Double/Debiased CoCoLASSO of Treatment Effects with Mismeasured High-Dimensional Control Variables

Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.

49,641 characters · 19 sections · 33 citation commands

Rendered from LaTeX for readability, not typeset faithfully. Citation keys are highlighted; maths is left as source; figures, tables and equation environments are summarised rather than reproduced; unrecognised commands are greyed out so nothing is silently dropped. Email addresses are removed.

abstractWe develop an estimator for treatment effects in high-dimensional settings with additive measurement error, a prevalent challenge in modern econometrics. We introduce the Double/Debiased Convex Conditioned LASSO (Double/Debiased CoCoLASSO), which extends the double/debiased machine learning framework to accommodate mismeasured covariates. Our principal contributions are threefold. (1) We construct a Neyman-orthogonal score function that remains valid under measurement error, incorporating a bias correction term to account for error-induced correlations. (2) We propose a method of moments estimator for the measurement error variance, enabling implementation without prior knowledge of the error covariance structure. (3) We establish the $\sqrt{N}$-consistency and asymptotic normality of our estimator under general conditions, allowing for both the number of covariates and the magnitude of measurement error to increase with the sample size. Our theoretical results demonstrate the estimator's efficiency within the class of regularized high-dimensional estimators accounting for measurement error. Monte Carlo simulations corroborate our asymptotic theory and illustrate the estimator's robust performance across various levels of measurement error. Notably, our covariance-oblivious approach nearly matches the efficiency of methods that assume known error variance.

Keywords: High-dimensional econometrics, measurement error, treatment effect estimation, Convex-Conditioned LASSO, Double/Debiased Machine Learning, Marchenko-Pastur distribution

JEL Classification: C13, C14, C21, C51, C55

Introduction

The increasing availability of large-scale datasets in economics has led to a growing interest in high-dimensional models, where the number of potential predictors may exceed the sample size. This paradigm shift has necessitated the development of sophisticated econometric methods capable of handling such complex datasets. However, the presence of measurement error in covariates--a pervasive issue in empirical economics--poses significant challenges to the accurate estimation of causal effects and model parameters in high-dimensional settings.

The intersection of high dimensionality and measurement error presents unique challenges that have attracted considerable attention in the econometric and statistical literature. loh2011high made a seminal contribution by proposing an error-corrected LASSO for high-dimensional linear models with mismeasured covariates. While groundbreaking, their approach suffered from a non-convex objective function. Building on this work, datta2017cocolasso introduced the Convex Conditioned LASSO (CoCoLASSO), which ensured a convex optimization problem by replacing the non-convex error-corrected sample covariance matrix with its positive-semidefinite projection.

The importance of addressing measurement error in high-dimensional models is further underscored by sorensen2015measurement, who demonstrated that ignoring measurement error in LASSO can lead to sign-inconsistent covariate selection. This implies that some important covariates might not be selected by the LASSO estimator, potentially leading to omitted variable bias. The consequences of such omissions in causal machine learning contexts have been explored by chernozhukov2022long and wuthrich2023omitted, highlighting the potential for biased estimates of causal effects.

Our paper contributes to this literature by developing a novel approach for estimating treatment effects in high-dimensional linear models with mismeasured covariates. We consider a model where the parameter of interest, $\theta_{0}$, represents the treatment effect, and we assume that the covariates are measured with additive error. Our work extends the CoCoLASSO framework in a crucial aspect: while datta2017cocolasso assume a known covariance matrix of the measurement error, we relax this stringent restriction and estimate the covariance matrix, which is assumed to be isotropic. This estimation boils down to estimating a constant factor of the identity matrix.

In this paper, we develop a Neyman-orthogonal score function that remains valid under measurement error and propose an error-corrected double CoCoLASSO estimator that efficiently combines information from multiple data splits while addressing both regularization and overfitting biases. We then derive theoretical guarantees for our estimator, establishing its asymptotic normality and providing finite-sample bounds on its performance under general conditions.

Our work contributes to several strands of the econometric literature. First, it extends the literature on robust inference in high-dimensional models: belloni2014high, belloni2014inference, van2014asymptotically, javanmard2014confidence, caner2018asymptotically, gautier2011high, and gold2020inference. Second, it builds on the rich literature on measurement error in econometrics, including the classical work of griliches1986errors and more recent contributions by schennach2004estimation, hu2008instrumental, and schennach2022measurement. Finally, our work contributes to the growing literature on causal inference in high-dimensional settings, including the work on double/debiased machine learning by chernozhukov2018double, chernozhukov2015valid, belloni2017program, fan2022estimation, chernozhukov2022automatic, and wager2018estimation.

By addressing the challenges of measurement error in high-dimensional causal inference, our work provides econometricians and applied researchers with a robust tool for estimating treatment effects in the presence of mismeasured covariates. This contribution is particularly relevant in the current era of big data, where the interplay between high-dimensionality and data quality issues is increasingly important in empirical economic research.

The rest of the paper is organized as follows: Section (ref) introduces the model setup and key assumptions. Section (ref) details our estimation procedure for the nuisance parameters of high-dimensional regression models and the variance of measurement error, including the construction of the Neyman-orthogonal score. In Section (ref), we construct the error-corrected Double/Debiased CoCoLASSO estimator for estimating the parameter of interest, the average treatment effect. Section (ref) presents the main theoretical results, including consistency and asymptotic normality of our estimator. Section (ref) presents Monte Carlo simulations to illustrate the finite-sample performance of our estimator.

