable.  An individual may select only one alternative.  The choices/categories are called alternatives and are coded as j =1, 2, …, m.  The numbers are only codes and their magnitude cannot be interpreted (use frequency for each category instead of means to summarize the dependent variable).  The data are usually recorded in two formats: a wide format and a long format.  When using the wide format, the data for each individual i is recorded on one row. The dependent variable is: U LF 4  When using the long format, the data for each individual i is recorded on j rows, where j is the number of alternatives. The dependent variable is: U ? L \1 if U L F 0 if U M F  Therefore, U ? L1 if the alternative j is the observed outcome and the remaining U ? L0. For each observation only one of U 5, U 6, …, U ? will be non-zero. Example for multinomial data in wide form Person ID (i) Dependent variable (y) Codes for y w i (income) x i1 (price of alternative 1) x i2 (price of alternative 2) 1 apple juice (alternative 1) y=1 40,000 2.5 1.5 2 orange juice (alternative 2) y=2 38,000 2.7 1.7 3 orange juice (alternative 2) y=2 50,000 2.9 1.6 5 Example for multinomial data in long form Person ID (i) Dependent variable (y j) Codes for y j w i (income) x ij (price) 1 apple juice (alternative 1) y 1 = 1 40,000 2.5 1 orange juice (alternative 2) y 2 = 0 40,000 1.5 2 apple juice (alternative 1) y 1 = 0 38,000 2.7 2 orange juice (alternative 2) y 2 = 1 38,000 1.7 3 apple juice (alternative 1) y 1 = 0 50,000 2.9 3 orange juice (alternative 2) y 2 = 1 50,000 1.6  The multinomial density for one observation is defined as: B: U; LL 5? - H… HL ?? ? L ? L ?? ? ? ?@5  The probability that individual i chooses the jth alternative is: L ?? Lpr>U ? LF? LF ?:  ?,?;  The functional form of F j should be selected so that the probabilities lie between 0 and 1 and sum over j to one. Different functional forms of F j lead to multinomial, conditional, mixed, and ordered logit and probit models. 6 Independent variables  Two types of independent variables.  Alternative-invariant or case-specific regressors –the regressors S ? vary over the individual i but do not vary over the alternative j. o Income, age, and education are different for each individual but they do not vary based on the type of a product that the individual selects. o Used in the multinomial logit model.  Alternative-variant or alternative-specific regressors – the regressors T ?? vary over the individual i and the alternative j. o Prices for products vary for each product and individuals may also pay different prices. o Salaries for occupation may be different between occupations and also for each individual. o Used in the conditional and mixed logit models. 7 Multinomial logit model  The multinomial logit model is used with alternative-invariant regressors.  The probability that individual i will select alternative j is: L ?? LL: U ? LF; Lexp : ?′??; ∑ exp : ?′??; k i@5  This model is a generalization of the binary logit model.  The probabilities for choosing each alternative sum up to 1, ∑ L ?? k h@5 L1  One set of coefficients needs to be normalized to zero to estimate the models (usually ? 5 L0;, so there are (j-1) sets of coefficients estimated. The coefficients of other alternatives are interpreted in reference to the base outcome.  Coefficient interpretation for alternative j: in comparison to the base alternative, an increase in the independent variable makes the selection of alternative j more or less likely. 8 Marginal effects  The marginal effect of an increase of a regressor on the probability of selecting alternative j is: ?L ?? ? ? ⁄ LL ?? :? ? F? * %;  The marginal effects do not necessarily correspond in sign to the coefficients (unlike the binary logit or probit model).  There are (j-1) sets of coefficients because one set is normalized to zero, but there are j sets of marginal effects.  Depending on which alternative we select as a base category, the coefficients will be different (in reference to the base category) but the marginal effects will be the same regardless of the base category.  The marginal effects of each variable on the different alternatives sum up to zero.  Marginal effects interpretation: each unit increase in the independent variable increases/decreases the probability of selecting alternative j by the marginal effect expressed as a percent. 9 Independence from Irrelevant Alternatives (IIA) property  The odds ratios in the multinomial logit models are independent of other alternatives. For choices j and k, the odds ratio only depends on the coefficients for choices j and k.  Odds ratio: L ?? L?? ⁄ Lexp @ ?