Specification testing for conditional moment restrictions under local identification failure
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Dovonon, Prosper; Gospodinov, Nikolaj Article Specification testing for conditional moment restrictions under local identification failure Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Dovonon, Prosper; Gospodinov, Nikolaj (2024) : Specification testing for conditional moment restrictions under local identification failure, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 15, Iss. 3, pp. 849-891, https://doi.org/10.3982/QE2242 This Version is available at: https://hdl.handle.net/10419/320313 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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 15 (2024), 849–891 1759-7331/20240849 Specification testing for conditional moment restrictions under local identification failure Prosper Dovonon Department of Economics, Concordia University, CIREQ, and CIRANO Nikolay Gospodinov Research Department, Federal Reserve Bank of Atlanta In this paper, we study the asymptotic behavior of specification tests in conditional moment restriction models under first-order local identification failure with dependent data. More specifically, we obtain conditions under which the conventional specification test for conditional moment restrictions retains its standard normal limit when first-order local identification fails but global identification is still attainable. In the process, we derive some novel intermediate results that include extending the firstand second-order local identification framework to models defined by conditional moment restrictions, establishing the rate of convergence of the GMM estimator and characterizing the asymptotic representation for degenerate U-statistics under strong mixing dependence. Importantly, the specification test is robust to first-order local identification failure regardless of the number of directions in which the Jacobian of the conditional moment restrictions is degenerate and remains valid even if the model is first-order identified. Keywords. GMM, conditional moment restrictions, test for overidentifying restrictions, local and global identification, first-order local identification failure, second-order local identification, U-statistics, strong mixing dependence, robustness. JEL classification. C01, C1, C14, G12. 1. Introduction While economic models are designed to be only partial and incomplete representations of real economic phenomena, it is still highly desirable to quantify the degree of model misspecification and the directions along which the model performance is unsatisfactory. Even if the model is rejected by the data, it can still be useful for policy analysis but Prosper Dovonon: [email protected] Nikolay Gospodinov: [email protected] We thank four referees and seminar and conference participants at LSE, Princeton University, UCL, the 2022 meeting of the Société Canadienne de Science économique (Montreal, Canada), and the 2023 Africa Meeting of the Econometric Society (Nairobi, Kenya) for insightful comments and suggestions. The first author acknowledges financial support from the Social Sciences and Humanities Research Council of Canada. The views expressed here are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE2242
850 Dovonon and Gospodinov Quantitative Economics 15 (2024) the inference and model comparison procedures should be adjusted for the underlying model uncertainty. For these reasons, it is now common practice to first subject the candidate models to specification testing before committing to a particular analytical and inference framework. There are at least two characteristics of economic models that make the development of fully robust and reliable specification testing procedures more challenging. First, economic models are typically defined by a set of conditional moment restrictions. The standard approach is to resort to the law of iterated expectations and reduce the conditional restrictions to unconditional moment restrictions that are then used to design the proper estimation and testing framework. When this is done in ad hoc manner, this approach could result in loss of efficiency and even in inconsistency of the estimator (see, e.g., Dominguez and Lobato (2004)). On the other hand, a transformation that preserves the information in the conditional moment restrictions leads to modified tests based on a continuum of moment conditions (see Bierens (1982), Bierens and Ploberger (1987), de Jong and Bierens (1994), Carrasco and Florens (2000), Kitamura, Tripathi, and Ahn (2004) among others). A common feature of all these tests is that they rely on root-n consistent estimators, which are readily available in models that are first-order locally identified. Second, it is often the case that the moment restriction model is locally underidentified. In linear models, for example, the lack of first-order local identification—rank deficiency of the Jacobian matrix of the moment conditions—implies global identification failure, which typically renders the standard specification tests invalid under both the null and alternative hypotheses as the power of the test, in certain contexts, is bounded by its size (Gospodinov, Kan, and Robotti (2017)). The intuition behind this result is that it is sometimes possible to recast the optimal specification test as a reduced rank test, which highlights the difficulty of determining if the reduced rank is induced by correct specification or identification failure. In nonlinear models, however, first-order identification is no longer a necessary condition for global identification. This paper builds on these two strands of literature to obtain conditions under which the conventional specification tests of conditional moment conditions remain valid under first-order local identification failure. It should be noted that this is not the case in unconditional moment restriction models, where the second-order local identification leads to overrejection of the standard specification test (Dovonon and Renault (2013)). By contrast, our proposed test is characterized by robust properties as it preserves its standard asymptotic limit irrespective of whether the model is first-order or secondorder identified. More specifically, the proposed conditional specification test retains its validity in the presence of possible uncertainty about the parameter values or moment conditions that determine first-order local identification. These robustness properties constitute the main advantage of our approach and warrant some additional remarks. Importantly, the proposed test is agnostic to the precise form of the local identification failure, that is, the degree of rank deficiency of the expected Jacobian matrix of the moment conditions, the values of the model parameters that give rise to this rank deficiency or if the Jacobian is exactly zero. In the latter case, and if prior knowledge of the zero Jacobian is available to the researcher, Lee and Liao (2018) propose to augment the set
Quantitative Economics 15 (2024) Robust specification testing 851 of moment conditions with this additional restriction that restores the first-order local identification and standard inference. However, if the source of the rank deficiency is different or first-order local identification holds, imposing these Jacobian restrictions would be invalid and lead to erroneous inference. By contrast, the implementation and validity of our proposed test obviates the need to take a stand on the form of the rank deficiency of the Jacobian matrix or, more generally, whether the model is first-order locally identified or not. When the model is indeed first-order locally identified, the conditional specification test continues to be characterized by the standard normal limit. To this end, we formalize the concepts of point identification and first-order local identification failure in conditional moment restriction models. Similar to pointidentified unconditional models, the first-order local identification failure allows only for a reduced number of directions of the parameter vector to be identified while identification of the remaining directions is obtained via a second-order expansion of the moment conditions. We then proceed with characterizing the limiting behavior of the estimator and the specification test in models with an expanding set of moment conditions when first-order local identification fails but global identification is still attainable. Our main contributions can be summarized as follows. First, we extend the test for validity of conditional moment restrictions (de Jong and Bierens (1994); Donald, Imbens, and Newey (2003)) to moment condition models that are first-order degenerate. We establish our results in a two-step generalized method of moments (GMM) framework with general forms of moment condition models and dependent data. While the test proposed by de Jong and Bierens (1994) is obtained in the context of nonlinear regression models, the test by Donald, Imbens, and Newey (2003) is also developed within the GMM framework for a specific choice of basis functions and cross-sectional data. Our results therefore show that the GMM-based test of Donald, Imbens, and Newey (2003) retains its size control with dependent data and under first-order local identification failure so long as second-order local identification holds. We should note that the extension to dependent data and characterizing the limiting behavior of the GMM estimator and the specification test in this context is nontrivial. We outline the conditions under which the specification test with an increasing number of unconditional moment restrictions is robust to the type of singularity arising from first-order local identification failure. More specifically, we extend the notion of second-order local identification to the setting of models defined by conditional moment restrictions. The limiting behavior of the GMM estimator and the specification test are studied in the setup where point identification holds, first-order local identification fails while local identification is maintained at second order. We show that the GMM estimator, based on the expanding moment restrictions, estimates the directions of the parameters that are locally first-order identified at the standard √n-rate while the remaining directions are estimated at a slower rate. Interestingly, this rate is faster than the n1/4-rate in second-order identified models with a fixed number of moment restrictions (Dovonon and Renault (2020)). In the conditional setting, the expanding number of moment restrictions enhances the identification signal and accelerates the rate of convergence. We also derive the asymptotic distribution of
852 Dovonon and Gospodinov Quantitative Economics 15 (2024) the GMM estimator in the scalar case, which highlights the highly nonstandard limiting behavior of the estimator. Despite this nonstandard asymptotic setup, we show that the test for validity of conditional moment restrictions is characterized by a standard normal limit even when the first-order local identification condition is compromised. Another important intermediate result that we develop in the paper is a central limit theorem (CLT) for degenerate U-statistics with linear kernels of increasing dimension under strong mixing dependence. The CLT is novel and of independent interest. Establishing the asymptotic normality of the test for overidentifying restrictions draws heavily on this CLT. The rest of the paper is structured as follows. Section 2introduces the main conditional moment restriction setup and the testing framework. It also presents the notions of firstand second-order local identification along with alternative characterizations in the context of conditional moment restrictions. In order to enhance the intuition behind the identification framework, Section 3discusses the model with common conditionally heteroskedastic features, which is later explored further in simulations and in the empirical application. This example also allows us to highlight the robustness of our testing approach to knowledge about the precise structure of the model. The asymptotic properties of the GMM estimator are analyzed in Section 4.Section5proposes a CLT for Ustatistics under strong mixing dependence and establishes the asymptotic normality of the specification test statistic under the null hypothesis. In addition, this section shows that this test is consistent against all alternatives. Section 6reports simulation results for the proposed specification test and provides an empirical application for the presence of common conditionally heteroskedastic features in portfolio bond returns. Section 7 concludes. Proofs and additional results are provided in Appendices A and B, and the Supplemental Material (Dovonon and Gospodinov (2024)). Throughout the paper, we use the following notation. For any matrix C,C2= √λmax(CC)denotes the spectral norm, where λmax(·)is the largest eigenvalue function. If Cis a vector, this amounts to its Euclidean norm as well. Also, let λmin(·)denote the smallest eigenvalue function, and Z,N,andRmsignify the set of all integers, the set of natural numbers, and the set of real m×1 vectors, respectively. Furthermore, Card(S) denotes the cardinality of a finite set S, defined to be the number of elements in the set S,vec(C)signifies column vectorization of a matrix C,a∨bdenotes the maximum of a and b,Rank(C)is the rank of a matrix C,andDiag(c11,c22,,cmm )denotes an m×m diagonal matrix with (c11,c22,,cmm )on its main diagonal. Convergence in probability and convergence in distribution are denoted by P →and d →, respectively, while the abbreviation a.s. stands for almost surely. Let {Xt:t∈Z}be a sequence of random variables and Fb abe the σ-algebra generated by {Xt:−∞≤a≤t≤b≤∞}.Then{Xt}is said to be strong mixing or α-mixing (Andrews (1984)) if sup −∞<t<∞ sup A∈Ft −∞,B∈F∞ t+sPr(A∩B)−Pr(A)Pr(B)=α(s)→0ass→∞. Finally, an=oP(1)denotes that the sequence antends to zero in probability and an= OP(1)signifies that anis bounded in probability.
