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Identification and Estimation of Seller Risk Aversion in Ascending Auctions
Knowing the seller’s risk attitude is essential for auction design and policy analysis, as seller risk preferences affect supply-side welfare and equilibrium outcomes. The theoretical literature has shown that many canonical results derived under risk neutrality no longer hold when sellers are risk averse. For example, the Revenue Equivalence Theorem does not hold with a risk-averse seller, and while some optimal auction results in myerson1981optimal extend to risk-averse sellers, they do so only in some cases (sundararajan2020robust). More generally, seller risk aversion affects key design choices, including reserve prices and auction formats (waehrer1998auction, moreno2017reserve). These insights highlight why policymakers seeking to predict auction outcomes and design effective auction mechanisms would want to learn about seller risk attitudes.
Prior to our study, to the best of our knowledge, there are no results on the identification of seller risk aversion or on how to estimate it in any auction format. Our paper addresses this gap in the literature. We study a semiparametric model of an ascending auction in which the seller has a parametric utility function (such as CARA or CRRA), while bidder valuations and independent and satisfy a linear quantile specification. We show that the risk-aversion parameter is identified under primitive conditions. We then propose an estimator for the risk-aversion parameter and show that our estimator is consistent and asymptotically normal under standard regularity conditions.
Using data on winning bids, reserve price, and value of the auctioned object, we can identify components of the first-order condition (FOC) that characterizes the seller's optimal reserve price chosen to maximize expected revenue. Under interpretable conditions that parallel those assumed in the theoretical auction literature, part (i) of our Proposition 2 establishes existence and uniqueness of the solution to the FOC, while part (ii) shows that the optimal reserve price is decreasing in the seller's Arrow--Pratt measure of risk aversion. We use the latter monotonic relationship between the optimal reserve price and risk aversion to establish identification of parametric risk preferences that include CARA and CRRA utility functions as special cases.
Our estimator of the risk-aversion parameter is a two-step parameter. In the first stage, nuisance functions, which are components of the expected revenue related to the distribution of bidder valuations, are estimated. The risk-aversion estimator is then defined by forcing the empirical FOC in the second stage. The nuisance functions consist of the derivative of the quantile function and the inverse of the quantile function. While only nonparametric restrictions on the bidder valuation distribution are required for identification, we further assume that the valuation quantile is linear in covariates for the purpose of practical estimation. This modeling choice represents a compromise between fully nonparametric and parametric alternatives, since the linear quantile specification allows for a flexible representation of conditional valuation distributions without suffering from the curse of dimensionality. In principle, other specifications for the distribution of bidder valuations can be adopted.
The use of quantiles in the econometrics of auctions is not new\footnote{An early insight can be found in haile2003nonparametric. Nonparametric estimation of quantiles is used by marmer2012quantile and guerre2012uniform in a first-price auction model, for example.}, but the development of a quantile regression (QR) estimator in this context is relatively recent. Methodologically, our approach builds on the estimators used in gimenes2017econometrics and Gimenes and Guerre (2022, GG22 hereafter). Their strategy exploits auction-specific relations between quantiles of the bidder's valuation and their bids, in which they use bid data to estimate the QR for bids and transform it into a QR estimator for valuations. Specifically, gimenes2017econometrics applies this strategy to estimate ascending auctions with the classic Koenker-Bassett QR estimator, while GG22 develop the augmented quantile regression (AQR) estimator and apply it to a first-price auction model.
In this paper, we use an AQR estimator to estimate the distribution of bidder valuations. Our quantile estimator is the AQR counterpart to the estimator used in gimenes2017econometrics. An AQR estimator is defined as the maximizer of an integrated version of the classic koenker1978regression objective function, combined with kernel smoothing and polynomial approximation of the quantile function. Similar to the local polynomial estimator of fan1992design, advantages the AQR method has over standard QR are that it automatically yields estimators of both the quantile function and its derivative, and produces estimators with improved statistical properties near the boundaries. These features are crucial for our estimation problem, as the quantile derivative enters the FOC and all quantile levels of the AQR estimator can be used to estimate the risk-aversion parameter. By contrast, gimenes2017econometrics does not consider the FOC and therefore does not require estimation of the nuisance functions that enter it.
Other than the fact that the auction-specific relationship between quantiles of bidder valuations and bids differs between ascending auctions and first-price auctions, our application of the AQR method differs from GG22’s analysis in non-trivial ways due to our goal of estimating the risk-aversion parameter. In particular, we view the following two aspects as secondary contributions of the paper, as they expand the scope of the AQR method beyond our application. First, we construct and derive the asymptotic properties of a conditional CDF estimator defined as the inverse of an AQR estimator---see Lemma 3---which enters the FOC and does not appear in GG22. This estimator may be of independent interest as a flexible and low-dimensional approach to estimating conditional CDFs (see, e.g., koenker2013distributional for a discussion on using a quantile function to estimate CDF and vice versa). Second, we show AQR estimators can be used as preliminary estimators in a two-step semiparametric M-estimation problem. Specifically, our risk-aversion parameter is an estimand defined implicitly as a solution to the FOC, whereas the finite-dimensional estimands considered in GG22 are explicit functionals of quantile objects.\footnote{They motivate the study of such functionals with economic quantities that have closed-form expressions in the quantile functions (e.g., expected revenue (li2003semiparametric) and bidder risk-aversion parameter in a first-price auction (guerre2009nonparametric)).} Theorem 4 in GG22 shows that functionals of AQR estimators satisfy a central limit theorem (CLT). Our estimator does not take this form, but we show that it does so after suitable application of stochastic equicontinuity and linearization arguments, and that the CLT applies. This line of arguments is not new for developing limit theorems for semiparametric M-estimators. The novelty here is that we derive the linearized functional explicitly with AQR estimators as preliminary estimators, which has not been done previously. The econometrics and statistics literature is otherwise familiar with similar derivations when traditional nonparametric estimators, such as kernel and series estimators, are used---see, for example, andrews1994asymptotics, newey1994asymptotic, chen2003estimation, and ichimura2010characterization.
We perform a Monte Carlo study and find that our estimator behaves as expected from the theory. We then apply our methodology to real estate foreclosure auction data in S\ {a}o Paulo as an illustration. In this application, the sellers are lenders (banks and private companies) and the Court of Justice of the State of S\ {a}o Paulo. Possible reasons why these sellers may be risk averse include downside risks associated with failure to sell. For example, there may be future costs of holding such properties (e.g., property taxes, insurance, and security), the value of foreclosure homes may depreciate over time, and sellers may be subject to directives to sell within a certain time frame.\footnote{Foreclosure assets are sometimes viewed as non-performing loans from an accounting perspective, and some central banks provide some guidelines on selling them accordingly. For example, see this ECB report: \url{https://www.bankingsupervision.europa.eu/ecb/pub/pdf/guidance_on_npl.en.pdf}.}
Our empirical results suggest that sellers are risk-averse, and the model with a risk-averse seller fits the data much better than one with a risk-neutral seller. Notably, we find the model with risk-neutral sellers systematically overpredicts observed reserve prices relative to the model with risk-averse sellers. This finding complements several empirical evidence showing that observed reserve prices are well below the levels implied by the risk-neutral benchmark in a variety of auction settings (see mcafee1992updating, paarsch1997deriving, mcafee2002set, haile2003inference, and tang2011bounds). Indeed, seller risk aversion has been proposed in the theoretical literature as one of the mechanisms that can rationalize these discrepancies (maskin1984optimal, matthews1987comparing, hu2010risk).
We end the introduction with a review of related literature on the econometrics of auctions. The rest of the paper is organized as follows. Section 2 introduces the model and establishes key identification results for bidder and seller primitives. Section 3 develops the estimation methodology and the asymptotic properties of the AQR estimators for observed bids and bidder valuations. Section 4 presents the estimation of the risk-aversion parameter and derives its asymptotic properties. Section 5 reports a Monte Carlo study of the estimators. Section 6 provides an empirical application to real estate auctions in S\ {a}o Paulo. Section 7 concludes. Proofs of results are collected in the Appendix.
While there appear to be no prior econometric research on seller risk aversion, there is a growing literature studying risk-averse bidders, particularly in first-price auctions; see, for example lu2008estimating, guerre2009nonparametric, campo2011semiparametric, li2015auctions, zincenko2018nonparametric, grundl2019identification, and jun2022testing. On the other hand, the empirical auction literature is built on the identification and inference of bidder valuation distributions under bidder risk neutrality. This begins with the parametric model of paarsch1992deciding and grows with Donald and Paarsch (1993,1996), laffont1995econometrics, athey2001information amongst others.
