Double robust inference for continuous updating GMM

Kleibergen, Frank; Zhan, Zhaoguo Article Double robust inference for continuous updating GMM Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Kleibergen, Frank; Zhan, Zhaoguo (2025) : Double robust inference for continuous updating GMM, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 16, Iss. 1, pp. 295-327, https://doi.org/10.3982/QE2347 This Version is available at: https://hdl.handle.net/10419/320333 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 16 (2025), 295–327 1759-7331/20250295 Double robust inference for continuous updating GMM Frank Kleibergen Amsterdam School of Economics, University of Amsterdam Zhaoguo Zhan Coles College of Business, Kennesaw State University We propose the double robust Lagrange multiplier (DRLM) statistic for testing hypotheses specified on the minimizer of the population continuous updating objective function. The (bounding) χ2limiting distribution of the DRLM statistic is robust to both misspecification and weak identification, hence its name. The minimizer is the so-called pseudo-true value, which equals the true value of the structural parameter under correct specification. To emphasize its importance for applied work where misspecification and weak identification are common, we use the DRLM test to analyze: the risk premia in Adrian et al. (2014) and He et al. (2017); the structural parameters in a nonlinear asset pricing model with constant relative risk aversion. Keywords. Weak identification, misspecification, robust inference, Lagrange multiplier. JEL classification. C12, C18, G12. 1. Introduction A little more than 20 years ago, inference procedures for analyzing possibly weakly identified structural parameters using the generalized method of moments (GMM) of Hansen (1982) were mostly lacking. Since then, huge progress has been made to develop such procedures; see, for example, Staiger and Stock (1997), Dufour (1997), Stock and Wright (2000), Kleibergen (2002,2005,2009), Moreira (2003), Andrews and Cheng (2012), Andrews and Mikusheva (2016a,2016b), and Han and McCloskey (2019). At present, we therefore have a variety of so-called weak identification robust inference methods. Given the prevalence of weak identification in applied work, a lot of emphasis has also been put in raising awareness among practitioners; see, for example, Kleibergen and Mavroeidis (2009), Beaulieu, Dufour, and Khalaf (2013), Mavroeidis, Plagborg-Moller, and Stock (2014), Andrews, Stock, and Sun (2019), and Kleibergen and Zhan (2020). When there is no so-called true value of the structural parameters where the GMM moment conditions exactly hold, the structural model is rendered misspecified, and GMM estimators provide inconsistent estimates of the true value. Early research on Frank Kleibergen: [email protected] Zhaoguo Zhan: [email protected] The research of Frank Kleibergen has been funded partly by the NWO (Dutch Research Council) Grant 401.21.EB.002: “Double robust inference for structural economic models (DRISEM)”. ©2025 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE2347 296 Kleibergen and Zhan Quantitative Economics 16 (2025) misspecification focused on inference on the true value by characterizing expressions for the bias and standard errors of inconsistent estimators; see, for example, Maasoumi and Phillips (1982)andMaasoumi (1990). Instead of focusing on the unattainable true value, recent work on misspecification analyzes the so-called pseudo-true value, which is the minimizer of the population objective function; see, for example, White (1994). For practical relevance of the pseudo-true value see, for example, Kan, Robotti, and Shanken (2013) who use the pseudo-true value for evaluating asset pricing models. The pseudotrue value depends on the population objective function at hand, so different objective functions can lead to distinct pseudo-true values. Hall and Inoue (2003), for example, develop inference methods for the pseudo-true value of the two-step GMM estimator, while Hansen and Lee (2021)do so for an iterated GMM estimator. We use the minimizer of the population continuous updating estimator (CUE) objective function of Hansen, Heaton, and Yaron (1996) as the pseudo-true value because of its invariance properties and since weak identification robust inference procedures lead to inference that is centeredaroundit. In case of misspecification, weak identification robust inference procedures for testing hypotheses specified on the pseudo-true value however become size distorted for just small amounts of misspecification. This would not sound as much of a problem if it was possible to efficiently detect such misspecification. This is not so since misspecification tests, like the Sargan–Hansen test (Sargan (1958)andHansen (1982)), are virtually powerless in settings of joint misspecification and weak identification; see Gospodinov, Kan, and Robotti (2017). Weak identification robust inference procedures thus came about to overcome the general critique of nonrobustness of traditional inference procedures to varying identification strengths (see, e.g., Staiger and Stock (1997)andDufour (1997)), but are similarly nonrobust to misspecification. While weak identification robust inference procedures are size distorted when misspecification is present, the misspecification robust inference procedures proposed by, for example, Hall and Inoue (2003), Kan, Robotti, and Shanken (2013), Lee (2018), and Hansen and Lee (2021), provide misspecification robust covariance matrix estimators to conduct Wald-based inference. Because the covariance matrix estimators are finite by