Monotonicity-Constrained Nonparametric Estimation and Inference for First-Price Auctions
aa r X i v : . [ ec on . E M ] S e p Monotonicity-Constrained Nonparametric Estimation andInference for First-Price Auctions ∗ Jun Ma † Vadim Marmer ‡ Artyom Shneyerov § Pai Xu ¶ Abstract
We propose a new nonparametric estimator for first-price auctions with independent private values that imposesthe monotonicity constraint on the estimated inverse bidding strategy. We show that our estimator has a smallerasymptotic variance than that of Guerre, Perrigne and Vuong’s (2000) estimator. In addition to establishingpointwise asymptotic normality of our estimator, we provide a bootstrap-based approach to constructing uniformconfidence bands for the density function of latent valuations.
Shape restrictions on infinite-dimensional parameters have received much attention in econometric research. The rolesof shape restrictions in the literature include facilitating identification, providing testable implications and improvingestimation and inference. See Chetverikov et al. (2018) for a recent review.This paper focuses on the first-price sealed-bid auction model with symmetric bidders, which is the same as thatstudied in Guerre et al. (2000, GPV, hereafter). GPV’s estimation strategy uses a nonparametrically estimated inversebidding function to generate pseudo valuations. However, the true bidding strategy must be strictly increasing and theplug-in nonparametric estimator of GPV ignores this shape restriction. Imposing such a constraint at the estimationstage nonparametrically is interesting. For this purpose, we may use the methods in the monotone nonparametricregression literature. Some of these methods can be easily adapted to producing constrained estimators for theinverse bidding strategy in the auction setting. E.g., Henderson et al. (2012) take the constrained re-weightingapproach pioneered by Hall and Huang (2001), which was originally used to build a monotone estimator for theconditional expectation function. Luo and Wan (2018) impose monotonicity by using the greatest convex minorant ofthe integrated quantile function of values.In this paper, we pursue a different approach. We investigate the asymptotic properties of a new monotonicity-constrained estimator based on the smooth rearrangement approach of Dette et al. (2006) and propose a uniformconfidence band around this monotonicity-constrained nonparametric estimator. We show that this rearrangement-based monotonicity-constrained estimator is asymptotically normal with an asymptotic variance smaller than theunconstrained estimator as in GPV. Since the asymptotic variance plays an important role in determining the widthof a uniform confidence band in large samples, the fact that the rearrangement-based estimator has a smaller asymptotic ∗ This version: October 1, 2019 † School of Economics, Renmin University of China ‡ Vancouver School of Economics, University of British Columbia § Department of Economics, Concordia University ¶ School of Economics and Finance, University of Hong Kong See Henderson and Parmeter (2009) for a comprehensive survey of the literature on monotone nonparametric regression. As a by-product, our method also produces a simple estimator for the true bidding function. Note that GPV’sprocedure is based on the inverse-bidding strategy, which has a simple form. On the other hand, the bidding functionhas an integral expression that depends on the unknown distribution of latent valuations and constructing its directplug-in type estimator would be cumbersome. Our simple estimator of the bidding function can be of interest on itsown, as it can be used in practical applications for computing counterfactual bids.We compare the finite sample performances of the confidence bands based on the rearrangement-based and theunconstrained estimators in Monte Carlo experiments. We find that the confidence band based on the rearrangement-based estimator tends to be narrower without sacrificing the coverage accuracy.The literature on structural econometrics of auctions is vast. See Gentry et al. (2018) for a recent review; see alsothe reviews of the literature in Athey and Haile (2007) and Hendricks and Porter (2007). In their seminal paper,GPV demonstrate nonparametric identification of the first-price auction model with independent private values, andpropose two-step nonparametric estimation of the density of latent valuations. This paper improves on the GPVestimator by incorporating the monotonicity constraint in the nonparametric estimation procedure.In a recent paper, Ma et al. (2018, MMS, hereafter) describe the asymptotic distribution of the GPV estimatorand propose a valid bootstrap procedure based on the GPV estimator. They also propose a procedure for constructinguniform confidence bands for the density of latent valuations. This paper builds on their results.The GPV estimator has been widely used in the empirical literature. For examples of applications, see theliterature review in Athey and Haile (2007) and Hendricks and Porter (2007). The GPV approach has been alsoutilized in models with risk aversion (Guerre et al., 2009, Zincenko, 2018), unobserved heterogeneity (Krasnokutskaya,2011), bidder asymmetry and affiliated values (Li et al., 2002), common values (Haile et al., 2003, Hendricks et al.,2003), and entry (Li and Zheng, 2009, Marmer et al., 2013, Gentry and Li, 2014).The recent related literature includes Liu and Vuong (2013), who propose a test for the monotonicity of the biddingfunction, Liu and Luo (2017), who propose a procedure for comparing valuation distributions, Marmer and Shneyerov(2012), Luo and Wan (2018), Gimenes (2017), who propose quantile based methods in the context of auctions. Ourpaper is also related to the econometrics literature on two-step nonparametric estimation. See, e.g., Mammen et al.(2012).The rest of the paper is organized as follows. Section 2 introduces the empirical auction model studied in thispaper and its estimation technique, including the GPV estimator and a new monotonicity-constrained estimator. InSection 3, we show that the new estimator is asymptotically normal with a smaller asymptotic variance, compared tothe unconstrained estimator. Section 4 provides an estimator for the asymptotic variance and a uniform confidenceband around the new monotonicity-constrained estimator. We extend the proposed estimation and inference methodto an auction model with observed auction heterogeneity in Section 5. Section 6 reports Monte Carlo simulationresults. Proofs are collected in the appendix.
Notation. “ a := b ” is understood as “ a is defined by b ”. “ a =: b ” is understood as “ b is defined by a ”. ( · ) denotesthe indicator function, and we also denote A := ( · ∈ A ) . Let ℓ ∞ ( A ) be the class of bounded functions defined on A . For any f ∈ ℓ ∞ ( A ) , let k f k A := sup x ∈ A | f ( x ) | be the sup-norm. See Dette and Pilz (2006) for simulation studies that compare the rearrangement and reweighting approaches to monotone regression.Dette and Pilz (2006) notice that the rearrangement approach has computational advantage since the reweighting approach requires solvingconstrained optimization. The Auction Model and Estimation
In this section, we consider an auction model for homogeneous goods and with a fixed number of bidders. A modelwith observed covariates capturing auction-specific heterogeneity will be considered in Section 5. The econometricianobserves bids from L auctions, with a fixed number of bidders in each auction: { B il : i = 1 , . . . , N, l = 1 , . . . , L } . (2.1)Bidders’ valuations { V il : i = 1 , . . . , N, l = 1 , . . . , L } are not unobservable to the econometrician. We assume the distribution of the valuations satisfy the following as-sumption. Assumption 1 ( Data Generating Process ) . (a). The unobserved valuations { V il : i = 1 , . . . , N, l = 1 , . . . , L } are i.i.d. with PDF f and CDF F . (b). f is strictly positive and bounded away from zero on its support, a compactinterval [ v, v ] ⊆ R + , and is twice continuously differentiable on ( v, v ) .Assumption 1(a) assumes that the bidders are symmetric and the auctions are identical. Assumption 1 is similarto Assumptions A1 and A2 of GPV and Assumption 1 of MMS. The object of interest is the PDF of the valuationsat interior points of [ v, v ] . Suppose that v l > v , v u < v and I := [ v l , v u ] is an inner closed sub-interval of [ v, v ] .We assume that the observed bids are generated from the valuations and by the Bayesian Nash equilibrium (BNE)bidding strategy: B il = s ( V il ) := V il − F ( V il ) N − Z V il v F ( u ) N − d u. (2.2)The BNE requires that the bidding strategy s is strictly increasing. Moreover, GPV show that s is at least three timescontinuously differentiable on ( v, v ) . The inverse of the BNE bidding strategy can be written as ξ ( b ) := s − ( b ) = b + 1 N − G ( b ) g ( b ) , (2.3)where G and g are CDF and PDF of the bids, respectively. Let b := s ( v ) and b := s ( v ) denote the boundaries ofsupport for the observed i.i.d. bids. GPV show that g is three-times continuously differentiable and also boundedaway from zero on its support (cid:2) b, b (cid:3) : C g := inf b ∈ [ b,b ] g ( b ) > . (2.4)Let b G be the empirical CDF of the bids: b G ( b ) := 1 N · L X i,l ( B il ≤ b ) and b g be the kernel density estimator of g : b g ( b ) := 1 N · L X i,l h g K g (cid:18) B il − bh g (cid:19) (2.5)3ith some bandwidth h g > and kernel K g . Therefore, the plug-in nonparametric estimator of the inverse biddingstrategy ξ is b ξ ( b ) := b + 1 N − b G ( b ) b g ( b ) . (2.6)For each B il , we construct a pseudo valuation by b V il := b ξ ( B il ) . There is a boundary bias issue when ordinary kerneldensity estimator as in (2.5) is used. Thus, the pseudo valuations corresponding to bids in boundary regions arecontaminated. GPV propose to trim off bids that lie in hb b, b b + h g (cid:17) ∪ (cid:16)b b − h g , b b i , where b b :=max { B il : i = 1 , ..., N, l = 1 , ..., L } b b :=min { B il : i = 1 , ..., N, l = 1 , ..., L } . We modify the kernel density estimator b g ( b ) in the boundary region b ∈ [ b, b + h g ) ∪ (cid:0) b − h g , b (cid:3) to avoid boundarybias and trimming. Another concern is that, in order to remove more bias, we need to make use of the fact thatthe bid density g is smoother than the valuation density f and use a higher-order kernel when estimating the inversebidding strategy. A suitable choice is the local quadratic minimum contrast estimator (MCE). See Bickel and Doksum(2015, Chapter 11.3). Similar to the local polynomial regression, the local quadratic MCE automatically adapts tothe boundary so that the rate of the bias is the same in the boundary region and the interior. The local quadraticMCE coincides with the kernel density estimator (2.5) with a fourth-order kernel, and therefore we achieve desiredbias removal in the interior region (cid:2) b + h g , b − h g (cid:3) . The GPV estimator is b f GP V ( v ) := 1 N · L X i,l h f K f b V il − vh f ! , for some bandwidth h f > and kernel K f . We note again that trimming can be avoided.Another important observation is that the plug-in estimator (2.6) may not be monotone in finite samples, althoughits population counterpart ξ is strictly increasing under the assumption that the empirical auction model is correctly-specified. We apply smooth rearrangement to build a new monotonicity-constrained estimator of ξ on the plug-inestimator b ξ . Define b s ( t ) := Z b b b b Z t −∞ h r K r b ξ ( b ) − uh r ! d u d b + b b, t ∈ R , (2.7)for some bandwidth h r > and (second-order) kernel function K r . Denote e K r ( u ) := R u −∞ K r ( t ) d t . Alternatively wecan write b s ( t ) = Z b b b b e K r t − b ξ ( b ) h r ! d b + b b, t ∈ R . (2.8)(2.8) contains less challenge in computation as the expression of e K r is always available for standard kernel functions. See Hickman and Hubbard (2014) for more discussion on why we should avoid trimming when estimating the auction model usingGPV approach. See Remark 2.3 of MMS. See Jales et al. (2017) and Ma et al. (2018) for more recent applications of MCE in econometrics. Note that the MCE requires knowledge of the locations of the endpoints b and b . Since the estimators b b and b b are super-consistent: b b = b + O p (log ( L ) /L ) and b b = b + O p (log ( L ) /L ) , we can replace the unknown endpoints in the MCE with these estimators withoutaffecting the validity of the asymptotic results.