Quantitative Economics 15 (2024) Robust specification testing 853 2. Model and identification 2.1 Conditional moment restriction setup In this paper, we consider a single conditional moment restriction model: Eu(yt,θ0)|xt=0a.s., (1) where uis a real-valued function, θ0∈⊂Rpis the parameter of interest, and {(xt,yt)}t is a sequence of Rkx×Rky-valued random vectors. Many economic equilibrium models take this conditional moment restriction form. A prominent example of the role of conditioning is the stochastic discount factor framework in asset pricing (see, for instance, Hansen (2014)).1In this setup, the null hypothesis of validity of the conditional moment restriction in (1)is H0:PrEu(yt,θ0)|xt=0=1(2) against the alternative H1:PrEu(yt,θ)|xt=0<1, for any θ∈.(3) While the single conditional restriction setup covers a wide range of practically relevant models, we focus on this case merely for the sake of notational simplicity. The main results in this paper carry over to higher-dimensional conditional moment restrictions at the cost of more cumbersome notation. Consistent estimation of θ0using (1) requires point identification, that is, for all θ∈ , ρ(xt,θ):=Eu(yt,θ)|xt=0, a.s. ⇔θ=θ0.(4) Moreover, inference about θ0hinges on the sharpness of the slope of the function θ→ ρ(x,θ)at θ0. The local behavior of this function determines the rate of convergence of the estimator of θ0. The standard approach to inference relies on a local identification condition, which states that Eρθ(x,θ0)ρθ(x,θ0)is nonsingular, (5) where ρθ(x,θ):=E(∇θu(y,θ)|x)with ∇θu(y,θ0)=∂u(yt,θ)/∂θ|θ=θ0. Following the literature on unconditional moment restriction models (Sargan (1983); Dovonon and Renault(2013); Dovonon and Hall (2018); among others), we shall refer to this condition as first-order local identification condition for conditional moment restriction models. This connection between the identification setups for unconditional and conditional restriction models is formalized in the next subsection. As pointed out in the Introduction, this paper considers a framework where the conditional moment model is point identified but there is a failure of the first-order local identification condition. 1For a comprehensive recent discussion of these issues in the context of asset pricing models, we refer the reader to Antoine, Proulx, and Renault (2020).
854 Dovonon and Gospodinov Quantitative Economics 15 (2024) The rest of our main analytical and testing framework can be summarized as follows. Consider the separable Hilbert space L2(P):=L2(Rkx,B(Rkx),P)of square Pintegrable real-valued functions defined on Rkx,wherePis the common probability distribution of xt’s—that are assumed to be stationary—and B(Rkx)is the Borel σ-algebra of Rkx.WeequipL2(P)with the inner product: f(·),h(·)=E(f(xt)g(xt)).Let(gl)l∈N be a countable basis (not necessarily orthonormal) of L2(P)and g(k):=(g1,,gk).2 We investigate the null hypothesis in (2) by proposing a test for the sequence of unconditional moment restrictions Egl(xt)u(yt,θ0)=0, l=1, ,k;t=1, ,n, or written more compactly as Eg(k)(xt)u(yt,θ0)=0, (6) where k=k(n)with k(n)→∞as n→∞. The GMM estimator, based on these restrictions, is defined as ˆ θ=argmin θ∈¯ fk(θ)ˆ Wk¯ fk(θ),(7) where ¯ fk(θ)=1 √n n t=1 fk(xt,yt,θ),fk(xt,yt,θ)=g(k)(xt)u(yt,θ), and ˆ Wkis a sequence of symmetric, positive definite weighting matrices. The specification test that we introduce next is expressed as a function of the two-step GMM estimator, which uses the weighting matrix ˆ Wk=ˆ V−1 k,ˆ Vk=1 n n t=1 fk(xt,yt,˜ θ)fk(xt,yt,˜ θ).(8) The preliminary (first-step) GMM estimator ˜ θused in (8) is obtained by commonly setting ˆ Wk=Ikor, more generally, to a nonrandom matrix sequence Wk,0. Furthermore, we will derive our results under the condition that the sequence (fk(xt,yt,θ0))tis serially uncorrelated. This is ensured by our maintained assumption that Eu(yt,θ0)|Ft=0, where Ft=σxt,u(yt−1,θ0),xt−1,u(yt−2,θ0),.(9) 2To enhance power, the sequence of functions (gl)lis chosen as an enumeration of some series expansion that does not depend on θ. Furthermore, as discussed by de Jong and Bierens (1994), the conditioning random variable xcanbeconsideredtobeboundedsince E(Y|x)=EY|(x) for any one-to-one function that maps Rkxinto a compact subset D⊂Rkx.Inthatrespect,ifx is not initially a bounded random variable, one can consider g((x)), instead of g(x). Examples of bounded transformations (·)include the componentwise arc tangent function, that is, x→ arctan(x)= (arctan(x1),,arctan(xkx)), while choices of enumeration of weight functions (gl)linclude polynomial, trigonometric, and flexible Fourier form families (de Jong and Bierens (1994); see also Andrews (1991), and Gallant (1981), for more details on these families).
Quantitative Economics 15 (2024) Robust specification testing 855 In fact, under this condition and for any k,(fk(xt,yt,θ0))tis a martingale difference sequence with respect to its natural filtration. The weighting matrix for the two-step GMM estimator is thus constructed as the inverse of ˆ Vk,where ˆ Vkis a sum of outer product of fk(xt,yt,˜ θ)as defined by (8). Our goal is to derive the asymptotic distribution of the test statistic of the null hypothesis in (2) ˆ Z=1 √2k¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ)−k(10) under first-order local identification failure. Characterizing the limiting distribution of ˆ Zrequires that we determine the limiting behavior of ˆ θunder (i) an expanding set of moment conditions (k(n)→∞as n→∞) and (ii) second-order local identification. We show that even in this highly nonstandard identification setup, the test ˆ Zretains its N(0, 1)limit under H0in (2)—which is the asymptotic distribution of the test in the standard identification setting (de Jong and Bierens (1994))—and is consistent under H1 in (3). 2.2 Identification Note that, for any kand any vector of instruments zt=g(xt)∈Rk, a function of xt,the conditional moment restriction in (1) implies the unconditional moment restriction: Ezt·u(yt,θ0)=0. (11) Following Sargan (1983)andDovonon and Renault (2013), among others, the unconditional moment restriction (11) locally identifies θ0at first order if RankEzt·∇θu(yt,θ0)=p, (12) whereas lack of first-order local identification occurs when RankEzt·∇θu(yt,θ0)<p. (13) Therefore, it is reasonable to conjecture that the conditional moment restriction (1) identifies θ0locally at first order if and only if there exists a set of instruments zisuch that (12) holds. Relatedly, first-order local identification fails if and only if (13)holdsregardless of the choice of instruments. The following proposition describes this property in terms of degeneracy of the expected Jacobian of u(yt,θ)at θ0. Proposition 2.1. The following two statements are equivalent: (i) For any kand any Rk-valued measurable function g,Rank(E(zt·∇θu(yt,θ0))) <p, where zt=g(xt)and assuming that the moment exists. (ii) There exists at least one linear combination of the elements of E(∇θu(yt,θ0)|xt)that is almost surely nil.3 3Note that, when uis a q-vector, “·” in part (i) should be replaced by the Kronecker product “⊗” and “elements” in part (ii) should be replaced by “columns,” with the understanding that ∇θu(yt,θ0)is a (q,p)- matrix.
856 Dovonon and Gospodinov Quantitative Economics 15 (2024) A proof of an alternative formulation of this proposition (Proposition A.1(ii))isprovided in Appendix B. The characterization in (ii) validates (5) as the first-order local identification condition in model (1). Furthermore, this highlights some similarities with the first-order local identification failure in parametric models as studied by Lee and Chesher (1986) and Rotnitzky, Cox, Bottai, and Robins (2000). In this setting, first-order local identification failure amounts to linear dependence of the elements of the score function of the model, evaluated at the true parameter value. With the basis functions g(k)(x)as defined in Section 2.1, Proposition A.1 in Appendix Aestablishes the connection between point identification (resp., first-order local identification failure) in conditional moment models such as (1) with point identification (resp., first-order local identification failure) in their corresponding sequences of unconditional moment models given by (6). While point identification by (1) amounts to point identification by (6)forsomek0,thefactthatg(k)(x)is an increasingly embedded sequence of vectors allows us to claim that (1) fails first-order local identification if and only if there exists rsuch that, for any klarge enough (say k≥k0), Rank(G(k))=r<p, where G(k)is the expected Jacobian matrix: G(k):=Eg(k)(x)·∇θu(y,θ0). By construction, the null space of G(k)and the range of its transpose are fixed for any klarge enough. This stability of range and null space will be key to second-order local identification that will be imposed on the moment restrictions in order to characterize the limiting behavior of estimators and specification tests. Indeed, the main consequence of first-order local identification failure, while global identification holds, is that only a certain number (r<p) of directions of the parameter vector are identified through first-order expansions of the moment function. Next, following Dovonon and Hall (2018)andDovonon and Renault (2020), we focus on configurations that allow the identification of the remaining directions via a secondorder expansion. Let k≥k0such that Rank(G(k))=r<p,R1be a (p,r)-matrix with columns spanning the range of G(k),andR2denote a (p,p−r)-matrix with columns spanning the null space of G(k). We say that the moment restriction (1) identifies θ0at second order if, for all a∈Rrand b∈Rp−r,wehave 4 G(k)R1a+bR 2Eg(k) l(x)∇θθu(y,θ0)R2b1≤l≤k=0⇔(a,b)=(0, 0), (14) where ∇θθu(y,θ0):=(∂2/∂θ∂θ)u(y,θ), evaluated at θ0. Letting M(k)be the matrix of orthogonal projection on the null space of G(k)(or equivalently the orthogonal of the column span of G(k)), Corollary 2.3 of Dovonon and Renault (2020) ensures that (14) is equivalent to the existence of γk>0suchthat,forany b∈Rp−r, M(k)bR 2Eg(k) l(x)∇θθu(y,θ0)R2b1≤l≤k 2≥√γkb2 2, with γk=inf b2=1 M(k)bR 2Eg(k) l(x)∇θθu(y,θ0)R2b1≤l≤k 2 2. (15) 4From our discussion above, if (14)holdsforagivenk,itholdsforallk≥k.