Model

Uncontaminated High-Dimensional Linear Model

First consider the uncontaminated high-dimensional linear model

align[align omitted — 125 chars of source]

where $Y$ is the outcome variable, $D \in \{0, 1\}$ is a binary variable indicating the receipt of treatment, $X \in {\mathbb{R}}^{p}$ consists of control variables, and $\theta_{0} \in {\mathbb{R}}$ is the parameter of interest. Here, we assume that the dimension $p$ of $X$ may exceed and grow along with the sample size $N$. Notice that we can directly interpret $\theta_{0}$ as the average treatment effect (ATE), $$ \theta_{0} = E[Y|D=1, X] - E[Y|D=0, X]. $$ The outcome equation (1) represents the causal relationship we are interested in estimating, while the treatment equation (2) models the process by which individuals or units are assigned to treatment. This structure is particularly relevant in policy evaluation, where we often want to estimate the effect of a policy or intervention $D$ on an outcome $Y$, while controlling for a large number of potential confounders $X$. Let $W = (Y, X', D)'$ be the vector of observables, and let $\eta = (\beta', \gamma')'$ be the vector of nuisance parameters. Based on the fact that $U$ and $V$ are uncorrelated, $E[UV] = 0$, chernozhukov2018double constructs a score function $$ \phi(W;\theta, \eta) = (Y-D\theta-X'\beta)(D-X'\gamma), $$ which is known to be robust to misspecification or estimation error in the nuisance parameters by satisfying the Neyman orthogonality. Neyman orthogonality is a key property that allows for robust estimation of the parameter of interest $\theta_0$ even when the nuisance parameters $\eta$ are estimated with some error. Specifically, the score function $\phi(W; \theta, \eta)$ is Neyman orthogonal, $$ \partial_\eta E[\phi(W; \theta_0, \eta_0)] = 0. $$ This property ensures that small errors in the estimation of $\eta$ do not lead to large errors in the estimation of $\theta_0$. In high-dimensional settings, where perfect estimation of all nuisance parameters is typically infeasible, this orthogonality property is crucial for obtaining valid inference on the parameter of interest.

The score function $\phi$ satisfies this property in the absence of measurement error. However, as we will show, this orthogonality is lost when we naively replace $X$ with the mismeasured $Z$. This will motivate our subsequent development of a new score function that maintains Neyman orthogonality even in the presence of measurement error.

Measurement Error Model

In many economic applications, the true covariates $X$ are not directly observable. Instead, we observe $Z$, which is a noisy measure of $X$. This scenario is particularly common in survey data, where respondent recall errors, data coding mistakes, or proxy variable usage can introduce measurement errors. To model this, we consider $$ Z = X+A, $$ where $A \in \mathbb{R}^p$ represents the measurement error. We make the following assumptions about the measurement error.

assumption[Measurement error] $\ $ \begin{enumerate} • $E[A] = 0$. • $E[AA'] = {\widehat{\mathbf{\Sigma}}_{a}} = \tau_{0} \cdot I$ for some $\tau_{0} < \infty$. • $A$ is uncorrelated with $X, D, U,$ and $V$. \end{enumerate}

The first assumption ensures that the measurement error does not introduce systematic bias. The second assumption simplifies our analysis by assuming homoscedastic measurement error across covariates, though this could potentially be relaxed in future work. The third assumption, independence of measurement error from other model components, is crucial for identification.

Under this measurement error setting, the score function $\phi$ constructed for the uncontaminated high-dimensional linear model no longer satisfies the Neyman orthogonality condition when $X$ is naively substituted with $Z$. This can be verified such that $$ E[\phi(W; \theta_{0}, \eta_{0})] = E[(Y - D\theta_{0} - Z'\beta_{0})(D - Z'\gamma_{0})] \neq 0 $$ and $$ \partial_{\eta} E[\phi(W; \theta_{0}, \eta_{0})] = \partial_{\eta} E[(Y - D\theta - Z'\beta)(D - Z'\gamma)] \big|_{\eta = \eta_{0}} \neq 0. $$ This violation of the Neyman orthogonality condition has important implications. In high-dimensional settings, where nuisance parameters are typically estimated with some error, the loss of Neyman orthogonality can lead to biased and inconsistent estimation of the treatment effect $\theta_0$.

Therefore, it is imperative to construct a new score function that accounts for the measurement error $A$ and satisfies the Neyman orthogonality condition. In the next section, we will introduce our proposed score function and demonstrate how it achieves Neyman orthogonality even in the presence of measurement error.

Notation

Let $P$ and $E$ denote the probability and expectation operators with respect to the probability measure that describes the distribution of the data. We use boldface capitalized letters to denote matrices. For a $p$-dimensional vector $x$, we write $\lVert x \rVert_{2}$ to be the Euclidean $\ell_{2}$ norm of $x$, $$ \lVert x \rVert_{2} = \sqrt{\sum_{i=1}^{p} x_{i}^{2}}, $$ and for $q>0$, a random element $W$, and a function $f$, we write $\lVert f(W) \rVert_{P,q}$ to denote the $L_{q}$ norm of $f(W)$ with respect to the probability measure $P$, $$ \lVert f(W) \rVert_{P,q} = E[|f(W)|^{q}]^{1/q} = \left( \int |f(W)|^{q} \, dP(w) \right)^{1/q}. $$

Nuisance Parameter Estimation

The presence of measurement error in high-dimensional settings poses significant challenges for consistent estimation of nuisance parameters, which are prerequisites for the treatment effect estimation. In this section, we develop a novel estimation approach that addresses these challenges by constructing a Neyman-orthogonal score function that remains valid under measurement error.

Our approach builds on the double/debiased machine learning framework of chernozhukov2018double, extending it to accommodate measurement error in covariates. The key innovation lies in the construction of a score function that satisfies the Neyman orthogonality condition even when the true covariates are observed with error.