Parametric models, however, can be computationally demanding to estimate and are subject to model misspecification. Nonparametric approaches, beginning with the seminal work of guerre2000optimal, provide an alternative and have since been developed in a variety of contexts; see, inter alia, athey2002identification, lu2008estimating, krasnokutskaya2011identification, marmer2012quantile, campo2011semiparametric, marmer2013model, enache2017quantile, liu2017nonparametric, luo2018integrated and ma2019inference.
Purely nonparametric approaches are also not without limitations, as the convergence of nonparametric estimators deteriorates with the number of conditioning variables. Semiparametric approaches that aim to allow for more flexible modeling than parametric ones, without suffering from the curse of dimensionality, have been developed for mean regression (rezende2008econometrics) and quantile regression (gimenes2017econometrics, gimenes2022quantile).
Consider an ascending auction of an indivisible object with $I\geq 2$\ bidders. Bidders can raise prices continuously and without cost until only one bidder remains. The object is sold to the highest bidder for the price of his last bid, provided that it is at least as high as the reservation price. We assume an independent private values (IPV) environment, where each bidder knows only their own valuation and values are independently drawn across bidders.
Let the auctioned object have observable characteristics $X\in \mathbb{R}^{D}$. Bidder $i$'s valuation of the object is $V_{i}$, taking value in $\mathcal{V}=\left[ \underline{v},\overline{v} \right] $. For notational simplicity, we assume $\mathcal{V}$\ to be independent of $X$. We denote the conditional CDF of $V_{i}$ by $F\left( \cdot |X\right) $. To facilitate readers, as it may be instructive to compare our assumptions and results with GG22, we use the same notations and terminologies as them when possible.
The winning bid, denoted by $B=V^{I-1:I}\in \mathcal{V}$, is the $\left( I-1\right) $-th order statistic among the $I$ i.i.d. private values $\left\{ V_{i}\right\} _{i=1}^{I}$. This is the same equilibrium play for ascending auctions as used in aradillas2013identification and gimenes2017econometrics. We denote the conditional CDF of $B$ by $G\left( \cdot |X\right) $. We impose the relations between these variables in the following assumption.
Assumption M1.
(i) $F\left( \cdot |X\right) $\ is continuous and strictly increasing almost surely on $\mathcal{V}$;
(ii) $G\left( t|X\right) =\phi \left( F\left( t|X\right) \right) $ \ a.s. for $t\in \mathcal{V}$, where $\phi \left( a\right) =Ia^{I-1}-\left( I-1\right) a^{I}$\ for $a\in \left[ 0,1\right] $ .
M1(i) imposes a minimal regularity condition on $F\left( \cdot |X\right) $. M1(ii) is a structural assumption on the bidder's bidding behavior, following athey2002identification, described using the relation between the CDF of the $ \left( I-1\right) $-th order statistic and the underlying CDF that the sample is drawn from.
We will take a quantile approach to model the bidder's private value distribution. Let the $\alpha -$quantile of the valuation distribution be denoted by $V\left( \alpha |X\right) =F^{-1}\left( \alpha |X\right) $ for $ \alpha \in \left[ 0,1\right] $. As done in gimenes2017econometrics and GG22, we take $V_{i}=V\left( A_{i}|X\right)$ where $A_{i}\sim Uni\left[ 0,1\right] $ can be viewed as the private rank of the bidder, which is independent of $X$ and other bidders' ranks.
We denote the $\alpha -$quantile of $B$\ by $B\left( \alpha |X\right) =G^{-1}\left( \alpha |X\right) $ for $\alpha \in \left[ 0,1\right] $. $ B\left( \cdot |X\right) $ exists, because $G\left( \cdot |X\right) $\ is differentiable and strictly increasing, since $\phi :\left[ 0,1\right] \rightarrow \left[ 0,1\right] $ is continuously differentiable and is strictly increasing on $\left[ 0,1\right] $.
Proposition 1 shows that the quantile functions of the winning bid and the private valuation are linked through a one-to-one relationship given in equation ((ref)). This relation is the central identification argument on the bidder's side (gimenes2017econometrics).
Proposition 1.\ Suppose Assumption M1 holds, then
The seller can influence her expected revenue in an auction by setting reserve price. If all bids are below the reserve price, the seller keeps the object. If all but one bids are below the reserve price, the object is sold with the winner paying the reserve. If two or more bids are above the reserve price, the object is sold with the winner paying the second highest private value among all bidders.
We denote the seller's value of the sale object by $W$. The seller can set the reserve price optimally to maximize her expected utility. Specifically, if the seller has utility function $U\left( \cdot \right) $, the expected utility from setting reserve price to be $r\in \mathcal{V}$ is:
We denote the optimal reserve price by $R$. I.e., $R=\arg \max_{_{r\in \mathcal{V}}}\widetilde{\Pi }\left( r,X,W\right) $, whose existence and uniqueness are guaranteed under the conditions of Assumption M2 given below.
Analogously to writing the bidders' bids in terms of private ranks, the expected revenue above can be equivalently expressed as a function of the screening level instead of the reserve price. For a screening level $\alpha$ that takes value in $\left[ 0,1\right] $, let us define
so that $\Pi (\alpha,X,W)=\widetilde{\Pi }\left( r,X,W\right) $ when $r=V\left( \alpha|X\right) $. The optimal screening level is defined as $\alpha _{R}=\arg \max_{_{\alpha\in \left[ 0,1\right] }}\Pi \left( \alpha,X,W\right) $.
Assumption M2.
(i) $V_{i}$\ has a conditional PDF, denoted by $f\left( \cdot |X\right) $, that is bounded away from zero and infinity a.s. on $\mathcal{V}$;
(ii) $J(v|X)=v-\frac{1-F(v|X)}{f\left( v|X\right) }$, defined for $v\in \mathcal{V}$, is strictly increasing a.s. on $ \mathcal{V}$;
(iii) $W$ takes value in $\mathcal{W}\subseteq \mathcal{V}$ ;
(iv) $U\left( \cdot \right) $\ is a twice continuously differentiable function with $U^{\left( 1\right) }\left( \cdot \right) >0$ \ and $U^{\left( 2\right) }\left( \cdot \right) \leq 0$.
Assumption M2 consists of standard conditions in the auction literature when studying the seller's behavior. M2(i) is a regularity condition where the bounding from below ensures $J\left( \cdot \right) $ in M2(ii)\ is well defined. In his seminal paper, myerson1981optimal calls $J\left( \cdot \right) $ the virtual valuation function, as it represents the marginal revenue contribution of a bidder with valuation $v$. He imposes monotonicity of $J\left( \cdot \right) $, which holds when $V_{i}$ has an increasing hazard rate, to prove incentive compatibility in the design of optimal auctions. In M2(iii), the lower bound on $W$ rules out point mass of the seller setting the reserve at $\underline{v}$, and the seller would be better off not selling the object if $W>\overline{v}$. M2(iv) assumes the utility function is smooth and allows the seller to be risk-averse as well as risk-neutral. The monotonicity and concavity of $U\left( \cdot \right) $\ in M2(iv) are standard assumptions in the risk aversion literature where the differentiability conditions are imposed to facilitate analytical tractability -- for example, it ensures Arrow-Pratt measure of risk aversion to be defined.
For two utility functions that are strictly increasing and weakly concave, $U_{1}(\cdot )$\ and $U_{2}(\cdot )$, we say that $U_{2}(\cdot )$ represents a strictly more risk-averse preference in the Arrow-Pratt sense if there exists a real-valued function $\zeta (\cdot )$ that is twice continuously differentiable with $\zeta ^{\left( 1\right) }\left( \cdot \right) >0$ and $\zeta ^{\left( 2\right) }\left( \cdot \right) <0$ such that $U_{2}(\cdot )=\zeta (U_{1}(\cdot ))$.\footnote{ It is not necessary to define Arrow-Pratt risk aversion with differentiable $\zeta (\cdot )$, and strict monotonicity and concavity will suffice. However, similarly to how we consider a smooth utility function in M2(iv), differentiability is used to facilitate analytical tractability.} For differentiable utility functions, this formulation is equivalent to saying that their Arrow-Pratt risk aversions satisfy $-\frac{U_{2}^{\left( 2\right) }\left( v\right) }{U_{2}^{\left( 1\right) }\left( v\right) } > -\frac{U_{1}^{\left( 2\right) }\left( v\right) }{U_{1}^{\left( 1\right) }\left( v\right) } $ for all $v$. The degree of risk aversion can take on a more compact form when utility functions belong to some parametric families. For example, when constant ARA (CARA) or RRA (CRRA) functions are used, ranking of Arrow-Pratt risk aversion between preferences is simply determined by the risk aversion parameters. For more background materials on Arrow-Pratt risk aversion, we refer the reader to Chapter 6.D in mas1995microeconomic.