Quantitative Economics 15 (2024) Robust specification testing 863 Note that D1and H(k)are nonzero matrices and the condition on their spectral norms in Assumption 4(ii) follows if each has at least one column with a number of nonzero elements that is proportional to k. The magnitude of γkfollows from the fact that (a) it is a nondecreasing sequence in k,and(b)itisoforderOP(H(k)2 2).Therequirement that the ratio of the extreme eigenvalues of D 1D1be bounded prevents this nonsingular matrix from being ill-conditioned. The condition on the (k,p−r)-matrix ¯ D2is not particularly restrictive since each component of this matrix is OP(1)by virtue of the central limit theorem. In Assumption 4(iii), the order of magnitude of Skfollows if λmax(W1/2 kVkW1/2 k)≤< ∞, which is the case, for example, if Vkand Wkhave bounded eigenvalues or if Wk=V−1 k. To see this, note that if ¯ λkis bounded, then for any unit vector c,wehave cVar(Sk)c=γ−1 kcH(k)W1/2 kM(k)W1/2 kVkW1/2 kM(k)W1/2 kH(k)c ≤γ−1 kcH(k)W1/2 kM(k)W1/2 kH(k)c≤λmax(Wk)γ−1 k H(k) 2 2=O(1), which is sufficient to claim that Sk=OP(1)since E(Sk)=0. Lastly, Assumption 4(iv) imposes that the eigenvalues of Wkare bounded and ˆ Wk is sufficiently close to Wkas ngrows. Note that these two conditions are fulfilled by the GMM estimator with nonrandom matrix having bounded eigenvalues such as Wk,0. These conditions are also satisfied for the two-step GMM estimator as we show in Appendix Bin the context of the rate-of-convergence results in the next subsection. 4.2 Limiting behavior of the GMM estimator Given the set of assumptions stated above, we now proceed to establishing the rate of convergence of the GMM estimator which, in turn, will be useful to characterize the asymptotic distribution of the specification test. In the standard case of a fixed number of moment restrictions (i.e., kis fixed), the GMM estimator is known to converge at a sharp rate of n1/4although a faster rate in some regions of the sample space is possible (Dovonon and Renault (2013)). This mixture of rates is essential for deriving the asymptotic distribution of the GMM overidentification test statistic as a mixture of chi-squared random variables. We show a similar rate behavior for the GMM estimator in the current context under local identification failure although the original rate needs to be adjusted in order to reflect the increasing number of moment restrictions. The next theorem states the rate of convergence of the parameter vector. Recall that R1denotes a (p,r)-matrix with columns spanning the range of G(k)and R2is a (p,p− r)-matrix with columns spanning the null space of G(k),whereRank (G(k))=r<p. Theorem 4.1. If Assumptions 1–4hold and k→∞as n→∞with k3/n →0, then ˆ θ−θ02=OPγ−1/4 kn−1/4, R 1(ˆ θ−θ0) 2=OPn−1/2,and R 2(ˆ θ−θ0) 2=OPγ−1/4 kn−1/4.
864 Dovonon and Gospodinov Quantitative Economics 15 (2024) Theorem 4.1 establishes that each of the components of the GMM estimator converges at least at a nonstandard rate of γ1/4 kn1/4while the standard √n-rate of convergence is possible in some directions. More specifically, the directions of the parameter vector that are identified at first order are √n-convergent while the directions that are second-order locally identified converge at a slower, γ1/4 kn1/4∼k1/4n1/4,rate.Interestingly, this rate is faster than the result in Dovonon and Renault (2020) who obtain, in a configuration of fixed number of moment restrictions, a slower rate n1/4for the directions identified at second order. The faster rate in our context is essentially due to to the increased information brought by the growing number of moment restrictions. This finding bears some similarities to Han and Phillips (2006) who show, in the context of weak instruments, that the GMM estimator may be consistent if the number of instruments is allowed to increase with the sample size (see also Chao and Swanson (2005), among others). The intuition behind this result is that the expanding number of moment conditions, if growing at an appropriate rate with the sample size, enhances the identification signal and renders a consistent estimator (in a location model) even with possibly irrelevant instruments. In our framework, point identification is maintained and consistent estimation is therefore possible even if the number of moment restrictions does not grow. But, as Theorem 4.1 shows, the second-order local identification also reaps important benefits from the expanding set of moment restrictions as the second-order identified parameters can be estimated at a faster rate. It is worth mentioning that since achieving consistent estimation requires the number of moment restrictions to grow at a slower rate than the sample size, it will not be possible to accelerate the convergence rate of second-order identified directions to the parametric √n-rate. Although the rates of convergence that are stated in Theorem 4.1 are sufficient to derive the asymptotic distribution of the specification test, it is interesting to further investigate the large sample properties of the GMM estimators. Unfortunately, characterizing the asymptotic distribution of the GMM estimator ˆ θin the general case proves difficult. For this reason, we restrict our attention to the simplest case of single parameter (p=1) models with a second-order local identification property. Theorem 4.2. Suppose that p=1and Assumptions 1–4hold.In addition,if k→∞as n→∞with k4/n →0, and γ−1/2 kH(k)Wk¯ fk(θ0)d →Z:=N(0, σ2)for some σ2>0, then √γkn(ˆ θ−θ0)2d −→ 1{Z≥0}(2Z). Theorem 4.2 first demonstrates that the slow rate of convergence derived in Theorem 4.1 is, in fact, sharp meaning that within the assumed model and identification framework, the estimator cannot converge at a faster rate. Furthermore, the asymptotic distribution in Theorem 4.2 can be readily used to conduct inference about the true parameter value θ0by replacing γkwith its sample counterpart. Note that this nonstandard asymptotic distribution with an atom mass of 1/2 at the origin is similar to the one derived by Dovonon and Hall (2018)forafixedk. As pointed out above, the characterization of the asymptotic distribution in the general case of p>1 appears to be quite involved and is beyond the scope of this paper.
