Sieve-Based Estimators

• Sieve-based estimators are nonparametric methods that approximate infinite-dimensional parameters by projecting them onto a sequence of expanding finite-dimensional spaces.
• They employ flexible basis functions—including polynomials, splines, wavelets, and neural networks—to balance approximation bias and estimation variance in complex models.
• Sieve methods achieve optimal convergence rates and support robust inference in econometrics, statistics, and machine learning through adaptive and penalized estimation techniques.

Sieve-based estimators are a foundational methodology for approximating infinite-dimensional parameters by projecting them onto a sequence of finite-dimensional spaces ("sieves"), which expand with the sample size. By leveraging flexible basis expansions—including polynomials, splines, wavelets, and increasingly, neural networks—sieve methods facilitate efficient estimation and inference in nonparametric, semiparametric, and structural models where classical parametric approaches are insufficient or inapplicable. Sieve-based techniques are especially central in modern econometrics, statistics, and machine learning for function estimation, structural modeling, and inference for functionals. The following exposition synthesizes the key principles, methodological frameworks, asymptotic theory, and applications of sieve-based estimators, referencing current research advances and empirical case studies.

1. Sieve Construction and General Principles

A sieve estimator operates by approximating an unknown infinite-dimensional object (function, distribution, solution path) by elements from a sequence of finite-dimensional spaces $\{\mathcal S_m\}$, called sieves, where the dimension $m$ increases with the sample size. This projection can be linear (e.g., polynomial, spline, or wavelet bases) or nonlinear (e.g., neural networks).

Formally, given data $\{(X_i, Y_i)\}_{i=1}^n$ and a target function $f_0$, the estimator solves: $$\hat{f}_n = \arg\min_{f \in \mathcal S_{m_n}} \frac{1}{n} \sum_{i=1}^n \ell(Y_i, f(X_i)) + \lambda_n J(f)$$ where $\ell$ is a loss (e.g., squared error, likelihood), $J$ is a penalty (e.g., $\ell_1$ or $\ell_2$ regularization), and $m_n$ is a sieve dimension diverging with $n$ at a controlled rate (Zhang et al., 2022, Shen et al., 2019).

The sieve space is typically constructed as: $$\mathcal S_{m_n} = \text{span}\{\phi_1, \ldots, \phi_{m_n}\}$$ where $\{\phi_j\}$ are basis functions such as polynomial, spline, trigonometric, or wavelet bases (Luo et al., 2022, Chen et al., 2013, Seri et al., 2019).

Linear sieves are suited to smooth function approximation, with error rates determined by the smoothness of $f_0$ and the growth of $m_n$. Adaptive sieves, including those based on neural networks (Shen et al., 2019, Chen et al., 2021), allow for nonlinearity and adaptivity in higher dimensions.

2. Penalized Sieve Estimation and High-dimensional Models

Sieve-based estimation in high dimensions leverages sparsity-inducing penalties: $$\hat{\beta} = \arg\min_{\beta} \frac{1}{n} \sum_{i=1}^n (Y_i - \psi(X_i)^\top \beta)^2 + \lambda_n \|\beta\|_1$$ where $\psi(X_i)$ collects multivariate (possibly tensor-product) basis terms, and $\|\beta\|_1$ encourages feature- or basis-selection (Zhang et al., 2022).

The tensor-product construction allows for flexible adaptation: $$\psi_{j_1,...,j_d}(x) = \prod_{k=1}^d \phi_{j_k}(x_k)$$ with orderings that prioritize lower-order interactions, and variable selection or support restrictions to avoid the curse of dimensionality. Sparse additive and ANOVA-type sieves further restrict the space to low-dimensional interactions (Zhang et al., 2022, Lu et al., 2015).

The sieve dimension $J_n$ and penalty $\lambda_n$ are tuned to balance approximation bias and estimation variance, with theoretical and practical selection via cross-validation or information criteria.

3. Asymptotic Theory: Consistency, Rates, and Inference

Sieve methods achieve consistency and optimal rates under standard conditions. For sufficiently rich basis and dimension growth (e.g., $J_n \asymp n^{1/(2s+1)}$ for $s$-smooth functions), mean-squared error decays at the minimax rate, up to logarithmic factors (Zhang et al., 2022, Chen et al., 2013, Shen et al., 2019). For $L^2$ and sup-norm risk,

$$E[\|\hat{f}_n - f_0\|^2] = O\left( \log(d) \log(n) \left( \frac{\log^{D-1} n}{n} \right)^{\frac{2s}{2s+1}} \right)$$

where $D$ is the effective number of active variables.

Uniform convergence (sup-norm rates) is established for wavelet and spline sieves, including in nonparametric IV settings and ill-posed inverse problems (Chen et al., 2013, Chen et al., 2021). Sieve estimators can achieve minimax-optimal uniform rates even in the presence of severe ill-posedness, and the adaptive selection of sieve dimension (e.g., via Lepski's method) is shown to yield bands and confidence intervals contracting at the optimal rate (Chen et al., 2021).

Semiparametric inference, including root-$n$ normality for finite-dimensional parameters nested in nonparametric models, is covered via the sieve M-theorem for bundled parameters, efficient influence functions, and plug-in procedures (Ding et al., 2012, Qiu et al., 2020). For certain functionals (e.g., average derivatives), sieve-based estimators, including those leveraging neural networks as nonlinear sieves, achieve semiparametric efficiency (Chen et al., 2021, Qiu et al., 2020).

