# Within Group Correlation of the Nested Logit Model

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).