An approximated exponentially tilted empirical likelihood estimator of moment condition models

Abstract.

This study proposes an approximated estimator for moment condition models. Our estimator approximates the exponentially tilted empirical likelihood (ETEL) estimator in Schennach (2007) by using a second-order approximation of implied probabilities (IPs) for the exponential tilting (ET). The resulting approximated ETEL estimator generates positive IPs. As the nested optimization of ETEL and EL is avoided, it is computationally simple. It is second-order asymptotically equivalent to the EL estimator. In particular, it has the same O(n⁻¹) bias as EL. It also has the same O(n⁻²) variance as EL. Like ET and ETEL estimators, it is $\sqrt{n}$ convergent under model misspecification, while the EL estimator may not be.

Keywords:

• Approximated estimator
• bias empirical likelihood
• exponential tilting
• misspecification

JEL Classification:

• C12
• C13
• C14
• C18
• C51

Acknowledgments

We are grateful for the helpful comments from the Editor (Esfandiar Maasoumi), an associate editor, an anonymous referee, Patrik Guggenberger, Guido W. Imbens, and Andres Santos.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. The analysis is based on Nagar (1959) type higher-order expansions of estimators. It has been criticized in the literature, as EL may even have no finite moments. See, for example, Guggenberger and Smith (2005) and Guggenberger (2008).

2. To solve this problem, all kinds of methods have been proposed in the literature; these include the penalized EL of Bartolucci (2007) and Lahiri and Mukhopadhyay (2012); the adjusted EL of Chen et al. (2008), Emerson and Owen (2009), Liu and Chen (2010), and Chen and Huang (2013); and the extended EL of Tsao (2013), and Tsao and Wu (2013, 2014).

3. Corcoran (1998), Jing and Wood (1996), and Lee and Young (1999) have considered the combination of ET and EL in the construction of likelihood ratio confidence regions for the mean.

4. This approximation method has some similarity to the two-step methods in Trognon and Gouriéroux (1990) and Frazier and Renault (2017) for extremum estimation, where there is an initial estimate of nuisance parameters or awkward occurrences of parameters of interest. Here, zero is a natural, consistent estimator for the auxiliary vector.

5. See Section 4 for a probit model with endogenous regressors, which is widely applied in practice. The sample size for the model is often very large, so various existing estimators are computationally intensive.

6. Other studies on k-step estimators of model parameters include, among others, Rothenberg (1984) and Andrews (2002).

7. For $\gamma =-1$ and 0, h(w) is defined to be the limiting value as γ approaches −1 and 0, respectively.

8. The poor behavior of EL under model misspecification relates to the first-order derivative $(-{\sum }_{i=1}^n\frac{1}{1-{\lambda }^{\prime }{g}_i(\theta )}\frac{\partial g_i^{\prime }(\theta )}{\partial \theta }\lambda )$ of the objective function with respect to θ, where the denominator 1−λ^′g_i(θ) can approach zero. By contrast, the corresponding derivative $(-{\sum }_{i=1}^n{e}^{{\lambda }^{\prime }{g}_i(\theta )}\frac{\partial g_i^{\prime }(\theta )}{\partial \theta }\lambda )$ of ET does not have such a feature (Imbens et al., 1998; Schennach, 2007).

9. If λ₂(θ) in (11(11) $$\begin{array}{l}{w}_{i,{\lambda }_2}(\theta )=\frac{e^{{\lambda }_2^{\prime }(\theta )g_i(\theta )}}{\sum _{j=1}^n{e}^{{\lambda }_2^{\prime }(\theta )g_j(\theta )}}\end{array}$$(11) ) is replaced by λ₁(θ), then the resulting estimator defined similarly as in (12(12) $$\begin{array}{l}{\hat{\theta }}_{\text{AETEL}}=\arg \underset{\theta }{max}\ln L_{\text{AETEL}}(\theta ),\end{array}$$(12) ) generally would not have the desirable higher-order properties in Theorems 4 and 5 below, although they would still have the properties in Theorems 1 and 2.

10. The identification condition and uniqueness of ${\hat{\theta }}_{\text{AETEL}}$ are studied later.

11. This argument on the identification uniqueness condition is similar to that for the exponentially tilted Hellinger distance estimator in Antoine and Dovonon (2021).

12. Note that $\hat{\theta }-{\theta }^{\ast }-n^{-1}\mathrm{E}(Q)=n^{-1/2}\mathrm{\Psi }+n^{-1}[Q-\mathrm{E}(Q)]+n^{-3/2}R+O_p(n^{-2})$, where E⁡(Ψ) = 0. Then, the variance of $\hat{\theta }$ up to order O(n⁻²) is $\mathrm{var}\{n^{-1/2}\mathrm{\Psi }+n^{-1}[Q-\mathrm{E}(Q)]\}+n^{-2}\mathrm{E}(\mathrm{\Psi }{R}^{\prime }+R{\mathrm{\Psi }}^{\prime })$. Thus, we have the O(n⁻²) variance in (17(17) $$\begin{array}{l}{n}^{-2}\mathrm{E}[\mathrm{var}(Q)+\mathrm{\Psi }\cdot (n^{1/2}{Q}^{\prime }+R^{\prime })+(n^{1/2}Q+R){\mathrm{\Psi }}^{\prime }].\end{array}$$(17) ).

13. Let the AETEL IPs be w_j(θ) for $j=1,\dots ,n$. Then, the empirical CDF of g(z, θ) obtained from AETEL’s IPs at θ puts a weight w_j(θ) on g(z_j, θ).

14. Estimation methods for the model include the limited information maximum likelihood (Godfrey and Wickens, 1982), the generalized two-stage simultaneous probit (Amemiya, 1978), the instrumental variables probit (Lee, 1981), the two-stage conditional ML (Rivers and Vuong, 1988), and the GMM (Wilde, 2008).

15. For ET and EL, the outer loop uses the derivative derived from the envelope theorem. All estimates are computed with the Matlab function “minFunc” written by Schmidt (2005), which is for unconstrained optimization using line-search methods and requires fewer function evaluations to converge than the “fminunc” function in Matlab on many problems. The default algorithm “lbfgs” is used. It calls for a quasi-Newton strategy with limited memory BFGS updating. For ET and EL, the inner loop uses a zero vector as the starting value. When ET and EL fail to converge in the inner loop, we follow the convention to set the function value to infinity for subsequent optimization.

16. This is computed as the median of estimates for a parameter minus its true parameter value.

17. The TSEEL estimate is the solution to an estimating equation. Then, the observed outliers might be due to multiple solutions to the estimating equation in finite samples.

18. The derivatives of AETEL’s objective function are provided in Appendix A. Owing to the relatively simple moments for the model in the Monte Carlo study, it does not take a very long time to compute various estimates.

19. For these two models, we have not observed outliers for all estimators. The means for all estimators are smaller than 0. 001 in absolute value, so the comparison of their SDs is meaningful. The pseudo-true values for CU, ET, and ETEL are indeed zero (Schennach, 2007), but we have not derived the pseudo-true value of AETEL theoretically.

Additional information

Funding

Fei Jin gratefully acknowledges the financial support from the National Natural Science Foundation of China (NNSFC) (No. 71973030 and No. 72333002). Yuqin Wang acknowledges the NNSFC financial support (No. 72103122).