4. Sieve Methods in Complex Models: Structural, Dynamic, and High-frequency Data

Sieve-based approaches are pivotal for models intractable for standard estimators:

Structural Models: Sieve-based efficient estimators (SEES) for structural models (e.g., games of entry) use linear combinations of basis functions to approximate equilibrium or value-function solutions, adding a penalty to enforce equilibrium conditions (Luo et al., 2022). These approaches avoid costly nested algorithms, guarantee consistency, root-$n$ normality, and efficient variance, and simplify standard error computation.

Time Series and Long Memory:

• Sieve VAR and AR approximations are used for forecasting and inference in VAR($\infty$) or fractionally integrated models, with attention to the subtlety that naive sieve-asymptotic confidence intervals can be too conservative in finite samples (Ballarin, 2021, Poskitt et al., 2014).
• Bootstrapped sieve procedures—such as the pre-filtered sieve bootstrap—offer near-pivotal inference and bias correction for long-memory parameters (Poskitt et al., 2014), balancing analytical adjustments and finite-sample properties.

Dynamic Latent Models:

• Sieve-SMM estimators flexibly approximate latent shock distributions in dynamic models using Gaussian-tail mixture sieves, ensuring robustness against parametric misspecification and enabling inference for both structural parameters and entire distributions (Forneron, 2019).
• Sieve-based strategies are used to estimate both constant and time-varying coefficients in nonlinear ODEs, carefully balancing the sieve approximation, numerical solver error, and statistical measurement error (Xue et al., 2010).

High-frequency and Lévy Process Estimation:

• For Lévy density estimation with discrete, high-frequency data, sieve-projection estimators achieve minimax rates and allow construction of both pointwise confidence intervals and uniform bands, employing Legendre-spline bases and Gumbel-limit results for extremal deviations (Figueroa-López, 2011).

5. Extensions: Online, Machine Learning, and Efficient Plug-in Methods

Recent advances extend sieve theory in several meaningful directions:

Online Estimation:

• Sieve stochastic gradient descent (Sieve-SGD) enables online nonparametric regression in Sobolev ellipsoids, achieving minimax rates with nearly minimal computational and memory overhead (Zhang et al., 2021).

Integration with Machine Learning:

• Neural network sieves provide universal approximators for regression and instrumental variables models, with sieve estimation theory guaranteeing consistency, optimal rates, and, when properly tuned, root-$n$ normal plug-in inference for smooth functionals (Shen et al., 2019, Chen et al., 2021).
• Hybrid estimators combine sieve and kernel approaches for high-dimensional additive models, producing valid uniform confidence bands even in large $d$, under sparsity assumptions (Lu et al., 2015).

Universal Efficient Estimation:

• Sieve-based methods underpin "universal" plug-in inference strategies utilizing highly adaptive lasso (HAL) or data-adaptive series around any $o_p(n^{-1/4})$ machine learning fit. Provided the initial estimator is sufficiently accurate, the sieve projection guarantees asymptotically efficient inference for a wide class of functionals without explicit derivation of influence functions (Qiu et al., 2020).
• Data-driven procedures for optimal sieve dimension yield minimax-optimal uniform bands, outperforming traditional undersmoothing approaches in adaptive inference (Chen et al., 2021).

6. Applications and Empirical Illustrations

Sieve-based estimators are applied in a range of substantive settings:

• Industrial Organization: Entry games estimated via SEES (e.g., Walmart–Kmart market entry) yield statistically significant and economically interpretable estimates of competitive effects (Luo et al., 2022).
• Optimal Transport: Sieve-based estimation for the entropic optimal transport problem provides consistency and finite-sample guarantees under minimal smoothness assumptions, contrasting empirical Sinkhorn estimators (Tabri, 26 Dec 2025).
• Econometric Demand Systems: Nonparametric IV estimation with neural net sieves recovers demand elasticities and average derivatives with valid inference, outperforming or matching spline-based competitors in empirical demand estimation (Chen et al., 2021).
• Experimental Psychology and Economics: Sieve least-squares estimators in multi-subject, multi-task experiments yield optimal rates for function recovery, optimal design balances for $n$ vs.\ $T$, and valid Wald tests for linear restrictions (Seri et al., 2019).

7. Computational Aspects and Practical Considerations

Sieve methods are computationally scalable:

• Linear sieves admit closed-form solutions or efficient coordinate-descent algorithms.
• Nonlinear sieves (neural networks) require gradient-based training with regularization and cross-fitting to control estimation error.
• Adaptive partitioning (e.g., in multivariate density estimation) leverages greedy heuristics to manage exponential partition complexity (Liu et al., 2014).
• Online settings utilize SGD updates to avoid repeated refitting (Zhang et al., 2021).

Tuning of sieve dimension, choice of basis, regularization parameters, and bootstrap or asymptotic variance estimation are crucial for practical effectiveness and theoretical guarantees.

---

In sum, sieve-based estimators represent a flexible, adaptive, and theoretically well-founded toolkit for nonparametric, semiparametric, and structural estimation across diverse statistical models. Their ongoing integration with machine learning and high-dimensional inference continues to expand the frontiers of practical statistically valid estimation and inference (Luo et al., 2022, Chen et al., 2021, Chen et al., 2021, Qiu et al., 2020).