Robust inference in deconvolution

Kato, Kengo; Sasaki, Yuya; Ura, Takuya Article Robust inference in deconvolution Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Kato, Kengo; Sasaki, Yuya; Ura, Takuya (2021) : Robust inference in deconvolution, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 12, Iss. 1, pp. 109-142, https://doi.org/10.3982/QE1643 This Version is available at: https://hdl.handle.net/10419/253609 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 12 (2021), 109–142 1759-7331/20210109 Robust inference in deconvolution Kengo Kato Department of Statistics and Data Science, Cornell University Yuya Sasaki Department of Economics, Vanderbilt University Takuya Ura Department of Economics, University of California, Davis Kotlarski’s identity has been widely used in applied economic research based on repeated-measurement or panel models with latent variables. However, how to conduct inference for these models has been an open question for two decades. This paper addresses this open problem by constructing a novel confidence band for the density function of a latent variable in repeated measurement error model. The confidence band builds on our finding that we can rewrite Kotlarski’s identity as a system of linear moment restrictions. Our approach is robust in that we do not require the completeness. The confidence band controls the asymptotic size uniformly over a class of data generating processes, and it is consistent against all fixed alternatives. Simulation studies support our theoretical results. Keywords. Deconvolution, measurement error, robust inference, uniform confidence band. JEL classification. C14, C57. 1. Introduction Empirical researchers are often interested in recovering features of unobserved variables in economic models. With an availability of repeated measurements or panel data, Kotlarski’s identity (Kotlarski (1967), see also Rao (1992)) is one of the most popular tools used to identify probability density functions of unobserved latent variables in additiveerror models. Examples of research topics that use Kotlarski’s identity include, but are Kengo Kato: [email protected] Yuya Sasaki: [email protected] Takuya Ura: [email protected] The first arXiv date: August 28, 2018. This paper supersedes the previous version which was entitled “Inference Based on Kotlsarki’s Identity.” K. Kato is partially supported by NSF grants DMS-1952306 and DMS- 2014636. We would like to thank numerous scholars, seminar participants at Aarhus University, Duke University, KU Leuven, Tinbergen Institute, Universidad Carlos III de Madrid, University of California Berkeley, University of California Davis, University of California Los Angeles, University of California San Diego, University of Illinois at Urbana-Champaign, University of Oxford, University of Texas Austin, and University of Wisconsin Madison, and conference participants at AMES 2019, NASM 2019, New York Camp Econometrics XVI, and 2019 Triangle Econometrics Conference for very helpful comments. The usual disclaimer applies. A replication file is posted (Kato, Sasaki, and Ura (2021)). ©2021 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1643 110 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) not limited to, empirical auctions (e.g., Li, Perrigne, and Vuong (2000), Krasnokutskaya (2011)), education and labor economics (e.g., Carneiro, Hansen, and Heckman (2003), Cunha, Heckman, and Navarro (2005), Cunha, Heckman, and Schennach (2010), Arcidiacono, Aucejo, Fang, and Spenner (2011), Bonhomme and Sauder (2011), Kennan and Walker (2011), Taber and Vejlin (2020)), and earnings dynamics (e.g., Bonhomme and Robin (2010), Botosaru and Sasaki (2018), Hu, Moffitt, and Sasaki (2019)). In these applications, researchers are interested in identifying the probability density function fXof a latent variable Xamong others. The variable Xof interest is not observed in data, but two measurements (Y1Y2)are available in data with classical errors, U1=Y1−Xand U2=Y2−X. Kotlarski’s identity is a nonparametric closed-form identifying restriction for the probability density function fXof Ximplied by this setup. The existing econometric literature on Kotlarski’s identity focuses on identification and consistent estimation of fXand related objects (e.g., Li and Vuong (1998), Li (2002), Schennach (2004a,2004b), Schennach (2008), Bonhomme and Robin (2010), Evdokimov (2010), Zinde-Walsh (2014), Song, Schennach, and White (2015), Firpo, Galvao, and Song (2017)); also see surveys on this literature by Chen, Hong, and Nekipelov (2011) and Schennach (2016). On the other hand, satisfactory inference methods for fXare missing in this literature—in fact, even the sharp rate of convergence is unknown for the estimators based on Kotlarski’s identity under unrestrictive assumptions,1and hence a limit distribution result is unavailable under such assumptions. Indeed some empirical papers implement nonparametric bootstrap without a theoretical guarantee. Other empirical papers, including many of those listed above, often consider parametric estimation and parametric inference given the ill-posedness of the deconvolution problem as well as the lack of available inference methods. In light of the current unavailability of theoretically supported methods of inference, we propose a method of inference for fX. Furthermore, we propose a method of inference that is robust against possible identification failure. This