Correct the continuous-treatment target transformation

[AliFozooni]

Summary

• derive the continuous-treatment transformed target using conditional treatment variance
• explain why dividing by an individual squared residual is unstable and does not recover CATE
• connect the normalized pseudo-outcome to the residual-on-residual R-loss
• give a clean sufficient condition under which omitting variance preserves treatment-effect ranking
• explain why square-root row scaling reproduces WLS only for linear least squares, not nonlinear learners
• recommend native sample weights or direct R-loss optimization for nonlinear final stages
• correct the treatment-variance notation and isoelastic-demand discussion

Why this matters

For continuous treatment, the normalizing term is v(x) = Var(T | X=x), not one observed squared residual. The latter turns the pseudo-outcome into an unstable ratio whose unweighted conditional mean is generally not the treatment effect. The revised derivation gives a conditionally unbiased target under a conditionally linear response, explains the equivalent weighted R-loss, and distinguishes the special randomized example from the general case. Constant conditional treatment variance guarantees unchanged ranking when the denominator is omitted; when variance changes with X, the ranking may change unless additional structure preserves it.

For a linear-in-parameters least-squares stage, multiplying design rows and targets by square-root weights is algebraically equivalent to WLS. Applying the same transformation before a nonlinear learner changes its inputs and hypothesis class, so it is not a generic substitute for sample weights.

The nonlinear-treatment section also distinguishes elasticity from a marginal effect and notes that a zero-cost isoelastic revenue curve has no unconstrained interior optimum.

Validation

• built the full Jupyter Book successfully after the final weighting and ranking clarifications
• validated notebook JSON and checked the diff for whitespace errors
• confirmed that no code cells or stored outputs changed

Closes #350
Closes #402
Closes #471