Neyman orthogonality

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 $Y={\theta }_{0}D+g_0(X)+U$, 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 $-\mathbb{E}[U{h}_m(X)]=0$. 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.^[ref]