Error-Corrected Neyman-Orthogonal Score Function

Recall that in the presence of measurement error, the naive score function $$ \phi(W; \theta, \eta) = (Y - D\theta - Z'\beta)(D - Z'\gamma) $$ no longer satisfies the Neyman orthogonality condition. This failure of orthogonality can lead to biased and inconsistent estimation of the treatment effect $\theta_0$. To address this issue, we propose a novel score function that incorporates a correction term to account for the measurement error. This score function is designed to satisfy the Neyman orthogonality condition even when the covariates are measured with error.

theorem[Neyman-Orthogonal Score Function under Measurement Error] Define the following score function: $$ \psi(W; \theta, \eta) = (Y - D\theta -Z'\beta)(D - Z'\gamma) - \tau\beta'\gamma. $$ Let $W = (Y, Z', D)'$ be the vector of observables with $Z = X+A$, where $A$ follows Assumption (ref), and let $\eta = (\beta', \gamma', \tau)'$ be the vector of nuisance parameters. Now, the score function satisfies the Neyman orthogonality condition.

The proofs of all the theorems stated in the paper are provided in the appendix. The key feature of this score function is the additional term $-\tau\beta'\gamma$, which serves as a bias correction. This term accounts for the correlation between the measurement errors in $Z$ when it appears in both factors of the expectation, effectively neutralizing the bias introduced by measurement error.

Covariance-Oblivious CoCoLASSO Estimator

We begin by considering the standard LASSO estimator in the absence of measurement error, which serves as a foundation for our subsequent development of the Covariance-Oblivious CoCoLASSO estimator.

In the absence of measurement error, where we observe the true covariates ${\mathbf{X}} = (x_{1}, x_{2}, \hdots, x_{N})'$ in the model $$ Y = {\mathbf{X}}\beta^{*} + e, $$ the LASSO estimator $\hat{\beta}_{lasso}$ is obtained by minimizing the objective function $$ \hat{\beta}_{lasso} = \text{argmin}_{\beta \in {\mathbb{R}}^{p}} \left\{ \frac{1}{2N} \lVert Y - {\mathbf{X}}\beta \rVert_{2}^{2} + \lambda \lVert \beta \rVert_{1} \right\} $$ where $\lambda > 0$ is a regularization parameter controlling the degree of sparsity in the solution. This formulation can be equivalently expressed as $$ \hat{\beta}_{lasso} = \text{argmin}_{\beta \in {\mathbb{R}}^{p}} \left\{ \frac{1}{2} \beta' {\mathbf{\Sigma}_{x}} \beta - \rho'\beta + \lambda \lVert \beta \rVert_{1} \right\}, $$ where we define $$ {\mathbf{\Sigma}_{x}} = \frac{1}{N}{\mathbf{X}}'{\mathbf{X}} \ \text{ and } \ \rho = \frac{1}{N} {\mathbf{X}}'Y. $$ Note that ${\mathbf{\Sigma}_{x}}$ can be viewed as the sample covariance matrix of ${\mathbf{X}}$. Now we assume that each $x_{i}$ is measured with additive error so that we observe $z_{i} = x_{i} + a_{i},$ where the measurement errors ${\mathbf{A}} = (a_{1}, \dots, a_{N})'$ satisfies Assumption (ref).

When measurement error is present, naively applying the LASSO estimator by replacing ${\mathbf{X}}$ with ${\mathbf{Z}}$ leads to biased and inconsistent estimates. To illustrate this, consider the expected values of the sample covariance and cross-covariance,

align*[align* omitted — 173 chars of source]

The bias in the sample covariance matrix introduces challenges for consistent estimation and variable selection.

To address the bias induced by measurement error, loh2011high proposed using unbiased surrogates $$ {\widehat{\mathbf{\Sigma}}} = \frac{1}{N}{\mathbf{Z}}'{\mathbf{Z}} - \mathbf{\Sigma}_{a} \ \text{ and } \ \tilde{\rho} = \frac{1}{N} {\mathbf{Z}}'y. $$

They further suggested solving the optimization problem $$ \hat{\beta}_{lasso} \in \text{argmin}_{\beta \in {\mathbb{R}}^{p}} \left\{ \frac{1}{2} \beta' {\widehat{\mathbf{\Sigma}}} \beta - \tilde{\rho}'\beta + \lambda \lVert \beta \rVert_{1} \right\}. $$ However, notice that ${\widehat{\mathbf{\Sigma}}}$ is not guaranteed to be positive semi-definite, implying that the objective function could be non-convex and the unique solution $\hat{\beta}_{lasso}$ is no longer guaranteed. Hence, datta2017cocolasso develops a straightforward solution by further replacing ${\widehat{\mathbf{\Sigma}}}$ by its positive semi-definite approximation $$ {\widetilde{\mathbf{\Sigma}}} = \text{argmin}_{\mathbf{\Sigma} \geq 0} \lVert {\widehat{\mathbf{\Sigma}}} - \mathbf{\Sigma} \rVert_{\text{max}} $$ obtained via the alternate direction method of multipliers (ADMM) algorithm to ensure computational efficiency under high-dimensional settings. The CoCoLASSO estimator is defined as the solution to $$ \hat{\beta}_{coco} = \text{argmin}_{\beta \in {\mathbb{R}}^{p}} \left\{ \frac{1}{2} \beta' {\widetilde{\mathbf{\Sigma}}} \beta - \tilde{\rho}'\beta + \lambda \lVert \beta \rVert_{1} \right\}. $$

The CoCoLASSO estimator provided a significant advancement in handling measurement error in high-dimensional settings. However, it relies on a critical assumption: the covariance matrix of the measurement error, $\mathbf{\Sigma}_{a}$, is known. In practice, this assumption is often unrealistic, as the true error covariance is rarely available to researchers. We assume an isotropic error structure, $\tau_{0} \cdot {\mathbf{I}}_p$, where $\tau_{0}$ is unknown and needs to be estimated. This structure simplifies the estimation problem while still capturing the essential features of measurement error in many practical scenarios.