construction, the resulting Wald-based tests cannot lead to unbounded confidence sets, which as shown by Dufour (1997), is a necessary condition for size correct testing in settings with potential identification failure. Hence, except for certain specifications of the structural parameter (see, e.g., Gospodinov, Kan, and Robotti (2014)),1these misspecification robust tests are potentially size distorted under weak identification. One of the first to emphasize the empirical relevance of misspecification in the presence of weak (or no) identification were Kan and Zhang (1999). With the surge in applied work on structural estimation, awareness of misspecification has grown further. In asset pricing models, for example, it is now generally accepted that misspecification, alongside weak identification, is an important empirical issue; see, for example, Kan, Robotti, 1Gospodinov, Kan, and Robotti (2014) show that the Wald/ttest for a zero risk premium in a linear asset pricing model with useless partialled out factors, which uses their adjusted standard error, is bounded by a χ2distribution and, therefore, size correct for the hypothesis of interest. Quantitative Economics 16 (2025) Double robust inference for GMM 297 and Shanken (2013)andKleibergen and Zhan (2020). Kan, Robotti, and Shanken (2013) therefore developed misspecification robust t-statistics for the Fama–MacBeth (1973) two-pass estimator, that is, the most commonly used estimator of the risk premia in linear asset pricing models. Similarly, in case of heterogeneous treatment effects, the local average treatment effects that result for different instruments can be distinct (see Imbens and Angrist (1994)), making the overidentified linear instrumental variables regression model that uses all instruments misspecified. Evdokimov and Kolesar (2018) and Lee (2018) therefore analyze misspecification robust tests on the treatment effect resulting from using such multiple instruments. These misspecification robust tests are however not robust to weak identification, so identical to the weak identification robust inference procedures, they cannot deal with the empirically relevant setting of both misspecification and weak identification. Yet the aforementioned weak identification and misspecification literature highlight that these two issues have to be taken into account in order to conduct valid inference on the structural parameters of interest. We extend the weak identification robust score or Lagrange multiplier (KLM) test from Kleibergen (2002,2005,2009) to a double robust Lagrange multiplier (DRLM) test, which can also be interpreted as the misspecification robust version of the KLM test. The DRLM test is size correct and robust to both misspecification and weak identification, hence its name. The DRLM statistic is a quadratic form of the score function, which equals zero at all stationary points of the CUE sample objective function. This is also the case for the KLM statistic and explains the power problems of the KLM test; see, for example, Andrews, Moreira, and Stock (2006). To overcome the power problems of the KLM test, the KLM statistic can be combined in a conditional or unconditional manner with the Anderson–Rubin (1949, AR) statistic; see, for example, Moreira (2003) and Andrews (2016). Andrews, Moreira, and Stock (2006) show that the conditional likelihood ratio (LR) test of Moreira (2003) provides the optimal manner of combining these statistics for the homoskedastic linear instrumental variables regression model with one included endogenous variable. In case of misspecification, it is however not obvious how to improve the power of the DRLM test by such combination arguments, since the statistics with which the DRLM statistic is to be combined to improve power, have non-central limiting distributions with parameters that cannot be consistently estimated under misspecification. We therefore show that the power of the DRLM test can be improved by exploiting the derivative property of the DRLM statistic. The rest of the paper is organized as follows. In Section 2, we discuss continuous updating GMM with misspecification and propose the DRLM test. To illustrate the size correctness of the DRLM test, we conduct a simulation experiment using linear moment equations in Section 3. A power study of the DRLM test and weak identification robust tests is also presented in Section 3. It shows that weak identification robust tests on the pseudo-true value of the structural parameters are size distorted for just small amounts of misspecification while the DRLM test is not. It also proposes power improvement and shows that the resulting test has generally good power. Section 4applies the DRLM test to risk premia using asset pricing data from Adrian, Etula, and Muir (2014)andHe, Kelly, and Manela (2017). Section 5extends the DRLM test to subvector inference, strong misspecification, which is dealt with by an additional component in the weight matrix, 298 Kleibergen and Zhan Quantitative Economics 16 (2025) and nonlinear moment equations from a nonlinear asset pricing model with constant relative risk aversion. Section 6concludes. Technical details and additional results are relegated to the Supplemental Appendix (Kleibergen and Zhan (2024b)) . 