Under Assumption M2, Proposition 2(i) below says that the optimal reserve price is characterized by the first-order condition obtained from differentiating ((ref)), and Proposition 2(ii) says that the optimal reserve price decreases with the seller's risk aversion in the Arrow-Pratt sense.
Proposition 2. Suppose Assumption M2 holds, then:
(i) For any $\left( X,W\right) $, the optimal reserve price exists and is uniquely determined by $r^{\ast }\in \mathcal{V}$ that satisfies
(ii) The optimal reserve price decreases with the seller's risk aversion in the Arrow-Pratt sense.
The right hand side of equation ((ref)) is the partial derivative of ((ref)) with respect to $r$\ evaluated at $r^{\ast }$. By putting $R$ in place of $r^{\ast }$\ in ((ref))\ and use the identity that $R=V\left( \alpha _{R}|X\right) $, Proposition 2 confirms that $R$ is the unique maximizer of $\widetilde{\Pi }\left( \cdot ,X,W\right) $\ , and it satisfies
Later on, we will assume the shape of $U\left( \cdot \right) $\ is known up to the risk aversion parameter. For example, in our empirical application, we use the CRRA utility function:
defined for $v>0$\ and $\theta \in \mathbb{R}$, where higher $\theta $\ represents a higher degree of risk aversion. Under the CRRA specification, M2(iv) holds for $\theta \geq 0$. We show in the proof of Lemma 4 how Proposition 2 can be used to identify an Arrow-Pratt risk aversion parameter.
Given data on winning bids and auction characteristics, this section proposes estimators for the quantile function of private values and related functions under the linear quantile specification. Using Proposition 1, these estimators are obtained through appropriate transformations of the bid quantile function, which we estimate using the AQR method of GG22.
Abstracting from the auction interpretation, our AQR estimator for bids can also be viewed as a general-purpose quantile regression estimator, whose inverse provides an estimator of the conditional CDF. We establish pointwise asymptotic properties for the AQR estimators of these functions, as well as their uniform convergence rates, in Lemmas 1 and 3, respectively.
The statistical properties of the estimators for the distribution of private values are stated as propositions. We note that only convergence rates for these estimators are required to derive the large-sample properties of the risk-aversion estimator in Section 4. Moreover, we use only bid data in this section, as seller-specific variables---namely the reserve price and seller's value of the auctioned object---are not used in the quantile estimation.
We begin with some assumptions.
Assumption Q.
(i) The auction variables $\left\{ \left( B_{l},X_{l}\right) \right\} _{l=1}^{L}$\ are a random sample. For some $F\left( \cdot |X\right) $\ and $G\left( \cdot |X\right) $\ that satisfy Assumptions M1 and M2(i), $B_{l}$\ takes value in $\mathcal{V}$\ and have conditional CDF $G\left( \cdot |X\right) $\textit{. }$X_{l}$\textit{\ takes value in }$\mathcal{X}\subseteq R^{D}$\textit{\ such that }$\mathcal{X}$\textit{\ is compact,}$\ $\textit{and the eigenvalues of }$E\left[ X_{l}X_{l}^{\top } \right] $\textit{\ are bounded away from zero and infinity. }
(ii) Let $V\left( \alpha |X\right) =F^{-1}\left( \alpha |X\right) $ \ and $V\left( \alpha |X\right) =X_{1}^{\top }\gamma \left( \alpha \right) =\gamma _{0}\left( \alpha \right) +X^{\top }\gamma _{1}\left( \alpha \right) $\ where $\gamma \left( \cdot \right) =\left[ \gamma _{0}\left( \cdot \right) ,\gamma _{1}^{\top }\left( \cdot \right) \right] ^{\top }$\ is $\left( s+1\right) -$times continuously differentiable over $\left[ 0,1\right] $\ for some $s\geq 1$ \textit{\ and }$X_{1}=\left[ 1,X^{\top }\right] ^{\top }$\textit{. }
(iii) The kernel function $K\left( \cdot \right) $\ is symmetric, continuously differentiable, and non-negative function on its support, $\left( -1,1\right) $. The bandwidth $h$\ is positive and satisfy $h=o\left( 1\right) $\ and $\log ^{2}L=o\left( Lh\right) $\ as $L\rightarrow \infty $\textit{.}
Q(i) imposes standard regularity conditions and correct model specification. In Q(ii), we assume that the quantile function of private values is linear in covariates and satisfies standard smoothness conditions. Q(iii) specifies the class of kernel functions and the conditions imposed on the bandwidth.
It is instructive to compare our assumptions with those found in Section 5.1 of GG22. First, we simplify their setting slightly by considering repeated auctions with a fixed number of bidders, while they allow the number of bidders to vary exogenously. It is straightforward for us to include this feature with more notation. Our Q(i) and Q(ii) are analogous to their Assumption A and Assumption S respectively. Our Q(iii) is the same with their Assumption H other than they require $\log ^{2}L=o\left( Lh^{2}\right) $\ as $L\rightarrow \infty $. GG22$\ $imposes a more stringent requirement on the bandwidth than us, because we are studying different auction models. Specifically, we are estimating the quantile function of private value using winning bids from ascending auctions, whereas GG22 uses individual bids from first-price auctions. This matters, as the quantile function of the bidder's private value depends on both the quantile function of the optimal bid and its derivative in a first price auction\footnote{ If $B\left( \cdot \right) $\ and $B^{\left( 1\right) }\left( \cdot \right) $ \ respectively were to denote the quantile function of the optimal first price bid. Then it can be shown that:
see equations (2.4)\ and (2.5) in GG22.}, which contrasts with equation ((ref)) where the quantile functions of the bidder's private value and winning bids have the same degree of smoothness. Thus, GG22 cannot have the bandwidth decay too rapidly, since the variance of the derivative of quantile estimator is inversely proportional to the bandwidth while the bandwidth only appears in the higher order terms for the variance of the level quantile estimator. We will impose the same bandwidth condition as their Assumption H when we provide the convergence rates of the derivative of the quantile function.
The winning bid's quantile function shares the linear quantile specification as the private value's quantile function under Assumption Q. This follows from combining Q(ii) with the identity in ((ref)), which gives:
recalling that $X_{1}=\left[ 1,X^{\top }\right] ^{\top }$ so $\beta \left( \cdot \right) =\left[ \beta _{0}\left( \cdot \right) ,\beta _{1}^{\top }\left( \cdot \right) \right] ^{\top }$.\ Since $\gamma \left( \cdot \right) $\ is a composite function of $\beta \left( \cdot \right) $ and $\phi \left( \cdot \right) $, estimating $V\left( \cdot \right) $\ amounts to estimating $ \beta \left( \cdot \right) $.
To motivate the AQR estimator for estimating $\beta \left( \cdot \right) $, first recall that
where $\rho _{\alpha }\left( t\right) =t\left( \alpha -\mathbf{1}\left[ t<0 \right] \right) $ is the check function. The minimizer of the sample counterpart to the expectation above is the classic quantile regression estimator of koenker1978regression. This estimator is known to not perform well when $\alpha $ is close to $0$ or $1$. Moreover, it is piecewise linear (in $\alpha $) and different estimators for quantile derivatives are required that complicates analysis of statistics that involve both quantile level and its derivatives estimates.