The Covariance-Oblivious CoCoLASSO estimator proceeds in two main steps.\\ (a) Estimate $\tau_{0}$ in a data-driven way, where we propose a method of moments estimator for $\tau_{0}$ based on the properties of the empirical spectral distribution of the sample covariance matrix in the following section.\\ (b) Solve the modified CoCoLASSO problem: Using the estimated $\hat{\tau}^2$, we define: $$ {\widehat{\mathbf{\Sigma}}} = \frac{1}{N}{\mathbf{Z}}'{\mathbf{Z}} - \hat{\tau}_{0} \cdot {\mathbf{I}}_p. $$ We then compute its positive semidefinite approximation ${\widetilde{\mathbf{\Sigma}}}$ as in the original CoCoLASSO $$ {\widetilde{\mathbf{\Sigma}}} = \text{argmin}_{\mathbf{\Sigma} \geq 0} \lVert {\widehat{\mathbf{\Sigma}}} - \mathbf{\Sigma} \rVert_{\text{max}}, $$ and we finally define the covariance-oblivious CoCoLASSO estimator as $$ \hat{\beta}_{coco} = \text{argmin}_{\beta \in {\mathbb{R}}^p} \left\{ \frac{1}{2} \beta' {\widetilde{\mathbf{\Sigma}}} \beta - \tilde{\rho}'\beta + \lambda \lVert \beta \rVert_1 \right\}. $$

Method of Moments Estimator for $\tau_{0}$

In the presence of measurement error, estimating the error covariance matrix is crucial for consistent estimation of our parameter of interest. We adopt an approach based on replicate measurements, building on the work of carroll1995measurement.

For each observation $z_i$, suppose we have $k_i$ replicate measurements $z_{i1}, z_{i2}, \hdots, z_{ik_i}$, and let $\bar{z}_i$ denote the mean of these replicate measurements. A natural estimator for the error covariance matrix $\boldsymbol{\Sigma}_{a}$ through replicate measurements is: $$ {\widehat{\mathbf{\Sigma}}_{a}} = \frac{\sum_{i=1}^N \sum_{j=1}^{k_i} (z_{ij} - \bar{z}_i)(z_{ij} - \bar{z}_i)'}{\sum_{i=1}^{N}(k_i-1)}, $$ where $\bar{z}_i$ denotes the mean across all replicate measurements for the $i$\textsuperscript{th} sample.

To simplify our analysis, we make the following assumption.

assumption[Replicate measurements] $z_{i}$ has exactly two replicate measurements for each $i \in \{1, 2, \hdots, N\}$.

Under Assumption (ref), with $k_i = 2$ for each $i \in \{1, 2, \hdots, N\}$, the estimator ${\widehat{\mathbf{\Sigma}}_{a}}$ can be simplified as

align*[align* omitted — 673 chars of source]

Letting $\tilde{z}_{i} = (z_{i1}-z_{i2})/\sqrt{2}$ and $\widetilde{\mathbf{Z}} = (\Tilde{z}_{1}, \Tilde{z}_{2}, \hdots, \Tilde{z}_{N})'$, we finally write

align*[align* omitted — 175 chars of source]

which is of standard form. From this point forward, we use ${\widehat{\mathbf{\Sigma}}_{a}}$ to denote the measurement error covariance estimate with two replicate measurements for each sample.

We now establish the key properties of the entries of $\widetilde{\mathbf{Z}}$ in the following theorem.

theoremThe entries of $\widetilde{\mathbf{Z}}$ are independent and identically distributed with mean zero and variance $\tau_{0}$.

To analyze the properties of ${\widehat{\mathbf{\Sigma}}_{a}}$, we introduce the concept of empirical spectral distribution (ESD). Let ${\widehat{\mathbf{\Sigma}}_{a}}$ have real eigenvalues $\{\Lambda_i\}_{i=1}^p$ with $\Lambda_1 \leq \cdots \leq \Lambda_p$. The ESD of ${\widehat{\mathbf{\Sigma}}_{a}}$ is defined as: $$ \mu_N(x) = \frac{1}{p} \sum_{i=1}^p \delta_{\Lambda_i}(x), $$ where $\delta_{\Lambda_i}$ denotes the Dirac measure centered at $\Lambda_i$. Recall that the $k$\textsuperscript{th} moment of $\mu_N$ is defined as $$ m_k(\mu_N) = \int_{-\infty}^{\infty} x^k \, d\mu_N(x), $$ and for $\mu_N$, we can write $$ m_k(\mu_N) = \frac{1}{p} \text{Tr}({\widehat{\mathbf{\Sigma}}_{a}}^k). $$

Given the structure of ${\widehat{\mathbf{\Sigma}}_{a}}$, we can leverage results from random matrix theory, specifically the Marchenko-Pastur law, to characterize the asymptotic behavior of its empirical spectral distribution.

theorem[Marchenko-Pastur Law for ${\widehat{\mathbf{\Sigma}}_{a}}$] As $N, p \to \infty$ with $p/N \to \kappa \in (0, \infty)$, $\mu_{N}$ converges to $\mu$ with the density $$ \frac{d\mu}{d\Lambda} = \frac{1}{2 \pi \tau_{0} \kappa \Lambda} \sqrt{(\Lambda^{+} - \Lambda)(\Lambda^{-} + \Lambda)} \mathbf{1}_{[\Lambda^{-}, \Lambda^{+}]}(\Lambda), $$ with probability $1$, where $\Lambda^{\pm} = \tau_{0}(1 \pm \sqrt{\kappa})$ and $\mu$ is the Marchenko-Pastur distribution with parameters $\tau_{0}$ and $\kappa$.

Given the results from the Marchenko-Pastur law and the properties of the empirical spectral distribution, we now introduce a method of moments estimator for the measurement error variance $\tau_{0}$. This estimator leverages the relationship between the moments of the Marchenko-Pastur distribution and the trace of our sample covariance matrix ${\widehat{\mathbf{\Sigma}}_{a}}$.