2. DRLM test for possibly misspecified GMM 2.1 Setup We analyze the m×1 parameter vector θ=(θ1θm)whose parameter region is the Rm. The kf×1 dimensional function f(θ,Xt)is a continuously differentiable function of the parameter vector θand a Borel measurable function of a data vector Xt,whichisobserved for time/individual t. Since we focus on misspecification, the model is overidentified, that is, there are more moment equations than structural parameters so kf>m. The population moment function of f(θ,Xt)equals μf(θ): EXf(θ,Xt)=μf(θ),(1) with μf(θ)akf-dimensional continuously differentiable function. Unlike regular GMM (see Hansen (1982)) , we do not request that there is a specific value of θ,sayθ0,atwhich μf(θ0)=0. Our analysis thus differs from a recent one proposed by Cheng, Dou, and Liao (2022), who construct a model selection procedure for evaluating potentially misspecified models with possibly weakly identified structural parameters, which explicitly uses a set of base moments contained in all considered models that are guaranteed to hold. We analyze θusing the continuous updating setting of Hansen, Heaton, and Yaron (1996). We use it because of its invariance properties and since it leads to inference based on identification robust statistics in standard GMM; see, for example, Stock and Wright (2000)andKleibergen (2005). For asset pricing studies, in particular, Peñaranda and Sentana (2015) recommend the continuous updating setting over two-step or iterated GMM procedures. The accompanying population continuous updating objective function is Qp(θ)=μf(θ)Vff (θ)−1μf(θ),(2) with Vff (θ)the covariance matrix of the sample moment:2fT(θ,X)=1 TT t=1f(θ,Xt), Vff (θ)=lim T→∞ETfT(θ,X)−μf(θ)fT(θ,X)−μf(θ),(3) so fT(θ,X)is the sample analog of μf(θ)for a data set of Tobservations: Xt,t=1, ,T. We define the pseudo-true value of θ,θ∗as the minimizer of the population objective function: θ∗=arg min θ∈RmQp(θ).(4) 2Throughout the paper, we use recentered covariance matrices while the continuous updating estimator is identical under a recentered or uncentered version of the same covariance matrix estimator; see Hansen and Lee (2021). Quantitative Economics 16 (2025) Double robust inference for GMM 299 Thus, the pseudo-true value θ∗equals the true value of θin correctly specified models. In misspecified models, the interpretation of θ∗depends on the context: for example, in asset pricing studies, Kan, Robotti, and Shanken (2013) interpret the pseudo-true value of risk premia as the value that minimizes pricing errors. The sample objective function for the CUE is ˆ Qs(θ)=fT(θ,X)ˆ Vff (θ)−1fT(θ,X),(5) with ˆ Vff (θ)a consistent estimator of Vff (θ),ˆ Vff (θ)→ pVff (θ),sotheCUE, ˆ θis ˆ θ=arg min θ∈Rmˆ Qs(θ).(6) At the CUE, the score or derivative of the CUE objective function equals zero:3 ˆ s(θ)=1 2 ∂ ∂θˆ Qs(θ)=fT(θ,X)ˆ Vff (θ)−1ˆ D(θ),(7) where ˆ D(θ)=qT(θ,X)−ˆ Vq1f(θ)ˆ Vff (θ)−1fT(θ,X) ˆ Vqmf(θ)ˆ Vff (θ)−1fT(θ,X),(8) with qT(θ,X)=∂fT(θ,X) ∂θ|θ=(q1T(θ)qmT (θ)),J(θ)=∂ ∂θμf(θ)=(J1(θ)Jm(θ)), ˆ Vqif(θ)is a consistent estimator of Vqif(θ)=limT→∞ E[T(qiT (θ)−Ji(θ))(fT(θ,X)− μf(θ))],i=1, ,m. Correspondingly, the population counterpart of the sample score ˆ s(θ)in (7)is s(θ)=1 2 ∂ ∂θQp(θ)=μf(θ)Vff (θ)−1D(θ),(9) where D(θ)is the recentered Jacobian: D(θ)=J(θ)−Vq1f(θ)Vff (θ)−1μf(θ)Vqmf(θ)Vff (θ)−1μf(θ).(10) Our proposed double robust Lagrange multiplier statistic on the pseudo-true value equals a quadratic form of the sample score ˆ s(θ)in (7), which involves the product of the sample moment and a recentered estimator of its Jacobian. While the sample moment fT(θ,X)is driven by the magnitude of misspecification, the recentered Jacobian estimator ˆ D(θ)reflects the strength of identification. Thus, both misspecification and weak identification are to be accounted for when we develop the explicit expression of the double robust Lagrange multiplier statistic based on ˆ s(θ)in (7). We focus on weak (local) misspecification in the sense that at the pseudo-true value θ∗,thereexists ˜μfθ∗=lim T→∞ √Tμfθ∗,(11) 3The construction of the derivative is in Kleibergen (2005) and also in Lemmas 3 and 4 in the Supplemental Appendix. 300 Kleibergen and Zhan Quantitative Economics 16 (2025) for ˜μf(θ∗)a finite valued kf-dimensional vector, which could equal zero. Thus, μf(θ∗) is allowed to depend on Twith the dependence kept implicit for the purpose of notational simplicity. This treatment thus differs from a strong misspecification case for which μf(θ∗)is considered a fixed nonzero vector. We discuss the strong misspecification case later in Section 5.2 as an extension. The proposed double robust Lagrange multiplier test (for a properly specified weight matrix) applies to both of these cases. Analogous to the treatment of μf(θ∗)in (11), the weak identification literature widely uses the so-called weak instrument asymptotics (see, e.g., Staiger and Stock (1997)), which in our context amounts to using a local to zero sequence for D(θ∗): ˜ Dθ∗=lim T→∞ √TDθ∗,(12) for ˜ D(θ∗)a finite valued kf×mdimensional matrix, which could be of lower rank. Although we initially use (12) for ease of exposition, it is worth noting that the double robust Lagrange multiplier test is equally applicable under strong identification as we show when simultaneously discussing strong misspecification in Section 5.2. Put differently, whether D(θ∗)is local to zero or fixed makes no difference to the double robust Lagrange multiplier statistic, so we allow for all strengths of identification. 