Long-Term Causal Inference with Many Noisy Proxies

Abstract: We propose a method for estimating long-term treatment effects with many short-term proxy outcomes: a central challenge when experimenting on digital platforms. We formalize this challenge as a latent variable problem where observed proxies are noisy measures of a low-dimensional set of unobserved surrogates that mediate treatment effects. Through theoretical analysis and simulations, we demonstrate that regularized regression methods substantially outperform naive proxy selection. We show in particular that the bias of Ridge regression decreases as more proxies are added, with closed-form expressions for the bias-variance tradeoff. We illustrate our method with an empirical application to the California GAIN experiment.

• The paper presents a novel latent variable errors-in-variables regression framework to recover long-term causal effects using many short-term, noisy proxies.
• It shows that increasing the number of proxies reduces attenuation bias and variance, with regularized methods like Ridge regression matching oracle estimators as proxy count increases.
• The study validates these theoretical results through simulation studies and an empirical application in the California GAIN experiment, offering practical insights for high-dimensional causal inference.

Long-Term Causal Inference with Many Noisy Proxies: An Expert Overview

Problem Formulation and Motivation

This paper addresses a crucial limitation in causal inference from randomized experiments on digital platforms: estimating long-term treatment effects when only high-dimensional, noisy, short-term proxy outcomes are available within the feasible experimental window. The fundamental setup considers an unobserved low-dimensional latent vector of surrogates, $S$, which fully mediates the effect of a randomly assigned binary treatment $W$ on a long-term outcome $Y$, but only noisy, high-dimensional proxies $P$ are observed. The density and noisiness of these proxies (where $k \gg p$) move the econometric problem well beyond conventional surrogate analyses, which generally presume low-dimensional, accurate surrogates.

The paper formalizes the computational problem as a latent variable errors-in-variables regression with partial identification. Critically, unlike much of the prior literature that assumes proxies adequately span the latent state $S$ and that mediation is complete, this work explicitly models the proxies $P$ as noisy projections of $S$ via a loading matrix, with additive noise. The identification challenge is therefore whether, and how, increased numbers of noisy proxies can compensate for lack of direct surrogate measurements in resolving the long-term treatment effect $\tau^* = \gamma^\top\beta$.

Analytical Contributions

Causal Model and Identification

The groundwork is a linear structural model:

• First Stage: $S = \gamma W + \epsilon_S$
• Proxy Generation: $P = L S + E$ (with $L \in \mathbb{R}^{k \times p}$, $E$ noise)
• Second Stage: $Y = S^\top\beta + \eta$

Causal identification follows from the complete mediation assumption, connecting first-stage proxy effects ($\tau_P = L\gamma$) and second-stage outcome associations.

A pivotal insight is the explicit characterization of measurement error bias: under perfect proxies ($E=0$), the estimator regains the conventional surrogacy result, but in realistic settings, estimation is attenuated. This manifests as classical errors-in-variables bias, with the key analytical novelty being closed-form expressions for bias and AMSE for high-dimensional ridge regression.

Bias and Variance With Many Proxies

An important result is the finding that attenuation bias on the long-term effect estimate monotonically decreases with the number of proxies. In the case $p=1$, $k$ i.i.d. proxies of a single scalar surrogate, the combined regression coefficient bias is:

$$\text{Total bias factor} = \frac{k\sigma_S^2}{k\sigma_S^2 + \sigma_E^2}$$

which converges to 1 as $k\to \infty$. Thus, high-proxy regimes permit recovery of the latent effect by “averaging out” proxy-specific noise, provided the proxies are sufficiently non-redundant. This is empirically supported (Figure 1).

Figure 1: Estimation accuracy (bias, RMSE) for scalar surrogate as the number of proxies increases demonstrates that regularized methods (especially Ridge and PLS) efficiently eliminate attenuation bias, matching the oracle estimator for large enough $k$.

For the general multivariate surrogate model, the paper provides a spectral decomposition of the bias in terms of the singular values of the loading matrix $L$ and the regularization parameter $\lambda$. The resulting bias for ridge regression with balanced loadings ($L^\top L = cI_p$) is

$$\text{Bias}_\lambda = - \left(\frac{\sigma_E^2 + \lambda}{c \sigma_S^2 + \sigma_E^2 + \lambda}\right)\tau^*$$

highlighting how increasing $c$—the effective number of proxies per true surrogate—shrinks bias, and regularization introduces additional attenuation.