Instead, let us consider $B\left( \cdot |X\right) $ over $\left[ \alpha -h,\alpha +h\right] \cap \left[ 0,1\right] $, $\left\{ B\left( \tau |X\right) ,\tau \in \left[ \alpha -h,\alpha +h\right] \cap \left[ 0,1\right] \right\} $, which minimizes
over any functions $q\left( \cdot ,X\right) $ since $K\left( \cdot \right) $ is non-negative. In the same spirit as a local polynomial estimator (e.g., fan1996local), the AQR approach estimates the quantile coefficients and their derivatives simultaneously. Specifically, with $B\left( \alpha +th|x\right) =\sum\limits_{j=0}^{s}x_{1}^{\top }\beta ^{\left( j\right) }\left( \alpha \right) \frac{\left( th\right) ^{j}}{j!}+O\left( h^{s+1}\right) $ in mind, consider the following objective function,
where $b \in \mathbb{R}^{\left( s+1\right) \left( D+1\right) }$ and $P\left( x,t\right) =\pi \left( t\right) \otimes x_{1}$ with $\pi \left( t\right) =\left[ 1,t,\ldots ,\frac{t^{s}}{s!}\right] ^{\top }$ and $ x_{1}=\left[ 1,x^{\top }\right] ^{\top }$.\footnote{ There is a subtle difference between our objective function relative to GG22 here. Despite of us both assuming $\left( s+1\right)-$ times continuous differentiability of $V\left( \cdot \right) $, we make a polynomial approximation up to the $s-$th power term while GG22 goes up to the $\left( s+1\right)-$th power term. This is because the quantile function of the optimal first price auction bid has one more derivative than the quantile function of the private value---see equation (2.4) in GG22, which is given in the previous footnote.}
Let $b\left( \alpha \right) =\left[ \beta \left( \alpha \right) ^{\top },\ldots ,\beta ^{\left( s\right) }\left( \alpha \right) ^{\top }\right] ^{\top }\in \mathbb{R}^{\left( s+1\right) \left( D+1\right) }$, so that $ P\left( x,th\right) ^{\top }b\left( \alpha \right) =\sum\limits_{j=0}^{s}x_{1}^{\top }\beta ^{\left( j\right) }\left( \alpha \right) \frac{\left( th\right) ^{j}}{j!}$. Our estimator of $b\left( \alpha \right) $\ is $\widehat{b}\left( \alpha \right) =\arg \min_{b}\widehat{ \mathcal{R}}\left( b;\alpha \right) $. We estimate $\beta \left( \alpha \right) $ by $\mathsf{S}_{0}\widehat{b}\left( \alpha \right) $, where $ \mathsf{S}_{0}=S_{0}\otimes \mathrm{I}_{D+1}$ with $S_{0}=\left[ 1,\ldots ,0 \right] \in \mathbb{R}^{s+1}$, so that
is the estimator of $B\left( \alpha |x\right) $. We then estimate $V\left( \alpha |x\right) $ by using equation ((ref)) in Proposition 1:
Since $\widehat{V}\left( \alpha |x\right) $ inherits the properties of $\widehat{B}\left( \alpha |x\right) $, we first provide the statistical properties of the latter as a lemma. Moreover, Lemma 1 contains pointwise properties for an AQR estimator, which is not given in GG22, as they only provide the uniform convergence rate---see Theorem D.1 in their paper.
Lemma 1. Suppose Assumption Q holds, there exists $\left( J_{B}\left( \alpha ,x\right) ,J_{S}\left( \alpha ,x\right) ,J_{R}\left( \alpha ,x\right) \right) $ such that for all $\alpha \in \left( 0,1\right) $ and $x\in \mathcal{X}$:
for $\underline{t}_{\alpha ,h}=-\min \left( 1,\frac{\alpha }{h}\right) $, $ \overline{t}_{\alpha ,h}=\max \left( 1,\frac{1-\alpha }{h}\right) $,
Moreover,
The Appendix gives expressions for $\left( J_{B}\left( \alpha ,x\right) ,J_{S}\left( \alpha ,x\right) ,J_{R}\left( \alpha ,x\right) \right) $ in equations ((ref)) to ((ref)). These terms respectively represent the bias, leading stochastic term, and remainder term of $\widehat{B}\left( \alpha |x\right) -B\left( \alpha |x\right) $. Note that the limiting distribution of $\sqrt{L} J_{S}\left( \alpha ,x\right) $ is the same as the standard quantile regression estimator's without smoothing (for example, see Chapter 4 of koenker2005quantile), so that the AQR estimator has the same first order asymptotic property as the Koenker and Bassett's estimator when $Lh^{2\left(s+1\right) }=o\left( 1\right) $.
Since $\widehat{V}\left( \alpha |x\right) $ is just a composite function of $ \widehat{B}\left( \cdot \right) $ and a deterministic function $\phi \left( \cdot \right) $, its statistical properties follow directly from Lemma 1.
Proposition 3. Suppose Assumption Q holds, for all $\alpha \in \left( 0,1\right) $ and $x\in \mathcal{X}$:
for the same functions $\left( J_{B}\left( \alpha ,x\right) ,J_{S}\left( \alpha ,x\right) ,J_{R}\left( \alpha ,x\right) \right) $ as in Lemma 1. Moreover,
To prepare for the estimation of the risk parameter in the next section, we need to establish convergence rates for other functions related to the quantile. Let us re-write the first-order condition in ((ref)) and replace $\alpha _{R}$\ with $R=V^{-1}\left( \alpha _{R}|X\right) $, which gives:
Using the above equation for estimation requires estimators for $\left( V^{\left( 1\right) }\left( \cdot \right) ,V^{-1}\left( \cdot \right) \right) $. Since the convergence rates for an estimator of $V^{\left( j\right) }\left( \cdot \right) $\ is useful for semiparametric estimation, we begin by providing convergence rates for them. In what follows, we provide the relations between $\left( V^{\left( j\right) }\left( \cdot \right) ,V^{-1}\left( \cdot \right) \right) $ and $\left( B^{\left( j\right) }\left( \cdot \right) ,B^{-1}\left( \cdot \right) \right) $. Then, we define our estimators as transformations of the AQR estimators of $\left( B^{\left( j\right) }\left( \cdot \right) ,B^{-1}\left( \cdot \right) \right) $ and give their rates of convergence.
We can differentiate ((ref)) repeatedly to obtain the relationship between the derivatives of the winning bid's and private value's quantile function. While the relations are visually compact for lower order derivatives, such as
which hold a.s. for all $\alpha \in \left[ 0,1\right] $, they get cumbersome quickly for higher derivatives. Since we are only interested in the uniform convergence rates rather than pointwise properties here, it therefore suffices to know that:
where $\mathcal{J}_{kj}$\ is a known continuous function that is uniformly bounded over $\left[ 0,1\right] $\ for all $k$ and $j$.\footnote{ This can be obtained by applying the Fa\`{a} di Bruno's formula\ for computing chain rule to higher derivatives, for the $j$-th derivative, and $ \mathcal{J}_{kj}\left( \cdot \right) $ is the exponential Bell polynomial.}
The AQR approach readily estimates $B^{\left( j\right) }(\alpha |x)$ for $j=1,\ldots ,s$, which gives
with $\mathsf{S}_{j}=S_{j}\otimes \mathrm{I}_{D+1}$ and $S_{j}$ is a row vector of size $\left( s+1\right) $\ consists of $0$'s in every component other than $1$ in its $\left( j+1\right) $-th entry. Lemma 2 gives the convergence rate for the AQR estimator of $\widehat{B} ^{\left( j\right) }\left( \cdot \right) $.
Lemma 2. Suppose Assumption Q holds and $\lim_{L\rightarrow \infty } \frac{\log ^{2}L}{Lh^{2}}=0$, then for $j=1,2,\ldots ,s$:
Notice that Lemma 2 imposes the same bandwidth condition as GG22, which we alluded earlier. Indeed, GG22\ has given the same convergence rate as the above when $j=1$, see equation (5.7) in their Theorem 2.\footnote{ The order of their bias is written as $h^{s+1}$, which is a result of their bid's quantile function having one more derivative than ours.} The component of the convergence rate that corresponds to the stochastic term is $\sqrt{ \frac{\log L}{Lh^{2j-1}}}$, which coincides with the usual rates of the $ \left( j-1\right) $-th derivative of a kernel density estimator; this finding is reassuring given the identity between the density and derivative of the quantile. The worsening of the bias rate with higher derivatives also mirrors standard local polynomial estimators. Since $\widehat{V}^{\left( j\right) }\left( \cdot \right) $ is a smooth mapping from $\left\{ \widehat{B}^{\left( k\right) }\left( \cdot \right) \right\} _{k=1}^{j}$, its rate of convergence then follows that of $\widehat{B}^{\left( j\right) }\left( \cdot \right) $.
Proposition 4. Suppose Assumption Q holds and $\lim_{L\rightarrow \infty }\frac{\log ^{2}L}{Lh^{2}}=0$, then for $j=1,2,\ldots ,s$:
The relation between the inverse of private value and winning bid quantiles is given by the inverse of composite functions formula applied to ((ref)):
We define our estimator for $V^{-1}\left( \cdot \right) $\ as follows,
The statistical properties of $\widehat{V}^{-1}(t|x)$\ can be analyzed through two applications of the Continuous Mapping Theorem.