2.2 Example: The linear IV regression model For illustration, we use a linear instrumental variables (IV) regression model:4 yt=βxt+εt, xt=zt+vt,(13) where βand are m×1andkf×mdimensional matrices containing unknown parameters, ytand xtare the scalar and m×1 dimensional endogenous variables, ztis the kf×1 dimensional instrumental variables, εtand vtare the scalar and m×1dimensional errors. Its population moment function reads μf(β)=σzy −zxβ =μf(0)+J(0)θ, (14) where, assuming that the observations over the individuals are i.i.d., σzy =E((zt− μz)(yt−μy)),zx =E((zt−μz)(xt−μx))=Qzz,Qzz =E((zt−μz)(zt−μz)),μy= E(yt),μx=E(xt),μz=E(zt). The last line of (14)usestheGMMnotationfrom(1)and (10)soμf(0)=σzy ,J(0)=−zx,andθ=β. Misspecification occurs when the structural error εtis correlated with the instruments zt. There is then no value of βat which the 4For expository purposes, we only discuss a simplified version of the linear IV regression model without so-called, included exogenous variables. We specify the linear IV regression model, and also the linear asset pricing model discussed later, using the notation generically used in the literature, and also show their connection to our earlier generic GMM notation. Quantitative Economics 16 (2025) Double robust inference for GMM 301 population moment function (14) is equal to zero. On the other hand, weak identification occurs when instruments ztare only weakly correlated with the endogenous xt,so the full rank condition of zx is at risk. The population objective function of the CUE for the linear IV model with homoskedastic errors is Qp(β)=1 ωuu −2ωuvβ+βvvβ(σzy −zxβ)Q−1 zz (σzy −zxβ),(15) where uv =(ωuu ωvu ωuv vv )=cov(ut,vt),forut=εt+v tβ. The CUE then corresponds with the so-called, limited information maximum likelihood (LIML) estimator, so we can use the well-known k-class notation of the LIML estimator (see, e.g., Hausman (1983)and Andrews (2019)) to express the pseudo-true value as β∗=arg min β∈RmQp(β)= zxQ−1 zz zx −τminvv−1 zxQ−1 zz σzy −τminωvu, (16) with τmin =minβ∈RmQp(β).5We note that different from the usual k-class notation for τmin =0 the pseudo-true value corresponds with the pseudo-true value of the two-stage least squares estimator; similarly for τmin =−1, the pseudo-true value corresponds with that of the least squares estimator. For the linear IV regression model, the double robust Lagrange multiplier test is designed for testing hypotheses on β∗, and it addresses weak identification and misspecification simultaneously. The test is based on the score or derivative of Qp(β),which equals zero at β∗. The quadratic form of the sample score constitutes the test statistic, whose explicit expression will be provided using the generic GMM notation in Section 2.4. 2.3 Assumptions We state the assumptions needed for constructing the large sample behavior of test statistics centered around the CUE for the generic GMM setting. These assumptions concern the components of the sample score ˆ s(θ)=fT(θ,X)ˆ Vff (θ)−1ˆ D(θ). We first make Assumption 1as in Kleibergen (2005) except that it concerns the large sample behavior of the sample moments and their derivative at the pseudo-true value θ∗instead of the true value. Assumption 1. The kf×1dimensional derivative of ft(θ)=f(θ,Xt)with respect to θi, qit(θ)=∂ft(θ) ∂θi :kf×1, i=1, ,m(17) is such that the joint limit behavior of the sums of the series ¯ ft(θ)=ft(θ)−E(ft(θ)) and ¯ qt(θ)=(¯ q1t(θ)¯ qmt(θ)),with ¯ qit(θ)=qit(θ)−E(qit(θ)),accords with the central limit 5The minimal value τmin equals the smallest root of |τuv −(σzy . . .zx)Q−1 zz (σzy . . .zx)|=0. 302 Kleibergen and Zhan Quantitative Economics 16 (2025) theorem at θ=θ∗,where θ∗is the minimizer of the continuous updating population objective function: 1 √T T  t=1¯ ftθ∗ ¯ qtθ∗→ dψfθ∗ ψqθ∗∼N0, Vθ∗,(18) where ψf:kf×1, ψq:kθ×1, kθ=mkf,and V(θ∗)is a positive semidefinite symmetric (kf+kθ)×(kf+kθ)matrix, Vθ∗=Vff θ∗Vfqθ∗ Vqf θ∗Vqqθ∗, (19) with Vqf (θ∗)=Vfq(θ∗)=(Vq1f(θ∗)Vqmf(θ∗)),Vqq(θ∗)=(Vqiqj(θ∗)) :i,j=1, ,m; Vff (θ∗),Vqif(θ∗),Vqiqj(θ∗)are kf×kfdimensional matrices for i,j=1, ,m,and Vθ∗=lim T→∞var√TfTθ∗,X vecqTθ∗,X. (20) Assumption 1requests a joint central limit theorem to hold at the pseudo-true value for the sample moments and their derivative. It is satisfied under mild conditions, which are listed in Kleibergen (2005) like, for example, finite rth moments for r>2incaseof i.i.d. data, mixing conditions for the sample moments in case of time-series data. Allowing for a positive semidefinite covariance matrix V(θ∗)is important for applications like, for example, dynamic linear panel data models. We next also use Assumption 2from Kleibergen (2005), which concerns the convergence of the covariance matrix estimator. Assumption 2. The convergence behavior of the covariance matrix estimator ˆ V(θ)toward V(θ)is such that (i)ˆ V(θ)→ pV(θ),and (ii)∂vec(ˆ Vff (θ)) ∂θ→ p ∂vec(Vff (θ)) ∂θ. Assumption 2(i) requests a consistent covariance estimator for V(θ). For Assumption 2(ii), the covariance matrix estimator ˆ Vff (θ)has to be such that the ˆ Vqf (θ),which results from the derivative of ˆ Vff (θ)with respect to θ, is also a consistent estimator of Vqf (θ). When using the same covariance matrix estimator for all elements of V(θ),Assumption 2holds under the conditions for consistency of (heteroskedastic autocorrelation consistent) covariance matrix estimators; see, for example, White (1980), Newey and West (1987). Lemma 11 in the Supplemental Appendix provides the low-level regularity conditions under which Assumption 2holds in the leading i.i.d. case. ThescoreoftheCUEobjectivefunctionin(7) factorizes as the product of the sample moment and a recentered estimator of its Jacobian (8). Under Assumptions 1and 2, the limit behaviors of the sample moment and this recentered Jacobian estimator are independent at the pseudo-true value θ∗: √TfTθ∗,X−μfθ∗→ dψfθ∗∼N0, Vff θ∗, √Tvecˆ Dθ∗−Dθ∗→ dψθθ∗∼N0, Vθθθ∗, (21) Quantitative Economics 16 (2025) Double robust inference for GMM 309 The mean of the limit behavior of the sample score in (31) equals zero when the normalized identification strength measure based on the Jacobian, ¯ D¯ D, and misspecification strength measure, ¯μ¯μ, are equal. It shows that the pseudo-true value θ∗is then not identified. 