Quantitative Economics 15 (2024) Robust specification testing 865 5. Asymptotic distribution of the specification test The characterization of the asymptotic distribution of our specification test statistic requires a central limit theorem for degenerate U-statistics with a linear kernel of the form hn(xt,xs):=f k(xt)V−1 kfk(xs),where(xt)t∈Zis a stationary and strong mixing process and (fk(xt))t∈Zis a martingale difference sequence with respect to its natural filtration. More specifically, we are interested in the asymptotic distribution of U-statistics of the form: Un=1 n t=s fk(xt)V−1 kfk(xs) √k, (21) where Vk:=Va r(fk(xt)).ThedegeneracyofUnarises from the fact that hn(x,y)dF(y)=0forallx,withFdenoting the marginal distribution of xt. While the asymptotic theory for degenerate U-statistics has been extensively studied in the literature (see the Supplemental Material), the available results are not well aligned with our framework, which features an inner product with an increasing dimension. For this reason, we develop a new CLT that is adapted to the form of the U-statistic in (21). Since this result may be of independent interest, we collect the sufficient conditions for establishing the CLT in the following assumptions. Assumption-clt 1. Assume that (xt)t∈Zis stationary and geometric strong mixing process,fk(xt)is an Rk-valued measurable function of xtsuch that the sequence (fk(xt))t∈Z is a martingale difference with respect to the σ-algebra σ(fk(xs):s≤t). Assumption-clt 2. Assume that k∼nαfor some α∈(0, 1)and there exists >0, such that sup k∈N 1 k k h=1 EV−1/2 kfk(xt)h 4+<∞, where [a]his the h-th element of the vector a. Assumption-clt 3. For some β≥0, Emax 1≤t≤n V−1/2 kfk(xt) /√k=Ologβnand Emax 1≤t=s≤nfk(xt)V−1 kfk(xs)/√k=Ologβn. Stationarity and mixing of (xt)t∈Zis already assumed above (see Assumption 1)and is restated in Assumption-clt 1to ensure that the results in Proposition S.2 in the Supplemental Material and Theorem 5.1 below, which could be of independent interest, are self-contained. Assumption-clt 2is used to obtain the limit variance of Unbecause its derivation requires dealing with fourth-order moments of fk(xt). Replacing these moments by their analogues under independence is a common approach in the literature. The remainder is then controlled by resorting to Lemma S.1 in the Supplemental Material, due to Roussas and Ioannides (1987), which can be applied if the condition on the
866 Dovonon and Gospodinov Quantitative Economics 15 (2024) moments in Assumption-clt 2is satisfied. Note that this condition is not particularly restrictive. It imposes the existence of moments of order higher than the fourth for the normalized components of fk(xt). The boundedness of the average of these moments means that no component dominates the others in terms of these moments. The first bound in Assumption-clt 3is not restrictive as it holds with β=1provided that the moment generating function of zt:=V−1/2 kfk(xt)2/√kexists. This holds regardless of the dependence structure.6The second bound in Assumption-clt 3is not too restrictive either. If fk(xt)and fk(xs)are independent, then E[fk(xt)V−1 kfk(xs)/ √k]2=1sothat|fk(xt)V−1 kfk(xs)|/√k=OP(1)and, as before, we can claim that the stated bound accommodates a large class of processes. We are now ready to state the following CLT for the scaled U-statistic in (21). Theorem 5.1. Under Assumptions-clt 1,2,and 3, Un √2 d −→ N(0, 1). The proof of Theorem 5.1 follows similar arguments as in Kim, Luo, and Kim (2011) and is provided in the Supplemental Material. We establish this CLT by showing that the moments of Unconverge to those of the normal distribution. Under Assumptionclt 3, we show that the summands of Unare essentially bounded by a slowly increasing function of the sample size, which turns out to be essential for controlling the difference between the moments Unand those of its Gaussian limit. Building on this central limit theorem, we now characterize the asymptotic distribution of the specification test statistic ˆ Z=1 √2k(¯ fk(xt,yt,ˆ θ)ˆ V−1 k¯ fk(xt,yt,ˆ θ)−k)under the null hypothesis that the conditional moment restriction (1)iscorrectlyspecified. Theorem 5.2. Suppose that Assumptions 1,2,3(i), 4(i,ii), A.1(ii,iii,iv), and Assumptions-clt 2–3with fk(x):=fk(x,y,θ0),hold.Also,assume that k=o(n1/5)and Wk,0 has bounded eigenvalues.Then,as n→∞, ˆ Zd −→ N(0, 1). Several remarks are warranted regarding the result in Theorem 5.2.First,itisimportant to underscore that the standard normal limit distribution in Theorem 5.2 is obtained in a highly non-standard setting. In particular, we have a lack of first-order local identification which, as discussed earlier, gives rise to nonstandard limiting behavior of the GMM estimator. The second-order local identification, in conjunction with the expanding set of moment conditions, ensures the consistency of the estimator and determines its rate of convergence. The conditions for the consistency of the two-step GMM estimator are collected in Assumption A.1 in Appendix Aand are used in establishing the 6For instance, β=1ifzthas a Gamma distribution, and β=1/2ifztis Gaussian. It is worth noting that zt=OP(1)since E(z2 t)=1. If zt’s are i.i.d. with common distribution F, it is known that this bound holds for a large class of Fbut rules out those with Paretian tail (Pereira (1983); see also Berman (1964) and Isaev, Rodionov, Zhang, and Zhukovskii (2020) for similar results for time-dependent processes).
Quantitative Economics 15 (2024) Robust specification testing 867 limit in Theorem 5.2. While the ˆ Ztest statistic is based on the two-step GMM estimator with ˆ Wk=[1 nn t=1fk(xt,yt,˜ θ)fk(xt,yt,˜ θ)]−1, stating explicitly that Wk,0 has bounded eigenvalues allows us to invoke Assumption 4(iii) in order to ensure the desired rate of convergence for the preliminary GMM estimator ˜ θ. (See Remark 1in Appendix B.) Also, as discussed earlier, Assumption 4(iii) holds provided that λmax(W1/2 kVkW1/2 k)≤<∞, which is trivially satisfied by the two-step GMM estimator that sets Wk=V−1 k. In the conventional framework where the conditional model is point identified, the properly recentered and standardized specification test with an increasing number of moment conditions converges, under some regularity conditions, to a standard normal limit (see, e.g., Carrasco and Florens (2000); Donald, Imbens, and Newey (2003); Tripathi and Kitamura (2003); among others). Theorem 5.2 establishes that the standard normal distribution continues to be the correct limit for the ˆ Ztest statistic under the null of correct specification, provided that k=o(n1/5)as n→∞. Unlike the regular setup, this limit is obtained within the second-order local identification framework in Assumption 2, which is characterized by first-order local identification failure. Intuitively, this is achieved by combining and balancing the benefits from the second-order local identification and the expanding number of moment conditions. Importantly, for theappropriatechoiceofk(as a function of n), inference for the correct specification of the conditional moment restriction model is straightforward in practice as it is based on the critical values from the standard normal distribution. In our simulations and empirical application, we set k∝n1/6. One may even choose kto grow arbitrarily slowly with n and the results in this paper would continue to hold. However, a kthat grows too slowly may compromise power. It is worth stressing that the construction and implementation of the test is agnostic about the precise form of first-order local identification failure. This robustness property is further enhanced by the fact that the test remains valid even if the model happens to be first-order identified. We complete our theoretical analysis by characterizing the limiting behavior of ˆ Z under the alternative hypothesis H1, specified in (3). With appropriate choices of series functions gl(·),H1implies that infθ∈E(g(k0)(x)u(y,θ))2>0forafixedk0so that the unconditional moment restriction E(g(k0)(x)u(y,θ)) =0 is misspecified. In this case, it is known that the Sargan–Hansen specification test for this unconditional restriction— albeit infeasible because k0is unknown—would be consistent. Theorem 5.3 shows that this result carries over to the feasible statistic ˆ Z, which makes our specification test consistent against all alternatives.7 Theorem 5.3. Let ˆ Vk(θ):=n−1n t=1fk(xt,yt,θ)fk(xt,yt,θ).Assume that k2=o(n),the gl(·)series are as in Lemma B.4 in Appendix B,and the conditions of that lemma are satisfied with u(θ):=u(y,θ).Assume further that there exists ¯ λ>0such that,with probability approaching one,supθ∈λmax(ˆ Vk(θ)) ≤¯ λk,and supθ∈|(1/n)n t=1gl(xt)u(yt,θ)− 7Studying the asymptotic behavior of the test under local alternatives proves to be very involved as it requires characterizing the limiting behavior of the GMM estimator in misspecified conditional restriction models under drifting sequences and first-order local identification failure. This analysis is beyond the scope of this paper.
868 Dovonon and Gospodinov Quantitative Economics 15 (2024) E(gl(xt)u(yt,θ))|=oP(1)for each l.Then,under H1, ∃δ>0: lim n→∞Prk3/2n−1|ˆ Z|>δ =1. The conditions of this theorem are essentially a subset of those of the main Theorem 5.2. The purpose of the condition on the bound of λmax(ˆ Vk)is to facilitate the proof as we can rely on more primitive conditions. Theorem 5.3 shows that |ˆ Z|diverges to infinity if kis such that k3/2=o(n). Note that in Theorem 5.2,whichstudies ˆ Zunder H0,we impose k5=o(n). This shows that the proposed test is consistent and has power against all alternatives. 6. Simulations and empirical analysis In this section, we provide simulation evidence on the empirical size and power of the standard normal asymptotic approximation of the specification test. We also apply the proposed testing framework to study the presence of a common CH factor in bond portfolio returns. 6.1 Simulations The simulation design for assessing the finite-sample properties of the specification test ˆ Zis tailored to the common CH factor example discussed in Section 3and used in the subsequent empirical application. More specifically, the data generating process has the form: Yt+1=Dtτ+ft+1+et+1, (22) where Yt+1and et+1∼iidN(0, κIm)are m×1vectors,andft+1is an m×1vectorof unobserved CH factors. The ith component fi,t+1of ft+1follows a GARCH(1,1) process: fi,t+1=σi,tεi,t+1,σ2 i,t=ωi,0 +ωi,1f2 i,t+ωi,2σ2 i,t−1, (23) with ωi,0,ωi,1,ωi,2 >0andεi,t+1∼iidN(0, 1).8Finally, is an m×mmatrix of factor loadings, Dt:=Diag(σ2 1,t,,σ2 m,t)and τis an m×1 vector of market prices of risk (see, e.g., King, Sentana, and Wadhwani (1994)). In all cases considered below, we set κ=0.1 and ωi,0 =1−ωi,1 −ωi,2 for i=1, 2, 3, where (ω1,1,ω1,2 )=(0.2, 0.6),(ω2,1,ω2,2)= (0.4, 0.4)and (ω3,1,ω3,2)=(0.1, 0.8). The parameterization of the factor loading matrix determines if the model is under the null (common CH features) or under the alternative. In evaluating the size properties of the specification tests, we consider two cases: (i) m=2, bivariate Yt+1with a single common CH factor, and (ii) m=3, trivariate Yt+1with two common CH factors. For case (i), =10 0.5 0 , and for case (ii), =100 110 0.5 0.5 0 . In assessing the power properties of the tests, we set to be the identity matrix. 8Bougerol and Picard (1992) derive the conditions for strict stationarity of GARCH processes.
