week_11__nominal_outcomes_multinomial_logit_updated

Week 11: Nominal Outcomes: M Chapter 6)
Tsun-Feng Chiang*
*School of Economics, Henan University, Kaifeng, China
June 9, 2014
Similarly, we can use NB and NC observations to estimate the binary logit and the probabilities of B versus C : Pr (B |X ) = exp(β0,B |C + β1,B |C x ) Pr (C |X ) Then select observations NA and NC for the logit: Pr (A|X ) = exp(β0,A|C + β1,A|C x ) Pr (C |X )
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Nominal Outcomes
Nominal Outcomes
When a variable is nominal, the categories cannot be ordered. But models for nominal outcomes are often used when the dependent variable is ordinal. Occupation Commuting: driving a car, taking the bus and walking to the bus stop, taking the bus and driving to the bus stop, carpooling, etc. The choice of a new vehicle: sports car, van, compact, Jeep, convertible, etc. Sometime, it is done to avoid the parallel regression assumption of the ordered model. Other times there may be uncertainty as to whether the dependent variable should be considered as ordinal. For example, the types of cars can be ordered by their prices, but we are not sure whether consumers’ preference move in the order of cars’ prices.
+ x i δ) + δ ])
=
exp(x i [βm + δ ])
J j =1 exp (x i [βj
While the value of the probabilities are unchanged, the original parameters βm have been replaced by different parameters βm + δ . Because δ can be any nonzero vector of parameters, βm + δ can be any nonzero vector of parameters too. To identify the model, we must impose constraints on the β ’s. The most common constraint is β1 = 0 Clearly, if we add a nonzero δ to β1 , the assumption that β1 = 0 is violated.
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Nominal Outcomes
The Multinomial Logit Model
The Multinomial Logit Model (MNLM)
There are several methods to derive the MNLM. The MNLM as a Probability Model Suppose y = j and j = 1, 2, · · · , m, · · · , J (the number is a notation, these categories are not assumed to be ordered). Let Pr (y = m|X ) be the probability of observing outcome m given X . Assume that Pr (y = m|X ) is a function of the linear combination X βm where the vector βm = (β0m , · · · , βlm , · · · , βkm ) includes the intercept β0m and the coefficient βl for the effect of xl on outcome m. In contrast to the ordered logit model, βm differs for each outcome. To ensure that the probabilities Pr (y = m|X ) are nonnegative, we take the exponential of X βm : exp(X βm ) > 0. The sum of all exponential functions from exp(X β1 ) to exp(X βJ ) is J j =1 exp (X βj ).
The coefficient β1,A|B can be interpreted as a unit increases in x , the log of the probabilities of A versus B changes β1,A|B . How to justify this statement? Let Ω(X ) = Pr (A|X ) ∂ lnΩ(X ) , so = βl ,A|B Pr (B |X ) ∂ xl
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Nominal Outcomes
The Multinomial Logit Model
The MNLM as a Probability Model (Continued) For every observation i , the probability of a category m is exp(x i βm ) divided by the sum, Pr (yi = m|x i ) = Now the probabilities sum to 1, i.e. exp(x i βm )
J j =1 exp (x i βj )
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×
exp(x i δ ) exp(x i δ )
Nominal Outcomes
The Multinomial Logit Model
Identification (Continued) = exp(x i βm + x i δ )
J j =1 exp (x i βj
J j =1 exp (x i βj ) J m=1 Pr (yi
(eq .2)
= m|x i ) = 1.
Identification (eq. 2) is unidentified since more than one set of parameters generates the same probabilities of the observed outcomes. To see this, multiply (eq. 2) by exp(x i δ )/exp(x i δ ), Pr (yi = m|x i ) = exp(x i βm )
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Nominal Outcomes
Introduction to the Multinomial Logit Model
However, this a binary logit redundant. If you know the probabilities of A versus B, and the probabilities of B versus C, then you can derive the probabilities of A versus C without doing the estimation. To see this, multiply two relative probabilities, Pr (A|X ) Pr (B |X ) Pr (A|X ) × = Pr (B |X ) Pr (C |X ) Pr (C |X ) ⇒ exp(β0,A|B + β0,B |C + (β1,A|B + β1,B |C )x ) = exp(β0,A|C + β1,A|C x ) Therefore, β0,A|B + β0,B |C = β0,A|C ; β1,A|B + β1,B |C = β1,A|C (eq .1) (eq. 1) describes necessary relationships among the parameters in the population. They will not hold with sample estimates from the three binary logits because the observations used in three models are different. In the MNLM, all of the logits are estimated simultaneously, so do the probabilities of one category versus another category.