3.4 Simulated power of DRLM Next, we illustrate the power of 5% significance DRLM tests of H0:θ∗=0comparedto other weak identification robust tests including GMM-AR, KLM, and LR tests; see, for example, Stock and Wright (2000), Moreira (2003), and Kleibergen (2005). We consider a simulation setting with kf=25 moment equations, weak misspecification, ¯μ¯μ=10, and varying identification strength. When there is no misspecification, Andrews, Moreira, and Stock (2006) show that the LR test is optimal, so we refrain from using that setting. Figures 3and 4show the power curves of KLM (panel 3.1), DRLM (panel 3.2), LR (panel 4.1), and GMM-AR (panel 4.2) tests of H0:θ∗=0 for various identification strengths and a fixed amount of misspecification. The power curves of the different tests in Figures 3and 4show that only the DRLM test is size correct for all settings of the identification strength. The size distortion of some of the other weak identification robust tests can be quite pronounced, which especially holds for the LR and GMM-AR tests. For the LR test, the rejection frequency at zero decreases from 30% to 8% when the identification strength increases, while for the KLM test, it decreases from 10% to 5%. The rejection frequency of the GMM-AR test at zero is equal to 36% for all identification strengths. Corollary 2shows that the pseudo-true value is not identified when the identification strength equals the amount of misspecification, ¯ D¯ D=¯μ¯μ. This holds for a value of around three (≈√10)on the “length ¯ D” axis, which explains why the power curves for all tests are flat in the opposite direction at this value. For the DRLM test, the power curve is flat with a rejection frequency, which is proven to be at most 5%, while for the other tests it exceeds 5%, and for the LR and GMM-AR tests even by a substantial amount. Figure 3. Power of 5% significance KLM and DRLM tests of H0:θ∗=0 with misspecification, ¯μ¯μ=10, kf=25, =I2. 310 Kleibergen and Zhan Quantitative Economics 16 (2025) Figure 4. Power of 5% significance LR and GMM-AR tests of H0:θ∗=0 with misspecification, ¯μ¯μ=10, kf=25, =I2. When ¯ D¯ Dis less than ¯μ¯μ, the minimal value of the population continuous updating objective function is no longer at θ∗but at θ1=−1 θ∗.9TheGMM-ARstatisticis the sample analog of the population continuous updating objective function, which explains why its power is minimal at zero and maximal at infinity when “length ¯ D” exceeds three, so ¯ D¯ D> ¯μ¯μ, and vice versa when “length ¯ D” is less than three, so ¯ D¯ D< ¯μ¯μ.In the latter case, the pseudo-true value is not identified because it is at infinity. Kleibergen and Zhan (2024a) prove that the pseudo-true value of the CUE is not identified when the population value of the traditional rank statistic of the Jacobian, which they label IS, equals the minimal value of the CUE population objective function, which they label MISS and equals the population value of the J-statistic. They also show that IS −MISS is always nonnegative and provides a measure of the identification strength in misspecified linear GMM whose sample analog is a (quasi-) likelihood ratio no-identification statistic. When ¯ D¯ Dis less than ¯μ¯μ,MISS =IS, which further indicates that the pseudotrue value is then not identified. Figure 5shows the distribution function of the misspecification J-statistic, which equals the minimal value of the GMM-AR statistic when the null hypothesis holds, so for values of θ∗equal to zero. It shows the distribution function for three different values of the identification strength ¯ D¯ D: 0, 10, and 100. Recognizing that the 95% critical value of the χ2(24)distribution, since kf−1=24, is about 36.42, Figure 5shows that we never reject no misspecification at the 5% significance level when ¯ D¯ Dequals 0, 7% of the times when ¯ D¯ D=10, and 33% when ¯ D¯ Dequals 100. This indicates the difficulty of detecting misspecification; see also Gospodinov, Kan, and Robotti (2017). The power curves of the LR and GMM-AR tests in Figure 4increase when moving away from the hypothesized value and when the identification strength exceeds 9Thefirstelementofthelimitbehaviorofthesamplescorein(31) equals zero at the two stationary points of the CUE population objective function: θ∗and −1 θ∗. The sign of the Hessian also results from it and is indicated by ¯μ¯μ−¯ D¯ D.Hence,when ¯μ¯μ−¯ D¯ D<0, the stationary points at θ∗and −1 θ∗are the minimum and maximum, respectively, and vice versa when ¯μ¯μ−¯ D¯ D>0; see Lemma 2(i) in the Supplemental Appendix. Quantitative Economics 16 (2025) Double robust inference for GMM 311 Figure 5. Distribution function of the J-statistic for misspecification when θ∗=0 holds, solid line: ¯ D¯ D=0, dash-dot: ¯ D¯ D=10 =strength of misspecification, dashed: ¯ D¯ D=100. the measure of misspecification, ¯ D¯ D> ¯μ¯μ. This however does not hold for the power curves of the two LM tests in Figure 3, which are eventually decreasing. To improve the power of the DRLM test, we propose to use a property of its derivative as discussed next. 3.5 Power improvement with m=1 The score is equal to zero at all stationary points of the CUE sample objective function, so the same holds for tests based on a quadratic form of it like, for example, the DRLM and KLM tests, as well. It explains the power decreasing away from the null hypothesis for the KLM and DRLM tests in Figure 3. Tests with better power properties therefore exist in correctly specified GMM that implicitly or explicitly combine the KLM test with an asymptotically independent J-test in