paper develops a confidence band for fX. Our construction of confidence bands works as follows. First, we derive linear complex-valued moment restrictions by modifying the proof of Kotlarski’s identity (Kotlarski (1967)); see also Rao (1992). Second, we let the Hermite orthogonal sieve (cf. Chen (2007)) approximate unknown probability density functions. Third, for a given sieve dimension and for a given class of probability density functions, we compute a bias bound for the linear complex-valued moment restrictions, and slack the linear complex-valued moment restrictions by this bias bound. Fourth, applying Chernozhukov, Chetverikov, and Kato (2019), we compute the supremum of the self-normalized process of the slacked linear complex-valued moment restrictions as the test statistic for each point in a set of sieve coefficients. Fifth, inverting this test statistic in the spirit of Anderson and Rubin (1949) yields a confidence set of sieve approximations to possible probability density functions. Sixth, for a given sieve 1By the unrestrictive assumptions, we specifically mean assumptions that do not impose either known error distribution or symmetric error distribution. Under such settings, the existing convergence rates are not shown to be sharp to the best of our knowledge. A recent paper by Kurisu and Otsu (2019) obtains improved convergence rates compared with those of Li and Vuong (1998) and Bonhomme and Robin (2010), although they are not shown to be sharp either. Quantitative Economics 12 (2021) Robust inference in deconvolution 111 dimension and for a given class for probability density functions, we compute a bias bound for sieve approximations of probability density functions, and the desired confidence band is obtained by uniformly enlarging the set of sieve approximations by this bias bound. The process of identifying fXin additive measurement error models is called deconvolution—for solving convolution integral equations. There are a number of existing papers on nonparametric inference in deconvolution. Bissantz, Dümbgen, Holzmann, and Munk (2007), Bissantz and Holzmann (2008), van Es and Gugushvili (2008), Lounici and Nickl (2011), and Schmidt-Hieber, Munk, and Dümbgen (2013) developed uniform confidence bands for fXunder the assumption of known error distributions.2In most economic applications, however, it is not plausible to assume that the error distributions are known. More recently, Kato and Sasaki (2018) and Adusumilli, Kurisu, Otsu, and Whang (2020) developed uniform confidence bands for fXand the distribution function, respectively, without assuming that the error distributions are known, but they both assume that at least one error distribution is symmetric.3Kotlarski’s identity is a powerful device for new identification results which require neither the known error distribution assumption nor the symmetric error distribution assumption. This useful feature attracts many economic applications including those listed above, but no econometrician has developed a method of inference in this framework for 20 years ever since its first introduction by Li and Vuong (1998) until our present paper. It is not surprising that such an inference method has been missing for long in the literature, given the technical difficulties of the problem. Deconvolution is an ill-posed inverse problem, and inference under this problem is known to be challenging; see Bissantz et al. (2007), Bissantz and Holzmann (2008), Lounici and Nickl (2011), Horowitz and Lee (2012), Hall and Horowitz (2013), Schmidt-Hieber, Munk, and Dümbgen (2013), Chen and Christensen (2018), Kato and Sasaki (2018), Babii (2019), Kato and Sasaki (2019), Adusumilli et al. (2020) for existing papers developing confidence bands in illposed inverse problems for example. We take a robust inference approach à la Anderson and Rubin (1949), and directly work with the moment restrictions based on Kotlarski’s identity. A positive side product of taking this approach is that we do not need to assume the nonvanishing characteristic functions (i.e., we do not need the completeness), which is commonly assumed for nonparametric identification or inversion. It is also worth mentioning that we chose to use the Hermite orthogonal sieve among other sieves in this paper. The Hermite orthogonal sieve has been in fact already known in the literature to be useful to approximate “smooth density with unbounded support” (Chen (2007)); also see her discussion of Gallant and Nychka (1987) therein. In addition to this known advantage, we also find this sieve particularly useful for the deconvolution problem. Note that the deconvolution problem involves applications of the Fourier 2These papers are based on the literature on deconvolution under known error distribution (e.g., Carroll and Hall (1988), Stefanski and Carroll (1990), Fan (1991b), Carrasco and Florens (2011)). Fan (1991a) develops a pointwise asymptotic inference result in this framework. 3These papers are based on the literature on deconvolution under unknown error distribution with auxiliary data or symmetric error distributions (e.g., Diggle and Hall (1993), Horowitz and Markatou (1996), Neumann and Hössjer (1997), Efromovich (1997), Delaigle, Hall, and Meister (2008), Johannes (2009), Comte and Lacour (2011), Delaigle and Hall (2015)). 