4t is clear that b s is increasing on R and b s ( t ) = b b if t ≤ inf b ∈ hb b, b b i b ξ ( b ) − h r b b if t ≥ sup b ∈ hb b, b b i b ξ ( b ) + h r It is easy to see that b s can be viewed as an estimator of the bidding function s (see Lemma 2). Note that thisestimator is of interest on its own, as the bidding function s is an important structural object. Moreover, constructingan estimator of s directly from its expression in (2.2) can be cumbersome.Let b s − denote the pseudo inverse of b s : b s − ( b ) := inf { u ∈ R : b s ( u ) ≥ b } . b s − is a rearrangement-based estimator of ξ with the monotonicity constraint imposed. A new modified GPV esti-mation procedure now can be proposed: First, we construct b ξ , the plug-in nonparametric estimator of the inversebidding strategy ξ . To avoid trimming, we use the local quadratic MCE instead of the ordinary kernel density es-timator. Then, we construct the monotonicity-imposed estimator of the inverse bidding strategy: b s − and generatemonotonicity-constrained pseudo valuations b V † il := b s − ( B il ) for i = 1 , ..., N , l = 1 , ..., L . A new estimator, therearrangement-based GPV (RGPV), is b f RGP V ( v ) := 1 N · L X i,l h f K f b V † il − vh f ! . The RGPV estimation procedure is computationally more involved than the standard GPV procedure. Whenimplementing it in practice, the integral in (2.8) can be approximated by an upper Riemann sum. Let M ∈ N be avery large number. Let d := (cid:16)b b − b b (cid:17) /M . We can approximate R b b b b e K r (cid:16)(cid:16) t − b ξ ( b ) (cid:17) /h r (cid:17) d b by M X i =1 e K r t − b ξ (cid:16)b b + i · d (cid:17) h r d. However, unlike the estimator in Henderson et al. (2012), it is not required to solve a constrained optimization problem.Thus the RGPV estimation procedure is computationally less demanding than Henderson et al. (2012)’s estimationprocedure, which is based on constrained reweighting.An alternative, and closely related, approach is to impose the monotonicity restriction through the “non-smooth”rearrangement. Instead of using (2.7), we can define b s ( t ) := Z b b b b (cid:16)b ξ ( b ) ≤ t (cid:17) d b + b b, t ∈ R , and use b s − as a monotonicity-constrained estimator of ξ to generated pseudo valuations. If the empirical auctionmodel is correctly specified so that ξ is strictly increasing, b s − is always an improvement over the plug-in estimator b ξ ,in the sense that b s − has a strictly smaller (finite-sample) integrated mean square error whenever b ξ is not monotonic.See Chernozhukov et al., 2009, Proposition 1. However, for the structural auction model, the parameter of interestis the density f . It is unclear whether the density estimator based on pseudo valuations generated by b s − is an5mprovement over the unconstrained GPV estimator theoretically. In this paper, we focus on smooth rearrangementand in the next section, we show that based on the RGPV estimator we could potentially achieve sharper inference inlarge samples, which can be regarded as theoretical advantage over the unconstrained GPV estimator. The following assumptions are imposed on the kernel functions and the bandwidths, respectively.
Assumption 2 ( Kernel ) . K f is a probability density function that is symmetric around 0, compactly supported on [ − , and has at least two Lipschitz continuous derivatives on R . Moreover, K r = K f and the same kernel functionis used in the local quadratic MCE of the bid density. Assumption 3 ( Bandwidth ) . Let h be a sequence { h L } ∞ L =1 satisfying h = L − γ for / ≤ γ ≤ / . h f = λ f h , h g = λ g h and h r = λ r h for some positive constants λ f , λ g and λ r .Assumption 2 implies that the kernel functions are of second order. When the kernel used in the local quadraticMCE of the bid density is second-order, at the interior points, the MCE is the same as the ordinary kernel densityestimator (2.5) with K g being fourth-order. Assumption 3 is similar to Assumption 3 in MMS.It is shown in MMS that under assumptions 1-3, (cid:0) Lh f h g (cid:1) / (cid:16) b f GP V ( v ) − f ( v ) (cid:17) → d N (0 , V GP V ( v )) , (3.1)where V GP V ( v ) := 1 N ( N − F ( v ) f ( v ) g ( s ( v )) Z (cid:26)Z K ′ f ( u ) K g (cid:18) w − λ f λ g s ′ ( v ) u (cid:19) d u (cid:27) d w. (3.2)MMS show that the asymptotic variance (3.2) can be consistently estimated by some estimator b V GP V ( v ) and anasymptotically valid confidence interval for f ( v ) can be constructed: " b f GP V ( v ) − z − α/ s b V GP V ( v ) Lh f h g , b f GP V ( v ) + z − α/ s b V GP V ( v ) Lh f h g , (3.3)where z − α/ denotes the − α/ quantile of the standard normal distribution. Note that b V GP V ( v ) and its probabilisticlimit V GP V ( v ) play important roles in determining the length of the confidence interval. Furthermore, a bootstrapuniform confidence band is given by CB GP V ( v ) := " b f GP V ( v ) − ζ GP V,α s b V GP V ( v ) Lh f h g , b f GP V ( v ) + ζ GP V,α s b V GP V ( v ) Lh f h g , for v ∈ I, where ζ GP V,α is some bootstrap critical value. It can be shown that
P [ f ( v ) ∈ CB GP V ( v ) , for all v ∈ I ] → − α, as L ↑ ∞ . Next, we show that a similar asymptotic normality result holds for the RGPV estimator, but with a smallerasymptotic variance. The proof uses the same arguments as in the proof of Theorem 2.1 in MMS. First, we derive the6ollowing asymptotic representation: b f RGP V ( v ) − f ( v ) = 1( N − · N · L ) X i,l X j,k M ( B il , B jk ; v ) + o p (cid:16)(cid:0) Lh (cid:1) − / (cid:17) , (3.4)where the remainder term is uniform in v ∈ I . Moreover, M ( b ′ , b ; v ) := − h f K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) Z bb h r K r (cid:18) ξ ( b ′ ) − ξ ( u ) h r (cid:19) G ( u ) g ( u ) (cid:18) h g K g (cid:18) b − uh g (cid:19) − g ( u ) (cid:19) d u. Note that this “kernel” is different from that of the unconstrained GPV estimator. See Equation (2.6) of MMS.Define M ( b ; v ) := Z M ( b, b ′ ; v ) d G ( b ′ ) , M ( b ; v ) := Z M ( b ′ , b ; v ) d G ( b ′ ) and µ M ( v ) := Z Z M ( b, b ′ ; v ) d G ( b ) d G ( b ′ ) . (3.5)Note that µ M ( v ) = E [ M ( B ; v )] = E [ M ( B ; v )] . Applying Hoeffding decomposition to the leading term in (3.4)and techniques from the theories of empirical processes and U processes, we can show that b f RGP V ( v ) − f ( v ) = 1 N − · N · L X i,l ( M ( B il ; v ) − µ M ( v )) + o p (cid:16)(cid:0) Lh (cid:1) − / (cid:17) , where the remainder term is uniform in v ∈ I . M ( B il ; v ) − µ M ( v ) , i = 1 , ..., N , l = 1 , ..., L are independent,zero-mean but dependent on the bandwidths. This term can be shown asymptotically normal. Theorem 1.
Suppose that Assumptions 1 - 3 are satisfied and v ∈ ( v, v ) . Also assume that λ r = λ f . Then, (cid:0) Lh f h g (cid:1) / (cid:16) b f RGP V ( v ) − f ( v ) (cid:17) → d N (0 , V RGP V ( v )) , (3.6)where V RGP V ( v ) := 1 N ( N − F ( v ) f ( v ) g ( s ( v )) Z (cid:26)Z Z K ′ f ( u ) K r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z d u (cid:27) d w. (3.7)Moreover, V RGP V ( v ) ≤ V GP V ( v ) (3.8)for all v ∈ ( v, v ) . Remark 3.1.
In the proof of Theorem 1, we show that E Lh f h g ( N − N · L X i,l ( M ( B il ; v ) − µ M ( v )) = 1 N ( N − h f h g Z (Z K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) Z bb h r K r (cid:18) ξ ( b ′ ) − ξ ( u ) h r (cid:19) G ( u ) g ( u ) K g (cid:18) b − uh g (cid:19) d u d G ( b ′ ) ) d G ( b )+ O (cid:0) h (cid:1) =:V M ( v ) + O (cid:0) h (cid:1) , (3.9)7here the remainder term is uniform in v ∈ I . The asymptotic variance is the limit of the leading term of (3.9) as h ↓ : lim h ↓ h f h g Z (Z K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) Z bb h r K r (cid:18) ξ ( b ′ ) − ξ ( u ) h r (cid:19) G ( u ) g ( u ) K g (cid:18) b − uh g (cid:19) d u d G ( b ′ ) ) d G ( b )= F ( v ) f ( v ) g ( s ( v )) Z (cid:26)Z Z K ′ f ( u ) K r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z d u (cid:27) d w. A consistent estimator of the asymptotic variance V RGP V ( v ) can be derived based on the sample analogue of V M ( v ) . Remark 3.2.
The proof of Theorem 1 also incorporates the bias term: (cid:0) Lh f h g (cid:1) / (cid:16) b f RGP V ( v ) − f ( v ) − ι ( v ) (cid:17) → d N (0 , V RGP V ( v )) , where ι ( v ) := 12 f ′′ ( v ) (cid:18)Z K f ( u ) u d u (cid:19) h f + 12 ( s ′′′ ( v ) f ( v ) + s ′′ ( v ) f ′ ( v )) s ′ ( v ) − s ′′ ( v ) f ( v ) s ′ ( v ) (cid:18)Z K r ( u ) u d u (cid:19) h r . A comparison of ι ( v ) with the bias given in Remark 2.3 of MMS shows that the smooth rearrangement incurs additionalbias. For inference, we take the “under-smoothing” approach to select sufficiently small bandwidths so that these biasterms become negligible. Remark 3.3.
MMS show that the GPV estimator has a smaller asymptotic variance than the quantile-based estimatorof Marmer and Shneyerov (2012). The proof of (3.8) uses similar arguments. It is easy to show that the inequality(3.8) is strict for all v ∈ ( v, v ) , if the kernel function satisfies K ′ ( u ) < for all u ∈ (0 , and K ′ ( u ) > for all u ∈ ( − , .Suppose that the valuations are drawn from the family of distributions F ( v ) = , v < v θ , ≤ v ≤ , v > supported on [0 , with some parameter θ > . The Bayesian Nash equilibrium bidding strategy in this example is s ( v ) = (cid:18) − θ ( N −
1) + 1 (cid:19) v, which is linear in v . Thus, the ratio V GP V ( v ) / V RGP V ( v ) is independent from v , by the definitions of V RGP V ( v ) and V GP V ( v ) . In this example, we choose the triweight kernel K ( u ) = 3532 (cid:0) − u (cid:1) ( | u | ≤ . We can analytically evaluate the multi-dimensional integrals in (3.2) and (3.7) and calculate V GP V ( v ) / V RGP V ( v ) inthis example. We experiment with different combinations of ( θ, N ) , and find that the ratio V GP V ( v ) / V RGP V ( v ) canbe quite large in my cases. For instance, in the case of ( θ, N ) = (1 , , V GP V ( v ) / V RGP V ( v ) is approximately 1.587.8 Inference
If the asymptotic variance V RGP V ( v ) in (3.7) can be consistently estimated by some estimator b V RGP V ( v ) , Theorem1 shows that we can construct an asymptotically valid confidence interval for f ( v ) : " b f RGP V ( v ) − z − α/ s b V RGP V ( v ) Lh f h g , b f RGP V ( v ) + z − α/ s b V RGP V ( v ) Lh f h g . (4.1)As shown in (3.8), our rearrangement-based estimator has a smaller asymptotic variance than the GPV estimator.Therefore, the confidence intervals based on our estimator in (4.1) should be shorter than those based on the GPVestimator in (3.3) in large samples.An estimator of V RGP V ( v ) is derived based on the sample analogue of V M ( v ) , (see (3.9)): b V RGP V ( v ) := 1 N ( N − h f h g N · L ) ( N · L −
1) ( N · L − X i,l X ( j,k ) =( i,l ) X ( j ′ ,k ′ ) =( i,l ) , ( j ′ ,k ′ ) =( j,k ) η il,jk ( v ) η il,j ′ k ′ ( v ) ,η il,jk ( v ) := K ′ f b V † jk − vh f ! b s ′ (cid:16) b V † jk (cid:17) Z b b b b h r K r b V † jk − b ξ ( u ) h r ! b G ( u ) b g ( u ) K g (cid:18) B il − uh g (cid:19) d u, where we use the fact ξ ′ ( b ) = 1 /s ′ ( ξ ( b )) and b s ′ ( v ) := Z b b b b h r K r b ξ ( b ) − vh r ! d b. The integral can be approximated by an upper Riemann sum in practice. The following result provides uniformconsistency of the variance estimator and also an estimate of its uniform rate of convergence. The proof uses the samearguments as in the proof of Theorem 3.1 of MMS.
Theorem 2.
