Dummy and Qualitative Dependent Variables

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Pr(0,1)Pr(0,1| sum = 1) =, orPr(0,1) + Pr(1,0)Pr(1,0)Pr(1,0 | sum = 1) =.Pr(0,1) + Pr(1,0)The conditional probability by using logit functions isPr(1) Pr(0)Pr(0) Pr(1) + Pr(1) Pr(0)=1α11+e+ β′xi1α1e1+eα1+e1+e+ β′x11β′xi 2i1α + β′xα + β′xi 2=i11α11+eα1+e+1+e+ β′x2β′x1i1α + β′x11α11+e+ β′xi 2α1+ β ′ xi1β ′ x1=′′=′ ′=α + β x α + β x β x β x β ′(x − xe1ei 2+ e1i1ee+ ei 2i1ei 21i1)+ 1= 1− Λ[β ′(xi2 − xi1)]Thus, in the fixed effects logit model, the fixed effect, α , is eliminated. Similarly, theiPr(1)Pr(0)1= 1−′( x 2 − 1Pr(0)Pr(1) + Pr(1)Pr(0)) i xie β + 1= Λ β ′(x − x )] .[i2 i1Thus, the log likelihood function for observation i isL = [ sum = 1]{ w ln( Λ(β ∆′ x ) + (1 − w )ln(1 − Λ(∆x))}i1iiiiwhere w i = 1 if (Y i1, Y i2 ) = (0, 1) and w i = 0 if (Y i1, Y i2 ) = (1, 0).The estimated coefficients of the fixed effects logit model can be interpreted as theeffects on log-odd ratio. But we are unable to find level effects of x i on the dependentvariables. This is partly the reasons that some researchers use the (liner) fixed effectsmodel on the binary response dependent variables to control for fixed effects.10
Multinomial Logit ModelThe logit model for binary response can be extended for more than two outcomes that arenot ordered. Let y denote a random variable taking on the values [0, 1, …, J]. Since theprobabilities must sum to unity, Pr(y = 0| x) is determined once we know the probabilitiesfor j = 1, …, J. Therefore, the probability that y takes j can be written asPr( YiPr( Yie= j)=J1+= 0) =1+β ′ xj∑k = 1J1∑k = 1ieeβ ′ xkifor j = 1,..., Jfor j = 0β ′ x.ki, andNote that for a binary response case, J = 2, this becomes the logit model. The modelimplies that we can compute J log-odd ratiosPijln[ ] = β ′jxi.Pi0This implies that the estimated coefficients of the Multinomial Logit should beinterpreted as the effects on the log-off ratios of choice j and the base choice, j = 0.The marginal effects on the particular outcome are∂Pj∂xi∑= P [ β − P β ].jjJk=1kkNote that this result suggests that the direction of the marginal effect could be differentfrom the sign of the estimated coefficient (of the odd-ratio).Quiz: Show that the sum of the marginal effects for all outcomes is zero.(STATA 9 provides the marginal effects.)Example 13-3: Primary and Secondary School Choice among Female Orphans and Nonorphansin Uganda (Yamano, Shimamura, and Sserunkuuma, 2005, EDCC, vol. 54 (4):833-856, Table 9)In this example, the dependent variable is a choice variable: Not attending school,attending primary school, and attending secondary school. The focus is on orphans who11
Page 1 and 2: Takashi YamanoFall Semester 2009Lec
Page 3 and 4: In the observations, we have n case
Page 5 and 6: Measuring the Goodness of FitThe pe
Page 7 and 8: Marginal EffectsThe coefficients of
Page 9: *** A Simulation in Pr(y) when lnpc

Dummy and Qualitative Dependent Variables

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