Quantitative Economics 15 (2024) Robust specification testing 869 As demonstrated in Section 3, the presence of common CH features amounts to testing the conditional moment restriction E(u1,t+1(θ)|Ft)) =0. In case (i), u1,t+1(θ)is parameterized as u1,t+1(θ):=(β1Y1,t+1+(1−β1)Y2,t+1)2−cwith θ=(β1,c)and for case (ii), u1,t+1(θ):=(β1Y1,t+1+β2Y2,t+1+(1−β1−β2)Y3,t+1)2−cwith θ=(β1,β2,c).9 For some choice of instruments zt∈Ft, the parameter vector θis estimated by the twostep GMM based on the moment conditions E[ztu1,t+1(θ)] =0. For the unconditional GMM approach and the corresponding J(Sargan–Hansen) test, we use zt=(Y t,Y2 t), with Y2 t:=(Y2 1,t,,Y2 m,t),whichgivesrisetok1−poveridentifying restrictions: with k1=4andp=2 for case (i), and k1=6andp=3 for case (ii). We report two versions of the Jtest: one based on critical values from χ2(k1−p)and one based on critical values from χ2(k1). As argued above, the former test is not valid when the model is not first-order locally identified while the latter remains valid but is conservative. The ˆ Ztest uses x:=Ytas conditioning variables in constructing the functions gl(·),l=1, ,k, so that the GMM estimation is based on E(ztu1,t+1(θ)) =E(gl(Yt)× u1,t+1(θ)) =0. In the case where kx:=size(x)=1, the tuning parameters for the test are (·):R→[−π,π],x→ 2arctan(x), series of bounded functions gl(·):[−π,π]→ [−1, +1],x→cos(lx)for l=1, k,andk=n1/6.10 Inthecasewherekx>1, (·)and gl(·)are applied componentwise to xleading to k=kx·n1/6moment restrictions.11 We report results for the one-sided test ˆ Zat nominal level α,ˆ Z>q 1−α,whereq1−αdenotes the (1−α)quantile of the N(0, 1)distribution. The fourth specification that we consider is also a Jtestbutitisbasedontheaugmented set of moment conditions E(ztu1,t+1(θ) u2,t+1(β))=0, where u2,t+1(β):=β1Y1,t+1+ (1−β1)Y2,t+1for case (i), and u2,t+1(β):=β1Y1,t+1+β2Y2,t+1+(1−β1−β2)Y3,t+1 for case (ii). The value of the parameter τin model (23) determines if the additional restriction E(ztu2,t+1(β)) =0 restores the first-order local identification (τ= 0) or not (τ=0). For case (i), we set τ=(0, 0)or τ=(0.1, 0.1), and for case (ii), τ=(0, 0, 0)or τ=(0.1, 0.1, 0.1).TheJstatistic in this augmented model is compared to critical values from χ2(k2−p),wherek2=8andp=2forcase(i)andk2=12 and p=3 for case (ii). The empirical rejection probabilities of the four specification tests are based on n=(2000, 5000, 10,000)and 10,000 Monte Carlo replications. Tables 1and 2present the results for case (i), m=2, and case (ii), m=3, respectively, with the top panel in 9We consider in this section a linear combination of assets with coefficients adding to one to mimic portfolio formation. This yields the same identifying properties for the resulting moment restrictions as those obtained using the weight (1, β), considered in Section 3, so long as the matrix of factor loadings does not contain a relevant column with equal elements. 10Even for large sample sizes, the choice k=n1/6may result in a relatively small number of instruments. It appears that further improvements can be obtained if we set k=const ·n1/6, where the constant “const”is calibrated to the particular setup (to the values of nand m). Ideally, it seems desirable to target a more datadriven choice of kvia subsampling or resampling methods. But, as argued in the concluding section, such methods may be difficult to implement due to the highly challenging nature of our setup: a conditional moment restrictions model with local first-order local identification failure, second-order local identification, and dependent data. Fortunately, our simulation and empirical results suggest that the properties of the test are not particularly sensitive to the values of “const”ink=const ·n1/6for const ≥1. 11We experimented with the construction of the basis functions as in Bierens (1990) but the results are broadly similar.
870 Dovonon and Gospodinov Quantitative Economics 15 (2024) Table 1. Empirical rejection rates of specification tests under the null (size) and alternative (power): case m=2. Panel A: τ=(0, 0)Panel B: τ=(0.1, 0.1) JTest JTest JAugment ˆ ZTest JTest JTest JAugment ˆ ZTest nχ 2(k1-p)χ2(k1)χ2(k2-p)N(0, 1)χ2(k1-p)χ2(k1)χ2(k2-p)N(0, 1) size 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 2000 8.1 4.0 1.6 0.7 5.3 2.2 6.6 4.0 8.2 4.3 1.7 0.7 5.2 2.3 6.7 4.2 5000 10.7 5.7 2.4 1.1 6.6 3.1 9.0 5.6 11.3 6.0 2.6 1.2 5.4 2.3 8.8 5.7 10,000 13.3 7.7 3.6 1.7 8.9 4.5 9.4 5.9 13.6 7.6 3.7 1.8 6.2 2.7 9.7 6.2 power 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 2000 10.8 5.5 2.3 0.9 10.7 5.6 99.1 98.4 14.9 8.0 3.3 1.4 89.4 82.6 99.0 98.5 5000 10.8 5.4 2.2 0.9 10.7 5.4 100 100 24.8 14.3 6.9 3.3 99.8 99.5 100 100 10,000 11.2 5.9 2.4 1.1 11.5 5.9 100 100 40.2 27.4 15.8 8.6 100 100 100 100 Note: In this simulation design (m=2), “size” corresponds to a bivariate Yt+1with a single common CH factor, and “power” corresponds to a bivariate Yt+1with two CH factors. The table presents the empirical size and power at 5% and 10% nominal level of three Jtests for overidentifying restrictions (p=2,k1=4,andk2=8)andthe ˆ Ztest, proposed in this paper. ‘‘J augment” stands for the Jtest, augmented with an additional conditional moment restriction. The value of τ(τ= 0or τ= 0) determines if the augmented model is first-order locally identified or not. The results are based on 10,000 Monte Carlo replications. each table reporting the empirical size of the tests and the bottom panel reporting their empirical power. In the setup where the model is not first-order locally identified but globally identified, the standard Jtest for overidentifying restrictions is known to overreject under the null (Dovonon and Renault (2013)). These overrejections are confirmed in Table 1 Table 2. Empirical rejection rates of specification tests under the null (size) and alternative (power): case m=3. Panel A: τ=(0, 0, 0)Panel B: τ=(0.1, 0.1, 0.1) JTest JTest JAugment ˆ ZTest JTest JTest JAugment ˆ ZTest nχ 2(k1-p)χ2(k1)χ2(k2-p)N(0, 1)χ2(k1-p)χ2(k1)χ2(k2-p)N(0, 1) size 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 2000 4.3 1.8 0.5 0.1 2.4 1.0 3.6 2.0 4.8 2.1 0.7 0.2 3.9 1.6 3.7 2.2 5000 6.5 2.8 0.7 0.3 3.3 1.5 5.2 2.9 7.0 3.3 0.9 0.4 5.3 2.5 5.4 3.2 10,000 8.1 3.7 1.1 0.5 4.3 1.8 6.1 3.9 9.4 4.7 1.3 0.4 6.7 2.9 6.1 3.5 power 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 10% 5% 2000 2.5 1.2 0.4 0.1 4.7 2.1 50.8 41.9 3.3 1.6 0.5 0.1 55.6 46.2 52.9 43.4 5000 1.8 0.6 0.1 0.0 2.4 1.0 97.3 95.7 3.0 0.9 0.3 0.1 95.2 91.0 97.8 96.4 10,000 5.2 1.9 0.3 0.1 4.6 2.3 100 100 9.7 3.9 0.8 0.3 100 99.9 100 100 Note: In this simulation design (m=3), “size” corresponds to a trivariate Yt+1with two common CH factors, and “power” corresponds to a trivariate Yt+1with three CH factors. The table presents the empirical size and power at 5% and 10% nominal level of three Jtests for overidentifying restrictions (p=3,k1=6,andk2=12)andthe ˆ Ztest, proposed in this paper. “J augment” stands for the Jtest, augmented with an additional conditional moment restriction. The value of τ(τ= 0or τ= 0) determines if the augmented model is first-order locally identified or not. The results are based on 10,000 Monte Carlo replications.