Suppose that Assumptions 1 - 3 are satisfied. Then, sup v ∈ I (cid:12)(cid:12)(cid:12) b V RGP V ( v ) − V M ( v ) (cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! . An alternative approach to constructing pointwise confidence intervals is based on bootstrapping. Let { B ∗ il : i = 1 , . . . , N, l = 1 , . . . , L } (4.2)denote a set of independent random variables drawn from the original sample (2.1) with replacement. b G ∗ and b g ∗ denote the bootstrap analogues of b G and b g respectively. Let b ξ ∗ be the bootstrap analogue of b ξ . b ξ ∗ is defined byreplacing b G and b g with b G ∗ and b g ∗ . We further define b s ∗ ( t ) := Z b b b b Z t −∞ h r K r b ξ ∗ ( b ) − uh r ! d u d b + b b, t ∈ R and bootstrap analogues of b V † il , denoted by b V †∗ il , i = 1 , . . . , N , l = 1 , . . . , L by using the pseudo inverse of b s ∗ . Lastly,9e construct a bootstrap analogue of b f RGP V : b f ∗ RGP V ( v ) := 1 N · L X i,l h f K f b V †∗ il − vh f ! . The percentile bootstrap pointwise confidence interval for f ( v ) is h q ∗ α/ ( v ) , q ∗ − α/ ( v ) i , where q ∗ τ ( v ) is the τ − th quantile of the conditional distribution of b f ∗ RGP V ( v ) given the original sample.The pointwise inference results for f ( v ) described above can be extended for the inference on the optimal reserveprice, as the latter is a function of the density at the reserve price. Such an extension is discussed in Section 8 inMMS. Similarly, a function of the density (1 − F ( v )) /f ( v ) is of interest in applications as it represents the markupof the bidder with value v . Again, since relatively to GPV’s our rearrangement estimator has a smaller asymptoticvariance, basing inference on optimal reserve price or the markup on our estimator would result in more powerful testsand shorter confidence intervals.Next, we show that a bootstrap-based uniform confidence band for { f ( v ) : v ∈ I } centered at the rearrangement-based estimator can be constructed by using intermediate Gaussian approximation pioneered by Chernozhukov et al.(2014b,a, 2016). Consider the following bootstrap process Z ∗ ( v ) := b f ∗ RGP V ( v ) − b f RGP V ( v ) (cid:16) Lh f h g (cid:17) − / b V RGP V ( v ) / , v ∈ I. (4.3)Let P ∗ [ · ] denote the probability conditional on the original sample and ζ RGP V,α := inf { z ∈ R : P ∗ [ k Z ∗ k I ≤ z ] ≥ − α } be the (1 − α ) -quantile of the conditional distribution of k Z ∗ k I given the original sample. A uniform confidence bandaround the rearrangement-based estimator is CB RGP V ( v ) := " b f RGP V ( v ) − ζ RGP V,α s b V RGP V ( v ) Lh f h g , b f RGP V ( v ) + ζ RGP V,α s b V RGP V ( v ) Lh f h g , for v ∈ I. The following theorem establishes the asymptotic validity of CB RGP V . Its proof uses the same arguments as in theproof of Corollary 4.3 of MMS.
Theorem 3.
Suppose that Assumptions 1 - 3 are satisfied. Then,
P [ f ( v ) ∈ CB RGP V ( v ) , for all v ∈ I ] → − α, as L ↑ ∞ . In previous sections, we focused on the case of identical auctions with a fixed number of bidders. In this section, weconsider auction models with auction-specific heterogeneity. The econometrician observes data from L auctions. Let10 l denote the d -dimensional relevant characteristics for the object in the l -th auction. Let N l denote the number ofbidders in the l -th auction. Let B il denote the bid submitted by the i -th bidder in the l -th auction. The data observedby the econometrician is given by { ( B il , X l , N l ) : i = 1 , ..., N l , l = 1 , ..., L } . Unobserved bidders’ valuations are denoted by { V il : i = 1 , ..., N l , l = 1 , ..., L } . Assume that { ( X l , N l ) : l = 1 , ..., L } are i.i.d. and for each l = 1 , ..., L , given X l = x and N l = n , the valuations { V il : i = 1 , ..., n } are i.i.d. with conditional PDF f ( ·| x ) . We follow the literature and assume that the valuations andthe number of bidders N l are conditionally independent given the observed characteristics X l . Also assume that theconditional probability mass function of N l given X l has a known support { n, ..., n } .The observed bid B il is assumed from the Bayesian Nash equilibrium bidding for risk-neutral bidder i submittedin the l -th auction. Let G ( ·| x , n ) denote the conditional CDF of B il given X l = x and N l = n . Let g ( ·| x , n ) be theconditional PDF. The inverse bidding strategy in this context becomes V il = ξ ( B il , X l , N l ) := B il + 1 N l − G ( B il | X l , N l ) g ( B il | X l , N l ) . (5.1)For estimation and inference, we can generate pseudo valuations from (5.1) by replacing the true conditionalCDF and PDF by their kernel estimators. The GPV estimator can be defined analogously in this general context.Asymptotically valid pointwise confidence intervals and uniform confidence bands for f ( ·| x ) can be constructed. SeeSection 5 of MMS for more details.Following Haile et al. (2003), we use a semi-parametric approach to homogenize the bids. Let { ǫ il : i = 1 , ..., N l , l = 1 , ..., L } (5.2)denote positive i.i.d. idiosyncratic values that are independent from the auction-specific characteristics X l , l = 1 , ..., L .Let F ǫ be its CDF and [ ǫ, ǫ ] be its support. Define e B il = s ( ǫ il , N l ) := ǫ il − F ǫ ( ǫ il ) N l − Z ǫ il ǫ F ǫ ( u ) N l − d u. Let e G ( b | n ) := P h e B il ≤ b | N l = n i be the conditional CDF and e g ( b | n ) be the corresponding conditional PDF. Note that we have ǫ il = s ( ǫ il , N l ) + 1 N l − e G ( s ( ǫ il , N l ) | N l ) e g ( s ( ǫ il , N l ) | N l ) . (5.3)The approach of Haile et al. (2003) assumes that, for some parametric function Υ to be specified below, V il = Υ ( X l ) ǫ il for i = 1 , ..., N l and l = 1 , ..., L . It then can be shown that the conditional CDF and PDF of Υ ( X l ) e B il For a test of this assumption, see Liu and Luo (2017). The homogenization approach is used extensively in the literature, e.g. Athey et al. (2011), Liu and Luo (2017), Luo and Wan (2018),and many others. e G Υ and e g Υ , respectively) satisfy e G Υ ( b | X l , N l ) := P h Υ ( X l ) e B il ≤ b | X l , N l i = e G (cid:18) bΥ ( X l ) | N l (cid:19) and e g Υ ( b | X l , N l ) = e g (cid:18) bΥ ( X l ) | N l (cid:19) Υ ( X l ) . It is clear from these results and (5.3) that Υ ( X l ) ǫ il = Υ ( X l ) s ( ǫ il , N l ) + 1 N l − e G Υ ( Υ ( X l ) s ( ǫ il , N l ) | X l , N l ) e g Υ ( Υ ( X l ) s ( ǫ il , N l ) | X l , N l ) , which implies that B il = Υ ( X l ) e B il , for i = 1 , ..., N l and l = 1 , ..., L .Now, we write log ( B il ) = α ( N l ) + log ( Υ ( X l )) + U il , (5.4)where α ( N l ) := E [log ( s ( ǫ il , N l )) | N l ] and U il :=log ( s ( ǫ il , N l )) − α ( N l ) . It is easy to check that
E [ U il | X l , N l ] = 0 . Since N l is discrete, we can write α ( N l ) = n X n = n α n ( N l = n ) . We assume that the function Υ is log-linear in parameters: log ( Υ ( X l )) = X T l β for some unknown β . Now (5.4) canbe written as log ( B il ) = n X n = n α n ( N l = n ) + X T l β + U il . Regressing the log-bids on the covariates and the indicators for the number of bidders yields an estimator b β of β .Then, the homogenized bids are given by B il := exp (cid:16) log ( B il ) + x T0 b β − X T l b β (cid:17) , for i = 1 , ..., N l and l = 1 , ..., L. (5.5)These bids can be interpreted as the bid that would have been submitted by the i -th bidder if the covariates wereequal to x , in the l -th auction. Suppose we are interested in inference on f ( ·| x ) for some fixed x . Let b G ( b, n ) := 1 L L X l =1 ( N l = n ) 1 N l N l X i =1 (cid:0) B il ≤ b (cid:1) , b g ( b, n ) := 1 L L X l =1 ( N l = n ) 1 N l N l X i =1 h g K g (cid:18) B il − bh g (cid:19) and b ξ ( b, n ) := b + 1 n − b G ( b, n ) b g ( b, n ) . b s ( · , n ) can also be defined analogously and b s − ( · , n ) is its pseudo inverse. Let b V † il := b s − (cid:0) B il , N l (cid:1) be the monotonicity- For example, x can be taken to be the sample mean L − P Ll =1 X l . See Haile et al. (2003). f ( ·| x ) is b f ( v | x ) := 1 L L X l =1 N l N l X i =1 h f K f b V † il − vh f ! , which is asymptotically normal. Its asymptotic variance can be consistently estimated by b V RGP V ( v ) := n X n = n n ( n − h f h g × L ( L −
1) ( L − L X l =1 X k = l X k ′ = k,k ′ = l ( N l = n, N k = n, N k ′ = n ) 1 N l N l X i =1 η il,k ( v ) η il,k ′ ( v ) , where η il,k ( v ) := 1 N k N k X j =1 K ′ f b V † jk − vh f ! b s ′ (cid:16) b V † jk , N k (cid:17) Z b b Nk b b Nk h r K r b V † jk − b ξ ( u, N k ) h r ! b G ( u, N k ) b g ( u, N k ) K g B jk − uh g ! d u, b s ′ ( v, n ) := Z b b n b b n h r K r b ξ ( b, n ) − vh r ! d b and b b n := max ( i,l ): N l = n B il and b b n := min ( i,l ): N l = n B il . A pointwise confidence interval for f ( v | x ) that is of the same form as (4.1) can be proved to be asymptoticallyvalid. For bootstrap resampling, we treat the homogenized bids (5.5) as our observed bids and apply the two-stepresampling procedure provided by Marmer and Shneyerov (2012). In each bootstrap replication, we first randomlydraw L observations from { N l : l = 1 , ..., L } with replacement. Next, we randomly draw bids with replacement frombids corresponding to the selected number of bidders. If for the l -th observation in the bootstrap sample, we have N ∗ l = N l ′ , then let (cid:8) B ∗ il : i = 1 , ..., N ∗ l (cid:9) be i.i.d. draws from all the bids in auctions with number of bidders being N l ′ with replacement. Now the bootstrap sample is (cid:0) B ∗ il , N ∗ l (cid:1) , i = 1 , ..., N ∗ l and l = 1 , ..., L . Then it is straightforwardto construct the bootstrap analogue of b f ( ·| x ) and a bootstrap-based uniform confidence band for f ( ·| x ) can beconstructed analogously. In this section, we assess the finite-sample performances of the uniform confidence bands based on both the uncon-strained GPV estimator and the rearrangement-based monotonicity-constrained estimator proposed in this paper.Our simulation design follows Marmer and Shneyerov (2012) and the DGP is described in Remark 3.3. We consider θ = 1 and draw 2100 independent valuations from F θ . In all these cases, the number of bidders N is constant. Thenumber of auctions is determined by N · L = 2100 . We choose K r = K f to be the second-order triweight kernel.For estimation of the inverse bidding strategy, we use the second-order triweight kernel in the MCE. In this case, thekernel K g in the expressions of the asymptotic variances is the fourth-order triweight kernel.We use the same bandwidths as in GPV. We take h g = 3 . · b σ b · ( N · L ) − / when estimating the inverse bidding13able 1: Coverage probabilities and the relative supremum width W GP V /W RP GV of the bootstrap-based uniformconfidence bands around the GPV and rearrangement estimators, θ = 1 GPV Rearrangement W GP V /W RP GV v ∈ [0 . , . N = 3 N = 5 N = 7 v ∈ [0 . , . N = 3 N = 5 N = 7 b σ b is the estimated standard deviation of the observed bids. We use h f = 3 . · b σ v · ( N · L ) − / as the second-step bandwidth, where b σ v is the estimated standard deviation of the (unconstrained) pseudo valuations. The constants . and . are Silverman’s rule-of-thumb constants corresponding to fourth-order and second-order triweight kernels.When imposing monotonicity, we take h r = h f . We consider different numbers of bidders N ∈ { , , } , and also thedensity function over the ranges v ∈ [0 . , . and v ∈ [0 . , . . When computing the bootstrap-based critical values,we set the number of bootstrap replications to 499 and use grid maximization over the grid [ v l : 0 .