′ k? ? F? ? oA  This weakness of the multinomial model is known as the red bus-blue bus problem. If the choice is between a car and a blue bus, according to the model the introduction of a red bus will not change the probabilities. Multinomial probit model  The multinomial probit model is similar to multinomial logit model, just like the binary probit model is similar to the binary logit model.  The difference is that it uses the standard normal cdf.  The probability that observation i will select alternative j is: L ?? LL: U ? LF; LΦ : ??′ ?;  It takes longer for a probit model to obtain results.  The coefficients are different by a scale factor from the logit model.  The marginal effects will be similar. 10 Conditional logit model  The conditional logit model is used with alternative-invariant and alternative-variant regressors.  The probability that observation i will choose alternative j is: L ?? LL: U ? LF; Lexp : ??′ ? E ?′??; ∑ exp : ??′ ? E ?′??; k i@5 where  ?? are alternative-specific regressors and  ? are case-specific regressors.  The conditional logit model has (j-1) sets of coefficients (? ?) (with one set being normalized to zero) for the case-specific regressors and only one set of coefficients (?) for the alternative- specific regressors.  The probabilities for choosing each alternative sum up to 1.  Coefficients for the alternative-invariant regressors ? ? (similar treatment as the multinomial logit model). o One set of coefficients for the alternative-invariant regressors is normalized to zero (say ? 5 L0;, this is the base outcome. The rest of coefficients are interpreted in relation to this base category. 11 o There are (j-1) sets of coefficients (corresponding to the number of alternatives minus 1 for the base). o Coefficient interpretation for alternative j: in comparison to the base alternative, an increase in the independent variable makes the selection of alternative j more or less likely.  Coefficients for the alternative-specific regressors (?). o No normalization is needed. o One set of coefficients across all alternatives. o Coefficient interpretation: an increase in the price of one alternative decreases the probability of choosing that alternative and increases the probability of choosing other alternatives. 12 Marginal effects  The marginal effect of an increase of a regressor on the probability of selecting alternative j is: ?L ?? ? ?? ⁄ LL ?? :? ??? FL ?? ;? where ? ??? L1 if j=k and 0 otherwise.  There are j sets of marginal effects for both the alternative-specific and case-specific regressors.  For each alternative-specific variable  ?? , there are jxj sets of marginal effects.  The marginal effects of each variable on the different alternatives sum up to zero.  Marginal effects interpretation: each unit increase in the independent variable increases the probability of selecting the kth alternative and decreases the probability of the other alternatives, by the marginal effect expressed as a percent. 13 Mixed logit model The mixed logit model (also called random parameters logit model) specifies the utility to the ith individual for the jth alternative to be: 7 ?? L ??′ ?? E ?′??? EA ?? L ??′ ? E ?′?? E ??′ ?? E ?′??? EA ?? where A ?? are iid extreme value (similar to the errors in the conditional logit model).  The mixed logit model allows for the parameters ? ? to be random. A common assumption is that ? ? L? E? ? where ? ?~Ν>0,Σ  ? and ? ?? L? ? E? ?? where ? ?? ~Ν>0,Σ  ? ?.  The introduction of the random parameters has the attractive property of inducing correlation across alternatives. The combined error  ??′ ?? E ?′??? EA ?? is now correlated across alternatives, say Cov c? ?? ,? ?? g L ??′ Σ  ?? .  The probability that individual i selects alternative j represents a mixed logit model: L ?? LL: U ? LF; Lexp : ??′ ? E ?′?? E ??′ ?? E ?′??? ; ∑ exp : ??′ ? E ?′?? E ??′ ?? E ?′??? ; k i@5 14  The mixed logit model relaxes the IIA assumption by allowing parameters in the conditional logit model to be normally (or log-normally) distributed.  When estimating the mixed logit model, the researcher needs to specify which parameters will be estimated as random. If a parameter is random, this implies that effect of a particular regressor on the chosen alternative varies across the individuals.  The mixed logit model produce random parameters coefficients for both the regressor (x i) and the standard deviation of the regressor (sd(x i)).  Coefficient interpretation for the regressors (x i): when the independent variable increases, the consumers are more or less likely to choose this alternative.  Coefficient interpretation on the standard deviation of a regressor (sd(x i)): there is a heterogeneity across individuals with respect to the effect of the independent variable on the alternative chosen.