112 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) transform operation and the inverse Fourier transform operation. To our convenience, the Hermite functions are eigenfunctions of the Fourier transform operator. While we deal with simultaneous restrictions in terms of density and characteristic functions, we can use the Hermite orthogonal sieve to approximate both the density and characteristic functions without having to apply the Fourier transform or the Fourier inverse because of the eigenfunction property. This convenient property saves computational time and resources as costly numerical integration within each iteration of a numerical optimization routine would be necessary if any other sieve were used. The rest of the paper is organized as follows. Section 2derives linear complexvalued moment restrictions based on Kotlarski’s identity. Section 3presents how to construct the confidence band for fXbased on the Hermite orthogonal basis. Section 4 presents asymptotic properties of the confidence band. Section 5applies our proposed method to wildcat auctions, in which we investigate the distribution of the ex post values, exp(X), of the mineral rights. Section 6illustrates simulation studies. The paper concludes in Section 7. All mathematical derivations and details are collected in the Appendix. 2. Linear complex-valued moment restrictions Consider the repeated measurement model Y1=X+U1 Y2=X+U2(1) where Y1and Y2are observed, but none of X,U1,orU2is observed. We are interested in making inference on the probability density function fXof X. We equip this model with the following assumption. Assumption 1. (i) XU1,and U2are continuous random variables with finite first moments. (ii) U1has mean zero,and XU1,and U2are mutually independent. This assumption is standard in the literature on identification and estimation based on Kotlarski’s identity (e.g., Li and Vuong (1998)). In fact, the existing literature imposes an additional assumption, namely the identification condition (nonvanishing characteristic function or the completeness); see Lemma 1ahead for a specific condition. We do not invoke such an identification assumption for the purpose of identification-robust inference; see Remark 1ahead for further details. Note also that this assumption (or any additional assumption that we make ahead) does not require a large support for either X,U1,orU2. This point reassures the identification-robustness. We now fix basic notation. In what follows, EPand VPdenote the expectation and variance operators, respectively, with respect to a joint distribution Pof (Y1Y2).Analogously, Enand Vndenote the expectation and variance operators, respectively, with respect the empirical distribution of nindependent copies of (Y1Y2).Weleti=√−1de- note the imaginary unit. For the set of absolutely integrable functions, L1, we define the Quantitative Economics 12 (2021) Robust inference in deconvolution 113 Fourier transform Fon L1by [Ff](t) =∞ −∞exp(itx)f(x)dx, and its inverse transform is [F−1φ](x) =1 2π∞ −∞exp(−itx)φ(t)dt;seeFolland (2007). In light of Assumption 1(i), we let fX,fU1,andfU2denote the density functions of X,U1,andU2, respectively. Further, we denote the characteristic functions of them by φX=FfX,φU1=FfU1,and φU2=FfU2. We first review the existing result of the identification. Lemma 1 (Kotlarski’s Identity). For every joint distribution Pof (Y1Y2)satisfying Assumption 1for (1)and EP[exp(itY2)]=0for all t∈R,we have φX(t) =expt 0 iEPY1exp(iτY2) EPexp(iτY2)dτ(2) This lemma presents Kotlarski’s identity due to Kotlarski (1967); see also Rao (1992). Since it is stated as a lemma, Kotlarski’s identity is also known as Kotlarski’s lemma or the lemma of Kotlarski in the econometrics literature. Li and Vuong (1998) first introduced it into econometrics and statistics, followed by a series of extensions (Li (2002), Schennach (2004a), Bonhomme and Robin (2010), Evdokimov (2010)). Some of these extensions relax the assumptions for identification and estimation in various ways. We do not need to rely on the prototypical assumptions for our purpose of inference, even though they are stated in Lemma 1for convenience of a concise review. Lemma 1shows that the characteristic function φXof Xis explicitly identified by the joint distribution of (Y1Y2).Under the additional assumption of absolutely integrable characteristic function φX,the formula fX=F−1φXin turn yields the identification of the probability density function fXof X. Uniform convergence rates for the estimator of fXbased on Kotlarski’s identity are discovered in the existing literature (Li and Vuong (1998), Li (2002), Schennach (2004a), Bonhomme and Robin (2010), Evdokimov (2010)), but the sharp rates under unrestrictive assumptions are still unknown; see footnote 1. In particular, limit distribution results under such assumptions are still unknown in the existing literature. This paper does not aim to derive a nondegenerate limit distribution for any estimator, but it aims to conduct an inference on fX. With this said, our proposed inference approach does not rely on an explicit identifying formula. We argue that rewriting Kotlarski’s identity in terms of moment restrictions suffices and serves even more conveniently for the sake of conducting inference. Here are the moment restrictions obtained from rewriting Kotlarski’s identity. Theorem 1 (Linear Complex-Valued Moment Restrictions). For every joint distribution Pof (Y1Y2)satisfying Assumption 1for (1), EPiY1φX(t) −φ(1) X(t)exp(itY2)=0(3) holds for every real t. A proof is provided in Appendix A.1. 