001 : v u ] . The main result of this paper is that imposing monotonicity using smooth rearrangement results in more efficientinference. Therefore, in addition to assessing coverage accuracy, we also report the ratio of the supremum widths ofthe confidence bands: W GP V := sup v ∈ I · ζ GP V,α s b V GP V ( v ) Lh f h g and W RGP V := sup v ∈ I · ζ RGP V,α s b V RGP V ( v ) Lh f h g . Table 1 reports the coverage probabilities as well as the relative supremum width of the confidence bands basedon the GPV and our rearrangement estimators. The coverage probabilities of both methods are similar and accurate.However, our approach can produce considerably smaller confidence with the reduction in supremum width rangingfrom 6.8% to 33.4%. More substantial reductions in supremum width are obtained for smaller numbers of bidders.This is due to the inverse relationship between the number of bidders and the asymptotic variance of the estimators.
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Appendix
Let > denote an inequality up to a universal constant that does not depend on the sample size L . For a sequence ofclasses of functions F L (that may depend on the sample size) defined on (cid:2) b, b (cid:3) d , for some d ≥ , let N (cid:16) ǫ, F L , k·k Q, (cid:17) denote the ǫ − covering number, i.e., the smallest integer m such that there are m balls of radius ǫ centered at pointsin F L , with respect to the metric induced by the norm k·k Q, , where k f k Q, := (cid:16)R | f | d Q (cid:17) / , f ∈ F L . A function F L : (cid:2) b, b (cid:3) d → R + is an envelope of F L if F L ≥ sup f ∈ F L | f | . We say that F L is a (uniform) Vapnik-Chervonenkis-type(VC-type) class with respect to the envelope F L (see, e.g., Chernozhukov et al., 2014b, Definition 2.1) if there existsome positive constant C and C that are independent of L such that N (cid:16) ǫ k F L k Q, , F L , k·k Q, (cid:17) ≤ (cid:18) C ǫ (cid:19) C , for all ǫ ∈ (0 , ,for all finitely discrete probability measure Q on (cid:2) b, b (cid:3) d . Note that the all function classes that appear later aredependent on L . We suppress the dependence for notational simplicity.“With probability approaching 1” is abbreviated as “w.p.a.1”. For notational simplicity, in the proofs, max i,l isunderstood as max ( i,l ) ∈{ ,...,N }×{ ,...,L } . P (2) is understood as P ( j,k ) =( i,l ) and P (3) is understood as X i,l X ( j,k ) =( i,l ) X ( j ′ ,k ′ ) =( i,l ) , ( j ′ ,k ′ ) =( j,k ) , i.e., summing over all distinct indices. ( N · L ) is understood as ( N · L ) ( N · L − and ( N · L ) is understood as ( N · L ) ( N · L −
1) ( N · L − .Fix δ := min { ( v − v u ) / , ( v l − v ) / } . A Proofs of the Main Results
Proof of Theorem 1.
It follows from Lemma 7 that (cid:0) Lh f h g (cid:1) / (cid:16) b f GP V ( v ) − f ( v ) (cid:17) = 1 N / ( N −
1) 1( N · L ) / X i,l h f h / g ( M ( B il ; v ) − µ M ( v )) + o p (1) . M ‡ ( b ′ , b ; v ) := − h f K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) Z bb h r K r (cid:18) ξ ( b ′ ) − ξ ( u ) h r (cid:19) G ( u ) g ( u ) h g K g (cid:18) b − uh g (cid:19) d u M ‡ ( b ; v ) := − Z h f K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) Z bb h r K r (cid:18) ξ ( b ′ ) − ξ ( u ) h r (cid:19) G ( u ) g ( u ) h g K g (cid:18) b − uh g (cid:19) d u d G ( b ′ ) and µ M ‡ ( v ) := Z Z M ‡ ( b, b ′ ; v ) d G ( b ) d G ( b ′ ) = Z M ‡ ( b ; v ) d G ( b ) . Then it is clear that M ( B il ; v ) − µ M ( v ) = M ‡ ( B il ; v ) − µ M ‡ ( v ) for all i = 1 , ..., N and L = 1 , ..., L and µ M ‡ ( v ) = µ M ( v ) − Z Z bb h f K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) 1 h f K r (cid:18) ξ ( b ′ ) − ξ ( u ) h f (cid:19) G ( u ) g ( u ) d u d G ( b ′ ) . Note that we assumed h r = h f . By change of variables, Z Z bb K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) 1 h f K r (cid:18) ξ ( b ′ ) − ξ ( u ) h f (cid:19) G ( u ) g ( u ) d u d G ( b ′ )= h f Z v − vhfv − vhf Z v − vhfv − vhf K ′ f ( w ) K r ( w − z ) G ( s ( h f z + v )) g ( s ( h f z + v )) s ′ ( h f z + v ) g ( s ( h f w + v )) d z d w = h f Z v − vhfv − vhf Z v − vhfv − vhf K ′ f ( w ) K r ( w − z ) G ( s ( v )) s ′ ( v ) g ( s ( v )) + n g ( s (¨ v )) s ′ (¨ v ) + G ( s (¨ v )) s ′′ (¨ v ) o g ( s (¨ v )) g ( s (¨ v )) h f z − G ( s (¨ v )) s ′ (¨ v ) g ′ ( s (¨ v )) g ( s (¨ v )) h f z ) { g ( s ( v )) + g ′ ( s ( ˙ v )) s ′ ( ˙ v ) h f w } d z d w, where ˙ v and ¨ v are mean values that are dependent on w and z with | ˙ v − v | ≤ h f | w | and | ¨ v − v | ≤ h f | z | .Since when h f is small enough, Z v − vhfv − vhf Z v − vhfv − vhf K ′ f ( w ) K r ( w − z ) d z d w = 0 , it follows that Z Z bb K ′ f (cid:18) ξ ( b ′ ) − vh f (cid:19) ξ ′ ( b ′ ) 1 h f K r (cid:18) ξ ( b ′ ) − ξ ( u ) h f (cid:19) G ( u ) g ( u ) d u d G ( b ′ ) = O (cid:0) h (cid:1) , uniformly in v ∈ I . sup v ∈ I | µ M ‡ ( v ) | = O (1) follows from this result and Lemma 6.Define U il ( v ) := 1 N / ( N −
1) 1( N · L ) / h f h / g (cid:16) M ‡ ( B il ; v ) − µ M ‡ ( v ) (cid:17) , and σ ( v ) := X i,l E h U il ( v ) i / = N ( N − h f h g E (cid:20)(cid:16) M ‡ ( B ; v ) − µ M ‡ ( v ) (cid:17) (cid:21)! / . (A.1)18y change of variables, E h h f h g M ‡ ( B ; v ) i = Z h g (Z bb Z bb h f K ′ (cid:18) ξ ( b ′ ) − vh f (cid:19) s ′ ( ξ ( b ′ )) 1 h f K (cid:18) ξ ( b ′ ) − ξ ( u ) h f (cid:19) G ( u ) g ( u ) × K (cid:18) b − uh g (cid:19) g ( b ′ ) d u d b ′ (cid:27) d G ( b )= Z b − s ( v ) hgb − s ( v ) hg ρ h ( y ) g ( h g y + s ( v )) d y. where ρ h ( y ) := Z v − vhfv − vhf Z v − vhfv − vhf K ′ f ( w ) K r ( w − z ) G ( s ( h f z + v )) g ( s ( h f z + v )) K g (cid:18) y − s ( h f z + v ) − s ( v ) h g (cid:19) × s ′ ( h f z + v ) g ( s ( h f w + v )) d z d w. (A.2)We can easily verify that the dominated convergence theorem applies to the integral on the right hand side of (A.2)(as h ↓ ) for each y ∈ R : lim h ↓ ρ h ( y ) = G ( s ( v )) s ′ ( v ) g ( s ( v )) Z Z K ′ f ( w ) K r ( w − z ) K g (cid:18) y − s ′ ( v ) λ f λ g z (cid:19) d z d w. Since K ′ f and K r are compactly supported on [ − , , by reverse triangle inequality, | ρ h ( y ) | > Z v − vhfv − vhf Z v − vhfv − vhf | K ′ ( w ) | | ( | z | ≤ | w | ) | (cid:18) | y | ≤ (cid:12)(cid:12)(cid:12)(cid:12) s ( h f z + v ) − s ( v ) h g (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) d z d w > (cid:18) | y | ≤ λ f λ g C s ′ (cid:19) , where C s ′ := sup v ∈ [ v,v ] s ′ ( v ) .Now the dominated convergence theorem applies and lim h ↓ Z b − s ( v ) hgb − s ( v ) hg ρ h ( y ) g ( h g y + s ( v )) d y = G ( s ( v )) s ′ ( v ) g ( s ( v )) Z (cid:26)Z Z K ′ ( w ) K ( w − z ) K (cid:18) y − s ′ ( v ) λ f λ g z (cid:19) d z d w (cid:27) d y = F ( v ) f ( v ) g ( s ( v )) Z (cid:26)Z Z K ′ ( w ) K ( w − z ) K (cid:18) y − s ′ ( v ) λ f λ g z (cid:19) d z d w (cid:27) d y where the second equality follows from the relation f ( v ) = g ( s ( v )) s ′ ( v ) . It follows from this result and sup v ∈ I | µ M ‡ ( v ) | = O (1) that E (cid:20) h f h g (cid:16) M ‡ ( B ; v ) − µ M ‡ ( v ) (cid:17) (cid:21) =E h h f h g M ‡ ( B ; v ) i − h f h g µ M ‡ ( v ) = F ( v ) f ( v ) g ( s ( v )) Z (cid:26)Z Z K ′ f ( w ) K r ( w − z ) K g (cid:18) y − s ′ ( v ) λ f λ g z (cid:19) d z d w (cid:27) d y + o (1) (A.3)19nd thus σ ( v ) = 1 N / ( N − ( F ( v ) f ( v ) g ( s ( v )) Z (cid:26)Z Z K ′ f ( w ) K r ( w − z ) K g (cid:18) y − s ′ ( v ) λ f λ g z (cid:19) d z d w (cid:27) d y ) / + o (1) . (A.4)By the c r inequality (see, e.g., Davidson, 1994, 9.28), we have X i,l E "(cid:12)(cid:12)(cid:12)(cid:12) U il ( v ) σ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) = σ ( v ) − ( N − − ( N · L ) − / E (cid:20) h f h / g (cid:12)(cid:12)(cid:12)(cid:16) M ‡ ( B ; v ) − µ M ‡ ( v ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:21) > σ ( v ) − ( N · L ) − / (cid:18) h f h / g E (cid:20)(cid:12)(cid:12)(cid:12) M ‡ ( B ; v ) (cid:12)(cid:12)(cid:12) (cid:21) + h f h / g | µ M ‡ ( v ) | (cid:19) . (A.5)Then it is easy to verify that the Lyapunov’s condition holds: lim L ↑∞ X i,l E "(cid:12)(cid:12)(cid:12)(cid:12) U il ( v ) σ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (A.6)By Lyapunov’s central limit theorem, X i,l U il ( v ) σ ( v ) → d N (0 , , as L ↑ ∞ . (A.7)For the second part, it suffices to show Z (cid:26)Z Z K ′ f ( u ) K r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z d u (cid:27) d w ≤ Z (cid:26)Z K ′ f ( u ) K g (cid:18) w − s ′ ( v ) λ f λ g u (cid:19) d u (cid:27) d w. For each w ∈ R , since K r and K ′ f are assumed to be bounded and compactly supported, the Fubini-Tonelli theoremapplies and therefore, Z Z K ′ f ( u ) K r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z d u = Z (cid:18)Z K ′ f ( u ) K r ( u − z ) d u (cid:19) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z. Since K r and K ′ f are supported on [ − , , by integration by parts, Z − K ′ f ( u ) K r ( u − z ) d u − Z − K f ( u ) K ′ r ( u − z ) d u = 0 . (A.8)Now, (cid:26)Z Z K ′ f ( u ) K r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z d u (cid:27) = (cid:26)Z (cid:18)Z K f ( u ) K ′ r ( u − z ) d u (cid:19) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z (cid:27) = (cid:26)Z Z K f ( u ) K ′ r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z d u (cid:27) ≤ Z K f ( u ) (cid:26)Z K ′ r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z (cid:27) d u, (A.9) It is easy to adapt the proofs to show that the remainder terms are indeed uniform in v ∈ I . See the proof of Theorem 2.1 of MMS. w ∈ R , where the first equality follows from (A.8), the second equality follows from the Fubini-Tonelli theoremand the inequality follows from Jensen’s inequality since K f is a probability density function.Since the inequality (A.9) holds for all w ∈ R , by the Fubini-Tonelli theorem, Z (cid:26)Z Z K ′ f ( u ) K r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z d u (cid:27) d w ≤ Z K f ( u ) Z (cid:26)Z K ′ r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z (cid:27) d w d u. (A.10)Now for any fixed ( u, w ) ∈ R , by change of variables, Z K ′ r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z = − Z K ′ r ( y ) K g (cid:18) w − λ f λ g s ′ ( v ) ( y + u ) (cid:19) d y. Then, Z K f ( u ) Z (cid:26)Z K ′ r ( u − z ) K g (cid:18) w − λ f λ g s ′ ( v ) z (cid:19) d z (cid:27) d w d u = Z K f ( u ) Z (cid:26)Z K ′ r ( y ) K g (cid:18) w − λ f λ g s ′ ( v ) ( y + u ) (cid:19) d y (cid:27) d w d u. (A.11)It follows from change of variables that Z (cid:26)Z K ′ r ( y ) K g (cid:18) w − λ f λ g s ′ ( v ) ( y + u ) (cid:19) d y (cid:27) d w = Z (cid:26)Z K ′ r ( y ) K g (cid:18) w − λ f λ g s ′ ( v ) y (cid:19) d y (cid:27) d w, for all u ∈ R . The conclusion follows from this result, (A.11) and the assumption K r = K f . (cid:4) Proof of Theorem 2.