either a conditional or unconditional manner; see Moreira (2003), Kleibergen (2005), Andrews, Moreira, and Stock (2006), Andrews (2016), and Andrews and Mikusheva (2016a,2016b). In our misspecified GMM setting, this is however not possible since the limiting distribution of the J-statistic is a noncentral χ2 distribution with an unknown noncentrality parameter. Hence, we cannot combine this limiting distribution with that of the DRLM statistic to obtain the (conditional) critical values for a combination test. To improve the power of the DRLM test, we aim to reject hypothesized pseudo-true values of θ, which are close to a stationary point of the CUE sample objective function other than the CUE. This would be similar to the conditional or unconditional identification robust combination tests in regular GMM, which use that while the KLM test does not reject at such values of θ,J, and/or GMM-AR tests (see Anderson and Rubin (1949) and Stock and Wright (2000)) likely do. For hypothesized values of θclose to the CUE, these combination tests put most weight on the KLM test, but shift the weight toward the Jand GMM-AR tests when θis close to other stationary points; see Andrews (2016) and Kleibergen (2007). Since the limiting distributions of the Jand GMM-AR statistics depend on unknown nuisance parameters in our misspecified GMM setting, it is not clear how we can use these statistics to improve power. 312 Kleibergen and Zhan Quantitative Economics 16 (2025) Stationary points of the sample CUE objective function lead to zero values of the DRLM statistic. The CUE is one of these stationary points and leads to the smallest value of the objective function. To improve the power of the DRLM test, we further reject values of θthat lead to nonsignificant values of the DRLM statistic, which result from stationary points different from the CUE. We can do so by rejecting values of θin a closed set of nonrejected values that does not contain the CUE, for which an illustrative application is presented later in Section 4(see Figure 7; a closed set of nonrejected values is on the left of Figure 7as discussed for that figure). To prove why the above power improvement rule leads to a size correct test procedure, we use the derivative of the DRLM statistic (see Lemma 12 in the Supplemental Appendix). In line with the previous Section 3.3, we also focus on m=1 when discussing power improvement in Theorem 3below. Theorem 3. When Assumptions 1and 2hold,m=1, fT(θ,X)is linear in θ,and testing H0:θ∗=θ∗ 0,with θ∗the pseudo-true value,at the 100 ×α%significance level,the power improved DRLM testing procedure that rejects a pseudo-true value of θboth if: 1. the DRLM statistic is significant at the 100 ×α%significance level or 2. the DRLM statistic is not significant but the hypothesized pseudo-true value of θlies in a closed set of nonsignificant values,which does not contain the CUE with some significant values in between the closed set and the CUE, is a test procedure of H0:θ∗=θ∗ 0with size 100 ×α%. Proof. See the Supplemental Appendix. While the generic specification of the DRLM test is for a stationary point of the population continuous updating objective function, the power improved DRLM test from Theorem 3explicitly tests for the minimizer. Later, Section 4contains an empirical application (see Figure 7) that illustrates the power improvement rule from Theorem 3,in particular, its Step 2. When computing the size of the test at the hypothesized value of, say, zero, we therefore have to ascertain that it is the minimizer of the population objective function. For the setup in Figures 1–2, which uses the limit expression of the DRLM statistic in (29), the population minimizer is at zero if the amount of misspecification is less than the strength of identification so the length of ¯μis less than that of ¯ D. When the length of ¯μexceeds that of ¯ D, the minimizer of the population objective function is at ±∞. In standard GMM, there is no misspecification, so the minimal value of the population objective function is equal to zero. The amount of misspecification is then always less than or equal to the identification strength, so the hypothesized value automatically corresponds with the minimizer of the population objective function. 3.6 Simulated power of power improved DRLM Figure 6shows the rejection frequency and power of the power improved DRLM test. Panel 6.1 contains the rejection frequency when the minimizer of the population continuous updating objective function equals the hypothesized value, which is zero. It therefore does not show the rejection frequency for values where the length of ¯μexceeds that Quantitative Economics 16 (2025) Double robust inference for GMM 313 Figure 6. Power improved 5% significance DRLM tests of H0:θ∗=0 using a conditional 95% critical value as a function of the lengths of ¯μand ¯ D(panel 6.1), ¯μ¯μ=10 (panel 6.2), kf=25, =I2. of ¯ D, since the hypothesized value does then not correspond with the minimizer of the population objective function, which is at ±∞. The rejection frequencies in panel 6.1 are computed using the calibrated conditional critical values. In line with Theorem 3, panel 6.1 shows that the power improvement does not affect the size of the DRLM test when the hypothesized value equals the minimizer of the continuous updating population objective function. Panel 6.2 of Figure 6shows power curves for the power improved DRLM test. It uses the same setup as for the power curves in Figures 3and 4.Atfirstglance,thepower improved DRLM test in panel 6.2 seems minorly-size distorted because its rejection frequency can reach 8% when the length of ¯ Dis below 3. At these values, the minimizer of the population CUE objective function is however at ±∞since the misspecification exceeds the identification strength, so the 8% is indicative of power and not of size distortion. When the minimizer is at zero, so the identification strength exceeds the amount of misspecification, that is, the length of ¯ Dexceeds 3, the rejection frequency of the DRLM test is at most 5% and reveals no size distortion. For the latter setting, panel 6.2 also shows that the power curves are (almost) monotonic and the decrease in power that we observed for the DRLM test in panel 3.2 is no longer present. 