114 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) Remark 1. Taking a few more steps beyond the claim in Theorem 1will lead us to the identification result of Lemma 1under the additional assumption of the invertibility or nonvanishing characteristic functions (also known as the completeness); see D’Haultfoeuille (2011).4Specifically, under the completeness, (3) has the unique solution equal to (2). For the purpose of inference, however, it is not essential to solve the inverse problem, and thus we stop short of obtaining the explicit formula (2), and only use the moment condition (3). This idea is analogous to that of Santos (2011,2012), where robust inference for functional parameters is conducted without assuming the completeness. Since the assumption of completeness is not testable in general (Canay, Santos, and Shaikh (2013)), we propose the robust inference approach based on (3) instead of (2). Remark 2. In addition to φX, Kotlarski’s identity also identities φU1and φU2by φY1/φX and φY2/φX, respectively. Analogously, we may augment the moment restriction (3) in Theorem 1with EP[exp(itY1)−φX(t)φU1(t)]=0to partially identify (φXφU1) jointly, and similarly, we may augment the moment restriction (3)inTheorem1with EP[exp(itY2)−φX(t)φU2(t)]=0to partially identify (φXφU2)jointly. These augmented moment restrictions can be obtained without strengthening Assumption 1. Remark 3. For the rest of the paper, we focus on the moment restriction in (3). This moment restriction is an implication from Assumption 1for (1) and it holds for the true characteristic function φXof X. It is worth noting that this characterization may not be sharp, that is, there could be other moment restrictions implied by Assumption 1. 3. Confidence band with the Hermite orthogonal basis In this section, we introduce a confidence band for fXusing the Hermite function basis.5 We consider a prespecified significance level α∈(01/2)throughout. 3.1 The Hermite orthonormal basis We recommend the Hermite orthonormal basis in particular for its convenient properties and its nice compatibility with the deconvolution framework—a Hermite function is an eigenfunction of the Fourier transform and the Fourier inverse.6The Hermite functions take the form ψj(x) =1 2jj!√π·exp−x2/2·Hj(x) (4) 4Evdokimov and White (2012) provided a relaxed assumption for the identification. 5We remark that our confidence band can be constructed using a different basis as well. For example, if we know that fXhas a bounded support, it is preferred to use a basis with a bounded support. For general discussions on sieve basis, see Chen (2007). 6The Hermite orthonormal sieve is not location invariant, and hence we recommend to location and scale normalize the observed data using the empirical moments. Quantitative Economics 12 (2021) Robust inference in deconvolution 115 for j=01,whereHjis the Hermite polynomial defined by Hj(x) =(−1)j·expx2·dj dxjexp−x2 As emphasized earlier, the Hermite functions are the eigenfunctions of the Fourier transform operator, and specifically, φj=Fψj=ij√2πψjholds. 3.2 Basis expansion and approximation for the density function Provided that fXbelongs to the L2space, we have the basis expansion fX=∞  j=0fXψj·ψj with the Hermite functions {ψj},where·· denotes the inner product defined by f1f2=Rf1(x)f2(x) dx (cf. Blanchard and Bruening (2002), Theorem 16.3.1). Instead of using all of the terms, we focus on the first (q +1)terms. In other words, we use q  j=0fXψj·ψj to approximate fX. We hereafter use θ=(θ0θq)Tfor a generic value for the unknown parameter vector (fXψj:j=01q)T.Foranyq∈N, we consider the set of sieve coefficients given by Θq+1=θ∈Rq+1:|θj|≤sup x∈Rψj(x)(5) In order to approximate fXwell by finite terms from the Hermite orthonormal basis, we impose the following restrictions on fX. Assumption 2. fXbelongs to a known set,L,of probability density function f’s with f∈L2and Ff∈L1satisfying |f ψj|≤j−3for every j≥q+1. We impose f∈L2to approximate Lby the Hermite orthonormal basis, whereas we impose Ff∈L1for applying the inverse Fourier transform. For the last part of the assumption, we restrict the coefficient behavior for the Hermite basis expansion of the density function fX. With this condition, we can bound the error from approximating the functions, φXand φ(1) X, which appear in the moment condition (3). This condition implies the differentiability of the density fXand, therefore, it excludes nondifferentiable density functions such as the uniform distribution and the triangular distribution. When we can take a sufficiently large value of q, a sufficient condition for |fXψj| ≤ j−3∀j≥q+1is that the function xκfX(x) and the first κth derivatives of fX(t) are bounded and integrable for some integer κ≥7.7 7Boyd (1984, p. 385) shows jκ/2|fXψj|=O(1)as j→∞. Since we can bound O(1)by j(κ−6)/2for sufficiently large j,wehave|f ψj|≤j−3for every sufficiently large j. 116 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) 3.3 Inequality constraints for (fXψj:j=01q) The moment restrictions of the form (3) impose equality restrictions. With this said, the finite-dimensional sieve approximation by Θq+1entails a truncation bias. This bias causes the equality restrictions into inequality restrictions with slackness by a uniform bias bound that can be determined by the restriction in Assumption 2.Inthissection,we present the resultant inequality restrictions and propose concrete