The proof is very similar to that of Theorem 3.1 of MMS. Let e V ( v ) := 1 N ( N − h f h g N · L ) X (3) T jk e η il,jk ( v ) T j ′ k ′ e η il,j ′ k ′ ( v ) , with e η il,jk ( v ) := K ′ f b V † jk − vh f ! s ′ (cid:16) b V † jk (cid:17) Z bb h r K r b V † jk − ξ ( u ) h r ! G ( u ) g ( u ) K g (cid:18) B il − uh g (cid:19) d u and V ( v ) := 1 N ( N − h f h g N · L ) X (3) η il,jk ( v ) η il,j ′ k ′ ( v ) , with η il,jk ( v ) := K ′ f V † jk − vh f ! s ′ (cid:16) V † jk (cid:17) Z bb h r K r V † jk − ξ ( u ) h r ! G ( u ) g ( u ) K g (cid:18) B il − uh g (cid:19) d u. By the arguments used in the proof of Lemma 3, b V RGP V ( v ) = 1 N ( N − h f h g N · L ) X (3) T jk b η il,jk ( v ) T j ′ k ′ b η il,j ′ k ′ ( v ) , for all v ∈ I , w.p.a.1.21y using Lemma 1, Lemma 2, (B.14), Taylor expansion, tedious algebra and empirical process techniques invoked inthe proof of Theorem 3.1 of MMS, one can show that sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ( N − h f h g N · L ) X (3) T jk b η il,jk ( v ) T j ′ k ′ b η il,j ′ k ′ ( v ) − e V ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! and sup v ∈ I (cid:12)(cid:12)(cid:12) V ( v ) − e V ( v ) (cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! . Now note that (cid:8)
V ( v ) : v ∈ I (cid:9) is a U process with V M ( v ) = E (cid:2) V ( v ) (cid:3) . One can apply Hoeffding decomposition to V ( v ) and apply techniques invoked in the proof of Theorem 3.1 of MMS from empirical process and U process theoryto derive the uniform rate of convergence of V ( v ) − V M ( v ) . Then the conclusion follows. (cid:4) Proof of Theorem 3.
First, it can be verified by standard arguments that the function class n M ‡ ( · ; v ) : v ∈ I o is(uniformly) VC-type with respect to a constant envelope that is a multiple of h − f h − g . Then it essentially follows fromLemma 7 and Theorem 2 that the process Z ( v ) := b f RGP V ( v ) − f ( v ) (cid:16) Lh f h g (cid:17) − / b V RGP V ( v ) / , v ∈ I (A.12)can be approximated by Γ ( v ) := 1( N · L ) / X i,l M ‡ ( B il ; v ) − µ M ‡ ( v )Var h M ‡ ( B ; v ) i / , v ∈ I uniformly in v ∈ I , with an estimated rate of uniform approximation error. See the proof of Lemma B.4 for details.By adapting the proofs of Lemmas B.5 - B.9 of MMS, we can show that the bootstrap process { Z ∗ ( v ) : v ∈ I } canbe approximated by the bootstrap analogue of Γ ( v ) uniformly in v ∈ I . The rest of the proof is identical to that ofCorollary 4.3 of MMS. We can show that the difference between the distribution of k Z k I and that of k Γ G k I , where { Γ G ( v ) : v ∈ I } is an intermediate Gaussian process that has the same covariance structure as that of Γ , convergesto zero uniformly. See Theorem 4.3 of MMS and its proof. Then it can be shown that the difference between thedistribution of k Z ∗ k I and that of k Γ G k I converges to zero uniformly in the bootstrap world. See Theorem 4.4 ofMMS and its proof. The conclusion follows easily from these observations. See the proof of Corollaries 4.2 and 4.3 ofMMS. (cid:4) B Lemmas
Lemma 1.
Suppose that Assumptions 1 - 3 are satisfied. Then we have sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12)b ξ ( b ) − ξ ( b ) (cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! and sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)b ξ ( b ) − ξ ( b ) + 1 N − G ( b ) g ( b ) ( b g ( b ) − g ( b )) − N − b G ( b ) − G ( b ) g ( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh + h (cid:19) . roof of Lemma 1. Standard arguments (see Lemma 1 of Marmer and Shneyerov, 2012) yield sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12) b G ( b ) − G ( b ) (cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) L (cid:19) / ! . (B.1)The bias of the local quadratic MCE E [ b g ( b )] − g ( b ) is O (cid:0) h (cid:1) uniformly over the entire support b ∈ (cid:2) b, b (cid:3) . The stochasticpart b g ( b ) − E [ b g ( b )] can be shown to be O p (cid:16) log ( L ) / ( Lh ) − / (cid:17) uniformly over the entire support b ∈ (cid:2) b, b (cid:3) , by usingstandard arguments (see, e.g., Newey, 1994). Therefore, sup b ∈ [ b,b ] | b g ( b ) − g ( b ) | = O p (cid:18) log ( L ) Lh (cid:19) / + h ! . (B.2)Applying the identity ab = ac − a ( b − c ) c + a ( b − c ) bc , b ξ ( b ) − ξ ( b ) = 1 N − ( − G ( b ) ( b g ( b ) − g ( b )) g ( b ) + b G ( b ) − G ( b ) g ( b ) + b G ( b ) b g ( b ) ( b g ( b ) − g ( b )) g ( b ) − (cid:16) b G ( b ) − G ( b ) (cid:17) ( b g ( b ) − g ( b )) g ( b ) . (B.3)By using (B.1), (B.2) and (2.4), sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ( b ) ( b g ( b ) − g ( b )) g ( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! (B.4)and sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b G ( b ) − G ( b ) g ( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) L (cid:19) / ! . (B.5)Since sup b ∈ [ b,b ] | b g ( b ) − g ( b ) | = o p (1) w.p.a.1, sup b ∈ [ b,b ] b g ( b ) − < (cid:0) C g / (cid:1) − , w.p.a.1 and consequently, sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b G ( b ) b g ( b ) ( b g ( b ) − g ( b )) g ( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > sup b ∈ [ b,b ] ( b g ( b ) − g ( b )) = O p (cid:18) log ( L ) Lh + h (cid:19) . The conclusion follows from this result, (B.3), (B.4) and (B.5). (cid:4)
Lemma 2.
Suppose that Assumptions 1 - 3 are satisfied. Then, (a). b s is strictly increasing on h ξ (cid:16)b b (cid:17) , ξ (cid:16)b b (cid:17)i , w.p.a.1;(b). b s (cid:16) ξ (cid:16)b b (cid:17) + h r (cid:17) − b b = O p ( h ) and b b − b s (cid:16) ξ (cid:16)b b (cid:17) − h r (cid:17) = O p ( h ) ; (c). sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i | b s ( z ) − s ( z ) | = O p (cid:18) log ( L ) Lh (cid:19) / + h ! ; d). sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i | b s ′ ( z ) − s ′ ( z ) | = O p (cid:18) log ( L ) Lh (cid:19) / + h ! ; (e). For any inner closed sub-interval [ b l , b u ] of (cid:2) b, b (cid:3) , sup b ∈ [ b l ,b u ] (cid:12)(cid:12)b s − ( b ) − ξ ( b ) (cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! . Proof of Lemma 2.
