theorem within_eliminates_fixed_effect (α_i ρ x_it x_bar_i ε_it ε_bar_i : ℝ) :
withinTransform (panelShareIdentity α_i ρ x_it ε_it)
(panelShareIdentity α_i ρ x_bar_i ε_bar_i) =
ρ * withinTransform x_it x_bar_i +
withinTransform ε_it ε_bar_i := by
simp only [withinTransform, panelShareIdentity]; ringCES Estimation Theory: Connecting Theory to Data