choices of the slackness parameters. To construct a confidence set for (fXψj:j=01q), we are going to derive the two types of inequality constraints about θ,whicharetruewhenθ=(fXψj:j= 01q)T. The first type derives from the moment restrictions (3)inTheorem1,and the second derives from natural restrictions on any density function. First, using the moment restrictions of the form (3)inTheorem1,weobtainthe moment inequality constraints of the form −δq≤ReEPiY1 q  j=0fXψj·φj(t) − q  j=0fXψj·φ(1) j(t)exp(itY2)≤δq(6) −δq≤ImEPiY1 q  j=0fXψj·φj(t) − q  j=0fXψj·φ(1) j(t)exp(itY2)≤δq(7) where Re(·)(resp., Im(·)) denotes the real (resp., imaginary) part of a complex number. Here, δqis the uniform approximation bound for the truncation bias, and is defined in (14) ahead. These inequalities are formally derived in Appendix A.2 as Lemma 2.Note that there are 4Linequalities in total if we use Lgrid points t1tLof frequencies to evaluate the moment restrictions—2Linequalities for the real part and 2Linequalities for the imaginary part. We can choose t1tLsuch that they are equally distributed in the interval [−h−1h−1]selected according to Appendix B.2. To economize our writings for (6)and(7), we introduce additional notation. For every function ψ∈Land for every frequency t∈R, define Rψt (y1y2) =−cos(ty2)y1Imφ(t)+Reφ(1)(t)−sin(ty2)y1Reφ(t)−Imφ(1)(t) (8) and Iψt (y1y2) =cos(ty2)y1Reφ(t)+Imφ(1)(t)−sin(ty2)y1Imφ(t)−Reφ(1)(t)(9) where φ=Fψ. Further, stack these functions across ψ∈{ψ0ψq}to in turn define the random vector Rt=Rψ0t(Y1Y2)Rψqt(Y1Y2)Tand (10) It=Iψ0t(Y1Y2)Iψqt(Y1Y2)T(11) Quantitative Economics 12 (2021) Robust inference in deconvolution 123 Corollary 1 (Consistency). Pick any P∈Pand f∈Lsuch that for some sequence t∗= t∗(n) ∈{t1tL}, √ninf θ∈Bq+1ηq(f) maxEP[Rt∗]Tθ−δqEP[It∗]Tθ−δq maxaP n(t∗)λP max(t∗)"log(1+L) Then PP(f /∈Cn(α)) →1provided that sup θ∈Θq+1 max l=1L maxθTVn(Rtl)−VP(Rtl)θθTVn(Itl)−VP(Itl)θ=oPλP max(t∗) A proof is provided in Appendix A.6. 5. Application to wildcat auctions In this section, we present an empirical application of our proposed method to wildcat auctions. The data set that we use is the Outer Continental Shelf (OCS) Auction Data. This data set records bids for mineral rights on oil and gas on offshore lands off the coasts of Texas and Louisiana in the gulf of Mexico. We refer interested readers to Hendricks, Porter, and Boudreau (1987), for example, for details of this data set. Among other types of sales, we focus on wildcat sales, that is, sales of rights for oil and gas tracts whose geological or seismic characteristics are unknown to bidding firms. The sales follows the first-price sealed-bid auction mechanism. In this mechanism, bidding firms simultaneously submit sealed bids. The bidder with the highest bid pays the price which they submitted to receive the right for the tract. Prior to an auction, firms which are planning to participating in the auction can carry out a seismic investigation to estimate the value of the rights. Results of these seismic investigation provide firm 1(resp., firm 2) with the ex ante value Y1(resp., Y2)ofthe mineral right in the logarithm of US dollars per acre. These ex ante values, Y1and Y2,are two measurements of the ex post value X, also known as the common component, in the logarithm of US dollars per acre, with ex ante assessment errors, U1and U2, respectively, also known as the private components, in the logarithm of US dollars per acre. In this setting, we obtain the system of repeated measurement error equations Y1=X+U1 Y2=X+U2 as is the case with the main equation (1) of our model framework. We assume that the ex post value Xand the two firms’ ex ante assessment errors, U1and U2, are continuously distributed and are mutually independent. Furthermore, we also assume the rational expectations, that is, E[U1]=0. These conditions satisfy our Assumption 1. Li, Perrigne, and Vuong (2000) applied the method of Li and Vuong (1998)tothis setup and this data set in order to estimate fX, but they do not conduct a statistical inference. Krasnokutskaya (2011) similarly estimated the density function fXin a different framework, and draws its confidence intervals via nonparametric bootstrap. They 124 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) first recover firms’ ex ante values (Y1Y2)from bid data through the method of Guerre, Perrigne, and Vuong (2000) based on an equilibrium restriction (Bayesian Nash equilibrium) for the first-price sealed-bid auction mechanism. Following their approach, we also directly take these ex ante values (Y1Y2)as the data to be used as an input for our analysis, in light of the faster convergence rate of the preliminary estimation than the convergence rate of the deconvolution estimation. The sample consists of 169 tracts with 2firms in each tract. Applying our proposed method, we draw the 95% confidence band Cn(005)for the density function fXof the ex post values Xon I=[416616]as follows. Following the settings for our simulations to be presented in Section 6,wesetL=50 for the grid size of frequencies, and q=2such that the dimensionality of θis 3.12 The frequency domain of integration for the Fourier transform is set to be the interval [−h−1h−1],whereh=0289 is chosen based on the procedure outlined in Appendix B.2. In other words, we construct Rtlas in (10)andItlas in (11) at each of the 50 equally spaced grid points t1t50 from −1/0289 to 1/0289.WiththeseRtland Itlfor t1t50, we