Quantitative Economics 15 (2024) Robust specification testing 871 and they continue to get larger as the sample size increases.12 While the Jtest based on χ2(k1)critical values is valid, it is very conservative, which results in loss of power. For τ=0, the standard Jtest also exhibits lack of power due to the fact that only the instruments Y2 tcarry information about the model parameters while the restrictions based on the instruments Ytare uninformative as these restrictions are correct under both the null and the alternative hypotheses and render the model just-identified. When τ=0, the power of the standard Jtest is somewhat improved (in the case m=2) but it remains low because this test does not exploit explicitly the moment restriction E(ztu2,t+1(β)) =0. This restriction is used by the Jtest based on the augmented model (“Jaugment” in Tables 1and 2) whose power approaches 100% when τ= 0. Because the augmented model regains its first-order local identifiability for τ=0, the standard inference (based on the χ2(k2−p)distribution) restores its validity under both the null and the alternative hypotheses. But when τ=0, the additional moment restriction E(ztu2,t+1(β)) =0isredundant and the model remains first-order locally unidentified. This should manifest itself in size distortions under the null (for n=50,000, the rejection rates of the augmented Jtest under the null are 14.9% and 8.6% at the 10% and 5% nominal level, respectively) and lack of power under the alternative. This highlights the need for a robust test that does not require prior knowledge of the true structure of the model and the value of τ. Indeed, the ˆ Ztest proposed in this paper offers precisely this type of robustness as it remains agnostic about the first-order identifiability of the model. Tables 1and 2 demonstrate that the ˆ Ztest controls size for both τ=0andτ=0. The minor underrejections of the test for m=3 arise from the fact that even for n=10,000, the finite-sample distribution of the ˆ Ztest is slightly asymmetric and it requires even larger sample sizes for the N(0, 1)asymptotics to fully assert itself. Further simulation evidence and discussion regarding the absolute and relative finite-sample performance of the ˆ Ztest is provided in the Supplemental Material. 6.2 Empirical application In this subsection, we investigate the presence of a common CH factor in U.S. bond returns of different maturities. After presenting some preliminary evidence on commonality in the GARCH-based volatility dynamics in bond returns, we subject these portfolio returns to the test of common CH features, which amounts to testing the validity of a version of the conditional moment restriction E(u(yt,θ0)|xt)=0. Let r(j) t+1denote the holding return, between periods tand t+1, on a bond with j years to maturity, in excess of the risk-free rate. Let Yt+1=(r(1) t+1,,r(m) t+1).Asinthe previous subsection, we posit that the m-vector of excess bond returns Yt+1,adaptedto the increasing filtration Ft, admits the representation (22) with common CH features.13 12In unreported results for n=50,000 and m=2, the empirical rejections for the standard Jtest based on χ2(k1−p)are 18.2% and 11.3% for τ=0, and 16.9 and 10.1% for τ=0 at 10% and 5% nominal levels, respectively. A similar increase in overrejections for the Jtest are observed for m=3 in sample sizes that exceed those reported in Table 2. 13The conventional term structure models impose no-arbitrage restrictions on the factor loading matrix . Recent research (Duffee (2011); Joslin, Singleton, and Zhu (2011); among others) casts doubt on the role
872 Dovonon and Gospodinov Quantitative Economics 15 (2024) Figure 1. Estimated GARCH(1,1) volatilities for portfolio bond excess returns of different maturities. This implies that there exists a vector β=0min Rmsuch that E((βYt+1)2|Ft)is constant, that is, E(ut+1(θ0)|Ft)=0withθ=(β,c)and ut+1(θ):=(βYt+1)2−c. In the empirical analysis, we use the Fama bond portfolio returns from the Center for Research in Security Prices (CRSP) (2023) with the following maturities: 1 to 2 years, 2 to 3 years, 3 to 4 years, 4 to 5 years, and 5 to 10 years.14 The data is at monthly frequency covering the period January 1952–December 2020. We construct excess bond returns by subtracting the 1-month risk-free rate (retrieved from Kenneth R. French—Data Library (2023)). We start by fitting a GARCH(1,1) to each of these excess bond returns. The filtered GARCH volatilities are plotted in Figure 1. As the graph reveals, there appears to be a strong comovement in these GARCH volatilities. This is probably not too surprising since the first principal component in these five bond returns explains in excess of 95% of their volatility. of no-arbitrage restrictions in modeling and forecasting bond yields. The forecasting properties are further deteriorated by incorporating stochastic volatility. As Joslin and Le (2021)demonstrate,thisislargely attributed to the fact that these models impose a tight link between risk compensation and interest rate volatility, and recommend the use of unrestricted factor models. This is the approach that we follow here. 14The data for the bond portfolio returns is obtained from the Wharton Research Data Services (WRDS), using database CRSP Treasuries—Fama bond portfolios ©2023 (CRSP).
Quantitative Economics 15 (2024) Robust specification testing 879 conditions imposed on the series functions gl(·)are as in de Jong and Bierens (1994). The continuity and dominance conditions on u(θ)are useful to guarantee the continuity of θ→E(gl(x)u(y,θ)) for each l. Continuity of these functions and compactness of are essential to claim the stated result in Lemma B.4. B.2 Proofs of main results Proof of Proposition A.1. (i) It suffices to show that (A.1)holdsfork0to claim that it holds for all k≥k0. Since αl≡0foralll≥k0,wehave Eu(y,θ)|x:=ρ(x,θ)= k0−1 l=1 αl(θ)gl(x)and ρ(x,θ)≡0⇔[αl=0, ∀l=1, ,k0−1]. By the law of iterated expectations, αl(θ)=E(gl(x)u(y,θ)) =0, ∀l=1, ,k0−1and this establishes the claim since [ρ(x,θ)≡0⇔θ=θ0]holds by assumption. To establish the second claim, recall that ρ(x,θ)=∞ l=1αl(θ)gl(x).Also,bythelaw of iterated expectations, E(g(k)(x)u(y,θ)) =E(g(k)(x)ρ(x,θ)) so that Eg(k)(x)u(y,θ)=α1(θ),,αk(θ). Hence, by the definition of θk, Eρ(x,θk)2=E l≥k+1 αl(θk)gl(x)2 = l≥k+1 αl(θk)2→0, as k→∞,(B.1) where the second equality holds by the stated assumption that (gl)lare orthonormal and the convergence follows from (d). Consider an arbitrary small and open neighborhood of Nof θ0and let = min\NE(ρ(x,θ))2. By the continuity assumption (c), the compactness of \N,and the identification property in (4), we can claim that >0. Also, from (B.1), it is clear that there exists k0∈Nsuch that E[ρ(x,θk)]2<for all k≥k0. It then follows that for k≥k0, we have θk∈N, which proves the claim. (ii) First, we establish the necessary condition. If the first-order local identification condition fails, then RankEE∇θu(y,θ0)|xE∇θu(y,θ0)|x<p, implying that there exists δ= 0∈Rpsuch that E(∇θu(y,θ0)|x)·δ=0almostsurely. Therefore, for any k∈N, Eg(k)(x)·∇θu(y,θ0)·δ=Eg(k)(x)·E∇θu(y,θ0)|x·δ=0. As a result, RankEg(k)(x)·∇θu(y,θ0)≤p−1, ∀k.
880 Dovonon and Gospodinov Quantitative Economics 15 (2024) Since k→ Rank(E(g(k)(x)·∇ θu(y,θ0))) takes integer values, it is nondecreasing and bounded from above, it reaches its maximum, say r≤p−1, as kincreases. This shows the necessary condition. Next, we establish the sufficient condition. Under the stated condition, there exists δ=0suchthat Egl(x)·∇θu(y,θ0)·δ=Egl(x)·E∇θu(y,θ0)|x·δ=0, for all l≥1. (B.2) Since E(∇θu(y,θ0)|x)∈(L2(P))p,itsith component can be written as l≥1αl,igl(x), with αl,i’s being scalars. Taking the relevant linear combinations (over l) of the equalities in (B.2), we have EE∇θu(y,θ0)|xE∇θu(y,θ0)|x·δ=0 and this completes the proof. Proof of Theorem 4.1.LetR=(R1|R2)and consider the transformation θ=Rη := R1η1+R2η2,withθ,η∈Rp,η1∈Rrand η2∈Rp−r,andset ˆ θ=Rˆη,andθ0=Rη0. Hence, ¯ fk(ˆ θ)=¯ fk(Rˆη)=¯ fk(R1ˆη1+R2ˆη2). By a first-order Taylor expansion of η1→ ¯ fk(R1η1+R2ˆη2)around η01 and a second-order Taylor expansion of η2→ ¯ fk(R1η01 + R2η2)around η02,wehave ¯ fk(ˆ θ)=¯ fk(θ0)+1 √n∇θ¯ fk(R1¯η1+R2ˆη2)R1√n(ˆη1−η01)+∇θ¯ fk(θ0)R2(ˆη2−η02) +1 2¯ H(k)(¯ θ)·√n·vecR2(ˆη2−η02)( ˆη2−η02 )R 2, where ¯η1∈(η01,ˆη1)and ¯ θ∈(θ0,ˆ θ)and both may differ from row to row. Let ˜ θ=R1¯η1+R2ˆη2,¯ D1=1 √n∇θ¯ fk(˜ θ)·R1,¯ D2=∇θ¯ fk(θ0)R2,z0n=√n·vec(R2(˜η2− η02)( ˜η2−η02)R 2)and we write ¯ fk(ˆ θ)=¯ fk(θ0)+¯ D1√n(ˆη1−η01)+¯ D2(ˆη2−η02)+1 2¯ H(k)(¯ θ)z0n.(B.3) By premultiplying this equation by ¯ D 1ˆ Wkand solving for √n(ˆη1−η01 ),weobtain √n(ˆη1−η01) =−¯ D 1ˆ Wk¯ D1−1¯ D 1ˆ Wk¯ fk(θ0)−¯ fk(ˆ θ)+¯ D2(ˆη2−η02)+1 2¯ H(k)(¯ θ)z0n.(B.4) Plugging this back into (B.3), we have ¯ M(k)ˆ W1/2 k¯ fk(ˆ θ) =¯ M(k)ˆ W1/2 k¯ fk(θ0)+¯ M(k)ˆ W1/2 k¯ D2(ˆη2−η02)+1 2¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n,(B.5)
Quantitative Economics 15 (2024) Robust specification testing 881 with ¯ M(k)=Ik−¯ P(k)and ¯ P(k)=ˆ W1/2 k¯ D1(¯ D 1ˆ Wk¯ D1)−1¯ D 1ˆ W1/2 k. Then multiplying each side of (B.5) by its own transpose and rearranging yields 1 4z 0n¯ H(k)(¯ θ)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n =¯ fk(ˆ θ)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ fk(ˆ θ)−¯ fk(θ0)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ fk(θ0) −(ˆη2−η02)¯ D 2ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ D2(ˆη2−η02)−2¯ fk(θ0)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ D2(ˆη2−η02) −¯ fk(θ0)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n−(ˆη2−η02)¯ D 2ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n. By definition, ¯ fk(ˆ θ)ˆ Wk¯ fk(ˆ θ)≤¯ fk(θ0)ˆ Wk¯ fk(θ0).Hence, 1 4z 0n¯ H(k)(¯ θ)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n =¯ fk(ˆ θ)ˆ W1/2 k¯ P(k)ˆ W1/2 k¯ fk(ˆ θ)−¯ fk(θ0)ˆ W1/2 k¯ P(k)ˆ W1/2 k¯ fk(θ0) −(ˆη2−η02)¯ D 2ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ D2(ˆη2−η02)−2¯ fk(θ0)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ D2(ˆη2−η02) −¯ fk(θ0)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n−(ˆη2−η02)¯ D 2ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n. We show in the Supplemental Material that ¯ fk(ˆ θ)ˆ W1/2 k¯ P(k)ˆ W1/2 k¯ fk(ˆ θ)−¯ fk(θ0)ˆ W1/2 k¯ P(k)ˆ W1/2 k¯ fk(θ0) =OP(¯ λkk/√n)+OP¯ λkkˆ θ−θ02(B.6) and, since ¯ λkis bounded and k/√n→0, only the second term matters. Using this fact and letting H:=H(k)(θ0)and 1n:=¯ H(k)(¯ θ)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)−HW1/2 kM(k)W1/2 kH, 2n:=¯ H(k)(¯ θ)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ fk(θ0)−HW1/2 kM(k)W1/2 k¯ fk(θ0), we can write 1 4z 0nHW1/2 kM(k)W1/2 kHz0n ≤OP(¯ λkk)ˆη−η02−(ˆη2−η02 )¯ D 2ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ D2(ˆη2−η02) −2¯ fk(θ0)ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ D2(ˆη2−η02)−¯ fk(θ0)W1/2 kM(k)W1/2 kHz0n− 2nz0n −(ˆη2−η02)¯ D 2ˆ W1/2 k¯ M(k)ˆ W1/2 k¯ H(k)(¯ θ)z0n−1 4z 0n1nz0n.(B.7) From (B.4), we can show that ˆη1−η012=OP(ˆη2−η022 2)so that ˆη−η02= OP(ˆη2−η022). Also, from the second-order local identification property, we have 1 4z 0nHW1/2 kM(k)W1/2 kHz0n≥1 4γkz0,n2 2=1 4γknˆη2−η024 2.
