In xtreg, is "rho" correctly computed?

Dear Statalists,

A perhaps very simple, basic question: when using "xtreg y x, fe", shouldn't the intra-class correlation coefficient (rho) be simply: (sigma_u^2 / sigma_e^2)

and NOT sigma_u^2 / (sigma_u^2+sigma_e^2)?

For example, if we specify the residual as e_ig=v_g+n_ig (as in Angrist & Pischke, Mostly Harmless Econometrics, page 231 ), Rho should be: sigma_v^2/(sigma_v^2+sigma_n^2)

and not sigma_v^2/(sigma_v^2+sigma_e^2) ?

Or it is the same somehow?!

Many thanks,
Laura