construct the test statistic T(θ)=√nmax 1≤l≤LmaxEn[Rtl]Tθ−δq θTVn(Rtl)θ En[Itl]Tθ−δq θTVn(Itl)θ where L=50 and (14) yields the approximation bound δq=0139 for the current application. Let c(005)denote the critical value obtained by setting α=005,L=50,and n=169 in c(α) =−11−α/(4L) 1−−11−α/(4L)2/n  Finally, using (13), we obtain the approximation error bound of ηq=0048 for the density function. With these settings, the lower bound of the confidence band Cn(005)is fL(x) − ηqwhere fL(x) =min θ∈Θq+1ψ(x)Tθsubject to T(θ)≤c(005) ψ(x)Tθ≥−ηqfor all x∈R √2πψ(0)Tdiag(10−10)θ−1≤ηq for all x∈I, and the upper bound of it is fU(x) +ηqwhere fU(x) =max θ∈Θq+1ψ(x)Tθsubject to T(θ)≤c(α) ψ(x)Tθ≥−ηqfor all x∈R 12Given that our sample size in this section is small, we use a small value of the sieve dimension and control the magnitude of the variance at the expense of the bias. In fact, the area of the light gray shade representing the bias is already much smaller than that of the dark gray area representing the stochastic part even with this small sieve dimension in Figure 1to be discussed ahead. Quantitative Economics 12 (2021) Robust inference in deconvolution 125 Figure 1. 95% confidence band Cn(005)of the density function fXof the ex post values Xof the mineral right in the logarithm of US dollars per acre. The dark gray shade represents the band based on the stochastic part, and the light gray shade represents the uniform bias bound ηq. √2πψ(0)Tdiag(10−10)θ−1≤ηq for all x∈I.13 We numerically solve these constrained optimization problems via Newton’s method with the penalty factor of 1000 and the penalty exponent of 2for the constraints. Note that the test statistic T(θ)in the constraint derives from objects in the frequency domain, and evaluation of it in general would require a numerical integration for Fourier transform within each iteration of the numerical optimization routine. We recommend to use the Hermite orthogonal sieve to substantially reduce this computational burden. Since the Hermite functions are the eigenfunctions of the Fourier transform and inverse operators, we only need to multiply by the eigenvalues, and thus do not need to conduct a numerical integration within each iteration thanks to the Hermite orthogonal sieve. Figure 1depicts the resultant confidence band Cn(005). Density functions per se are often of research interest in the auction literature (e.g., Li, Perrigne, and Vuong (2000), Krasnokutskaya (2011)). With this said, many parameters of economic interest can be also obtained as functionals of the density function fX.We next use Cn(005)to obtain confidence intervals for parameters of economic interest as functionals of the density function fX. Specifically, we are interested in the average ex post values of the mineral rights in US dollars per acre. Note that Xis the logarithm of the pecuniary unit, and hence, parameters of more economic interest would be statistics about exp(X).(IftheaverageofXwere the parameter of interest, one could simply estimate its mean by E[X]=E[Y1]and obtain its confidence interval without relying on 13See footnote 10 for the definition of diag(10−10). 126 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) Table 1. 95% confidence intervals of the mean and quantiles of the ex post values of mineral rights on oil and gas on offshore lands off the coasts of Texas and Louisiana in the Gulf of Mexico in US dollars per square acre. The first column shows the 95% confidence interval for the mean. The remaining five columns show the 95% confidence intervals for the 01-th, 03-th, 05-th, 07- th, and 09-th quantiles. Mean Quantiles E[exp(X)]τ=01τ=03τ=05τ=07τ=09 Upper bound 219 114 150 190 267 362 Lower bound 146 81 100 131 166 218 the deconvolution approach.) We obtain the 95% confidence interval for the average ex post values, E[exp(X)], of mineral rights in US dollars per acre by exp(x)f(x) dx|f∈Cn(005) Similarly,wecanobtainthe95% confidence interval for the τth quantile of the ex post values exp(X) of mineral rights in US dollars per acre by infexp(x)|x −∞ fxdx≥τf∈Cn(005) Table 1summarizes the 95% confidence interval for the mean E[exp(X)]and the 95% confidence interval for the quantile Qexp(X)(τ) with τ=0103050709.The 95% confidence interval for the average ex post value E[exp(X)]in US dollars per acre is [146219]. This is close to, but a little above the 95% confidence interval for the median, which is [131190].The95% confidence intervals for the five quantile points highlight a large variation of the ex post values of the mineral rights across tracts. 6. Simulation studies In this section, we present and discuss finite-sample performance of the proposed method by simulation studies. Simulation outcomes that we present include the size under the null of the true distribution, the power under alternative distributions, and the lengths of confidence bands. The lengths will be further decomposed into the bias bound ηqand the remaining lengths due to the stochastic part. 6.1 Simulation setting We employ three distribution families to generate the latent variable X—the normal distribution, the skew normal distribution, and the tdistribution. We employ the skew normal distribution and the tdistribution to see whether our method is effective for asymmetric distributions and super-Gaussian tails, respectively. Specifically, we generate a random sample of (X U1U2)mutually independently according to the marginal Quantitative Economics 