The $k^{th}$ moment of the Marchenko-Pastur distribution with parameters $\tau_{0}$ and $\kappa$ is given by $$ m_{k}^{*} = \tau_{0}^{k} \sum_{r = 0}^{k-1} \frac{1}{r+1} {k \choose r} {k-1 \choose r} \kappa^{r}. $$

This relationship suggests that we can estimate $\tau_0$ by calculating the average of the eigenvalues of ${\widehat{\mathbf{\Sigma}}_{a}}$, which is equivalent to computing the trace of ${\widehat{\mathbf{\Sigma}}_{a}}$.

definition[Method of moments estimator for $\tau_{0}$] Let $\{\Lambda_{i}\}_{i=1}^{p}$ be the eigenvalues of ${\widehat{\mathbf{\Sigma}}_{a}}$ with $\Lambda_{1} \leq \dots \leq \Lambda_{p}$. We define the method of moments estimator for $\tau_{0}$ as $$ \hat{\tau}_{0} = \frac{1}{p}\sum_{i=1}^{p} \Lambda_{i}, $$ which can be equivalently formulated as $$ \hat{\tau}_{0} = \frac{\text{Tr}({\widehat{\mathbf{\Sigma}}_{a}})}{p}. $$

Error-Corrected Double/Debiased CoCoLASSO

We now turn to the main estimation procedure of our paper, the Error-Corrected Double/Debiased CoCoLASSO, to provide consistent estimates of treatment effects in the presence of measurement error. This method builds upon the Convex Conditioned LASSO (CoCoLASSO) of datta2017cocolasso and the double/debiased machine learning framework of chernozhukov2018double, incorporating our measurement error correction.

The following algorithm defines the Double/Debiased CoCoLASSO (Double CoCo) estimator.

algorithm[algorithm omitted — 1,030 chars of source]

Theoretical Properties

In this section, we establish the theoretical properties of nuisance parameter estimators and Double/Debiased CoCoLASSO.

Theoretical Properties of $\hat{\tau}_0$

We begin by introducing a distributional assumption on $\tilde{z}_{i}$.

assumption[Sub-Gaussian Entries] The entries of $\tilde{z}_{i}, \left\{\tilde{z}_{i}^{(j)}\right\}_{j=1}^{p},$ are sub-Gaussian with $\lVert \tilde{z}_{i}^{(j)} \rVert_{\psi_{2}} = K$ for all $i \in \{1,2, \hdots, N\}$.

We first establish that our method of moments estimator for the error variance is unbiased.

theorem[Unbiasedness of $\hat{\tau}_{0}$] The method of moments estimator $\hat{\tau}_{0}$ is unbiased, $$ {\text{E}}[\hat{\tau}_{0}] = \tau_{0}. $$

This result ensures that, on average, our error covariance estimator correctly identifies the true parameter, providing a solid foundation for our subsequent analysis. Moreover, we establish the asymptotic normality of the error covariance estimator.

theorem[$\sqrt{N}$-consistency of $\hat{\tau}_{0}$] Suppose Assumption (ref) holds. Then, $$\sqrt{N}(\hat{\tau}_{0} - \tau_{0}) \xrightarrow{d} \mathcal{N}(0,V)$$ where $V$ is the finite asymptotic variance.

Consistency result ensures that the error covariance estimator $\hat{\tau}_{0}$ converges to the true parameter fast enough.

Under high-dimensional regime, the actual orientation of data vectors might be less interpretable due to the large number of variables. The following rotational invariance property will ensure that the variance estimation is robust and not dependent on the specific alignment of covariate space.

theorem[Rotational invariance of $\hat{\tau}_{0}$] The estimate $\hat{\tau}_{0}$ is invariant under any unitary matrix $\mathbf{Q} \in {\mathbb{C}}^{p \times p}$ such that $\hat{\tau}_{0}$ estimated from the rotated error covariance matrix $\mathbf{Q} {\widehat{\mathbf{\Sigma}}_{a}} \mathbf{Q}^{\dag}$ is identical to the estimate $\hat{\tau}_{0}$ from the true error covariance matrix ${\widehat{\mathbf{\Sigma}}_{a}}$.

Finally, we provide a non-asymptotic result that bounds the probability of large deviations of our estimator from the true error covariance matrix.

theorem[Finite sample tail bound] Suppose Assumption (ref) holds. For all $\varepsilon > 0$, $$ {\text{P}}\left(\left\lVert \hat{\tau}_{0} \cdot \mathbf{I}_{p} - {\widehat{\mathbf{\Sigma}}_{a}} \right\rVert_{op} > \varepsilon\right) \leq 2 \exp\left(-c\min\left\{\frac{N\varepsilon^{2}}{C^{2}_{0}K^{2}}, \frac{N\varepsilon}{C_{0}K}\right\}\right). $$

This theorem provides a non-asymptotic guarantee on the performance of our estimator. It shows that the probability of large deviations decays exponentially with the sample size $N$, providing strong finite-sample guarantees even in high-dimensional settings.

Theoretical Properties of Covariance-Oblivious CoCoLASSO

In this section, we establish the theoretical properties of our Covariance-Oblivious CoCoLASSO estimator. We begin by stating a set of key assumptions from datta2017cocolasso.

assumption[Closeness] Assume that the distribution of ${\widehat{\mathbf{\Sigma}}}$ and $\tilde{\rho}$ are identified by a set of parameters. Then there exists universal constants $C$ and $c$ such that for every $\varepsilon \leq \tau_{0}$, ${\widehat{\mathbf{\Sigma}}}$ and $\hat{\rho}$ satisfy the probability bounds $$ P(|{\widehat{\mathbf{\Sigma}}}_{ij} - \mathbf{\Sigma}_{x,ij}| \geq \varepsilon) \leq C \exp\left( -\frac{cN\varepsilon^{2}}{\max\{\tau_{0}^{2}, \xi^{4}, 1\}} \right) $$ and $$ P(|\hat{\rho}_{j} - \rho_{j}| \geq \varepsilon) \leq C \exp\left( -\frac{cN\varepsilon^{2}}{s^{2}\max\{\tau_{0}^{2}, \xi^{4}, 1\}} \right) $$ where $\xi^{2}$ is the variance of the error term of the CoCoLASSO model, $e$, and $s$ is the number of true support of $\beta_{coco}$ .