By the definition of b s , we have b s ′ ( t ) = Z b b b b h r K r b ξ ( b ) − th r ! d b. For any t ∈ h ξ (cid:16)b b (cid:17) , ξ (cid:16)b b (cid:17)i , b s ′ ( t ) > if the measurable set n b ∈ hb b, b b i : (cid:12)(cid:12)(cid:12)b ξ ( b ) − t (cid:12)(cid:12)(cid:12) ≤ h r o has positive measure. Let r ξ := sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12)b ξ ( b ) − ξ ( b ) (cid:12)(cid:12)(cid:12) . Clearly, n b ∈ hb b, b b i : | ξ ( b ) − t | + r ξ ≤ h r o ⊆ n b ∈ hb b, b b i : (cid:12)(cid:12)(cid:12)b ξ ( b ) − t (cid:12)(cid:12)(cid:12) ≤ h r o . By Lemma 1, r ξ = o p ( h ) . Consequently, r ξ ≤ h r / w.p.a.1, and n b ∈ hb b, b b i : | ξ ( b ) − t | ≤ h r / o ⊆ n b ∈ hb b, b b i : (cid:12)(cid:12)(cid:12)b ξ ( b ) − t (cid:12)(cid:12)(cid:12) ≤ h r o , for all t ∈ h ξ (cid:16)b b (cid:17) , ξ (cid:16)b b (cid:17)i , w.p.a.1. (B.6)Since ξ is continuous and strictly increasing on hb b, b b i , n b ∈ hb b, b b i : (cid:12)(cid:12)(cid:12)b ξ ( b ) − t (cid:12)(cid:12)(cid:12) ≤ h r o has positive measure.For Part (b), by change of variables and Fubini-Tonelli theorem, we have b s (cid:16) ξ (cid:16)b b (cid:17) + h r (cid:17) − b b = Z b b b b Z ∞−∞ w ≤ ξ (cid:16)b b (cid:17) + h r − b ξ ( b ) h r K r ( w ) d w d b ≤ Z ∞−∞ Z b b b b (cid:16) ξ ( b ) ≤ r ξ + ξ (cid:16)b b (cid:17) + (1 − w ) h r (cid:17) d bK r ( w ) d w = Z ∞−∞ max n s (cid:16) r ξ + ξ (cid:16)b b (cid:17) + (1 − w ) h r (cid:17) − b b, o K r ( w ) d w. By a mean value expansion, we have Z ∞−∞ max n s (cid:16) ξ (cid:16)b b (cid:17) + (1 − w ) h r + r ξ (cid:17) − b b, o K ( w ) d w ≤ Z ∞−∞ (cid:12)(cid:12)(cid:12) s (cid:16) ξ (cid:16)b b (cid:17) + (1 − w ) h r + r ξ (cid:17) − b b (cid:12)(cid:12)(cid:12) K ( w ) d w > Z ∞−∞ | (1 − w ) h r + r ξ | K ( w ) d w = O p ( h ) , where the last equality holds since r ξ = o p ( h ) . Therefore, b s (cid:16) ξ (cid:16)b b (cid:17) + h r (cid:17) − b b = O p ( h ) . The proof of b b − b s (cid:16) ξ (cid:16)b b (cid:17) − h r (cid:17) = p ( h ) is similar.For Part (c), by the triangle inequality and a second-order Taylor expansion, we have | b s ( z ) − s ( z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b e K r z − b ξ ( b ) h r ! d b − Z b b b b e K r (cid:18) z − ξ ( b ) h r (cid:19) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b e K r (cid:18) z − ξ ( b ) h r (cid:19) d b + b b − s ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b h r K r (cid:18) z − ξ ( b ) h r (cid:19) (cid:16)b ξ ( b ) − ξ ( b ) (cid:17) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z b b b b h r K r z − ˙ ξ ( b ) h r ! (cid:16)b ξ ( b ) − ξ ( b ) (cid:17) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b e K r (cid:18) z − ξ ( b ) h r (cid:19) d b + b b − s ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B.7)where ˙ ξ ( b ) is the mean value satisfying (cid:12)(cid:12)(cid:12) ˙ ξ ( b ) − ξ ( b ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)b ξ ( b ) − ξ ( b ) (cid:12)(cid:12)(cid:12) for each b ∈ hb b, b b i . Then we have sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b h r K r (cid:18) z − ξ ( b ) h r (cid:19) (cid:16)b ξ ( b ) − ξ ( b ) (cid:17) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i Z b b b b h r K r (cid:18) z − ξ ( b ) h r (cid:19) d b r ξ ≤ sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i Z b b b b h r ( | z − ξ ( b ) | ≤ h r ) d b r ξ = O p (cid:18) log ( L ) Lh (cid:19) / + h ! , (B.8)where the equality follows from Lemma 1 .For all z ∈ h ξ (cid:16)b b (cid:17) + h r , ξ (cid:16)b b (cid:17) − h r i , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b h r K z − ˙ ξ ( b ) h r ! (cid:16) ˆ ξ ( b ) − ξ ( b ) (cid:17) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z b b b b h r K r z − ˙ ξ ( b ) h r ! d b r ξ > Z b b b b h r ( | z − ξ ( b ) | ≤ h r + r ξ ) d b r ξ ≤ Z b b b b h r ( | z − ξ ( b ) | ≤ h r ) d b r ξ , (B.9)where the last inequality holds w.p.a.1 since r ξ = o p ( h ) . Now by (B.7), (B.8) and (B.9), sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b e K r z − b ξ ( b ) h r ! d b − Z b b b b e K r (cid:18) z − ξ ( b ) h r (cid:19) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! . (B.10)For all z ∈ h ξ (cid:16)b b (cid:17) + h r , ξ (cid:16)b b (cid:17) − h r i , since e K r ( u ) = R u −∞ K r ( t ) d t and K r is supported on [ − , , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b e K r (cid:18) z − ξ ( b ) h r (cid:19) d b + b b − s ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s ( z − h r ) b b e K r (cid:18) z − ξ ( b ) h r (cid:19) d b + Z s ( z + h r ) s ( z − h r ) e K r (cid:18) z − ξ ( b ) h r (cid:19) d b + b b − s ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s ( z − h r ) + Z s ( z + h r ) s ( z − h r ) e K r (cid:18) z − ξ ( b ) h r (cid:19) d b − s ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By change of variables and integration by parts, Z s ( z + h r ) s ( z − h r ) e K r (cid:18) z − ξ ( b ) h r (cid:19) d b = Z − e K r ( − u ) h r s ′ ( h r u + z ) d u = − s ( z − h r ) + Z − s ( z + h r u ) K r ( u ) d u. Now it follows that sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b e K r (cid:18) z − ξ ( b ) h r (cid:19) d b + b b − s ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i (cid:12)(cid:12)(cid:12)(cid:12)Z − K r ( u ) s ( z + h r u ) d u − s ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:0) h (cid:1) , (B.11)where the second equality follows from Taylor expansion and the fact R uK r ( u ) d u = 0 . The conclusion of Part (c)follows from (B.7), (B.10) and (B.11).For Part (d), note | b s ′ ( z ) − s ′ ( z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b h r K r b ξ ( b ) − zh r ! d b − Z b b b b h r K r (cid:18) ξ ( b ) − zh r (cid:19) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b h r K r (cid:18) ξ ( b ) − zh r (cid:19) d b − s ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By arguments that are similar to those used to prove (B.10), sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b h r K r b ξ ( b ) − zh r ! d b − Z b b b b h r K r (cid:18) ξ ( b ) − zh r (cid:19) d b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! . By integration by parts and Taylor expansion, sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z b b b b h r K r (cid:18) ξ ( b ) − zh r (cid:19) d b − s ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i (cid:12)(cid:12)(cid:12)(cid:12)Z − K r ( u ) s ′ ( z + h r u ) d u − s ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:0) h (cid:1) . The conclusion of Part (d) therefore follows.For Part (e), Part (b) implies that [ b l , b u ] is contained in the interior of h ξ (cid:16)b b (cid:17) + h r , ξ (cid:16)b b (cid:17) − h r i w.p.a.1. ThenPart (a) implies that b s − ( b ) ∈ h ξ (cid:16)b b (cid:17) + h r , ξ (cid:16)b b (cid:17) − h r i and b = b s (cid:0)b s − ( b ) (cid:1) , for all b ∈ [ b l , b u ] , w.p.a.1 . (B.12)Therefore, for any b ∈ [ b l , b u ] , s (cid:0)b s − ( b ) (cid:1) − s ( ξ ( b )) = s ′ ( ˙ v ) (cid:0)b s − ( b ) − ξ ( b ) (cid:1) for some mean value ˙ v with | ˙ v − ξ ( b ) | ≤ (cid:12)(cid:12)b s − ( b ) − ξ ( b ) (cid:12)(cid:12) . Then, since s ′ is bounded away from zero (see Lemma A126f GPV), sup b ∈ [ b l ,b u ] (cid:12)(cid:12)b s − ( b ) − ξ ( b ) (cid:12)(cid:12) > sup b ∈ [ b l ,b u ] (cid:12)(cid:12) s (cid:0)b s − ( b ) (cid:1) − s ( ξ ( b )) (cid:12)(cid:12) ≤ sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i | s ( z ) − b s ( z ) | = O p (cid:18) log ( L ) Lh (cid:19) / + h ! , where the second inequality follows from (B.12) and holds w.p.a.1. (cid:4) Lemma 3.
Suppose that Assumptions 1 - 3 hold. Let T il := ( V il ∈ [ v − δ , v + δ ]) . Then we have b f RGP V ( v ) − f ( v ) = 1 N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) (cid:16) b V † il − V il (cid:17) + 12 f ′′ ( v ) (cid:18)Z K f ( u ) u d u (cid:19) h f + O p log ( L ) Lh + (cid:18) log ( L ) Lh (cid:19) / + h ! , where the remainder term is uniform in v ∈ I . Proof of Lemma 3.
Write b f RGP V ( v ) = 1 N · L X i,l ( T il h f K f b V † il − vh f ! + (1 − T il ) 1 h f K f b V † il − vh f !) . Now for any v ∈ h ξ (cid:16)b b (cid:17) + h f + h r , ξ (cid:16)b b (cid:17) − h f − h r i , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L X i,l (1 − T il ) 1 h f K f b V † il − vh f !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N · L X i,l h − f (1 − T il ) (cid:16)(cid:12)(cid:12)(cid:12) b V † il − v (cid:12)(cid:12)(cid:12) ≤ h f (cid:17) ≤ N · L X i,l h − f ( B il > s ( v + δ )) ( B il ∈ [ b s ( v − h f ) , b s ( v + h f )])+ 1 N · L X i,l h − f ( B il < s ( v − δ )) ( B il ∈ [ b s ( v − h f ) , b s ( v + h f )]) ≤ N · L X i,l h − f ( B il > s ( v + δ )) ( B il ∈ [ s ( v − h f ) − r s , s ( v + h f ) + r s ])+ 1 N · L X i,l h − f ( B il < s ( v − δ )) ( B il ∈ [ s ( v − h f ) − r s , s ( v + h f ) + r s ]) , where r s := sup z ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i | b s ( z ) − s ( z ) | and the second inequality holds w.p.a.1. Therefore, b f RGP V ( v ) = 1 N · L X i,l T il h f K f b V † il − vh f ! , for all v ∈ I , w.p.a.1.27et e f denote the infeasible estimator that uses the unobserved true valuations: e f ( v ) = 1 N · L X i,l h f K f (cid:18) V il − vh f (cid:19) . It now follows that b f RGP V ( v ) − e f ( v ) = 1 N · L X i,l T il h f K f b V † il − vh f ! − K f (cid:18) V il − vh f (cid:19)! , for all v ∈ I , w.p.a.1.By a second-order Taylor expansion of the right-hand side of the above equality, b f RGP V ( v ) − e f ( v ) = 1 N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) (cid:16) b V † il − V il (cid:17) + 12 · N · L X i,l T il h f K ′′ f ˙ V il − vh f ! (cid:16) b V † il − V il (cid:17) , (B.13)for some mean value ˙ V il that lies on the line joining b V † il and V il .Lemma 2(e) implies that sup v ∈ I max i,l T il (cid:12)(cid:12)(cid:12) b V † il − V il (cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + h ! . (B.14)Since K ′′ f is compactly supported on [ − , and bounded, by the triangle inequality, sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L X i,l T il h f K ′′ f ˙ V il − vh f ! (cid:16) b V † il − V il (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > sup v ∈ I N · L X i,l T il h − f (cid:16)(cid:12)(cid:12)(cid:12) ˙ V il − v (cid:12)(cid:12)(cid:12) ≤ h f (cid:17) (cid:26) sup v ∈ I max i,l T il (cid:16) b V † il − V il (cid:17) (cid:27) ≤ sup v ∈ I N · L X i,l T il h − f ( | V il − v | ≤ h f ) (cid:26) sup v ∈ I max i,l T il (cid:16) b V † il − V il (cid:17) (cid:27) , (B.15)where the last inequality holds w.p.a.1, since sup v ∈ I max i,l T il (cid:12)(cid:12)(cid:12) ˙ V il − V il (cid:12)(cid:12)(cid:12) = o p ( h ) . It follows that sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L X i,l T il h f K ′′ f ˙ V il − vh f ! (cid:16) b V † il − V il (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh + h (cid:19) . (B.16)By standard arguments for kernel density estimation, e f ( v ) − E h e f ( v ) i = O p (cid:18) log ( L ) Lh (cid:19) / ! and E h e f ( v ) i − f ( v ) = 12 f ′′ ( v ) (cid:18)Z K f ( u ) u d u (cid:19) h f + o (cid:0) h (cid:1) , (B.17)where the remainder terms are uniform in v ∈ I . The conclusion follows. (cid:4) Lemma 4.
Suppose that Assumptions 1 - 3 hold. Let e s ( t ) := Z bb Z t −∞ h r K r (cid:18) ξ ( b ) − uh r (cid:19) d u d b + b, t ∈ R . hen, b f RGP V ( v ) − f ( v ) = − N · L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) b s ( V il ) − e s ( V il ) s ′ ( V il ) + 12 f ′′ ( v ) (cid:18)Z K f ( u ) u d u (cid:19) h f + 12 ( s ′′′ ( v ) f ( v ) + s ′′ ( v ) f ′ ( v )) s ′ ( v ) − s ′′ ( v ) f ( v ) s ′′ ( v ) s ′ ( v ) (cid:18)Z K r ( u ) u d u (cid:19) h r + O p log ( L ) Lh + (cid:18) log ( L ) Lh (cid:19) / + h ! , where the remainder term is uniform in v ∈ I . Proof of Lemma 4.
It follows from Lemma 2(b) that [ s ( v l − δ ) , s ( v u + δ )] is an inner closed sub-interval of hb s (cid:16) ξ (cid:16)b b (cid:17) + h r (cid:17) , b s (cid:16) ξ (cid:16)b b (cid:17) − h r (cid:17)i w.p.a.1. By Lemma 2(a), b s is strictly increasing on h ξ (cid:16)b b (cid:17) + h r , ξ (cid:16)b b (cid:17) − h r i w.p.a.1and for all B il satisfying B il ∈ [ s ( v l − δ ) , s ( v u + δ )] , we have the following expansion by Dette et al. (2006, LemmaA.1): b V † il − V il = − (cid:18) b s − ss ′ (cid:19) ◦ ξ ( B il ) + χ ,il + χ ,il where χ ,il := − (cid:18) b s − ss ′ + λ il ( b s ′ − s ′ ) · b s ′ − s ′ s ′ + λ il ( b s ′ − s ′ ) (cid:19) ◦ ( s + λ il ( b s − s )) − ( B il ) and χ ,il := ( b s − ss ′ + λ il ( b s ′ − s ′ ) · ( b s − s ) ( s ′′ + λ il ( b s ′′ − s ′′ ))( s ′ + λ il ( b s ′ − s ′ )) ) ◦ ( s + λ il ( b s − s )) − ( B il ) for some λ il ∈ (0 , that depends on B il . By Lemma 2(b), ( s + λ il ( b s − s )) − ( B il ) ∈ h ξ (cid:16)b b (cid:17) + h r , ξ (cid:16)b b (cid:17) − h r i , w.p.a.1,for all B il satisfying B il ∈ [ s ( v l − δ ) , s ( v u + δ )] . Next, write N L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) (cid:16) b V † il − V il (cid:17) = − N L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) b s ( V il ) − e s ( V il ) s ′ ( V il ) − N L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) e s ( V il ) − B il s ′ ( V il )+ 1 N L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) χ ,il + 1 N L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) χ ,il . (B.18)By Lemma 2(c) and Lemma 2(d), sup i,l T il | χ ,il | = O p (cid:18) log ( L ) Lh + h (cid:19) . For χ il , since K ′ r is supported on [ − , , sup t ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i | b s ′′ ( t ) | > sup t ∈ h ξ ( b b ) + h r ,ξ (cid:16)b b (cid:17) − h r i Z b b b b h r ( | t − ξ ( b ) | ≤ h r + r ξ ) d b = O p (cid:0) h − (cid:1) .
