Does the nested logit model impose homogeneous within group correlation?

Consider a nested logit situation where there are $J$-many limbs (the first level) and each limb has $K_j$-many brances.

Here, I will use the index $j$ for limbs and index $k$ for branches. Also, the utility of a person for the alternative in the $j$th of $J$ limbs and $k$th of $K_j$ branches is $U_{jk}=V_{jk}+\epsilon_{jk}$.

As we know, the nested logit model of (McFadden, D. (1978) "Modelling the Choice of Residential Location," in Spatial Interaction Theory and Plannin Models) arises when the error temrs $\epsilon_{jk}$ have the Generalized Extreme Value joint distribution function $$F(\epsilon_{11},\ldots,\epsilon_{1K_1},\ldots,\epsilon_{J1},\ldots,\epsilon_{JK_J})=\exp[-G(\exp(-\epsilon_{11}),\ldots,\exp(-\epsilon_{JK_J}))]$$ where $G(\exp(-\epsilon_{11}),\ldots,\exp(-\epsilon_{JK_J}))=\sum_{j=1}^J\left[\sum_{k=1}^{K_j}\exp(-\epsilon_{jk})^{1/\rho_j}\right]^{\rho_j}$.

Here, the parameter $\rho_j$ can be shown to equal $\sqrt{1-Corr(\epsilon_{jk}, \epsilon_{jl})}$, so $\rho_j$ is inversely related to the within $j$-group correlation of the error terms $\epsilon_{jk}s$.

Therefore, I think, the nested logit model impose homogeneous within group correlation assumption because $\rho_j=\sqrt{1-Corr(\epsilon_{jk}, \epsilon_{jl})} \Rightarrow Corr(\epsilon_{jk}, \epsilon_{jl})=1-\rho_j^2$.

Is it correct? If it is, is there any other multinomial model that allows relative flexible correlations? (except the multinomial probit).



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