Culling the Herd of Moments with Penalized Empirical Likelihood

Abstract

Models defined by moment conditions are at the center of structural econometric estimation, but economic theory is mostly agnostic about moment selection. While a large pool of valid moments can potentially improve estimation efficiency, in the meantime a few invalid ones may undermine consistency. This article investigates the empirical likelihood estimation of these moment-defined models in high-dimensional settings. We propose a penalized empirical likelihood (PEL) estimation and establish its oracle property with consistent detection of invalid moments. The PEL estimator is asymptotically normally distributed, and a projected PEL procedure further eliminates its asymptotic bias and provides more accurate normal approximation to the finite sample behavior. Simulation exercises demonstrate excellent numerical performance of these methods in estimation and inference.

Keywords:

• Empirical likelihood
• Estimating equations
• High-dimensional statistical methods
• Misspecification
• Moment selection
• Penalized likelihood

Supplementary Materials

The supplementary materials consists of three parts. Part A provides the proofs and technical details about the methods developed in the present article. Part B reports additional simulation results concerning the liner IV model in the main text and an additional dynamic panel data model, respectively. Part C checks the robustness of PEL in the empirical application.

Acknowledgments

The three coauthors contributed to this article equally. We thank the editor, the associate editor, and two anonymous referees for their constructive comments and suggestions.

Notes

1 Identification of the simple linear IV model requires the IVs satisfying the orthogonality condition and the relevance condition. However, the more relevant an IV to the endogenous variables, the more likely it is that the so-called “IV” is correlated with the structural error, thereby violating orthogonality. There is a thin line between a valid IV and an invalid one.

2 In GMM estimation under a fixed r, Liao (2013) shrinks $\mathit{\xi }$ toward zero using the adaptive Lasso (Zou 2006).

3 If (3.6) is violated, the asymptotic normality of ${\widehat{\mathit{\theta }}}_{\text{PEL}}$ in Theorem 3.1 will still hold under (3.4) along with more complicated notations to spell out the restrictions.

4 See the RMSEs in Table 2, the length of confidence intervals in Figure 2, and the standard deviations in Table 4.

5 See the arguments below (3.5) for the validity of these assumptions.

6 This is a generic property shared by procedures of the oracle properties, for example SCAD (Fan and Li 2001) and the adaptive Lasso (Zou 2006).

7 Leng and Tang (2012) employ this BIC to choose the tuning parameter for the high-dimensional parameter $\mathit{\psi }$, and Chang, Tang, and Wu (2018) apply it to select multiple tuning parameters. We follow their practice as (6.2) remains valid in our setting where two tuning parameters are included.

8 The oracle EL in Section 2.2 is for low-dimensional parameters and moments. To handle the large pool of orthogonal IVs, the oracle estimator here is a 2SLS estimator taking advantage of a few most relevant IVs—those with the top $0.1n$ big coefficients ${\gamma }_{\mathit{w}2,j}$ in the reduced-form equation.

Additional information

Funding

Chang and Zhang were supported in part by the National Natural Science Foundation of China (grant nos. 72125008, 71991472, 11871401 and 72003150). Chang was also supported by the Center of Statistical Research at Southwestern University of Finance and Economics.