Semiparametric estimation of moment condition models with weakly dependent data

ABSTRACT

This paper develops the asymptotic theory for the estimation of smooth semiparametric generalized estimating equations models with weakly dependent data. The paper proposes new estimation methods based on smoothed two-step versions of the generalised method of moments and generalised empirical likelihood methods. An important aspect of the paper is that it allows the first-step estimation to have an effect on the asymptotic variances of the second-step estimators and explicitly characterises this effect for the empirically relevant case of the so-called generated regressors. The results of the paper are illustrated with a partially linear model that has not been previously considered in the literature. The proofs of the results utilise a new uniform strong law of large numbers and a new central limit theorem for U-statistics with varying kernels that are of independent interest.

KEYWORDS:

• Alpha-mixing
• empirical processes
• empirical likelihood
• stochastic equicontinuity
• uniform law of large numbers

Acknowledgements

We thank an associate editor and two anonymous referees for various useful suggestions that improve the readability and clarity of the paper. We also acknowledge the usage of the np and gmm packages by Hayfield and Racine (2008) and Chaussé (2010) respectively in the statistical computing environment R. Chu and Jacho-Chávez gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. For an asymptotically equivalent approach based on blocking techniques, see, e.g. Kitamura (1997).

2. We note that Yoshihara (1992) uses an alternative approach to the one we follow to obtain the CLTs (and more generally invariance principles) for α-mixing sequences. His approach relies on the Karhunen–Loève expansion of the kernel and is based on a set of regularity conditions that are not imposed directly on the kernel and thus could be very hard to verify in practice.

3. This implies that a consistent estimator of the efficient metric is given by an appropriately standardised version of the outer product of the smoothed estimating equations , viz. (16) see the proof of Theorem 3.2 for more details.

4. Note that condition (9(7) ) combined with Equation (10(8) ) for for some would suffice to prove a weaker version of the uniform law of large numbers given in Theorem 3.1.

5. This follows since , for and .

6. In fact Lemma C.1 in Smith (2011) states that . However, an examination of the proof reveals that, actually, .

Additional information

Funding

This work was supported by the Social Science and Humanities Research Council of Canada grant (MBF Grant 410-2011-1700).