12 (2021) Robust inference in deconvolution 127 laws: Model 1: X∼Nξ1ξ2 2U 1∼N0σ2 U1U 2∼N0σ2 U2 Model 2: X∼SN(ξ1ξ2ξ3) U1∼N0σ2 U1U 2∼N0σ2 U2 Model 3: X∼tξ4U 1∼N0σ2 U1U 2∼N0σ2 U2 Here, N(ξ1ξ2 2)denotes the normal distribution with mean ξ1and variance ξ2 2,SN(ξ1 ξ2ξ3)denotes the skew normal distribution with location ξ1,scaleξ2,andshapeξ3,and tξ4denotes the tdistribution with ξ4degrees of freedom. The distribution parameters for the latent variable Xare set to (ξ1ξ2)=(01)for Model 1, (ξ1ξ2ξ3)=(011)for Model 2, and ξ4=10 for Model 3. The choice of the normal error distribution, which is an instance of supersmooth distributions, imposes a difficult case in deconvolution; see Li and Vuong (1998). The error variance parameters are set to σU1=σU2=05in each of the three models. We conduct experiments with three sample sizes n=250, 500, and 1000,andrun2500 Monte Carlo iterations for each set of simulations. In the simulation studies, we use q=2. We have experimented with alternative sieve dimensions q∈{46}, but the results are qualitatively similar. The number of frequency grid points is set to L=50. The interval on which the confidence band is formed is set to I=[E[X]−2√Var(X) E[X]+2√Var(X)],whereE[X]and Var(X) are the theoretical mean and the theoretical variance, respectively, of Xunder the relevant model. Throughout, we use the critical value c(αθ)based on 1000 multiplier bootstrap iterations. The level is set to α=005 throughout. 6.2 Simulation results Figure 2(A) shows the simulated frequencies that the confidence band formed under Model 1 covers alternative probability density functions for N(ξ1ξ2 2)indexed by location parameter values ξ1∈[0010]while the scale parameter is fixed at the true value ξ2=10. The coverage frequency under ξ1=00indicates (the complement of) the size, whereas the coverage frequencies under ξ1∈(0010]indicate (the complement of) the power. Similarly, Figure 2(B) shows the simulated frequencies that the confidence band formed under Model 1 covers alternative probability density functions for N(ξ1ξ2 2)indexed by scale parameter values ξ2∈[1020]while the location parameter is fixed at thetruevalueξ1=00. These results show the correct size and increasing power. The size entails overcoverage, which is still consistent with our theory on size control. Although we present these coverage and power results only for Model 1, we observe similar qualitative patterns for Models 2 and 3. We display instances of confidence bands in Figure 3. The gray shades indicate the confidence bands including the bias bound and the stochastic parts together. The internal dark gray shades include only the stochastic parts. We also plot the true density functions and Li–Vuong estimates (with the choice of tuning parameter according to Appendix B.2) as solid and dashed curves, respectively. While such instances of confidence bands will not tell us any evidence on the statistical properties, they at least inform how a confidence band may look in applications. 128 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) Figure 2. The simulated frequencies that the confidence band formed under Model 1 covers alternative probability density functions for N(ξ1ξ2 2). Panel (A) runs across alternative location parameter values ξ1∈[0010]while the scale parameter is fixed at the true value ξ2=10.Panel (B) runs across alternative scale parameter values ξ2∈[1020]while the location parameter is fixed at the true value ξ1=00. 7. Conclusion Since its introduction to econometrics by Li and Vuong (1998), Kotlarski’s identity (Kotlarski (1967)) (see also Rao (1992)) has been widely used in empirical economics. Examples include applications to empirical auctions (e.g., Li, Perrigne, and Vuong (2000), Krasnokutskaya (2011)), education and labor economics (e.g., Carneiro, Hansen, and Heckman (2003), Cunha, Heckman, and Navarro (2005), Cunha, Heckman, and Schennach (2010), Arcidiacono et al. (2011), Bonhomme and Sauder (2011), Kennan and Walker (2011), Taber and Vejlin (2020)), and earnings dynamics (e.g., Bonhomme and Robin (2010), Botosaru and Sasaki (2018), Hu, Moffitt, and Sasaki (2019)). Despite its popular use in applications, a method of inference based on Kotlarski’s identity has long been missing in the literature. After 20 years since Li and Vuong (1998), we now propose a method of inference based on Kotlarski’s identity. Specifically, we develop confidence bands for the probability density function fXof Xin the repeated measurement model where two measurements (Y1Y2)of unobserved variable Xare available in data with additive independent errors, U1=Y1−Xand U2=Y2−X. Our construction of confidence bands can be summarized as follows. First, we derive linear complex-valued moment restrictions based on Kotlarski’s identity. Second, we Quantitative Economics 12 (2021) Robust inference in deconvolution 129 Figure 3. Instances of confidence bands. The gray shades indicate the confidence bands. The internal dark gray shades indicate the stochastic parts of the confidence bands. The solid and dashed curves indicate the true density functions and Li–Vuong estimates, respectively. let the Hermite orthogonal sieve approximate unknown probability density functions. Third, for a given sieve dimension and for a given class for probability density functions, we compute a bias bound for the linear complex-valued moment restrictions, and slack the linear complex-valued moment restrictions by this