--- title: Proximal Linear Structural Equations Model url: https://www.emergentmind.com/topics/proximal-linear-structural-equations-model type: topic --- # Proximal Linear Structural Equations Model A proximal linear structural equations model is a linear SEM used to analyze causal effects in the presence of hidden confounding by exploiting observed proxies, typically treatment-inducing proxies \(Z\) and outcome-inducing proxies \(W\). In the proximal causal inference literature, such models supply bridge equations, moment restrictions, and estimators that recover the causal effect of a treatment \(A\) or \(D\) on an outcome \(Y\) even when an unobserved confounder \(U\) is present, provided that proxy validity, completeness, and relevance conditions hold or are replaced by structured restrictions on proxy invalidity [2208.00105; 2507.19623]. In a separate SEM literature, the term “proximal” refers instead to proximal optimization algorithms for convex relaxations of regularized path-analysis models, not to proxy-based identification [1809.06156]. ## 1. Canonical model and notation In proximal causal inference, the observed data may be written as i.i.d. draws \[ D_i=(A_i,Y_i,Z_i,W_i,X_i), \] where \(U\) is an unobserved confounder, \(X\) denotes observed covariates, \(A\in\{0,1\}\) or \(D\in\mathbb R\) is the treatment, \(Z\) is a negative-control exposure proxy or treatment-inducing proxy, \(W\) is a negative-control outcome proxy or outcome-inducing proxy, and \(Y\) is the outcome [2208.00105; 2507.19623]. A mathematically explicit LSEM used for bias analysis specifies that \(A\) follows an \(\mathrm{expit}\) model in \((U,X)\), that \(Z\) and \(W\) depend linearly on \(U\) and \(X\), and that the potential outcome \(Y(a)\) depends linearly on \(a\), \(X\), \(U\), and the interaction \(aU\) [2208.00105]. This formulation is designed to study what occurs when proximal identification assumptions fail. A canonical proximal linear SEM used for identification with potentially invalid treatment proxies is \[ E[Y_i\mid D_i,Z_i,U_i] = \beta D_i + \alpha^T Z_i + \beta_u U_i, \] \[ E[W_i\mid D_i,Z_i,U_i] = \eta_u U_i. \] Here \(\beta\) is the causal effect of \(D\) on \(Y\), \(\alpha=(\alpha_1,\dots,\alpha_{p_z})^T\) captures direct effects of the \(Z\)-proxies on \(Y\), and \(\beta_u,\eta_u\) are nuisance loadings of \(U\) [2507.19623]. From the second equation, \[ U_i = E[W_i\mid D_i,Z_i]/\eta_u, \] so the model implies \[ E[Y_i\mid D_i,Z_i] = \beta D_i + \alpha^T Z_i + \gamma\,E[W_i\mid D_i,Z_i], \qquad \gamma=\beta_u/\eta_u. \] This leads to the observed regression form \[ Y_i = \beta D_i + Z_i^T\alpha + W_i\gamma + \varepsilon_i, \qquad E[\varepsilon_i\mid D_i,Z_i]=0, \] with \(\operatorname{Var}(\varepsilon_i\mid D_i,Z_i)=\sigma_\varepsilon^2\) [2507.19623]. Within this formulation, a \(Z\)-proxy \(j\) is valid iff \(\alpha_j=0\); otherwise it is invalid. \(W\) is a valid outcome-inducing proxy because its conditional mean does not depend on \(D\) or \(Z\) beyond \(U\) [2507.19623]. ## 2. Identification via proxy restrictions and bridge functions The proximal identification program rests on a set of assumptions that extend standard causal conditions by introducing proxy restrictions [2208.00105]. Beyond consistency and positivity, the key assumptions are negative-control proxy validity, \[ Y(a,z)=Y(a)\quad\forall a,z,\qquad W(a,z)=W\quad\forall