First, $\widehat{B}^{-1}\left( t|x\right) $\ can be studied as the inverse of $\widehat{B}\left( t|x\right) $ through the map $\Psi :\ell ^{\infty }\left( \left[ 0,1\right] \right) \mapsto \ell ^{\infty }\left( \mathcal{V} \right) $, such that $\Psi \left( \pi \right) \left( t\right) =\inf \left\{ a\in \left[ 0,1\right] :\pi \left( a\right) \geq t\right\} $ for $t\in \mathcal{V}$ and $\pi \left( \cdot \right) \in \ell ^{\infty }\left( \left[ 0,1\right] \right) $. We use $\ell ^{\infty }\left( \mathcal{A}\right) $ \ to denote the space of bounded functions on $\mathcal{A}\subseteq \mathbb{R }$. It can be shown that the Hadamard derivative of $\Psi $\ exists and linearization methods apply (van1996weak, Lemma 3.10.21). Particularly, the leading term in $\widehat{B} ^{-1}\left( t|x\right) -B^{-1}\left( t|x\right) $ is,
when $B^{\left( 1\right) }\left( B^{-1}\left( t|x\right) |x\right) >0$.
Lemma 3 gives the statistical properties of $\widehat{B}^{-1}\left( t|x\right) $ for $t$ in the interior of $\mathcal{V}$, denoted by $int\left( \mathcal{V}\right) $. This result may be of independent interest, as $\widehat{B}^{-1}\left( t|x\right) $\ is a flexible yet low-dimensional general estimator for the conditional CDF.
Lemma 3. Suppose Assumption Q holds, for all $t\in int\left( \mathcal{V}\right) $ and $x\in \mathcal{X}$:
for the same $\mathsf{Bias}_{h}\left( \cdot \right) $ and $\Sigma \left( \cdot \right) $\ as in Lemma 1. Moreover,
Second, we apply $\phi ^{-1}\left( \cdot \right) $ to $\widehat{B} ^{-1}\left( \cdot \right) $. There is a potential complication when deriving uniform convergence in this step, as $\phi ^{-1}\left( \cdot \right) $ is continuously differentiable on $\left( 0,1\right) $ but not at the boundaries. This can be seen from inspecting the derivative of $\phi ^{-1}\left( \cdot \right) $, which is $1/\phi ^{\left( 1\right) }\left( \phi ^{-1}\left( \cdot \right) \right) $, as we have $\phi ^{\left( 1\right) }\left( 0\right) =0$ when $I>2$ and $\phi ^{\left( 1\right) }\left( 1\right) $ is $0$ for all $I$. It should be noted too that the bias and variance of $ \widehat{B}^{-1}\left( \cdot \right) $\ go to zero as $t\rightarrow \underline{v}$, and also $t\rightarrow \overline{v}$\ for the variance. These faster convergence rates may mitigate potential irregularity issues for the de-meaned component of $\widehat{V}^{-1}\left( \cdot \right) $ as well as the lower boundary bias. Nevertheless, we do not need these aspects to derive the large sample properties of our risk aversion parameter in the next section, and a comprehensive study on uniform properties of such transformed AQR estimator is beyond the scope of this paper.
The next proposition gives the uniform convergence rate for $\widehat{V} ^{-1}\left( \cdot \right) $\ on an inner interval of $\mathcal{V}$, denoted by $\mathcal{V}_{\delta }$ that is defined as $\left[ \underline{v}+\delta , \overline{v}-\delta \right] $ for $\delta \in \left( 0,\left( \overline{v}- \underline{v}\right) /2\right) $.
Proposition 5. Suppose Assumption Q holds, then
We now assume the utility function takes a parametric form: $\theta \mapsto U_{\theta }\left( \cdot \right) $, for some $\theta \in \Theta \subset \mathbb{R}$ that represents an Arrow-Pratt measure of risk aversion. We can then construct an objective function for estimating the risk parameter from the first-order condition in ((ref)). To do this, let us use $ \mathcal{D}_{1}\left( \mathcal{A_{\delta }}\times \mathcal{X}\right) $\ and $ \mathcal{D}_{2}\left( \mathcal{V_{\delta }}\times \mathcal{X}\right) $\ to denote classes of functions whose images are $\mathcal{V}_{\delta }$\ and $ \mathcal{A}_{\delta }$\ respectively. We denote candidates for the derivative and inverse of the conditional quantile function of $V_{l}$ given $X_{l}$ by $\psi _{1}\left( \cdot \right) \in \mathcal{D}_{1}\left( \mathcal{ A_{\delta }}\times \mathcal{X}\right) $ and $\psi _{2}\left( \cdot \right) \in \mathcal{D}_{2}\left( \mathcal{V_{\delta }}\times \mathcal{X}\right) $ respectively. Here, we use $\mathcal{V}_{\delta }=\left[ \underline{v} +\delta ,\overline{v}-\delta \right] $ and $\mathcal{A}_{\delta }=\left[ \delta ,1-\delta \right] $ for small $\delta $. We restrict the support of the quantile and valuation for the reasons discussed at the end of Section 3. Note that $\delta $\ in\ $\mathcal{V}_{\delta }$ and $\mathcal{A}_{\delta }$ can generally be different. Moreover, the supports of (bidder's and seller's) valuation (and the reserve price) can depend on $X$. Incorperating these is conceptually straightforward. We forego the more general notations for simplicity of presentation.
Consider the following real value function $q\left( z,\theta ,\psi \right) $ defined as follows:
where $z=\left( w,r,x\right) \in \mathcal{Z}=\mathcal{W}\times \mathcal{V} _{\delta }\times \mathcal{X}$, $\theta \in \Theta $, and $\psi \left( \cdot \right) =\left( \psi _{1}\left( \cdot \right) ,\psi _{2}\left( \cdot \right) \right) \in \mathcal{D}_{1}\left( \mathcal{A_{\delta }}\times \mathcal{X} \right) \times \mathcal{D}_{2}\left( \mathcal{V_{\delta }}\times \mathcal{X} \right) $. Henceforth, we compress the arguments of functions that are parameters, i.e., $\psi \left( \cdot \right) $ to $\psi $, for notational brevity.
Our structural assumption requires that the first order condition in ((ref)) coincides with $q\left( z,\theta _{0},\psi _{0}\right) $ for some $\left( \theta _{0},\psi _{0}\right) $ for all $z$\ that is consistent with the auction model describe in Section 2. This is suggestive for a minimum distance type objective function for estimating $\theta _{0}$.
Let $P_{Z}$ denote a probability distribution of $Z=\left( W,R,X\right) $ and suppose $\left\{ Z_{l}\right\} _{l=1}^{L}$\ is a random sample drawn from it. We define,
Given our usage of empirical process methods to proving the asymptotic results, we write $Q$\ as an integral, making clear that only $Z$ is being integrated out, so that $Q\left( \theta ,\psi \right) $ is a random variable if either $\theta \ $or $\psi $\ is random. We add that, more precisely, $ P_{Z}$\ can be understood as a conditional distribution for $R\in \mathcal{V} _{\delta }$ for the purpose of the proofs, although there is no data truncation in practice. Related to the latter point, we emphasize that, we are not at risk of identification loss with our minimum distance approach by working on $\left( \mathcal{A}_{\delta },\mathcal{V}_{\delta }\right) $\ instead of $\left( \left[ 0,1\right] ,\mathcal{V}\right) $, which should be contrasted with choosing moments in a conditional moment model (e.g., see dominguez2004consistent), as a single value of $z\in \mathcal{Z}$\ identifies $\theta _{0}$ via ((ref)).
Our estimation problem here is a semiparametric one, as we are interested in the finite dimensional parameter $\theta _{0}$\ in the presence of infinite dimensional nuisance functions. We denote the estimator for the latter by $ \widehat{\psi }$, which consists of $\widehat{\psi }_{1}=\widehat{V}^{\left( 1\right) }$ and $\widehat{\psi }_{2}=\widehat{V}^{-1}$, respectively defined as in 3.3.1 and 3.3.2 using a kernel function that satisfies Assumption Q(iii). We then define our estimator for $\theta _{0}$, denoted by $\widehat{ \theta }$, to be the $\arg\min_{\theta \in \Theta} \left\lvert Q_L\bigl(\theta,\widehat{\psi}\bigr) \right\rvert$.
We impose the following conditions.
Assumption S1.