4. Applications To show the importance and relevance of the DRLM test for applied research where linear moment equations are common, we briefly revisit the linear models considered in Adrian, Etula, and Muir (2014)andHe, Kelly, and Manela (2017) using our DRLM test and the identification robust GMM-AR, KLM, and LR tests; see also Kleibergen (2009) and Kleibergen and Zhan (2020). For the linear IV regression model, the Supplemental Appendix contains an empirical study using the data of Card (1995). 314 Kleibergen and Zhan Quantitative Economics 16 (2025) Adrian, Etula, and Muir (2014) propose a leverage risk factor (“LevFac”) for asset pricing. The leverage level is the ratio of total assets over the difference between total assets and liabilities, and the leverage risk factor equals its log change. The empirical study of Adrian, Etula, and Muir (2014) uses quarterly data between 1968Q1 and 2009Q4. Following Lettau, Ludvigson, and Ma (2019), we extend the time period to 1963Q3– 2013Q4 and use N=25 size and book-to-market sorted portfolios as test assets. Adrian, Etula, and Muir (2014) show that the leverage factor prices the cross-section of many test portfolios, as reflected by the significant Fama–MacBeth (FM) (1973) and Kan–Robotti– Shanken (KRS) t-statistics on the risk premium reported in Table 1.TheKRSt-statistic is robust to misspecification but not to weak identification; see Kan, Robotti, and Shanken (2013). He, Kelly, and Manela (2017) propose the banking equity-capital ratio factor (“EqFac”) for asset pricing. We consider one of their specifications with “EqFac”and the market return “Rm” as the two factors. The significant FM and KRS t-statistics for the risk premium on “EqFac”inTable1show that this factor is considered to be priced by the test assets. DRLM: Adrian, Etula, and Muir (2014) Using the same data as for Table 1, Figure 7shows the p-values for testing the risk premium on the leverage factor (horizontal line) using the DRLM, GMM-AR, KLM, and LR tests. We also apply the LR no-identification test of Kleibergen and Zhan (2024a), whose test statistic 3.55 is considerably below its 95% conditional critical value of 83.3 (for conditioning statistic equal to 119), so we cannot reject the hypothesis that the pseudotrue value is not identified at the 5% (and much larger) significance level. Most of the p-values in Figure 7are therefore above the 5% level, which implies that none of the DRLM, GMM-AR, KLM, and LR tests leads to tight 95% confidence intervals for the risk premium on the leverage factor as shown in Table 1. Given the smallish p-value of the J-test, 0.20, and the likely weak identification of the risk premium on the leverage factor reflected by the unbounded 95% confidence sets, misspecification could be present, so it would be appropriate to use the DRLM test. The p-values of the DRLM test in Figure 7are equal to one at two different points. The p-values of the GMM-AR test show that one of these two points relates to the minimal value of the GMM-AR test and the other one to the maximal value of the GMM-AR test. Using the power enhancement rule for DRLM stated in Theorem 3,wecanreject nonsignificant values resulting from the DRLM test that lie within the closed interval indicated by the significant maximizer of the GMM-AR statistic (around the left peak in Figure 7, which does not contain the CUE), so the nonsignificant p-values of the DRLM test, which occur around the maximizer of the GMM-AR test can all be categorized as significant ones according to the power enhancement rule. The resulting 95% confidence set for the DRLM test rejects a zero value of the risk premium of the leverage factor and is reported in Table 1alongside the one which results from just applying the DRLM test. The FM and KRS t-statistics reported in Table 1also reject a zero value of the risk premium, but these tests are not reliable because of the potentially weak identification of the risk premium of the leverage factor and the likely misspecification reflected Quantitative Economics 16 (2025) Double robust inference for GMM 315 Table 1. Inference on risk premia λFin Adrian, Etula, and Muir (2014)andHe, Kelly, and Manela (2017). The test assets are the N=25 size and book-to-market portfolios from 1963Q3 to 2013Q4 taken from Lettau, Ludvigson, and Ma (2019). “LevFac” is the leverage factor of Adrian, Etula, and Muir (2014). “EqFac” is the banking equity-capital ratio factor of He, Kelly, and Manela (2017). “Rm” is the market return. The estimate of λF and the FM t-statistic result from the Fama–MacBeth (1973) two-pass procedure. The KRS t-statistic is based on the KRS t-test of Kan, Robotti, and Shanken (2013). The point estimates of λFare identical to those reported in Lettau, Ludvigson, and Ma (2019). The F-statistic rank test is from Kleibergen and Zhan (2020) for testing H0:rank(β)=m−1, with mthe number of risk factors. The LR no-identification test and 95% conditional critical value are from Kleibergen and Zhan (2024a). Adrian, Etula, and