This assumption ensures that our sample quantities concentrate around their population counterparts with high probability. Factors are chosen in accordance with Lemma 1 of datta2017cocolasso, which states that the additive-error-corrected sample covariance matrix satisfies the closeness condition.

Now we impose the restricted eigenvalue condition assumption on ${\mathbf{\Sigma}_{x}}$, the assumption ensuring that the ${\mathbf{\Sigma}_{x}}$ is well-behaved on sparse subsets, which is necessary for consistent estimation of CoCoLASSO.

assumption[Restricted Eigenvalue Condition] Let $S$ be the true support of the high-dimensional regression coefficient of the CoCoLASSO model with $|S|=s$. We assume that for any nonzero $u \in {\mathbb{R}}^{p}$ with $\lVert u_{S^{c}} \rVert_{1} \leq 3 \lVert u_{S} \rVert_{1}$, we have $$ \Omega = \min_{u \in {\mathbb{R}}^{p}} \frac{u' {\widehat{\mathbf{\Sigma}}}_{x} u}{u'u} > 0. $$

We now restate the main theorem on the $\ell_{2}$ error bound of the CoCoLASSO estimator, which equivalently applies to the Covariance-Oblivious CoCoLASSO estimator.

theorem[$\ell_{2}$ Error Bound of CoCoLASSO Estimator] Suppose Assumptions (ref) and (ref) hold. Then, for $s\sqrt{\max\{\tau_{0}^{2}, \xi^{4}, 1\} \log p/N} < \lambda < s\sqrt{\zeta \log p/N} < \min\{\tau_{0}, 12\tau_{0} \lVert \beta_{S} \rVert_{\infty} \}$, we have $$ P\left(\lVert \hat{\beta}_{coco} - \beta^{*} \rVert_{2} \leq \frac{C \lambda \sqrt{s}}{\Omega} \right) \geq 1-C \exp(-c \log p), $$ where $\zeta$ is some positive function depending on $\tau_{0}$ and $\xi^{2}$.

Theoretical Properties of Double/Debiased CoCoLASSO

In this section, we establish the theoretical properties of our Double/Debiased CoCoLASSO estimator. We begin by stating the key assumptions and then present lemmas that lead to our main inferential result.

assumption[Regularity Conditions for the Double/Debiased CoCoLASSO Estimator] Let $c, C > 0$ be finite constants with $c \leq C$, and let $q>2$. Suppose that the following conditions hold. \begin{enumerate} • $\lVert X'\beta_{0} \rVert_{P,q} + \lVert X'\beta \rVert_{P,q} + \lVert X'\gamma_{0} \rVert_{P,q} + \lVert X'\gamma \rVert_{P,q} \leq C$ • $\lVert A'\beta_{0} \rVert_{P,q} + \lVert A'\beta \rVert_{P,q} + \lVert A'\gamma_{0} \rVert_{P,q} + \lVert A'\gamma \rVert_{P,q} \leq C$ • $E[DV], E[U^{2}], E[V^{2}] \geq c$ • $\lVert D \rVert_{P,q}, \lVert Y \rVert_{P,q} \leq C.$ \end{enumerate}

We now establish key properties of our score function, which is central to the Double/Debiased Machine Learning approach.

lemma[Linear Score Function with Exact Neyman Orthogonality] For all $N \geq 3$, the following conditions hold. \\ (a) The true parameter $\theta_{0}$ satisfies the moment condition $E[\psi(W; \theta_{0}, \eta_{0})] = 0$. \\ (b) The score function $\psi$ is linear in the sense that $$ \psi(W; \theta, \eta) = \psi^{a}(W; \eta) \theta + \psi^{b}(W;\eta). $$ (c) The map $\eta \mapsto E[\psi(W; \theta, \eta)]$ is twice continuously differentiable on $\mathcal{T}$, where $\mathcal{T}$ is a convex nuisance parameter space. \\ (d) The score function $\psi$ satisfies the exact Neyman orthogonality condition at $(\theta_{0}, \eta_{0})$. \\ (e) $\theta_{0}$ is identified, i.e. $c < |E[\psi^{a}(W;\eta_{0})]| < 2C^{2}$.

Through this lemma, we have established that our error-corrected score function satisfies the key properties required for the Double/Debiased Machine Learning framework, effectively mitigating the effect of measurement error.