a,z, \] latent unconfoundedness, \[ (Z\perp (Y(a),W)\mid U,X), \qquad (W\perp A\mid U,X), \] completeness, and \(U\)-relevance [2208.00105]. Under these assumptions, there exists an outcome-bridge \(h(w,a,x)\) satisfying \[ E[Y\mid Z,A,X]=\int h(w,A,X)\,dF(w\mid Z,A,X), \] which yields the proximal g-formula \[ E[Y(a)] = E[h(W,a,X)]. \] This bridge representation is central because it translates latent-confounding adjustment into an estimable functional equation involving observables [2208.00105]. In the canonical linear SEM with many treatment-inducing proxies, identification can be expressed through population moments. Let \(M=[Z\mid D]\). Then the model imposes \[ E\!\left[M^T(Y-D\beta-Z\alpha-W\gamma)\right]=0. \] Define \[ \Gamma^*=(E[M^TM])^{-1}E[M^TY],\qquad \delta^*=(E[M^TM])^{-1}E[M^TW]. \] Partition \(\Gamma^*=(\Gamma_Z^*,\Gamma_D^*)\) and similarly \(\delta^*\). The moment equations imply \[ \Gamma_Z^*=\alpha+\delta_Z^*\gamma,\qquad \Gamma_D^*=\beta+\delta_D^*\gamma. \] If \(A=\{j:\alpha_j\neq 0\}\) is the set of invalid \(Z\)-proxies and \(s_z=|A|\), Theorem 1 states that there is a unique solution \((\alpha,\gamma)\) iff for every subset \(C\subseteq\{1,\dots,p_z\}\) with \(|C|=p_z-I+1\), there exists a constant \(q_C\) such that \(\delta_j^* q_C=\Gamma_j^*\) for all \(j\in C\), and \(q_C\) must be the same for all such \(C\) [2507.19623]. A simple corollary is the majority rule: if \(I\le p_z/2\), meaning fewer than half of the \(Z\)-proxies are invalid, the solution is automatically unique. Once \((\alpha,\gamma)\) is identified, \[ \beta=\Gamma_D^*-\delta_D^*\gamma. \] A common misconception is that adding more proxies automatically strengthens identification. The invalid-proxy formulation shows that this is not generally true: with many proxies, exclusion violations can introduce bias unless uniqueness restrictions such as the majority rule are satisfied [2507.19623]. ## 3. Violations of completeness and \(U\)-relevance A distinctive contribution of the proximal LSEM literature is the derivation of closed-form bias formulas when proximal identification assumptions fail [2208.00105]. Cobzaru et al. study a simplified setting with \(X\equiv 0\), \(\gamma_{au}=0\), \(p=2\), and \(m=n=1\), and analyze two minimal failures. Under a completeness violation, let \(U=(U_1,U_2)^T\), let both \(Z\) and \(W\) depend on \(U_1\) and \(U_2\), and assume \(\theta_{u2}\mu_{u2}\neq 0\) so that \(\dim U>\dim Z,W\) and completeness fails. If one fits the wrong linear bridge \[ h(w,a;b)=b_0+b_a a+b_w w+b_{aw}aw \] by Method-of-Moments, the asymptotic bias in the ATE estimate \(\hat\psi_{\rm POR}\) is \[ \delta_{\rm POR} = \frac{E[AU_1]}{E[A](1-E[A])} \frac{\theta_{u2}\mu_{u2}}{\theta_{u1}\mu_{u1}} \left\{ \frac{(1-E[A])S_2}{\mu_{u1}+S_2\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}} + \frac{E[A]S_1}{\mu_{u1}+S_1\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}} \right\}\gamma_{u1}, \] where \[ S_1=\frac{(1-E[A])^2}{(1-E[A])(1-E[AU_1^2])-E[AU_1]^2}, \qquad S_2=\frac{E[A]^2}{E[A]E[AU_1^2]-E[AU_1]^2}. \] Under a \(U\)-relevance violation, completeness is retained but the proxies fail to “see” an outcome-relevant confounder \(U_2\). In that case, \[ \delta_{\rm POR} = - \frac{\bigl(E[AU_1^2](1-E[AU_1^2])-E[AU_1]^2\bigr)\,E[AU_2]} {\bigl(E[A]E[AU_1^2]-E[AU_1]^2\bigr)\bigl((1-E[A])(1-E[AU_1^2])-E[AU_1]^2\bigr)} \,\gamma_{u2}. \] These formulas make the