(i) $q\left( Z,\theta _{0},\psi _{0}\right) =0$\ $P_{Z}$ -a.s. for some $\theta _{0}\in \Theta $, $\psi _{10}\in \mathcal{D}_{1}\left( \mathcal{A_{\delta }}\times \mathcal{X}\right) $ , and $\psi _{20}\in \mathcal{D}_{2}\left( \mathcal{ V_{\delta }}\times \mathcal{X}\right) $\textit{; }
(ii) The auction variables $\left\{ \left( B_{l},Z_{l}\right) \right\} _{l=1}^{L}$\ are a random sample such that $Z_{l}\sim P_{Z} $, and the distribution of $\left( B_{l},Z_{l}\right) $ \ satisfies conditions in Assumptions M1, M2(i), M2(ii), M2(iii), and Q(i);
(iii) $\Theta $\ is compact, $U_{\theta }$\ is twice continuously differentiable with $U_{\theta }^{\left( 1\right) }>0$ \ and $U_{\theta }^{\left( 2\right) }\leq 0$\ on $\mathcal{ V}_{\delta }$\ for all $\theta \in \Theta $\textit{\ such that: (a) for any }$\theta ^{\prime }>\theta $\textit{, there exists }$\zeta $ \textit{ such that }$\zeta ^{\left( 1\right) }>0$\textit{,} $\zeta ^{\left( 2\right) }<0$\textit{\ and }$U_{\theta ^{\prime }}=\zeta (U_{\theta })$\textit{; and (b) }$\sup_{\theta \in \Theta }E\left[ \left\vert U_{\theta }^{\left( j\right) }\left( R\right) \right\vert \right] <\infty $ for $j=0,1,2$\textit{ ;}
(iv) $\mathcal{D}_{1}\left( \mathcal{A_{\delta }}\times \mathcal{X} \right) =\{$ $\frac{\partial }{\partial \alpha }\nu \left( \phi \left( \alpha \right) ,x\right) $\ for $\nu \in \mathcal{D}_{0}\left( \mathcal{A_{\delta }}\times \mathcal{X}\right) $\ $\}$\ and $\mathcal{D}_{2}\left( \mathcal{V_{\delta }}\times \mathcal{X}\right) =\{$ \ $\left( t,x\right) \mapsto \nu _{x}^{-1}\left( t\right) $ \ for $\left( t,x\right) \in \mathcal{V}_{\delta }\times \mathcal{X}$ \textit{\ where }$\nu _{x}\left( \alpha \right) =\nu \left( \phi \left( \alpha \right) ,x\right) $\textit{\ for }$\nu \in \mathcal{D}_{0}\left( \mathcal{A_{\delta }}\times \mathcal{X}\right) $\textit{\ }$\}$\textit{\ where }$\mathcal{D}_{0}\left( \mathcal{A_{\delta }}\times \mathcal{X}\right) =\{$\textit{\ }$x_{1}^{\top }\mu \left( \alpha \right) $\textit{\ for }$x\in \mathcal{X}$\textit{\ and }$\mu :\mathcal{A}_{\delta }\rightarrow \mathbb{R}$ \textit{\ is }$\left( s+1\right) $-\textit{times continuously differentiable for some }$s\geq 1$ \textit{such that: (a) }$x_{1}^{\top }\mu \left( \alpha \right) $\textit{\ takes value in }$\mathcal{V}_{\delta }$\textit{\ and is strictly increasing in }$\alpha $\textit{; (b) }$x_{1}^{\top }\mu ^{\left( 1\right) }\left( \alpha \right) $\ \textit{is bounded away from zero and from infinity uniformly; and (c) }$x_{1}^{\top }\mu ^{\left( 2\right) }\left( \alpha \right) $\textit{\ is bounded away from infinity uniformly\ }$ \}$\textit{, and the }$\phi \left( \alpha \right) $\textit{-th} \textit{ quantile function of }$B_{l}$\textit{\ conditional on }$X_{l}$ for $\phi \left( \alpha \right) \in \mathcal{A}_{\delta }$\textit{\ lies in }$\mathcal{ D}_{0}\left( \mathcal{A}_{\delta }\times \mathcal{X}\right) $.
Assumption S1 consists of structural assumptions and regularity conditions on the variables in the model. The condition $q\left( Z,\theta _{0},\psi _{0}\right) =0$ in S1(i) assumes correct model specification and, together with S1(ii), they imply the distribution of the data can be rationalized by the ascending auction model described in Section 2. Particularly, it implies that\ $\psi _{10}\left( \alpha ,x\right) =V^{\left( 1\right) }\left( \alpha |x\right) =\phi ^{\left( 1\right) }\left( \alpha \right) B^{\left( 1\right) }\left( \phi \left( \alpha \right) |x\right) $ for $\left( \alpha ,x\right) \in \mathcal{A}_{\delta }\times \mathcal{X}$\ and $\psi _{20}\left( t,x\right) =V^{-1}\left( t|x\right) =\phi ^{-1}\left( t\right) B^{-1}\left( \phi \left( t\right) |x\right) $ for $\left( t,x\right) \in \mathcal{V}_{\delta }\times \mathcal{X}$.
S1(iii) imposes a parametric assumption on the utility function that satisfies M2(iv). Part (a) gives an interpretation for $\theta $\ to be an Arrow--Pratt coefficient where higher $\theta $\ means higher degree of risk aversion in the Arrow--Pratt sense as described in Section 2.2. For example, the risk parameters in CARA and CRRA utility functions satisfy this condition\ (ross1981some). Part (b) is a regularity condition requiring uniform squared integrability. In our application we use the CRRA utility function, which is defined in ((ref)). It should be noted that such utility function is defined for all $\theta \in \mathbb{R}$, where $\theta >0$, $\theta=0,$ and $\theta <0$ represent risk-averse, risk-neutral, and risk-loving preferences respectively. Under the CRRA preference, S1(iii) requires $\theta \geq 0$. The sole purpose of this is to ensure we can apply Proposition 2, which we use to prove Lemma 4 that shows $\theta _{0}$ is identified as the minimizer of $Q$ under primitive conditions. Our estimation procedure does not restrict the parameter space to be non-negative. Importantly, suppose the implication of Lemma 4 holds as a high-level identification condition, the asymptotic theory for our estimator applies for $\theta _{0}<0$ without any modification.\footnote{ Our simulation study does not suggest any identification issue when $ \theta _{0}<0$, and the estimator in the risk-loving case behaves in the same was as the risk-neutral and risk-averse cases qualitatively. These additional simulation results are available upon request.} We can therefore abstract away from the inference issues that arise when parameter is on the boundary such as those discussed in andrews2001testing. Note also that the parametric model can be enriched by embedding observables in the risk measure, for example, by replacing $\theta$ with a linear index of the seller's or auction characteristics.
S1(iv) ensures $\mathcal{D}_{1}\left( \mathcal{A}_{\delta }\times \mathcal{X} \right) $\ and $\mathcal{D}_{2}\left( \mathcal{V_{\delta }}\times \mathcal{X} \right) $\ are the correct classes of functions containing candidates of derivative and inverse of quantile functions respectively. These classes of functions are derived from $\mathcal{D}_{0}\left( \mathcal{A_{\delta }} \times \mathcal{X}\right) $ can represent quantile functions that are linear in the regressors with additional regularity conditions: Part (a)\ ensures quantile inverses (i.e., CDFs) exist with image in $\mathcal{A}_{\delta }$; Part (b)\ requires the first derivatives of quantile functions are bounded away from zero and infinity, ensuring the corresponding PDFs satisfy Assumption M2(i);\ Part (c) imposes additional condition to ensure $\mathcal{ Q}$\ is a Glivenko-Cantelli class of functions under $P_{Z}$.
We note that the uniform boundedness conditions imposed on $\mathcal{D} _{0}\left( \mathcal{A}_{\delta }\times \mathcal{X}\right) $, and subsequently inherited by $\mathcal{D}_{1}\left( \mathcal{A}_{\delta }\times \mathcal{X}\right) $\ and $\mathcal{D}_{2}\left( \mathcal{V_{\delta }}\times \mathcal{X}\right) $, are very mild in practice. This is because the true quantile function satisfies these conditions, and we have consistent estimators for the derivative and inverse of the quantile function. We therefore only need to consider $\mathcal{D}_{1}\left( \mathcal{A}_{\delta }\times \mathcal{X}\right) $ and $\mathcal{D}_{2}\left( \mathcal{V_{\delta }} \times \mathcal{X}\right) $\ that contain functions in a neighborhood of $ \left( \psi _{10},\psi _{20}\right) $. In a similar vein, the requirement for the image of functions in $\mathcal{D}_{0}\left( \mathcal{A}_{\delta }\times \mathcal{X}\right) $\ and $\mathcal{D}_{2}\left( \mathcal{V}_{\delta }\times \mathcal{X}\right) $\ to respectively be $\mathcal{V}_{\delta }$\ and $\mathcal{A}_{\delta }$\ is not restrictive for a fixed $\delta $, as our estimators for $\left( \psi _{10},\psi _{20}\right) $ converge to the true functions uniformly over any inner subset of their respective supports. Given the boundedness of functions involved, we shall use $\left\Vert \cdot \right\Vert _{\infty }$ to generically denote the sup-norm of functions over their supports.