We now present the regularity conditions for the score function and nuisance parameter estimation with respect to our score function $\psi$.

lemma[Score and Nuisance Parameter Estimation Regularity Conditions] For all $N \geq 3$ , the following conditions hold. \\ (a) Fix $\lambda$ from Theorem (ref) and fix $\varepsilon>0$. Let $s_{0}$ and $s_{1}$ be the number of true support sets of $\beta_{0}$ and $\gamma_{0}$ respectively, and let $\Omega$ be the restricted eigenvalue constant of ${\mathbf{Z}} = (Z_{1}, Z_{2}, \hdots, Z_{N})'$. Given a random subset $I \subseteq \{1, 2, \hdots, N\}$ of size $n = N/K$, the nuisance parameter estimator $\hat{\eta}_{0,I^{c}} = \hat{\eta}_{0}((W_{i})_{i \in I^{c}}) = (\hat{\beta}_{0, I^{c}}', \hat{\gamma}_{0, I^{c}}', \hat{\tau}_{0, I^{c}})'$ belongs to the realization set $$ \mathcal{T}_{N} = \{ (\beta', \gamma', \tau)' \subseteq {\mathbb{R}}^{2p+1} : \beta \in B_{\beta}, \gamma \in B_{\gamma}, \text{ and } \tau \in B_{\tau} \}, $$ where $B_{\beta}, B_{\gamma},$ and $B_{\tau}$ are Euclidena balls defined as \begin{align*} B_{\beta} &= \lVert \beta - \beta_{0} \rVert_{2} \leq \frac{C_{\beta}\lambda\sqrt{s_{0}}}{\Omega} \\ B_{\gamma} &= \lVert \gamma - \gamma_{0} \rVert_{2} \leq \frac{C_{\gamma}\lambda\sqrt{s_{1}}}{\Omega} \end{align*} and $$ B_{\tau} = |\tau - \tau_{0}| > \varepsilon $$ for some absolute constants $C_{\beta}, C_{\gamma} > 0$. Note that by construction, $\hat{\eta}_{0, I^{c}} \in \mathcal{T}_{N}$ with probability at least $$ \min\left\{1-C \exp(-c \log p), 1-2 \exp\left(-c\min\left\{\frac{N\varepsilon^{2}}{C^{2}_{0}K^{2}}, \frac{N\varepsilon}{C_{0}K}\right\}\right)\right\}. $$ \\ (b) The following uniform moment conditions hold. \begin{align*} \sup_{\eta \in \mathcal{T}_{N}}(E[\lVert \psi(W;\theta_{0}, \eta) \rVert^{q}])^{1/q} \leq c_{1}; \\ \sup_{\eta \in \mathcal{T}_{N}}(E[\lVert \psi^{a}(W; \eta) \rVert^{q}])^{1/q} \leq c_{1}. \end{align*} (c) The following statistical rates with respect to the score function $\psi$ hold. \begin{align*} \sup_{\eta \in \mathcal{T}_{N}} \lVert E[\psi^{a}(W;\eta)] - E[\psi^{a}(W;\eta_{0})] \rVert = o_{p}(1); \\ \sup_{\eta \in \mathcal{T}_{N}} E[\lVert \psi^{a}(W;\theta_{0}, \eta) - \psi^{a}(W;\theta_{0}, \eta_{0}) \rVert^{2}]^{1/2} = o_{p}(1) \\ \sup_{r \in (0,1), \eta \in \mathcal{T}_{N}} \lVert \partial_{r}^{2} E[\psi^{a}(W;\theta_{0}, \eta_{0} + r(\eta - \eta_{0}))] \rVert = o_{p}(1). \end{align*} (d) The variance of the score $\psi$ is non-degenerate in the sense that $$ E[\psi(W; \theta_{0}, \eta_{0})\psi(W; \theta_{0}, \eta_{0})'] \geq 0. $$

We finally state the main theorem on the inference of the Double/Debiased CoCoLASSO estimator.

theorem[Double/Debiased CoCoLASSO Inference] Suppose that Assumption (ref) holds. Then, the Double/Debiased CoCoLASSO estimator $\hat{\theta}_{0}$ constructed in Algorithm (ref) using the error-corrected score function $\psi$ obeys $$ \sqrt{N} (\hat{\theta}_{0} - \theta_{0}) \rightsquigarrow N(0, \sigma^{2}) $$ uniformly over $P \in \mathcal{P}$, where $$ \sigma^{2} = (E[DV])^{-1}(E[U^{2}V^{2}] + \tau_{0}(\lVert \gamma_{0} \rVert_{2}^{2} E[U^{2}] + \lVert \beta_{0} \rVert_{2}^{2} E[V^{2}] + \tau_{0} \lVert \beta_{0} \rVert_{2}^{2} \lVert \gamma_{0} \rVert_{2}^{2}))(E[DV])^{-1}. $$ The asymptotic variance $\sigma^{2}$ can be estimated as $$ \hat{\sigma}^{2} = (\hat{J}_{0})^{-1} \frac{1}{K} \sum_{k=1}^{K} E_{n,k} [\psi(W; \hat{\theta}_{0}, \hat{\eta}_{0,k})\psi(W; \hat{\theta}_{0}, \hat{\eta}_{0,k})'] (\hat{J}_{0})^{-1}, $$ where $$ \hat{J}_{0}^{-1} = \frac{1}{K} \sum_{k=1}^{K} E_{n,k}[\psi^{a}(W; \hat{\eta}_{0,k})]. $$

Simulation Results

To evaluate the finite-sample performance of the Double/Debiased CoCoLASSO (Double CoCo) estimator in the presence of measurement error, we conducted a series of Monte Carlo simulations. Our simulations are designed to assess the estimator's performance under various levels of measurement error and to compare it with alternative estimation strategies.

Simulation Design

We generated data according to the following model,

align*[align* omitted — 109 chars of source]

for $i \in \{1, 2, \hdots, N\}$, where $\beta_{0}, \gamma_{0}$ are sparse and $U_{i}, V_{i} \sim N(0, \xi^{2})$. We have generated the measurement error $A_{i}$ with a covariance matrix $\tau_{0} \cdot {\mathbf{I}}_{p}$ to introduce the observed covariate vector $Z_{i} = X_{i} + A_{i}$, where we varied the magnitude of $\tau_{0}$ from 1.0 to 2.0 throughout the simulations to assess the impact of measurement error.

In our primary simulations, we set the sample size $N$ to $500$ and the number of covariates $p$ to $750$. The true parameter of interest, $\theta_{0}$, was set to 10 for all simulations. To ensure sparsity, we set 50 randomly chosen components of both $\beta_{0}$ and $\gamma_{0}$ to randomly generated numbers, with the remaining coefficients set to 0.