The AMSE is derived, decomposing error into bias and variance, and the optimal regularization parameter $\lambda^*$ is solved in closed form (Appendix), though the optimal tuning depends on data-generating process parameters, thereby warranting empirical cross-validation.

Figure 2: Extending to 5-dimensional surrogates, increasing the number of proxies per surrogate steadily shrinks both RMSE and bias, again with regularized estimators (Ridge, PLS) dominating OLS screening and naive methods.

Methodological and Empirical Evaluation

Simulation Studies

Extensive simulations systematically explore the effect of increasing the number of proxies ($k$), the number of latent surrogates ($p$), and the choice of estimator (OLS screening, LASSO, Ridge, PLS). Key empirical findings include:

• Regularized estimators (Ridge, PLS, LASSO) consistently outperform naive screening (selection on maximal regression coefficients).
• As $k$ increases (more proxies per surrogate), regularized estimators converge rapidly to the oracle estimator’s RMSE and bias, validating theoretical results.
• As $p$ increases with $k/p$ fixed, estimation becomes more challenging for all methods, but regularization retains robustness compared to the instability and high error of OLS screening.

  Figure 3: Increasing the number of underlying surrogates (with fixed ratio of proxies per surrogate) degrades estimator accuracy for all methods, but regularized estimators maintain lower RMSE and bias compared to OLS.

This difference is particularly acute for OLS screening/seleciton, which, in high-$k$ high-$p$ regimes, suffers from model selection instability and consequent large error—in some configurations, bias exceeds 30–40% of the true effect (see Figure 4).

Figure 4: OLS screening shows high RMSE and bias relative to the oracle solution, especially as the number of proxies and surrogates increases.

Empirical Application: California GAIN Experiment

Applying these methods to the California GAIN experiment, the analysis shows that ridge regression surrogate indices yield long-term effect estimates that closely approach the true experimental benchmark, with improved precision as the number and diversity of proxies increases. In contrast, naive extrapolation and OLS fail to achieve similar accuracy or precision.

Figure 5: Ridge estimator tracks the true long-term employment effect as more lagged proxies and diverse surrogates are incorporated, outperforming naive approaches.

Theoretical and Practical Implications

The analysis establishes a rigorous framework for surrogate-based causal inference with high-dimensional, noisy proxies, reinforcing several key insights:

• Regularization is essential in high-dimensional proxy settings; both theoretical and empirical evidence support penalized regression (especially Ridge) as the preferred approach.
• Increasing the number of conditionally independent, non-redundant proxies reliably reduces both bias and variance of long-term effect estimates, provided adequate coverage of the latent state.
• The closed-form expressions for bias and AMSE under Ridge regression offer actionable guidance on the design of digital experiments and on the optimal aggregation of abundant behavioral metrics.

Practically, these results motivate both (i) the collection and utilization of as many informative short-term behavioral proxies as feasible, and (ii) the application of appropriately regularized estimators to mitigate both classical measurement error and model selection noise. The utility of these techniques is buttressed in scenarios endemic to modern online platforms—e.g., A/B testing for product, recommendation, or intervention effects—where direct observation of long-term value is infeasible.

Theoretically, the findings prompt future inquiries into nonlinear measurement-error settings, the analysis of regularization via low-rank methods (such as PLS), scenarios with partial mediation rather than strict surrogacy, and optimal proxy selection under budget and experiment size constraints.

Conclusion

This work systematically analyzes and solves the problem of long-term causal effect estimation using many short-term noisy proxies. By modeling the problem as regression with latent surrogates under high-dimensional measurement error and leveraging regularization, it delivers explicit characterizations of estimator bias and variance, validated by simulations and experiment data. Regularized methods, especially Ridge regression, are shown to outperform naive approaches, with estimation error decreasing as the number of proxies rises. The implications are broadly applicable to digital experimentation and high-dimensional causal inference, and the analytic framework opened here suggests several promising directions for methodological generalization and theory.