= Neyman orthogonality
{c}
An estimating score has <Neyman orthogonality> when its population expectation has zero first derivative in every admissible nuisance direction at the truth. Writing $M(\theta,\eta)=\mathbb E\psi(W;\theta,\eta)$, this means $D_\eta M(\theta_0,\eta_0)[h]=0$. It is a property of the score and model, not ordinary perpendicularity of observed data vectors.
For $Y=\theta_0D+g_0(X)+U$, with $\mathbb E[U\mid X,D]=0$ and $m_0(X)=\mathbb E[D\mid X]$, take $\psi=(Y-\theta D-g(X))(D-m(X))$. A perturbation of $g$ contributes $-\mathbb E[h_g(X)(D-m_0(X))]=0$; a perturbation of $m$ contributes $-\mathbb E[Uh_m(X)]=0$. Suitable second-order smoothness then makes small nuisance errors enter the population equation at higher order. Estimation still needs identification, moment and convergence-rate conditions.<Double Debiased Machine Learning for Treatment and Structural Parameters by Victor Chernozhukov and others>[Section 2.1, Definition 2.1; partially linear model score and derivative calculation]{ref}
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