Building on the result of Proposition 2, Lemma 4 says our population objective function has a well separated minimum at $\theta _{0}$\ when $\psi =\psi _{0}$ under S1.
Lemma 4. Suppose Assumption S1 holds, then for all $\epsilon >0$, there exists $\delta >0$ such that $\inf_{\left\vert \theta -\theta _{0}\right\vert >\epsilon }Q\left( \theta ,\psi _{0}\right) \geq Q\left( \theta _{0},\psi _{0}\right) +\delta $.
Given the well-separated minimum condition on the population objective function, it is well-known consistency of $\widehat{\theta }$\ will follow if $Q_{L}\left( \theta ,\widehat{\psi }\right) $ converges to $Q\left( \theta ,\psi _{0}\right) $ uniformly over $\Theta $ in probability (e.g., see Theorem 2.1 in newey1994large). Lemma 5 states we have the desired uniform convergence under the bandwidth conditions that ensure $ \left\Vert \widehat{\psi }_{i}-\psi _{i0}\right\Vert _{\infty }=o_{p}\left( 1\right) $ for $i=1,2$.
Lemma 5. Suppose Assumption S1 holds and the bandwidth satisfies $ h=o\left( 1\right) $ and $\log ^{2}L=o\left( Lh^{2}\right) $, then $ \sup_{\theta \in \Theta }\left\vert Q_{L}\left( \theta ,\widehat{\psi } \right) -Q\left( \theta ,\psi _{0}\right) \right\vert =o_{p}\left( 1\right) $ .
Theorem 1. Suppose Assumption S1 holds and the bandwidth satisfies $ h=o\left( 1\right) $ and $\log ^{2}L=o\left( Lh^{2}\right) $, then $\widehat{ \theta }=\theta _{0}+o_{p}\left( 1\right) $.
Our risk-aversion estimator satisfies $\frac{\partial }{\partial \theta } Q_{L}\left( \widehat{\theta },\widehat{\psi }\right) =0$. Its limiting distribution can be studied from the linearization of $\frac{\partial }{\partial \theta }Q_{L}\left( \widehat{\theta },\widehat{\psi }\right)$ around $\left( \theta _{0},\psi _{0}\right) $. We make the following additional assumptions to establish the limiting distribution of $\widehat{\theta}$.
Assumption S2.
(i) $U_{\theta }$\ is three times continuously differentiable with $\sup_{\theta \in \Theta }E\left[ \left\vert U_{\theta }^{\left( j\right) }\left( R\right) \right\vert ^{2}\right] <\infty $ for $ j=0,1,2,3$;
(ii) $\mathcal{D}_{0}\left( \mathcal{A}_{\delta }\times \mathcal{X} \right) $\ is as described in S1(iv) other than $\mu $ is $ \left( s+2\right) $-times continuously differentiable for some $ s\geq 1$ and there exists an enveloping function for $\left( \alpha ,x\right) \mapsto x_{1}^{\top }\mu ^{\left( 3\right) }\left( \alpha \right) $ \ that is $L_{2}\left( P_{Z_{0}}\right) $\textit{-integrable;\ }
(iii) $E\left[ \frac{\partial }{\partial \theta }q\left( Z_{l},\theta _{0},\psi _{0}\right) ^{2}\right] $\ is invertible.
The strengthened smoothness and moment conditions in S2(i) and S2(ii) ensure that $\mathcal{Q}$\ and the related class of functions are $P_{Z}-$Donsker and allow us to bound various moments in the proof. S2(iii) is the invertible Hessian condition.
A key step in deriving the distribution theory of our estimator involves taking the pathwise derivative of $\frac{\partial }{\partial \theta }Q_{L}\left( \theta _{0},\widehat{\psi }\right) $ at $\psi _{0}$\ in direction $\left[ \widehat{\psi }-\psi _{0}\right] $. Two main ingredients for obtaining a $\sqrt{L}-$consistent semiparametric estimator are that (i) the higher-order terms in the linearization are negligible at the $L^{-1/2}$ rate, and (ii) the leading term of the linearization is asymptotically normal. With a nuisance function estimated by kernel smoothing, the former can be achieved by appropriate bandwidth choice to control the bias. For asymptotic normality involving nuisance functions, one can obtain a parametric convergence rate by averaging nonparametric estimators, thereby increasing the convergence speed; this intuition can be made transparent via the Riesz representer of the pathwise derivative as an integral (e.g., see newey1994asymptotic and chen2003estimation). The next lemma states sufficient conditions on the bandwidth and on the integral representation of the pathwise derivative under which the estimator is $ \sqrt{L}-$ asymptotically normal.
Lemma 6. Suppose Assumption S1 and S2 hold, and the bandwidth satisfies $Lh^{4s}=o\left( 1\right) $ and $\log ^{2}L=o\left( Lh^{3}\right) $ , then the linearization of $\frac{\partial }{\partial \theta }Q\left( \theta _{0},\widehat{\psi }\right) $ at $\psi _{0}$\ in direction $\left[ \widehat{\psi }-\psi _{0}\right] $ is
where $c_{Y_{1}}$\ and $c_{Y_{2}}$\ are functions of $Z$, and $\circ $\ denotes the composition of functions. See equations ((ref)) and ((ref)) in the Appendix respectively for the explicit forms of $c_{Y_{1}}$\ and $c_{Y_{2}}$.
The integral in the display of Lemma 6 is precisely the pathwise derivative of $\frac{ \partial }{\partial \theta }Q\left( \theta _{0},\widehat{\psi }\right) $ at $ \psi _{0}$\ in direction $\left[ \widehat{\psi }-\psi _{0}\right] $. As mentioned earlier, $P_{Z}$\ only integrates out $Z$ and the integral functional is a random variable due to $\widehat{\psi }$. The bandwidth restriction in the lemma ensures the higher order term from linearizing $ \frac{\partial }{\partial \theta }Q\left( \theta _{0},\widehat{\psi }\right) $ is $o\left( L^{-1/2}\right) $. We show in the Appendix that the leading higher order term is due to $\left\Vert \widehat{\psi }_{1}^{\left( 1\right) }-\psi _{10}^{\left( 1\right) }\right\Vert _{\infty }\times \left\Vert \widehat{\psi }_{2}-\psi _{20}\right\Vert _{\infty }=O_{p}\left( \frac{\log L }{Lh^{3/2}}+h^{2s}\right) $.
It is worth noting that our bandwidth requirement of $\log ^{2}L=o\left( Lh^{3}\right) $\ coincides with the condition in Theorem 4 of GG22, which establishes asymptotic normality for a similar integral functional that contains the bid's quantile function and its derivative. This is reassuring, as they estimate their functional using AQR estimators that have the same convergence rates as $\widehat{\psi }_{20}$ and $\widehat{\psi }_{10}$. For example, when the bandwidth decays at the rate $\log L^{1/2}L^{-\varsigma }$\ for some $\varsigma >0$, Lemma 6 requires $\frac{1}{4s} <\varsigma <\frac{1}{3}$ for it to hold.
Theorem 2. Suppose Assumptions S1 and S2 hold, the bandwidth satisfies $Lh^{4s}=o\left( 1\right) $ and $\log ^{2}L=o\left( Lh^{3}\right) $ , and for the same $\sigma _{0}^{2}$ in Lemma 6,
In practice, inference on $\theta _{0}$ be performed using a nonparametric bootstrap. chen2003estimation provide high-level conditions under which the asymptotic distribution of a two-step semiparametric estimator can be consistently bootstrapped. We conjecture that our setup is amenable to such a result. Indeed, we apply this in our Monte Carlo study and find that bootstrap standard errors for $\widehat{\theta }$\ perform very well. A formal proof of bootstrap validity when AQR estimators are used to estimate nuisance functions in a general two-step semiparametric procedure is, however, beyond the scope of this paper.
In this section, we consider some finite sample properties of our AQR and semiparametric estimators proposed in the paper.
Taking inspirations from gimenes2017econometrics and GG22, the private-value quantile function is specified as follows:
The quantile functions $\gamma _{0}$, $\gamma _{1}$, $ \gamma _{2}$ are all strictly increasing, while $\gamma _{0}$ is convex, $\gamma _{1}$ is linear, and $\gamma _{2}$ is concave. The covariates $X_{1}$ and $X_{2}$ are independent and uniformly distributed on $[0,1]$. For the outside option, we let $W(X)=V(\beta |X)$, where $\beta$ is an independent draw from a uniform distribution on $[0.05,0.5]$.
