Should we go one step further? An accurate comparison of one-step and two-step procedures in a generalized method of moments framework

Hwang, Jungbin; Sun, Yixiao

doi:10.1016/j.jeconom.2018.07.006

Cited by 93 publications

(27 citation statements)

References 29 publications

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“…In finite sample, g n (θ 0 ) = 0 because g n (θ) = 0 for all θ, and this causes extra variations through the terms in (24) and (26). Similar to the Windmeijer correction, by taking into account for these (asymptotically negligible) terms in estimating the variance we can make more accurate inference.…”

Section: Assume Thatmentioning

confidence: 99%

A Doubly Corrected Robust Variance Estimator for Linear GMM

2019

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We propose a new finite sample corrected variance estimator for the linear generalized method of moments (GMM) including the one-step, two-step, and iterated estimators. Our formula additionally corrects for the over-identification bias in variance estimation on top of the commonly used finite sample correction of Windmeijer (2005) which corrects for the bias from estimating the efficient weight matrix, so is doubly corrected. Formal stochastic expansions are derived to show the proposed double correction estimates the variance of some higher-order terms in the expansion. In addition, the proposed double correction provides robustness to misspecification of the moment condition. In contrast, the conventional variance estimator and the Windmeijer correction are inconsistent under misspecification. That is, the proposed double correction formula provides a convenient way to obtain improved inference under correct specification and robustness against misspecification at the same time.1 two-step linear GMM. Specifically, his formula corrects for the bias arising from using the efficient weight matrix being evaluated at an estimate, rather than the true value. The correction formula (the Windmeijer correction, hereinafter) has been routinely used in practice. 1 However, the Windmeijer correction does not take into account for the over-identification bias, which is another important source of bias in the GMM standard error. The over-identification bias arises from the fact that the over-identified sample moment condition is nonzero while it converges in probability to zero under correct specification.We propose a new finite sample correction which takes into account for the over-identification bias for the variance of the linear one-step, two-step, and iterated GMM estimators. For one-step GMM such as the two-stage least squares (2SLS) estimators, the proposed finite sample correction is new as the Windmeijer correction does not cover the one-step GMM. For two-step and iterated GMM, the proposed correction improves upon the Windmeijer correction by additionally correcting for the over-identification bias. Thus, we doubly correct the finite sample bias of the linear GMM variance estimator. We provide an additional formal justification of the proposed double correction by deriving the stochastic expansions of the GMM estimators under local misspecification where the moment condition is modeled as a local drifting sequence around zero. The result shows that the double correction estimates the (co)variance of some higher-order terms which increase with the overidentification bias. Rothenberg (1984) provides a general treatment of the stochastic expansion and Newey and Smith (2004) derive the stochastic expansion of GMM and the generalized empirical likelihood (GEL) estimators under correct specification. The order of our double correction equals the order of the sample moment condition. Under correct specification or local misspecification, these terms are O p (n −1/2 ) so that the double correction is a finite-sample corr...

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Section: Assume Thatmentioning

confidence: 99%

A Doubly Corrected Robust Variance Estimator for Linear GMM

2019

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“…The location model is an ideal framework to present the basic ideas and intuition, as it abstracts away the unnecessary details and complications. For more discussions, see Hwang and Sun (2015). At the mechanical level, the parameter 0 can be estimated using the GMM.…”

Section: Understanding the Asymptotic F And T Testsmentioning

confidence: 99%

AsymptoticFandttests in an efficient GMM setting

Hwang

Sun

2017

Journal of Econometrics

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This paper considers two-step e¢ cient GMM estimation and inference where the weighting matrix and asymptotic variance matrix are based on the series long run variance estimator. We propose a simple and easy-to-implement modi…cation to the trinity of test statistics in the two-step e¢ cient GMM setting and show that the modi…ed test statistics are all asymptotically F distributed under the so-called …xed-smoothing asymptotics. The modi…cation is multiplicative and involves the J statistic for testing over-identifying restrictions. This leads to convenient asymptotic F tests that use standard F critical values. Simulation shows that, in terms of both size and power, the asymptotic F tests perform as well as the nonstandard tests proposed recently by Sun (2014b) in …nite samples. But the F tests are more appealing as the critical values are readily available from standard statistical tables. Compared to the conventional chi-square tests, the F tests are as powerful, but are much more accurate in size.JEL Classi…cation: C12, C32

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“…In DGPs 2 and 3 that lack slope heterogeneity, the IV estimator is the most efficient when errors are Gaussian (and n is large enough), but not with Cauchy errors, reflecting the greater efficiency of the median (over the mean) when errors are heavy-tailed. Perhaps the additional variance of the two-step estimator due to its use of a long-run variance estimator (for the weighting matrix) makes it less efficient in these cases, a phenomenon explored in non-quantile GMM by Hwang and Sun (2015). Alternatively, perhaps future work can improve the long-run variance estimator's precision, in turn improving the twostep estimator's precision.…”

Section: Monte Carlo Simulationsmentioning

confidence: 99%

“…This is not always true with time series under fixed-smoothing asymptotics(Hwang and Sun, 2015). 10 There are other approaches to achieve efficiency without explicitly estimating the long-run variance, like the (Bayesian) exponentially tilted empirical likelihood ofSchennach (2007, Thm.…”

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Smoothed GMM for quantile models

Castro

Galvao

Kaplan

et al. 2019

Journal of Econometrics

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This paper develops theory for feasible estimators of finite-dimensional parameters identified by general conditional quantile restrictions, under much weaker assumptions than previously seen in the literature. This includes instrumental variables nonlinear quantile regression as a special case. More specifically, we consider a set of unconditional moments implied by the conditional quantile restrictions, providing conditions for local identification. Since estimators based on the sample moments are generally impossible to compute numerically in practice, we study feasible estimators based on smoothed sample moments. We propose a method of moments estimator for exactly identified models, as well as a generalized method of moments estimator for over-identified models. We establish consistency and asymptotic normality of both estimators under general conditions that allow for weakly dependent data and nonlinear structural models. Simulations illustrate the finite-sample properties of the methods. Our in-depth empirical application concerns the consumption Euler equation derived from quantile utility maximization. Advantages of the quantile Euler equation include robustness to fat tails, decoupling of risk attitude from the elasticity of intertemporal substitution, and log-linearization without any approximation error. For the four countries we examine, the quantile estimates of discount factor and elasticity of intertemporal substitution are economically reasonable for a range of quantiles above the median, even when two-stage least squares estimates are not reasonable.

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Should we go one step further? An accurate comparison of one-step and two-step procedures in a generalized method of moments framework

Cited by 93 publications

References 29 publications

A Doubly Corrected Robust Variance Estimator for Linear GMM

A Doubly Corrected Robust Variance Estimator for Linear GMM

AsymptoticFandttests in an efficient GMM setting

Smoothed GMM for quantile models

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