Estimation Procedures

We provide results for four different estimation procedures, which are distinguished as

enumerate• Double/Debiased LASSO with naive score function (Naive): high-dimensional regression coefficients $\beta_{0}$ and $\gamma_{0}$ are estimated using the classical LASSO estimator that does not account for measurement error, then $\theta_{0}$ is estimated via error-prone score function $\phi = (Y-D\theta-X'\beta)(D-X'\gamma)$. • Oracle Double/Debiased CoCoLASSO (Oracle): the oracle estimator makes use of the true regression coefficients $\beta_{0}$ and $\gamma_{0}$, which are unavailable to researchers in practice, to estimate $\theta_{0}$ using the error-corrected score function $\psi$ following the Double/Debiased CoCoLASSO procedure in Algorithm (ref) with the CoCoLASSO estimation steps omitted. • Covariance-aware Double/Debaised CoCoLASSO (CA-Double CoCo): $\theta_{0}$ is estimated using the Double/Debiased CoCoLASSO procedure, where $\tau_{0}$ of the covariance matrix of $A$ is assumed to be known, following the assumption of the original CoCoLASSO paper datta2017cocolasso. • Covariance-oblivious Double/Debaised CoCoLASSO (CO-Double CoCo): $\theta_{0}$ is estimated using the Double/Debiased CoCoLASSO procedure, where $\tau_{0}$ of the covariance matrix of $A$ is assumed to be unknown and is estimated via the method of moments estimator introduced in Definition (ref).

We have used 5-fold cross-validation for selecting the regularization parameter $\lambda$ in the LASSO and CoCoLASSO estimators, and we have used 5-fold cross-fitting procedure for the Double/Debiased CoCoLASSO estimator.

Results and Discussion

Table (ref) presents the mean squared error (MSE), bias, and variance of the estimators across 1000 Monte Carlo replications for different levels of measurement error.

table[table omitted — 2,057 chars of source]

First, we observe that as $\tau_{0}$ increases from 1.0 to 2.0, the performance of all estimators deteriorates, but to varying degrees. This degradation in performance is expected, as measurement error with a greater variance introduces more noise into the estimation process, making it more challenging to recover the true parameter values. The naive estimator, which ignores measurement error, performs poorly across all scenarios. Its bias and MSE are substantially larger than those of the other estimators, demonstrating the critical importance of accounting for measurement error in high-dimensional settings. Moreover, the naive estimator's performance worsens more rapidly as $\tau_{0}$ increases, demonstrating its lack of robustness to measurement error.

In contrast, the oracle estimator, which has perfect knowledge of $\beta_{0}$ and $\gamma_{0}$, provides the best performance across all scenarios, as expected. It serves as a lower bound on the achievable MSE and bias, showing what would be possible with perfect information about the nuisance parameters. Notably, the oracle estimator's performance remains relatively stable across different levels of measurement error, showing only slight increases in MSE and bias as $\tau_{0}$ increases.

Our proposed Double CoCo estimators, both the covariance-aware (CA) and covariance-oblivious (CO) versions, perform similarly across all scenarios. The CO-Double CoCo shows only slightly larger MSE and bias compared to the CA-Double CoCo, despite not knowing the true $\tau_{0}$. This suggests that our method of moments estimator for $\tau_{0}$ is highly effective, nearly matching the performance achieved when the true measurement error variance is known.

Both Double CoCo estimators demonstrate remarkable robustness to increasing measurement error. While their MSE and bias do increase as $\tau_{0}$ grows, this increase is much more modest compared to the naive estimator. This stability underscores the effectiveness of our error-corrected, Neyman-orthogonal score function in mitigating the impact of measurement error in high-dimensional settings.

To further illustrate the performance of the naive estimator, Figure (ref) shows the sampling distribution of $\hat{\theta}_{0} - \theta_{0}$ for the naive estimator and the covariance-oblivious Double/Debiased CoCoLASSO estimator when $\tau_{0} = 1.5$.

figure[figure omitted — 512 chars of source]

The rightward shift clearly demonstrates the naive estimator's tendency to systematically overestimate the true parameter value when measurement error is present but unaccounted for. The shape of the distribution deviates significantly from the theoretical normal distribution represented by the dashed line in the figure. The observed distribution is notably left-skewed with a long left tail. This departure from normality is particularly concerning as it indicates that the naive estimator is not only biased but is also asymptotically non-normal, which undermines the validity of inference that necessitates asymptotic normality.

Conclusion

In this paper, we have introduced the Double/Debiased (Covariance-Oblivious) Convex Conditioned LASSO (Double/Debiased CoCoLASSO) estimator for treatment effects in high-dimensional settings with measurement error. Our key contributions are threefold. First, we develop a Neyman-orthogonal score function valid under measurement error, incorporating a bias correction term for error-induced correlation. Second, we propose a method of moments estimator for the measurement error variance, enabling implementation without prior knowledge of error covariance. Third, we establish theoretical results demonstrating $\sqrt{N}$-consistency and asymptotic normality of our estimator.

Simulation studies corroborate our theoretical findings, demonstrating the estimator's robustness to varying levels of measurement error and its superiority over naive approaches. Notably, our covariance-oblivious approach nearly matches the performance of methods assuming known error variance, highlighting its practical utility.

This work opens new avenues for empirical research in economics by providing a framework for dealing with measurement error in high-dimensional settings. Future research directions include extending the approach to nonlinear models and GMM settings, developing methods for selecting the regularization parameter $\lambda$ under measurement error, and applying the error-corrected score function to other machine learning contexts. Further investigation of the estimator's performance under different error structures, such as heteroscedastic or correlated errors, and empirical applications to real-world economic data would provide additional insights into its practical benefits and limitations.

\printbibliography