29t follows from this result, Lemma 2(c) and Lemma 2(d) that sup i,l T il | χ ,il | = O p (cid:18) log ( L ) Lh + h (cid:19) . Now it is clear that sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) χ ,il (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup v ∈ I N · L X i,l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h f K ′ f (cid:18) V il − vh f (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup i,l T il | χ ,il | ! = O p (cid:18) log ( L ) Lh + h (cid:19) (B.19)and similarly, sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) χ ,il (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh + h (cid:19) . (B.20)By the definition of e s , when h f is small enough, N L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) e s ( V il ) − B il s ′ ( V il )= 1 N L X i,l h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb e K r (cid:18) V il − ξ ( b ) h r (cid:19) d b + b − s ( V il ) ! . By change of variable and integration by parts, Z bb e K r (cid:18) V il − ξ ( b ) h r (cid:19) d b = Z vv e K r (cid:18) V il − uh r (cid:19) s ′ ( u ) d u = b · e K r (cid:18) V il − vh r (cid:19) − b · e K r (cid:18) V il − vh r (cid:19) + Z vv h r K r (cid:18) V il − uh r (cid:19) s ( u ) d u = Z vv h r K r (cid:18) V il − uh r (cid:19) s ( u ) d u − b, when h r is small enough, for all V il satisfying V il ∈ [ v l − δ , v u + δ ] .Denote β s ( w ) := Z vv h r K r (cid:18) w − uh r (cid:19) s ( u ) d u − s ( w ) and write N · L X i,l h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z vv h r K r (cid:18) V il − uh r (cid:19) s ( u ) d u − s ( V il ) ! = N · L X i,l h f K ′ f (cid:18) V il − vh f (cid:19) β s ( V il ) s ′ ( V il ) − E " h f K ′ f (cid:18) V − vh f (cid:19) β s ( V ) s ′ ( V ) + E " h f K ′ f (cid:18) V − vh f (cid:19) β s ( V ) s ′ ( V ) . (B.21)By standard argument for kernel density estimation (see, e.g., Newey, 1994), since s is three-times continuously30ifferentiable, β s ( w ) = h r s ′′ ( w ) Z u K r ( u ) d u + o (cid:0) h (cid:1) β ′ s ( w ) = h r s ′′′ ( w ) Z u K r ( u ) d u + o (cid:0) h (cid:1) , (B.22)where the remainder terms are uniform in v ∈ I .By change of variables, E " h f K ′ f (cid:18) V − vh f (cid:19) β s ( V ) s ′ ( V ) = Z v − vhfv − vhf h f K ′ f ( z ) β s ( h f z + v ) s ′ ( h f z + v ) f ( h f z + v ) d z = Z v − vhfv − vhf h f K ′ f ( z ) ( β s ( v ) f ( v ) s ′ ( v ) + ( β ′ s ( ˙ v ) f ( ˙ v ) + β s ( ˙ v ) f ′ ( ˙ v )) s ′ ( ˙ v ) − β s ( ˙ v ) f ( ˙ v ) s ′′ ( ˙ v ) s ′ ( ˙ v ) h f z ) d z, (B.23)where ˙ v is the mean value depending on z with | ˙ v − v | ≤ h f | z | . It is clear that for small enough h f , β s ( v ) f ( v ) s ′ ( v ) Z v − vhfv − vhf K ′ f ( z ) d z = 0 for all v ∈ I. Now it follows from (B.22) and (B.23) that E " h f K ′ f (cid:18) V − vh f (cid:19) β s ( V ) s ′ ( V ) = h r s ′′′ ( v ) f ( v ) + s ′′ ( v ) f ′ ( v )) s ′ ( v ) − s ′′ ( v ) f ( v ) s ′′ ( v ) s ′ ( v ) Z u K r ( u ) d u + o (cid:0) h (cid:1) , (B.24)where the remainder term is uniform in v ∈ I .Denote S ( z ; v ) := 1 h f K ′ f (cid:18) z − vh f (cid:19) β s ( z ) s ′ ( z ) and thus N · L X i,l h f K ′ f (cid:18) V il − vh f (cid:19) β s ( V il ) s ′ ( V il ) − E " h f K ′ f (cid:18) V − vh f (cid:19) β s ( V ) s ′ ( V ) = 1 N · L X i,l S ( V il ; v ) − E [ S ( V ; v )] . By standard arguments (see the proof of Lemma B.3 of MMS), it can be easily verified that {S ( · ; v ) : v ∈ I } is(uniformly) VC-type with respect to a constant envelope F S which satisfies F S > h − f sup v ∈ [ v l − δ ,v u + δ ] | β s ( v ) | when h f is small enough. The applying a maximal inequality (van der Vaart and Wellner, 1996, Theorem 2.14.1) yields E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L X i,l S ( V il ; v ) − E [ S ( V ; v )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L − / | F S | = O (cid:16) L − / (cid:17) , when h is small enough. The conclusion follows from this result, (B.18), (B.19), (B.20) and (B.24). (cid:4) emma 5. Suppose that Assumptions 1 - 3 hold. We have b f RGP V ( v ) − f ( v ) = 1( N −
1) 1( N · L ) X i,l X j,k M ( B il , B jk ; v ) + 12 f ′′ ( v ) (cid:18)Z K f ( u ) u d u (cid:19) h f + 12 ( s ′′′ ( v ) f ( v ) + s ′′ ( v ) f ′ ( v )) s ′ ( v ) − s ′′ ( v ) f ( v ) s ′ ( v ) (cid:18)Z K r ( u ) u d u (cid:19) h r + O p log ( L ) Lh + (cid:18) log ( L ) Lh (cid:19) / + h ! , where the remainder term is uniform in v ∈ I . Proof of Lemma 5.
First we show − N · L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) b s ( V il ) − e s ( V il ) s ′ ( V il )= − N − N · L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) ξ ( b ) − V il h r (cid:19) G ( b ) g ( b ) ( b g ( b ) − g ( b )) d b + O p log ( L ) Lh + h + (cid:18) log ( L ) Lh (cid:19) / ! . Since by the Borel-Cantelli lemma we have (cid:12)(cid:12)(cid:12)b b − b (cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) L (cid:19) , (cid:12)(cid:12)(cid:12)b b − b (cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) L (cid:19) , therefore, N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) b s ( V il ) − e s ( V il ) s ′ ( V il )= 1 N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb e K r V il − b ξ ( b ) h r ! − e K r (cid:18) V il − ξ ( b ) h r (cid:19)! d b + O p (cid:18) log ( L ) Lh (cid:19) , (B.25)where the remainder term is uniform in v ∈ I .By a second-order Taylor expansion, we have N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb e K r V il − b ξ ( b ) h r ! − e K r (cid:18) V il − ξ ( b ) h r (cid:19)! d b = 1 N − N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) V il − ξ ( b ) h r (cid:19) G ( b ) g ( b ) ( b g ( b ) − g ( b )) d b − N − N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) V il − ξ ( b ) h r (cid:19) b G ( b ) − G ( b ) g ( b ) d b + I ( v ) + I ( v ) (B.26)32here I ( v ) := − N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) V il − ξ ( b ) h r (cid:19) × ((cid:16)b ξ ( b ) − ξ ( b ) (cid:17) + 1 N − G ( b ) g ( b ) ( b g ( b ) − g ( b )) − N − b G ( b ) − G ( b ) g ( b ) ) d b and I ( v ) := 1 N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K ′ r V il − ˙ ξ ( b ) h r ! (cid:16)b ξ ( b ) − ξ ( b ) (cid:17) d b for some mean value ˙ ξ ( b ) with (cid:12)(cid:12)(cid:12) ˙ ξ ( b ) − ξ ( b ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)b ξ ( b ) − ξ ( b ) (cid:12)(cid:12)(cid:12) for each b ∈ (cid:2) b, b (cid:3) .Then, by Lemma 1, sup v ∈ I | I ( v ) | > sup v ∈ I N · L X i,l h f (cid:12)(cid:12)(cid:12)(cid:12) K ′ f (cid:18) V il − vh f (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( sup v ∈ [ v l − δ ,v u + δ ] Z bb h r K r (cid:18) v − ξ ( b ) h r (cid:19) d b ) × sup b ∈ [ b,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)b ξ ( b ) − ξ ( b ) (cid:17) + 1 N − G ( b ) g ( b ) ( b g ( b ) − g ( b )) − N − b G ( b ) − G ( b ) g ( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh + h (cid:19) (B.27)and sup v ∈ I | I ( v ) | > sup v ∈ I N · L X i,l h f (cid:12)(cid:12)(cid:12)(cid:12) K ′ f (cid:18) V il − vh f (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( sup v ∈ [ v l − δ ,v u + δ ] Z bb h r K v − ˙ ξ ( b ) h r ! d b ) r ξ = O p (cid:18) log ( L ) Lh + h (cid:19) , (B.28)when the inequalities hold when h r is small enough.Define G ( b, b ′ ; v ) := 1 h f K ′ f (cid:18) ξ ( b ) − vh f (cid:19) s ′ ( ξ ( b )) Z bb h r K r (cid:18) ξ ( b ) − ξ ( z ) h r (cid:19) ( b ′ ≤ z ) − G ( z ) g ( z ) d z. By the definition of G , we have N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) V il − ξ ( b ) h r (cid:19) b G ( b ) − G ( b ) g ( b ) d b = 1( N · L ) X (2) G ( B il , B jk ; v ) + 1( N · L ) X i,l G ( B il , B il ; v ) , for all v ∈ I, (B.29)when h f is small enough. The kernel G satisfies G ( b ; v ) := Z G ( b, b ′ ; v ) d G ( b ′ ) = 0 and µ G ( v ) := Z Z G ( b, b ′ ; v ) d G ( b ′ ) d G ( b ) = 0 , for all v ∈ I. Also define G ( b ; v ) := Z G ( b ′ , b ; v ) d G ( b ′ ) . N · L ) X (2) G ( B il , B jk ; v ) = 1 N · L X i,l G ( B il ; v ) + 1( N · L ) X (2) {G ( B il , B jk ; v ) − G ( B il ; v ) } . (B.30)By standard arguments (see the proof of Lemma B.2 of MMS for details), it can be easily verified that {G ( · , · ; v ) : v ∈ I } is (uniformly) VC-type with respect to a constant envelope F G which satisfies F G > h − f . This implies sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L ) X i,l G ( B il , B il ; v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:16)(cid:0) Lh (cid:1) − (cid:17) . (B.31)Application of a maximal inequality (Chen and Kato, 2017, Corollary 5.6) and Markov’s inequality gives sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L ) X (2) {G ( B il , B jk ; v ) − G ( B il ; v ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:16)(cid:0) Lh (cid:1) − (cid:17) . (B.32)Next, we show that sup v ∈ I E h G ( B ; v ) i > h − , (B.33)when h is small enough. Denote τ ( z ) := Z bb h f K ′ f (cid:18) ξ ( b ) − vh f (cid:19) ξ ′ ( b ) 1 h r K r (cid:18) ξ ( b ) − ξ ( z ) h r (cid:19) g ( b ) d b. Then by change of variables and the Fubini-Tonelli theorem, Z G ( b ; v ) d G ( b ) ≤ Z bb (Z bb τ ( z ) ( b ≤ z ) g ( z ) d z ) g ( b ) d b = Z bb Z bb τ ( z ) g ( z ) τ ( z ′ ) g ( z ′ ) G (min { z, z ′ } ) d z d z ′ = h r Z v − vhrv − vhr Z v − vhrv − vhr τ ( s ( h r w + v )) g ( s ( h r w + v )) τ ( s ( h r w ′ + v )) g ( s ( h r w ′ + v )) G (min { s ( h r w + v ) , s ( h r w ′ + v ) } ) s ′ ( h r w + v ) s ′ ( h r w ′ + v ) d w d w ′ = 2 h f Z v − vhrv − vhr s ′ ( h r w ′ + v ) g ( s ( h r w ′ + v )) Z v − vhrv − vhr K ′ f ( u ′ ) K r (cid:18) λ f λ r u ′ − w ′ (cid:19) g ( s ( h f u ′ + v )) d u ′ × Z w ′ v − vhr s ′ ( h r w + v ) G ( s ( h r w + v )) g ( s ( h r w + v )) Z v − vhrv − vhr K ′ f ( u ) K r (cid:18) λ f λ r u − w (cid:19) g ( s ( h f u + v )) d u d w d w ′ , (B.34)where the last equality follows from symmetry. It follows from integration by parts that Z Z w ′ −∞ (cid:26)Z K ′ f ( u ′ ) K r (cid:18) λ f λ r u ′ − w ′ (cid:19) d u ′ (cid:27) (cid:26)Z K ′ f ( u ) K r (cid:18) λ f λ r u − w (cid:19) d u (cid:27) d w d w ′ = 0 . Then (B.33) follows from this result, (B.34) and Taylor expansion.Since {G ( · , · ; v ) : v ∈ I } is (uniformly) VC-type with respect to a constant envelope F G which satisfies F G > h − f ,34t follows from (B.33) and a maximal inequality (Chernozhukov et al., 2014a, Corollary 5.1) that sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L X i,l G ( B il ; v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:18) log ( L ) Lh (cid:19) / + log ( L ) Lh ! . Now it follows from this result, (B.29), (B.30), (B.31) and (B.32) that N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) V il − ξ ( b ) h r (cid:19) b G ( b ) − G ( b ) g ( b ) d b = O p (cid:18) log ( L ) Lh (cid:19) / + log ( L ) Lh ! , uniformly in v ∈ I . Then it follows from this result, (B.25), (B.26), (B.27) and (B.28) that N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) b s ( V il ) − e s ( V il ) s ′ ( V il )= 1 N − N · L X i,l T il h f K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) V il − ξ ( b ) h r (cid:19) G ( b ) g ( b ) ( b g ( b ) − g ( b )) d b + O p (cid:18) log ( L ) Lh (cid:19) / + log ( L ) Lh + h ! , where the remainder term is uniform in v ∈ I .By the definition of the MCE, b g ( b ) = 1 N · L X i,l h g K g (cid:18) B il − bh g (cid:19) , for b ∈ hb b + h g , b b − h g i . Therefore, since K ′ f and K r are compactly supported on [ − , , it is easy to verify that − N − N · L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) ξ ( b ) − V il h r (cid:19) G ( b ) g ( b ) ( b g ( b ) − g ( b )) d b = − N − N · L X i,l h f T il K ′ f (cid:18) V il − vh f (cid:19) s ′ ( V il ) Z bb h r K r (cid:18) ξ ( b ) − V il h r (cid:19) G ( b ) g ( b ) × N · L X j,k h g K g (cid:18) B jk − bh g (cid:19) − g ( b ) d b, for all v ∈ I , w.p.a.1.The conclusion follows from this result, (B.25), (B.26), (B.27), (B.28) and Lemma 5. (cid:4) Lemma 6.