882 Dovonon and Gospodinov Quantitative Economics 15 (2024) Let z1n=γ1/4 kn1/4(ˆη2−η02). By the Cauchy–Schwarz inequality, (B.7)yields 1 4z1n4 2≤1 γ1/4 kn1/4OP(¯ λkk)z1n2+1 √nγk ¯ D 2ˆ Wk¯ D2 2·z1n2 2 +2 (nγk)1/4 ˆ W1/2 k¯ fk(θ0) 2 ˆ W1/2 k¯ D2 2·z1n2 + γ−1/2 kHW1/2 kM(k)W1/2 k¯ fk(θ0) 2·z1n2 2+1 √γk2n2·z1n2 2 +1 γ3/4 kn1/4 ˆ W1/2 k¯ D2 2 ˆ W1/2 k¯ H(k)(¯ θ) 2·z1n3 2+1 4γk1n2·z1n4 2. Since γk/k=O(1), by Lemma B.3,wehave 1 √nγk ¯ D 2ˆ Wk¯ D2 2=OP(¯ λkk/n), 1 (nγk)1/4 ˆ W1/2 k¯ fk(θ0) 2 ˆ W1/2 k¯ D2 2=OP¯ λkk3/4/n1/4, 1 γ3/4 kn1/4 ˆ W1/2 k¯ D2 2 ˆ W1/2 k¯ H(k)(¯ θ) 2=OP¯ λkk1/4/n1/4, 1 γk1n2=OPˆ Wk−Wk2+OP(¯ λk/√n)+OP¯ λkˆ θ−θ02, and 1 √γk2n2=OP(¯ λkk/n)+OP¯ λk√kˆ θ−θ02+OP√kˆ Wk−Wk2. Since ¯ λkis bounded, k3/n →0and√kˆ Wk−Wk2=oP(1), it follows that 1 γk1n2=oP(1),and 1 √γk2n2=OPk1/4/n1/4z1n2=oP(1)z1n2. Hence, z1n4 2≤ γ−1/2 kHW1/2 kM(k)W1/2 k¯ fk(θ0) 2·z1n2 2 +oP(1)·z1n2+oP(1)·z1n2 2 +oP(1)·z1n3 2+oP(1)·z1n4 2.(B.8) Since γ−1/2 kHW1/2 kM(k)W1/2 k¯ fk(θ0)=OP(1), we can readily claim that z1n=OP(1). Indeed, (B.8) amounts to 1+oP(1)z1n2≤OP(1) z1n2+oP(1) z1n2 2+oP(1) z1n2+oP(1). Hence, if z1n2>1, this inequality implies (1+oP(1))z1n2≤OP(1)+oP(1).Thus, we either have (z1n2<1)or (1+oP(1))z1n2≤OP(1), which ensures that z1n2=
Quantitative Economics 15 (2024) Robust specification testing 883 OP(1),thatis, γ1/4 kn1/4(ˆη2−η02)=OP(1). Using (B.4), we obtain that √n(ˆη1−η01 )=OP(1). Recalling that ˆ θ−θ0=R1(ˆη1−η01)+ R2(ˆη2−η02),wehave ˆ θ−θ02=OPn−1/2+OPγ−1/4 kn−1/4=OPγ−1/4 kn−1/4. Also, by the definition of R1and R2as spanning the range of the transpose a matrix and the null space of that same matrix, respectively, we have R 1R2=0. Hence, R 1(ˆ θ−θ0)=R 1R1(ˆη1−η01)=OPn−1/2 and R 2(ˆ θ−θ0)=R 2R2(ˆη2−η02)=OPγ−1/4 kn−1/4. To complete the proof, it only remains to establish (B.6), which is done in the Supplemental Material. The validity of the results in Theorem 4.1 hinges on verifying Assumptions 3(ii) and 4(iii, iv). For this reason, some further remarks on the rates of convergence in Theorem 4.1, specialized to the first-step and two-step GMM estimators, are warranted. Remark 1. For the first-step GMM estimator, the weighting matrix ˆ Wk:=Wk,0 is nonrandom with bounded eigenvalues from above and away from zero, and Assumptions 3(ii) and 4(iv) are trivially verified. Assumption 4(iii) is also satisfied if, for instance, Vkhas bounded eigenvalues and this estimator, say ˜ θ, is characterized by ˜ θ−θ0= OP(k−1/4n−1/4). Note that the eigenvalues of Vkare bounded under Assumption A.1(iv) and if Vkhas uniformly bounded diagonal elements. This latter condition is implied by Assumption A.1(ii). Remark 2. For the two-step estimator with ˆ Wk:=ˆ V−1 k, we assume that the smallest eigenvalue of Vkis bounded away from 0, that is, λmin(Vk)≥λ>0forallk.Thisisa reasonable assumption since λmax(Vk)is an increasing sequence and we require that λmin(Vk)and λmax(Vk)areofthesameorderofmagnitudetopreserveVkfrom being ill-conditioned. In this case, λmaxV−1 k=1/λmin(Vk)≤1/λ. Furthermore, Lemma B.2(ii, v) ensures that λmaxˆ V−1 k/λmaxV−1 k=1+oP(1),andλminˆ V−1 k/λminV−1 k=1+oP(1). Note that in this lemma, vn=(k3/n)1/4=o(1). This shows that Assumption 3(ii) holds. Remark 3. Finally, from Lemma B.2(iv), we have that √kˆ V−1 k−V−1 k2=OP(k5/4/n1/4), which ensures that Assumption 4(iv) holds if k5/n →0asn→∞.
884 Dovonon and Gospodinov Quantitative Economics 15 (2024) Proof of Theorem 4.2. Since θ0is in the interior of and ˆ θconverges in probability to θ0,ˆ θis also an interior optimum with probability approaching one. Therefore, this estimator solves ∇θ¯ fk(ˆ θ)ˆ Wk¯ fk(ˆ θ)=0. (B.9) By a mean-value expansion of ∇¯ fk(ˆ θ)and a second-order Taylor expansion ¯ fk(ˆ θ) around θ0,wehave ∇θ¯ fk(ˆ θ)=∇θ¯ fk(θ0)+¯ H(k)(˙ θ)√n(ˆ θ−θ0) and ¯ fk(ˆ θ)=¯ fk(θ0)+∇θ¯ fk(θ0)( ˆ θ−θ0)+1 2¯ H(k)(¨ θ)√n(ˆ θ−θ0)2, with ˙ θ,¨ θ∈(θ0,ˆ θ).Let ˙ h:=¯ H(k)(˙ θ),¨ h:=¯ H(k)(¨ θ),and ¯ D2:=∇ θ¯ fk(θ0).Thefirst-order condition (B.9)yields n−1/4∇¯ fk(ˆ θ)ˆ Wk¯ fk(ˆ θ) =n−1/4¯ D 2ˆ Wk¯ fk(θ0)+n−1/4¯ D 2ˆ Wk¯ D2(ˆ θ−θ0)+1 2¯ D 2ˆ Wk¨ hn1/4(ˆ θ−θ0)2 +˙ hˆ Wk¯ fk(θ0)n1/4(ˆ θ−θ0)+˙ hˆ Wk¯ D2n1/4(ˆ θ−θ0)2 +1 2˙ hˆ Wk¨ hn3/4(ˆ θ−θ0)3=0. In this framework, γk=HWkH.WithH:=H(k)(θ0),wecanwrite 1 2(γkn)3/4(ˆ θ−θ0)3+1 2γk˙ hˆ Wk¨ h−HWkH(γkn)3/4(ˆ θ−θ0)3 +3 2γ3/4 kn1/4Hˆ Wk¯ D2√γkn(ˆ θ−θ0)2 +1 γ3/4 kn1/4(˙ h−H)ˆ Wk¯ D2√γkn(ˆ θ−θ0)2+1 √γk HWk¯ fk(θ0)(γkn)1/4(ˆ θ−θ0) +1 √γk˙ hˆ Wk−HWk¯ fk(θ0)(γkn)1/4(ˆ θ−θ0)+1 2γ3/4 kn1/4¯ D 2ˆ Wk(¨ h−H)√γkn(ˆ θ−θ0)2 +1 √γkn¯ D 2ˆ Wk¯ D2(γkn)1/4(ˆ θ−θ0)+n−1/4¯ D 2ˆ Wk¯ fk(θ0)=0. Similar to the lines of the proof of Lemma B.3, it is not hard to see that 1 γk˙ hˆ Wk¨ h−HWkH=oP(1),Hˆ Wk¯ D2=OP(k), ˙ h−H2=OP√kn−1/2∨ˆ θ−θ02=OPk1/4/n1/4, (˙ h−H)ˆ Wk¯ D2=OPk3/4/n1/4,and 1 √γk˙ hˆ Wk−HWk¯ fk(θ0)=oP(1).