source of bias explicit. \(\theta_{u2}\mu_{u2}\) measures the dependence of \(Z\) and \(W\) on the “extra” confounder \(U_2\); \(E[AU_1]\) and \(E[A]\) measure the confounding of treatment by \(U_1\); \(S_1\) and \(S_2\) are normalization constants reflecting how well \(U_1\) can be recovered from \((Z,W)\); and in the \(U\)-relevance failure, \(\gamma_{u2}\) and \(E[AU_2]\) enter linearly [2208.00105]. Because completeness and \(U\)-relevance are empirically untestable, these closed forms provide a direct basis for quantitative bias analysis. The proposed workflow is to estimate or fix observable terms, specify plausible ranges for latent structural parameters, compute the implied range of \(\delta_{\rm POR}\), and, if desired, place priors on the latent parameters and propagate uncertainty to a posterior on \(\delta_{\rm POR}\) [2208.00105]. ## 4. Penalized estimation and adaptive proxy selection When some treatment-inducing proxies may be invalid, a direct estimator targets \((\beta,\alpha,\gamma)\) through penalized GMM: \[ \min_{\beta,\alpha,\gamma} \left\|P_M\!\left[Y-D\beta-Z\alpha-W\gamma\right]\right\|_2^2 +\lambda\|\alpha\|_1, \] where \(P_M\) projects onto the column space of \(M=[Z\mid D]\) and only \(\alpha\) is \(\ell_1\)-penalized [2507.19623]. An equivalent two-step implementation computes \(\widehat W=P_MW\), forms orthogonalized regressors \[ \widetilde D=P_{\widehat W^\perp}D,\qquad \widetilde Z=P_{\widehat W^\perp}Z, \] and solves \[ \hat\alpha= \arg\min_\alpha \frac12\left\|Y-P_{\widetilde D^\perp}(\widetilde Z\alpha)\right\|_2^2+\lambda\|\alpha\|_1. \] Then \[ \hat\beta= \frac{\widetilde D^T\left[Y-\widetilde Z\hat\alpha\right]}{\|\widetilde D\|_2^2}. \] Because the LASSO shrinks some \(\hat\alpha_j\) to zero, it selects a subset of \(Z\)-variables treated as valid proxies [2507.19623]. To recover \(\gamma\), the model uses the ratio identity \[ \gamma=\operatorname{median}_j(\Gamma_j^*/\delta_j^*), \] since \[ \Gamma_j^*/\delta_j^*=\gamma+\alpha_j/\delta_j^*. \] Under the majority rule, the median of these ratios consistently estimates \(\gamma\) [2507.19623]. Standard LASSO may fail to recover the invalid set \(A\) unless the irrepresentable condition \[ \|C_{A^c,A}C_{A,A}^{-1}\operatorname{sign}(\alpha_A)\|_\infty<1 \] holds, where \(C=\operatorname{plim}(1/n)\widetilde Z^T\widetilde Z\). Because the \(\widetilde Z\)’s can remain highly correlated, the adaptive LASSO is proposed as a remedy. It first constructs a \(\sqrt n\)-consistent initial estimator using the median-of-ratios method, \[ \gamma^m=\operatorname{median}(\widehat\Gamma_j/\widehat\delta_j), \qquad \alpha^m=\widehat\Gamma_Z-\widehat\delta_Z\gamma^m, \] and then solves \[ \hat\alpha_{\mathrm{ad}} = \arg\min_\alpha \left\|Y-P_{\widetilde D^\perp}(\widetilde Z\alpha)\right\|_2^2 + \lambda_n\sum_{j=1}^{p_z}\frac{|\alpha_j|}{|\alpha_j^m|}. \] After selecting \[ \widehat A_{\mathrm{ad}}=\{j:\hat\alpha_{\mathrm{ad},j}\neq 0\}, \] the method treats those variables as confounders and re-estimates \(\beta\) by two-stage least squares: \[ \hat\beta_{\mathrm{post}} = \frac{D^TP_{[Z_{\widehat A_{\mathrm{ad}}}\mid W]^\perp}Y} {D^TP_{[Z_{\widehat A_{\mathrm{ad}}}\mid W]^\perp}D}. \] This procedure is designed to jointly select valid proxies and estimate the causal effect [2507.19623]. ## 5. Large-sample properties and many invalid outcome proxies Under the majority rule \(s_z