Suppose that Assumptions 1 - 3 hold. Let β g ( b ) := Z (cid:26) h g K g (cid:18) b ′ − bh g (cid:19) − g ( b ) (cid:27) d G ( b ′ ) (B.35) be the bias of the kernel density estimator of g ( b ) . Then µ M ( v ) := − Z ( h f K ′ f (cid:18) ξ ( u ) − vh f (cid:19) s ′ ( ξ ( u )) Z bb h r K r (cid:18) ξ ( b ) − ξ ( u ) h r (cid:19) G ( b ) β g ( b ) g ( b ) d b ) d G ( u ) = o (cid:0) h (cid:1) , See the proof of Lemma B.2 of MMS for more details. niformly in v ∈ I . Proof of Lemma 6.
By change of variables,
Z ( h f K ′ f (cid:18) ξ ( u ) − vh f (cid:19) s ′ ( ξ ( u )) Z bb h r K r (cid:18) ξ ( b ) − ξ ( u ) h r (cid:19) G ( b ) β g ( b ) g ( b ) d b ) d G ( u )= Z v − vhfv − vhf Z v − vhrv − vhr h f K ′ f ( w ) K r (cid:18) z − λ f λ r w (cid:19) G ( s ( h r z + v )) β g ( s ( h r z + v )) g ( s ( h r z + v )) s ′ ( h r z + v ) g ( s ( h f w + v )) d z d w. (B.36)Let ψ ( z ) := G ( s ( z )) s ′ ( z ) /g ( s ( z )) . By a mean value expansion, the second line of (B.36) is equal to h f Z v − vhfv − vhf Z v − vhrv − vhr K ′ f ( w ) K r (cid:18) z − λ f λ r w (cid:19) (cid:8) ψ ( v ) β g ( s ( v )) + (cid:0) ψ ′ ( ˙ v ) β g ( s ( ˙ v )) + ψ ( ˙ v ) β ′ g ( s ( ˙ v )) s ′ ( ˙ v ) (cid:1) h r z (cid:9) × { g ( s ( v )) + g ′ ( s (¨ v )) s ′ (¨ v ) h f w } d z d w = ψ ( v ) β g ( s ( v )) g ( s ( v )) 1 h f Z v − vhfv − vhf Z v − vhrv − vhr K ′ f ( w ) K r (cid:18) z − λ f λ r w (cid:19) d z d w + ψ ( v ) β g ( s ( v )) Z v − vhfv − vhf Z v − vhrv − vhr K ′ f ( w ) K r (cid:18) z − λ f λ r w (cid:19) g ′ ( s (¨ v )) s ′ (¨ v ) w d z d w + g ( s ( v )) 1 h f Z v − vhfv − vhf Z v − vhrv − vhr K ′ f ( w ) K r (cid:18) z − λ f λ r w (cid:19) (cid:0) ψ ′ ( ˙ v ) β g ( s ( ˙ v )) + ψ ( ˙ v ) β ′ g ( s ( ˙ v )) s ′ ( ˙ v ) (cid:1) h r z d z d w + h r Z v − vhfv − vhf Z v − vhrv − vhr K ′ f ( w ) K r ( z ) (cid:0) ψ ′ ( ˙ v ) β g ( s ( ˙ v )) + ψ ( ˙ v ) β ′ g ( s ( ˙ v )) s ′ ( ˙ v ) (cid:1) g ′ ( s (¨ v )) s ′ (¨ v ) zw d z d w, (B.37)where ˙ v and ¨ v are mean values that are dependent on z and w with | ˙ v − v | ≤ h r | z | and | ¨ v − v | ≤ h f | w | . When h issmall enough, Z v − vhfv − vhf Z v − vhrv − vhr K ′ f ( w ) K r (cid:18) z − λ f λ r w (cid:19) d z d w = 0 , for all v ∈ I. (B.38)By standard arguments for the bias of kernel estimators for the density (see, e.g., Newey, 1994), since K g issupported on [ − , , for each b ∈ [ s ( v l − δ ) , s ( v u + δ )] , | β g ( b ) | ≤ h g b ′ ∈ [ b − h g ,b + h g ] | g ′′′ ( b ′ ) | Z (cid:12)(cid:12) u K g ( u ) (cid:12)(cid:12) d u, (B.39)when h g is small enough. By change of variable and Taylor expansion, we have sup b ∈ [ s ( v l − δ ) ,s ( v u + δ )] (cid:12)(cid:12) β ′ g ( b ) (cid:12)(cid:12) = sup b ∈ [ s ( v l − δ ) ,s ( v u + δ )] (cid:12)(cid:12)(cid:12)(cid:12)Z h g K g (cid:18) b ′ − bh g (cid:19) g ′ ( b ′ ) d b ′ − g ′ ( b ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup b ∈ [ s ( v l − δ ) ,s ( v u + δ )] h g (cid:12)(cid:12)(cid:12)(cid:12)Z K g ( u ) u (cid:16) g ′′′ (cid:16) ˙ b (cid:17) − g ′′′ ( b ) (cid:17) d u (cid:12)(cid:12)(cid:12)(cid:12) , (B.40)when h g is small enough, where ˙ b is the mean value depending on u with (cid:12)(cid:12)(cid:12) ˙ b − b (cid:12)(cid:12)(cid:12) ≤ h g | u | . Since g ′′′ is uniformlycontinuous on any inner closed subset of (cid:2) b, b (cid:3) , the assumption that K g is supported on [ − , and (B.40) implythat β ′ g ( b ) = o (cid:0) h (cid:1) uniformly in b ∈ [ s ( v l − δ ) , s ( v u + δ )] . The conclusion follows from these results, (B.37) and(B.38). (cid:4) emma 7. Suppose that Assumptions 1 - 3 hold. Then b f RGP V ( v ) − f ( v ) = 1( N −
1) 1( N · L ) X i,l ( M ( B il ; v ) − µ M ( v )) + 12 f ′′ ( v ) (cid:18)Z K f ( u ) u d u (cid:19) h f + 12 ( s ′′′ ( v ) f ( v ) + s ′′ ( v ) f ′ ( v )) s ′ ( v ) − s ′′ ( v ) f ( v ) s ′ ( v ) (cid:18)Z K r ( u ) u d u (cid:19) h r + O p log ( L ) Lh + (cid:18) log ( L ) Lh (cid:19) / + h ! . Proof of Lemma 7.
Hoeffding decomposition gives N · L ) X i,l X j,k M ( B il , B jk ; v ) = µ M ( v ) + N · L X i,l M ( B il ; v ) − µ M ( v ) + N · L X i,l M ( B il ; v ) − µ M ( v ) + 1( N · L ) X (2) {M ( B il , B jk ; v ) − M ( B il ; v ) − M ( B jk ; v ) + µ M ( v ) } + 1( N · L ) X i,l M ( B il , B il ; v ) − N · L ) ( N · L ) X (2) M ( B il , B jk ; v ) . (B.41)By standard arguments used in the proofs of Lemma B.2 and Lemma B.3 of MMS, it can be easily verified that theclass {M ( · , · ; v ) : v ∈ I } is (uniformly) VC-type with respect to a constant envelope F M that satisfies F M = O (cid:0) h − (cid:1) .Therefore, it is easy to check that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L ) X i,l M ( B il , B il ; v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:16)(cid:0) Lh (cid:1) − (cid:17) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L ) ( N · L ) X (2) M ( B il , B jk ; v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:16)(cid:0) Lh (cid:1) − (cid:17) , (B.42)uniformly in v ∈ I , and sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L ) X (2) {M ( B il , B jk ; v ) − M ( B il ; v ) − M ( B jk ; v ) + µ M ( v ) } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O p (cid:16)(cid:0) Lh (cid:1) − (cid:17) (B.43)follows from the maximal inequality Chen and Kato (2017, Corollary 5.6) and Markov’s inequality.By change of variables, M ( b ; v ) = 1 h f K f (cid:18) ξ ( b ) − vh f (cid:19) ξ ′ ( b ) Z bb h r K r (cid:18) ξ ( b ) − ξ ( z ) h r (cid:19) G ( z ) g ( z ) β g ( z ) d z = 1 h f K f (cid:18) ξ ( b ) − vh f (cid:19) ξ ′ ( b ) Z v − ξ ( b ) hrv − ξ ( b ) hr K r ( u ) G ( s ( h r u + ξ ( b ))) β g ( s ( h r u + ξ ( b ))) s ′ ( h r u + ξ ( b )) g ( s ( h r u + ξ ( b ))) d u. By the arguments used in the proof of Lemma B.3, it can be verified that {M ( · ; v ) : v ∈ I } is (uniformly) VC-typewith respect to a constant envelope F M that satisfies F M > h − f sup b ∈ [ s ( v l − δ ) ,s ( v u + δ )] | β g ( b ) | , h is small enough. The maximal inequality van der Vaart and Wellner (1996, Theorem 2.14.1) yields E sup v ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N · L X i,l M ( B il ; v ) − µ M ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > L − / F M = O (cid:16) L − / h (cid:17) . The conclusion follows from this result, Markov’s inequality, (B.41), (B.42), (B.43) and Lemma 6. (cid:4)(cid:4)