bias bound. Fourth, we compute the uniform norm of the self-normalized process of the slacked linear complex-valued moment restrictions as the test statistics for each point in a set of sieve coefficients. Fifth, inverting this test statistic yields a confidence set of sieve approximations to possible 130 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) probability density functions. Sixth, for a given sieve dimension and for a given class for probability density functions, we compute a bias bound for sieve approximations of probability density functions, and the desired confidence band is obtained by uniformly enlarging the set of sieve approximations by this bias bound. We not only provide a method that is guaranteed to work theoretically, but also care for its practicality. The Fourier transform and the inverse Fourier transform operations are known to be computationally costly in the deconvolution literature. By exploiting the property of the Hermite functions as eigenfunctions of the Fourier transform operator, we propose to let the Hermite orthogonal sieve approximate both the density and characteristic functions without having to implement numerical integrations within each iteration of a numerical optimization routine. This convenient feature of the proposed method saves computational resources. Simulation studies indeed conclude reasonably fast with informative inference results. The results evidence the efficacy of the proposed method. Since latent-variable models with repeated measurements and panel data are of use in a number of applied fields, including empirical auctions, income dynamics, and labor economics, we hope that our method will contribute to the practice of economic analyses in these and other topics. We conjecture that the proposed methodology can be extended to related models. For example, Cunha, Heckman, and Schennach (2010) considered a nonlinear factor model for the evolution of unobserved multidimensional skills. Our current methodology is not readily applicable to their model because we focus on the univariate analysis for X. We speculate that our proposed strategy will work for the inference in a more complicated model, such as that of Cunha, Heckman, and Schennach (2010). Namely, as in this paper, it could be possible to construct a confidence set for the sieve coefficient of the unknown functions and use a bias bound (ηqin this paper) to enlarge the confidence set. Since it is beyond the scope of this paper, we leave such extensions for future work. Appendix A: Proofs for the results in the main text A.1 Proof of Theorem 1(linear complex-valued moment restrictions) Proof. By Assumption 1(ii), we have the following three equations: EPiX exp(itX) exp(itU2)=EPiX exp(itX))EPexp(itU2) EPiU1exp(itX)exp(itU2)=EPiEP[U1|XU2]exp(itX)exp(itU2)=0 EPexp(itX) exp(itU2)=EPexp(itX)EPexp(itU2) for every real t,whereweuseE[U1|XU2]=0and the independence between Xand U2.By(1), EPiY1φX(t) −φ(1) X(t)exp(itY2) =EPiX exp(itX) exp(itU2)φX(t) +EPiU1exp(itX)exp(itU2)φX(t) Quantitative Economics 12 (2021) Robust inference in deconvolution 131 −EPexp(itX) exp(itU2)φ(1) X(t) =EPiX exp(itX))EPexp(itU2)φX(t) −EPexp(itX)EPexp(itU2)φ(1) X(t) =0 for every real t, where the last equality follows from φ(1) X(t) =EP[iX exp(itX))]. A.2 Proof of Lemma 2 Proof. Regarding the first statement, the Hermite basis expansion for the pdf fX= ∞ j=0fXψj·ψj(x) implies ψ(x)dxT θ−1=ψ(x)dxT θ−fX(x)dx=∞  j=q+1fXψj·ψj(x) dx By the triangle inequality, Assumption 2, and the definition of ηq, ψ(x)dxT θ−1≤∞  j=q+1fXψjψj(x)dx ≤∞  j=q+1 j−3ψj(x)dx ≤ηq Regarding the second statement, the Hermite basis expansion implies inf x∈Rψ(x)Tθ≥inf x∈Rψ(x)Tθ−fX(x)=−sup x∈R ∞  j=q+1fXψj·ψj(x) By the triangle inequality, Assumption 2, and the definition of ηq, inf x∈Rψ(x)Tθ≥− ∞  j=q+1fXψj·sup x∈Rψj(x)≥− ∞  j=q+1 j−3sup x∈Rψj(x)≥−ηq Regarding the last two statements, EP[(iY1φX(t) −φ(1) X(t)) exp(itY2)]=0implies EP[Rtl]Tθ2+EP[Itl]Tθ2 =EPiY1 q  j=0fXψj·φj(t) − q  j=0fXψj·φ(1) j(t)exp(itY2) =EPiY1 ∞  j=q+1fXψj·φj(t) −∞  j=q+1fXψj·φ(1) j(t)exp(itY2) By the triangle inequality, Assumption 2, and the definition of δq, EP[Rtl]Tθ2+EP[Itl]Tθ2 132 Kato, Sasaki, and Ura Quantitative Economics 12 (2021) ≤EP|Y1|∞  j=q+1fXψj·φj(t)+∞  j=q+1fXψj·φ(1) j(t)exp(itY2) =EP|Y1|∞  j=q+1fXψj·φj(t)+∞  j=q+1fXψj·φ(1) j(t) ≤∞  j=q+1 j−3EP|Y1|φj(t)+φ(1) j(t) ≤δq A.3 Proof of Lemma 3 Proof. There is some constant ε>0such that EPiY1φ(¯ t)−φ(1)(¯ t)exp(i¯ tY2)>2ε where φ=Ff. Suppose qand Lh are sufficiently large so that ε≥∞  j=q+1 j−3EP|Y1|sup t∈Rφj(t)+sup t∈Rφ(1) j(t)=δq and that ε≥sup t∈Rφ(2)(t)+EP|Y1|+|Y2|+1sup t∈Rφ(1)(t)+EP|Y1]+|Y1Y2|2 Lh Note that supt∈R|φ(1)(t)|and supt∈R|φ(2)(t)|are finite by Assumption 2.Bythesecond inequality, we have  ∂ ∂t EPiY1φ(t) −φ(1)(t)exp(itY2) ≤EP|Y1|+1φ(1)(t) +EP|Y1|+φ(2)(t) +EP|Y1Y2|+EP|Y2|φ(1)(t) =φ(2)(t)+EP|Y1|+|Y2|+1φ(1)(t)+EP|Y1]+|Y1Y2| ≤Lh 2ε Since |tl−¯ t|≤2/(Lh) for some l=1L,wehave EPiY1φ(tl)−φ(1)(tl)exp(itlY2) ≥EPiY1φ(¯ t)−φ(1)(¯ t)exp(i¯ tY2) Quantitative Economics 12 (2021) Robust inference in deconvolution 139 Canay, I. A., A. Santos, and A. M. Shaikh (2013), “On the testability of identification in some nonparametric models with endogeneity.” Econometrica, 81, 2535–2559. [114] Carneiro, P., K. T. Hansen, and J. J. 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Manuscript received 23 May, 2020; final version accepted 31 August, 2020; available online 15 September, 2020.