Quantitative Economics 15 (2024) Robust specification testing 885 Then, letting z1n:=(γkn)1/4(ˆ θ−θ0)and Zn:=1 √γkHWk¯ fk(θ0),wehave z1nz2 1n+2Zn=oP(1). (B.10) Since (z1n,Zn)=OP(1), by the Prokhorov’s theorem, each subsequence of has a further subsequence that converges in distribution to, say, (V,Z). Thus, along this converging subsequence, (B.10) implies that VV2+2Z=0. Therefore, it is not difficult to see that |V|=1Z≤0√−2Zand, since as a Gaussian random variable Zhas a symmetric distribution, |V|=1Z≥0√2Z. The fact that this limit distribution is not specific to the subsequence implies that the whole sequence converges to (V,Z). By the continuous mapping theorem, it follows that z2 1n d →V2=1Z≥0(2Z)and this completes the proof. Proof of Theorem 5.2. We proceed in two steps by showing first that ˆ Zis bounded by two statistics ˆ Z1and ˆ Z2,thatis, ˆ Z1+oP(1)≤ˆ Z≤ˆ Z2+oP(1). (B.11) We then show in the second step that ˆ Z1and ˆ Z2converge in distribution to N(0, 1), which establishes the stated result. Step 1: By definition, ¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ)≤¯ fk(θ0)ˆ V−1 k¯ fk(θ0)=¯ fk(θ0)V−1 k¯ fk(θ0)+¯ fk(θ0)ˆ V−1 k−V−1 k¯ fk(θ0). Note that ¯ fk(θ0)ˆ V−1 k−V−1 k¯ fk(θ0)≤ ˆ V−1 k−V−1 k 2 ¯ fk(θ0) 2 2. From Theorem 4.1, the first-step GMM estimator ˜ θis such that ˜ θ−θ0=OP(k−1/4n−1/4). Hence, Lemma B.2(iv) implies that ˆ V−1 k−V−1 k2=OP(k3/4n−1/4).Thus, ˆ V−1 k−V−1 k 2 ¯ fk(θ0) 2 2=OPk7/4n−1/4=√kOPk5/4n−1/4=oP(√k). As a result, we have ˆ Z=¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ)−k √2k≤¯ fk(θ0)V−1 k¯ fk(θ0)−k √2k+oP(1):=ˆ Z2+oP(1). (B.12) On the other hand, using (B.5), we can write ¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ) =¯ fk(ˆ θ)ˆ V−1/2 k¯ P(k)ˆ V−1/2 k¯ fk(ˆ θ)−¯ fk(θ0)ˆ V−1/2 k¯ P(k)ˆ V−1/2 k¯ fk(θ0)+¯ fk(θ0)ˆ V−1 k¯ fk(θ0) +(ˆη2−η02)¯ D 2ˆ V−1/2 k¯ M(k)ˆ V−1/2 k¯ D2(ˆη2−η02)
886 Dovonon and Gospodinov Quantitative Economics 15 (2024) +1 4z 0n¯ H(k)(¯ θ)ˆ V−1/2 k¯ M(k)ˆ V−1/2 k¯ H(k)(¯ θ)z0n +2¯ fk(θ0)ˆ V−1/2 k¯ M(k)ˆ V−1/2 k¯ D2(ˆη2−η02)+¯ fk(θ0)ˆ V−1/2 k¯ M(k)ˆ V−1/2 k¯ H(k)(¯ θ)z0n +(ˆη2−η02)¯ D 2ˆ V−1/2 k¯ M(k)ˆ V−1/2 k¯ H(k)(¯ θ)z0n :=(1)+(2)+(3)+(4)+(5)+(6)+(7). From (B.6), and Lemma B.2,wehave (1)=OPk3/4n−1/4=oP(1),(3)=OPk1/2n−1/2=oP(1), (4)=1 4z 0nHV−1/2 kM(k)V−1/2 kHz0n+oP(1),(5)=OPk3/4n−1/4=oP(1), (6)=¯ fk(θ0)V−1/2 kM(k)V−1/2 kHz0n+oP(1),(7)=OPk1/4n−1/4=oP(1) and from the lines above, (2)=¯ fk(θ0)V−1 k¯ fk(θ0)+oP(√k).Asaresult,wewrite ¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ)=¯ fk(θ0)V−1 k¯ fk(θ0)+¯ fk(θ0)V−1/2 kM(k)V−1/2 kHz0n +1 4z 0nHV−1/2 kM(k)V−1/2 kHz0n+oP(√k). (B.13) Let the rank factorization of M(k)V−1/2 kHbe M(k)V−1/2 kH=H1H2,whereH1and H2are a (k,rh)-matrix and a (rh,p2)-matrix, respectively, with the same rank rh= Rank(M(k)V−1/2 kH)≤p2. By second-order local identification, rh= 0sothat M(k)V−1/2 kH=0. Thus, (B.13) can be written as ¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ)=¯ fk(θ0)V−1 k¯ fk(θ0)+¯ fk(θ0)V−1/2 kH1H2z0n +1 4z 0nH 2H 1H1H2z0n+oP(√k). Letting m(u):=¯ fk(θ0)V−1 k¯ fk(θ0)+¯ fk(θ0)V−1/2 kH1u+1 4uH 1H1u,andM1:=Ik− H1(H 1H1)−1H 1, we can claim that min u∈Rrh m(u)+oP(√k)=¯ fk(θ0)V−1/2 kM1V−1/2 k¯ fk(θ0)+oP(√k)≤¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ). Letting ˆ Z1:=¯ fk(θ0)V−1/2 kM1V−1/2 k¯ fk(θ0)−k √2k,weobtain(B.11). Step 2: We now show that both ˆ Z1and ˆ Z2are asymptotically standard normal. We first consider ˆ Z2.Wehave ˆ Z2=1 n√2k n t=s:t,s=1 fk(xs,ys,θ0)V−1 kfk(xt,yt,θ0) + 1 n n t=1 fk(xt,yt,θ0)V−1 kfk(xt,yt,θ0)−k √2k:=U1n+U2n.
Quantitative Economics 15 (2024) Robust specification testing 887 The asymptotic normality of U1nfollows readily from the central limit theorem stated by Theorem 5.1. In addition, it is not hard to see that E(U2n)=0. Using similar arguments as in the proof of Proposition S.2 in the Supplemental Material, we can show that E(U2 2n)= o(1). This establishes that U2n=oP(1). We can then conclude that ˆ Z2is asymptotically standard normal. We now consider ˆ Z1. Note first that, since M1is an orthogonal projection matrix on a space of dimension k−rh,thereexistsa(k,k−rh)-matrix S1such that S 1S1=Ik−rhand M1=S1S 1. In that respect, ¯ fk(θ0)V−1/2 kM1V−1/2 k¯ fk(θ0)=¯ fk(θ0)V−1/2 kS1S 1V−1/2 k¯ fk(θ0). Also, Var (S 1V−1/2 k¯ fk(θ0)) =Ik−rh. Using Theorem 5.1 and similar to the lines above for ˆ Z2, we can claim that ˆ Z3:=¯ fk(θ0)V−1/2 kS1S 1V−1/2 k¯ fk(θ0)−(k−rh) 2(k−rh) d →N(0, 1). Since 0 ≤rh≤p2with pfixed, we can see that ˆ Z1=ˆ Z3+oP(1). Therefore, ˆ Z1 d →N(0, 1). Proof of Theorem 5.3. From Lemma B.4,thereexistk0and δ0>0suchthatk>k 0 and E(fk0(xt,yt,ˆ θ))E(fk0(xt,yt,ˆ θ)) ≥δ0.Wehave k3/2n−1|ˆ Z|≥2−1/2kn−1¯ fk(ˆ θ)ˆ V−1 k¯ fk(ˆ θ)−k ≥2−1/2kn−1¯ fk(ˆ θ)¯ fk(ˆ θ)/λmax(ˆ Vk)+21/2k2n−1 ≥2−1/2/¯ λn−1¯ fk0(ˆ θ)¯ fk0(ˆ θ)+o(1), with probability approaching one. Also, n−1¯ fk0(ˆ θ)¯ fk0(ˆ θ)≥Efk0(xt,yt,ˆ θ)Efk0(xt,yt,ˆ θ) +21 n n t=1 fk0(xt,yt,ˆ θ)−Efk0(xt,yt,ˆ θ) Efk0(xt,yt,ˆ θ) ≥δ0−2 1 n n t=1 fk0(xt,yt,ˆ θ)−Efk0(xt,yt,ˆ θ) 2 Efk0(xt,yt,ˆ θ) 2 =δ0+oP(1)OP(1). It follows that, with probability approaching one, k3/2n−1|ˆ Z|≥(2−1/2/¯ λ)δ0+oP(1)and this concludes the proof by setting δ:=(2−1/2/¯ λ)δ0. Proof of nondecreasing γkin Equation (15). We need to show that for k≥k0, γk≤γk+1. For this, we note that R2is fixed for k≥k0. Therefore, it suffices to show that for any a∈Rkand b∈R, and letting v:=(a,b)∈Rk+1, aM(k)a≤vM(k+1)v.
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