In the GMM framework, researchers rely on the assumption that the moments they use are valid and they test this assumption by means of Hansen’s over-identification test. If the null hypothesis of consistency is rejected, it becomes very challenging to find the max-imum number d0 of available moments that are asymptotically zero. There exist many articles dealing with this issue in the econometric literature. [Ghysels and Hall, 1990]
present a test to discriminate between two non-nested sets of Euler conditions based on GMM estimators. [Smith, 1992] provides a procedure to compare non-nested GMM esti-mators. [Andrews, 1999] derives a class of Moment Selection Criteria (MSC), including MSC-BIC, MSC-HQIC, Downward (DT) and Upward Testing (UT) which asymptoti-cally maximizes the number of moments that are asymptotiasymptoti-cally zero. The latter is extended by [Andrews and Lu, 2001] to moment and model selection and dynamic panel data models. [Donald and Newey, 2001] derive the optimal set of instruments in the
in-1This chapter is co-authored by Elvezio Ronchetti.
strumental variables setting based on the asymptotic approximation of the MSE for the corresponding estimator. In the same spirit as in [Andrews, 1999], [Hong et al., 2003]
derive a procedure based on the Generalized Empirical Likelihood estimator. Follow-ing the concept of canonical correlations, [Hall and Peixe, 2003] develop the Canonical Correlation Information Criterion (CCIC) which finds the non-redundant instruments among the valid moments, as defined by [Breusch et al., 1999]. This idea is extended by [Hall et al., 2007] to derive a procedure based on long run canonical correlations that can choose moments avoiding redundancy and weak identification, where the latter is de-fined in [Nelson and Startz, 1990]. [Caner, 2009] derives a LASSO-type GMM estimator that, under suitable conditions, chooses the right number of non-zero parameters and estimates them given a set of moments. [Hall and Pelletier, 2011] study the behavior of some measures of goodness of fit to discriminate between non-nested GMM estimators and possibly local misspecification. [Carrasco, 2012] considers a possibly infinite number of instruments in the linear IV setting and proposes three types of regularizations that are asymptotically efficient and normally distributed. [Belloni et al., 2012] consider LASSO-type estimators in the linear IV framework that can handle a large number of instruments, even larger than the sample size. Their estimators can also hande non-gaussian and het-eroscedastic errors. [Liao, 2013] develops a penalized criterion that detects invalid and redundant moments and provides estimation of the structural parameters simultaneously.
[Cheng and Liao, 2015] propose a LASSO-based GMM estimators that select the valid and relevant moments with a possibly divergent number of moments. The selection and estimation of the structural parameters is done simultaneously.
In spite of the properties mentioned above, small departures from the underlying data generating process can drastically affect classical MSC. We defined classical MSC as any criterion relying on established GMM estimators in the econometric literature. In this chapter, we focus on the methodology developed by [Andrews, 1999] and investigate its robustness properties by applying the notion of local robustness in the context of GMM moment selection. In the context of IV we provide evidence that the classical consistent procedures are badly affected in the presence of a small contamination in the sample. For
1.1. Introduction 3 example, with a 5% contamination, the probability of choosing d0 for consistent criteria drops by more than 50% compared to the non-contaminated case (from around 90% to over 30%); see section 1.8.1. In constrast, the robust criteria developed in this chapter are less vulnerable to local misspecifications and still select d0 with about 90% probability in the presence of a 5% contamination.
Nowadays, the need for robust statistical methods for estimation and testing is well rec-ognized both in the statistical and econometric literature; cf. for instance, the books [Huber, 1981], 2nd edition by [Huber and Ronchetti, 2009], [Hampel et al., 1986],
[Maronna et al., 2006] in the statistical literature and [Peracchi, 1990], [Peracchi, 1991], [Ronchetti and Trojani, 2001], [Gagliardini et al., 2005], and [Koenker, 2005] in the econo-metric literature. More specifically some papers propose robust methods for model selec-tion in the setting of M-estimators; cf. for instance [Ronchetti, 1985], [Ronchetti, 1997], [Hurvich and Tsai, 1990], [Machado, 1993], [Ronchetti and Staudte, 1994],
[Victoria-Feser, 1997], [Shi and Tsai, 1998], [Cantoni et al., 2005].
This chapter combines both strands of the literature in the following directions.
First we investigate the local robustness properties of Moment Selection Criteria by study-ing their influence function (IF) [Hampel, 1974]. An unbounded IF implies an unbounded behavior of the MSC when the underlying distribution lies in a small neighborhood of the reference model. Therefore, a small outlying proportion of the data can drastically change the moment selection. On the other hand, boundedness of the IF implies that the MSC will display a stable behavior not only at the model but also in a neighborhood of it. In the Moment Selection framework, the boundedness of IF prevents a small fraction of the data from changing the rank between d0 and any other set of orthogonality conditions.
Secondly, we derive robust moment selection criteria, in particular robust versions of MSC-AIC, MSC-BIC, MSC-HQIC, DT and UT, and show that, at the model, they asymp-totically behave like the classical criteria. In contrast to MSC, RMSC display a stable behavior under departures from the reference model. This stability is achieved by impos-ing a bound on the influence function underlyimpos-ing the correspondimpos-ing GMM estimator.
The results are quite general and the new robust moment selection techniques can be
applied in many contexts in econometrics including, for instance, nonlinear and linear instrumental variables and some time series models.
The approach of this chapter for moment selection is rather empirical based, in the sense that there is not necessarily an economic theory supporting the choice of mo-ments. This methodology does not apply when we want to test an economic theory as in [Hansen and Singleton, 1982]. In their framework there is an underlying set of mo-ments that will give rise to the most efficient and consistent estimation of the structural parameters. If this set of moments turns out to be invalid, then the economic theory behind the model is not verified. In this chapter, we do not make any assumption on any theory behind the choice of set of moments. We aim at maximizing the number of moments that are asymptotically zero by taking into account possible departures from the ideal underlying distribution.
This chapter is organized as follows. In the next section, we define the setup of MSC and derive the local robustness properties for valid and invalid moments. In section 1.3, we define a robust GMM estimator (RGMM) based on the local robustness results for both types of moments. In addition, we derive the corresponding RMSC and in section 1.4 we study their properties. In section 1.5, we propose an algorithm to perform robust moment selection. Section 1.6 sheds more light on the relationship between robustness and the properties of the estimators associated with MSC and RMSC. We then apply our methodology in section 1.7 in the linear IV case. Section 1.8 compares the behavior of different MSC and RMSC in a linear IV setup through Monte-Carlo simulations and in an empirical application. Additional assumptions and proofs of the Propositions are collected in the last sections of the chapter.