Muir (2014)He, Kelly, and Manela (2017) LevFac RmEqFac Estimate of λF13.91 1.19 6.88 FM t3.58 0.81 2.14 KRS t2.55 0.77 2.10 CUE of λF51.77 23.22 94.02 95% confidence set FM t(6.29, 21.54) (−1.67, 4.05) (0.57, 13.19) KRS t(3.22, 24.60) (−1.84, 4.22) (0.46, 13.30) DRLM (−∞,−91.4)∪(−9.2, 1.7)∪(17.8, +∞)( −∞,+∞)( −∞,+∞) DRLM (power enh.) (−∞,−91.4)∪(17.8, +∞)( −∞,+∞)( −∞,+∞) GMM-AR (−∞,−101.3)∪(18.3, +∞)( −∞,−64.6)∪(8.1, +∞)( −∞,−244.1)∪(37.7, +∞) KLM (−∞,−185.5)∪(−5.7, −0.9)∪(20.8, +∞)( −∞,−7.2)∪(−4.7, −0.3)∪(1.0, +∞)( −∞,−23.8)∪(−8.1, 1.7)∪(11.8, +∞) LR (−∞,−274.2)∪(21.8, +∞)( −∞,−9.7)∪(2.2, +∞)( −∞,−33.8)∪(16.2, +∞) IS χ2-statistic (p-value) 31.97 (0.13) 35.88 (0.04) Rank F-statistic (p-value) 1.17 (0.28) 1.33 (0.16) LR no-identification (95%) 3.55 (83.3) 0.57 (56.3) 316 Kleibergen and Zhan Quantitative Economics 16 (2025) Figure 7. Adrian, Etula, and Muir (2014). p-value from the DRLM (dashed red), GMM-AR (dashed blue), KLM (solid black), LR (dash-dotted green), and the 5% level (dotted black). J-statistic (=minimum GMM-AR) equals 28.42, with p-value of 0.20 resulting from χ2(N−2), IS =31.97. by the smallish p-value of the J-test. Since the zero risk premium is rejected by DRLM, the leverage factor does appear to have its pricing ability; on the other hand, since the confidence set of the risk premium is wide, we cannot precisely infer the leverage risk premium by using the adopted data. DRLM: He, Kelly, and Manela (2017) Figure 8shows the joint 95% confidence sets (shaded areas) of the risk premia on the banking equity-capital ratio factor “EqFac” and the market return “Rm,” from using the DRLM, GMM-AR, KLM, and LR tests. The p-value of the J-test shows that misspecification is likely present, so it is appropriate to use the DRLM test for the confidence set of the minimizer of the population continuous updating objective function. The LR noidentification statistic of Kleibergen and Zhan (2024a), 0.57 is far below its 95% conditional critical value, 56.3 (for conditioning statistic equal to 114), so we cannot reject the hypothesis that the pseudo-true value is not identified. The 95% confidence sets of the DRLM and KLM tests consist of two or three rather disjoint sets. The power enhancement rule for the DRLM test shows that the smaller disjoint closed set can be discarded for the joint 95% confidence set that results from the DRLM test. The resulting 95% confidencesetfromtheDRLMtestincludesazerovaluefortheriskpremiumon“EqFac”and is also unbounded, which indicates that the pricing ability of “EqFac”isunderdoubt. Gospodinov and Robotti (2021) also criticize the two-factor model of He, Kelly, and Manela (2017). They warn that “EqFac”and“Rm” are closely related, so jointly using them could lead to a reduced rank of the beta matrix. To compare with Figure 8,wereplace the “EqFac” risk factor with the “SMB” (small minus big) factor from Fama and French (1993) and similarly construct Figure 9. The GMM-AR test now indicates misspecification, since it rejects every hypothesized risk premia as shown in panel 9.2, so the 95% confidence set that results from the GMM-AR test is empty. The LR noidentification statistic is now 69 with a 5% conditional critical value of 56.4 (for conditioning statistic equal to 214, see Kleibergen and Zhan (2024a)), so we reject the hypothesis that the pseudo-true value is not identified at the 5% significance level. Our Quantitative Economics 16 (2025) Double robust inference for GMM 317 Figure 8. He, Kelly, and Manela (2017). 95% confidence sets from DRLM, GMM-AR, KLM, and LR. J-statistic (minimum of GMM-AR) equals 35.32, with p-value of 0.036 resulting from χ2(N−3),IS =35.88. DRLM test, which allows for misspecification therefore yields a tight confidence set in panel 9.1. This tight confidence set, in contrast with the wide one in panel 8.1, indicates that the pricing ability of “EqFac” differs substantially from “SMB,” that is, “SMB” leads to stronger identification of risk premia than “EqFac.” Because of the misspecification, the 95% confidence sets resulting from the KLM and LR tests are not representative for the minimizer of the population objective function. 5. Extensions We next briefly discuss how the DRLM test can be used for subvector inference. Following up, we state the DRLM test that also allows for strong misspecification. Thereafter, we illustrate the size properties of the DRLM test for a setting with nonlinear moment equations and discuss an application to it. 5.1 Subvector inference The expression of the DRLM statistic applies to settings where the structural parameter vector potentially has multiple elements, so the pseudo-true value θ∗=(θ∗ 1θ∗ m)with m≥1. Many times, practitioners are interested in constructing confidence sets on a sub- 318 Kleibergen and Zhan Quantitative Economics 16 (2025) Figure 9. Rmand SMB. 95% confidence sets from DRLM, GMM-AR, KLM, and LR. J-statistic (minimum of GMM-AR) equals 59.34, with p-value of 0.00 resulting from χ2(N−3),IS =128.35. vector or individual element of the structural parameter vector. We discuss how to use the DRLM test for constructing such a confidence set in the Supplemental Appendix. 5.2 DRLM test for strong misspecification For the strong misspecification case, that is, μf(θ∗)is considered as a fixed nonzero vector instead of a drifting sequence to zero, the weight matrix involved in the DRLM statistic needs to be modified. For the general robustness of the DRLM test, we extend Assumption 1to Assumption 1∗, which concerns the joint limit behavior of the sample moment, its derivative, and the covariance matrix estimator. 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