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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 $b' = (6, 7, 8)$?

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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 $$\exp(a_j+b_j z_i)\exp(1+2z_i)=\exp((a_j+1)+(b_j+2)z_i)=\exp(a'_j+b'_j z_i).$$

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