Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am using a multinomial logit to estimate $a_1, a_2, a_3, b_1, b_2$, and $b_3$ in a paper. My dependent variable takes 3 possible values, $y = \{1, 2, 3\}$; my independent variable is $z_i$; and the probability that $y$ takes a value of $j$ is $$Pr(y=j) = \frac{\text{exp}[a_j + b_j z_i]}{\sum_{k=1}^3 \text{exp}[a_k + b_k z_i]}$$

My estimate of $a = (1, 2, 3)$ and the estimate for $b = (4, 5, 6)$. How can I show that the probability $y=1$ is identical to the case where $a' = (2, 3, 4)$ and $b' = (6, 7, 8)$?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

By multiplying numerator and denominator of the formula for ${\Bbb P}[y=j]$ by $\exp (1+2z_i)$. This changes each term $\exp(a_j+b_j z_i)$ to $$\exp(a_j+b_j z_i)\exp(1+2z_i)=\exp((a_j+1)+(b_j+2)z_i)=\exp(a'_j+b'_j z_i).$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.