[Paper Review] Inference in partially identified models with many moment inequalities using Lasso

Federico A. Bugni, Mehmet Caner+2 more|arXiv (Cornell University)|Apr 8, 2016

Advanced Causal Inference Techniques4 citations

TL;DR

This paper proposes a two-step inference procedure for partially identified models with many moment inequalities, using Lasso for moment selection in the first step to improve power while maintaining uniform asymptotic size control. The method achieves better power than existing approaches across much of the parameter space, with implementation via thresholding standardized sample averages.

ABSTRACT

This paper considers inference in a partially identified moment (in)equality model with many moment inequalities. We propose a novel two-step inference procedure that combines the methods proposed by Chernozhukov, Chetverikov and Kato (2018a) (CCK18, hereafter) with a first step moment inequality selection based on the Lasso. Our method controls asymptotic size uniformly, both in underlying parameter and data distribution. Also, the power of our method compares favorably with that of the corresponding two-step method in CCK18 for large parts of the parameter space, both in theory and in simulations. Finally, we show that our Lasso-based first step can be implemented by thresholding standardized sample averages, and so it is straightforward to implement.

Motivation & Objective

• Address inference in partially identified econometric models with a large number of moment inequalities, where traditional methods fail due to high-dimensional moment conditions.
• Extend the framework of Chernozhukov et al. (2018a) to include many unstructured moment equalities, broadening applicability to real-world models.
• Develop a two-step inference procedure that enhances power compared to existing methods while maintaining uniform asymptotic size control over the parameter space and data distribution.
• Provide a computationally feasible implementation of the first-step moment selection using Lasso via thresholding standardized sample averages.
• Ensure the method remains valid under weak dependence and heavy-tailed moment conditions, with theoretical guarantees under minimal assumptions.

Proposed method

• Propose a two-step inference procedure: first, use Lasso to select active moment inequalities based on standardized sample averages, forming a data-driven subset of inequalities.
• In the second step, apply the CCK18 (2018a) multiplier bootstrap or empirical bootstrap to construct critical values for hypothesis testing.
• Leverage the Lasso’s ability to consistently select relevant moment inequalities even when the number of inequalities $ p $ grows faster than the sample size $ n $.
• Control asymptotic size uniformly over the parameter space and data distribution by combining Lasso selection with bootstrap-based critical values.
• Use the Borell–Cirelson–Sudakov inequality and moment bound conditions to derive uniform bounds on the quantiles of the maxima of Gaussian processes.
• Establish theoretical validity through a three-part proof: (1) showing that the Lasso-selected set is contained in the bootstrap-selected set, (2) relating bootstrap critical values to theoretical quantiles, and (3) proving uniform size control via concentration inequalities.

Experimental results

Research questions

• RQ1 Can Lasso-based moment selection improve the power of inference in partially identified models with many moment inequalities without sacrificing uniform asymptotic size control?
• RQ2 How does the proposed method perform relative to the CCK18 (2018a) two-step procedure in terms of power across different regions of the parameter space?
• RQ3 Is the Lasso-based first step implementable in practice, and can it be computed via simple thresholding of standardized sample averages?
• RQ4 Under what conditions does the Lasso selection consistently identify the true set of binding moment inequalities in high-dimensional settings?
• RQ5 Can the method maintain uniform asymptotic size control when the number of moment inequalities $ p $ grows at exponential rates relative to sample size $ n $?

Key findings

• The proposed method controls asymptotic size uniformly over both the parameter space and the data distribution, ensuring valid inference under weak regularity conditions.
• The power of the method is strictly greater than that of the CCK18 (2018a) two-step procedure over large parts of the parameter space, both theoretically and in simulations.
• The Lasso-based first step can be implemented by thresholding standardized sample averages, making the method computationally straightforward and accessible.
• Theoretical guarantees are established under minimal assumptions, including moment bounds and weak dependence, allowing for $ p $ growing at exponential rates relative to $ n $.
• The method remains valid even when moment inequalities are unstructured and not conditionally derived, extending the scope of existing inference frameworks.
• The proof establishes that $ P(T_n o c_n^{B,2S}(eta_n)) o 1 $ uniformly, with error bounds decaying at polynomial rates, ensuring consistency of the inference procedure.