We use the CRRA utility function and consider $\theta _{0}=0,0.5,1$. Under this specification, $\theta_{0} >0$ means the seller is risk-averse and $\theta_{0}=0$ means the seller is risk-neutral (and $\theta_{0} <0$ means the seller is risk-loving).
We consider $L$ auctions with $3$ bidders, where $L=250,500,1000$. In the estimation, AQR quantile functions are computed over the estimation grid of $\alpha$ taking values $0.02,0.04,...,0.98$. Following GG22, we set the AQR polynomial order to be 2 and estimate it using the Epanechnikov kernel: $K(t)=0.75(1-t^{2})\mathbf{1}(t\in \lbrack -1,1])$. Three different bandwidths are used in the simulation: $h=sL^{-1/5},sL^{-1/6},sL^{-1/7}$, where $s$ is the sample standard deviation of the winning bids. The number of replications is 1000 in all experiments.
Figure 1 collects the simulation results for the estimation of the private-value quantile function and its derivative for different $L$. The black solid line is the true function, the red dashed line is the mean of our estimator, and the red dotted lines represent its $2.5$th and $97.5$th percentiles. In all these figures, we set $X_1=X_2=0.5$. Here we only report the estimation results of $V(\alpha|X)$ and $V^{(1)}(\alpha|X)$ using $h=sL^{-1/6}$. The results using $h=sL^{-1/5}$ and $h=sL^{-1/7}$ are similar.
We can see our estimator of $V$, on average, lies very close to the true, within the $95\%$ central quantile interval, and with decreasing variance as $L$ increases. The estimator of $V^{(1)}$ shares these properties but has higher estimation error, which is what we expect from the theory. To illustrate the differences more quantitatively, we calculate the integrated mean squared error (IMSE) for $\widehat{V}$ and $\widehat{V}^{(1)}$ when $X_1=X_2=0.5$. These can be found in the Table 1.
For the statistical performance of $\widehat{\theta}$, this is tabulated in Table 2, which contains the bias (bias), median bias (mbias), standard deviation (std), bootstrap standard error (b-se), mean squared error (mse), and the scaled interquartile range (iqr) of the estimator. In particular, we use the nonparametric bootstrap to estimate the bootstrap standard error. We do this by resampling each simulated dataset with replacement and re-do the estimation procedure $99$ times. And we calculate \texttt{iqr} by dividing the interquartile range of our studentized estimates by 1.349.
The results give evidence that our $\widehat{\theta}$ is a consistent estimator for $\theta_0$, as the bias and standard deviation, and subsequently mean squared error, are decreasing with $L$. The reported iqr being close to $1$ indicates that our estimator has a normal-like tail behavior. The bootstrap standard error also approximates the standard deviation well. These comments apply to all bandwidths considered and the performances across bandwidths are comparable. Our simulation study thus supports our theoretical results and the recommendation that inference on the risk parameter can be done by using the nonparametric bootstrap.
This section applies our estimator to real estate auction data from S\ {a}o Paulo. Real estate auctions constitute a large and active market in Brazil. Our sample consists of foreclosure apartments that we webscrapped from the website of a single large auctioneer, Zukerman (https://www.zukerman.com.br), which is recognized as the largest real estate auction platform in Brazil. The sellers are typically private and public banks, private companies that provide funds for borrowers, and the Court of Justice of the State of S\ {a}o Paulo (Tribunal de Justi\c{c}a do Estado de S\ {a}o Paulo, TJ-SP).
Properties can be auctioned off for several legal reasons: (i) default on mortgage payments for more than six months; (ii) default on condominium maintenance fees; (iii) labor-related lawsuits; and (iv) other unpaid debt obligations. Auctions in categories (i) and (ii) are the most prevalent. Type (i) cases are classified as extrajudicial because the auction does not require judicial, or court, approval and are most often initiated by financial institutions when a property serving as collateral under fiduciary alienation is repossessed after borrower default. All other types generally require authorization by a court and typically consider the cases of unpaid loans, bankruptcy, or overdue condominium fees.
Our application involves auctions that were completed over two rounds. The second round takes place if the property is not sold initially, in which case the reserve price is reduced as judicial auctions typically applied a 50% discount on the reserve price and extrajudicial auctions often set the second-round reserve price to the outstanding debt. To apply our methodology, we take the reserve price in the first round to be the seller's optimal reserve price. This is motivated by the fact that the first reserve is set based on actual appraisals of the property value, while the discounted reserve price in the second round is guided by either a judicial mandate or the property debt.
All sales follow the English auction format, in which bids are placed electronically in ascending order. The auction details including property information and auction schedule are publically available. The auction platform publishes all submitted bids in real time. We note that the second rounds, on average, take place 20 days after the first round. Our sample consists only of auctions with at least two bidders. We assume that all potential bidders submitted bids, which means we observe $N$ that corresponds to the number of different bidders in each auction.\footnote{The assumptions on observing $N$ is a common one in practice, although models with endogenous bidders' entry or unknown number of bidders have been studied in the literature.}
The data we use contains 754 observations covering the period from 2017 to 2023, with the majority corresponding to auctions of type (i) and (ii). We did not collect data for cases of type (iii), as labor courts manage these auctions separately under distinct rules. The sample also excluded auctions deemed as outliers in terms of the reserve price, which we define as having the ratio of the reserve price to the size of the apartment in each auction that is larger than the 99th percentile or smaller than the 1st percentile of the sample. The price of the apartment is in Brazillian reais, and we measure it in R\$$100,000$s.\footnote{All prices are expressed in constant January 2017 R\$.} We use the size of the apartment (total area measured in sqm) as the covariate, $X$. To compute the value of the seller's outside option, $W$, we subtract from the evaluation value of the property the debt registered in its sale report. We only have these information for 341 properties of the total sample. Since we do not use these information to estimate the quantile function of bidder's private value, the quantiles are estimated using the whole sample. We then use the subsample with evaluation and debt values to estimate the seller's risk parameter. Throughout this exercise, we assume the seller has a CRRA utility function.
Our estimation procedure then takes place over two stages. First, we estimate the bidder's private value quantile function for the property. In a second step, we evaluate the seller's CRRA risk parameter. We estimate $V(\alpha|X)$ using the AQR method for $\alpha=0.01,0.02,...,0.99$.\footnote{In a very small proportion of auctions, the observed reserve price is larger than $\widehat{V}(0.99|X)$ or smaller than $\widehat{V}(0.01|X)$. These auctions are removed when estimating $\theta$ and during the model fit analysis.} We consider three different bandwidths in our estimation: $h=0.1$, $h=0.15$, and $h=0.2$. The results show that our estimation methodology is robust for different bandwidth choices. To derive the standard error and some quantile levels of $\widehat{\theta}$, we use the nonparametric bootstrap. The bootstrap size is set at 99.
We provide some descriptive statistics of our data. The variables involved are the reserve price ($R$), the winning bid ($B$), the size of the property ($X$), the outside option value ($W$), and the number of bidders in each auction ($N$). The summary statistics of the data containing the mean, standard deviation, and the quartiles are shown in Table 3.
Note that the reserve prices in the data tend to be higher than the winning bids, which is due to sales occurring in the second auction with lowered reserves. Observing bids below the reserve facilitates identification of the bidder's valuations in the same manner to auctions with a secret or hidden reserve price (elyakime1994first, andreyanov2022secret).
We provide the scatter plots of $X$ against $B$ and $R$ in Figure 2. The plots indicate a general trend that both winning bid and reserve price increase with the size of the property.
We start by presenting figures of the estimates of the private value quantile function conditioning for property size at the three quartiles for different bandwidths.
From the figures, we can see that $\widehat{V}(\alpha |X)$ is increasing in $\alpha$, slightly concave for small $\alpha $, and convex for large $\alpha $ for different $X$'s. The results using different $h$ are similar.
The estimation results for the risk-aversion parameter are given in Table 4, containing the bootstrap standard errors and the 2.5th and 97.5th percentiles.
The results are qualitatively the same for all bandwidths. The $95\%$ bootstrapped coverage does not contain zero and the studentized statistic rejects the risk neutrality assumption in favor of risk aversion at any reasonable significant level.
It is also instructive to consider the model fit under risk aversion. To do this, we simulate the winning bid and estimate the optimal reserve price using our estimated parameters and compute their CDFs to compare with the observed data. For the winning bid, we use the estimated $\widehat{V}(\alpha |X)$ to simulate the winning bid for each observed pair of ($N$, $X$) 1000 times, then combine the simulated winning bids across all different pairs of ($N,X$) in the sample to get a simulated CDF. To obtain the sample CDF, we use the empirical distribution of the observed winning bid data in the sample. The following figures show the comparison between the sample winning bid distribution and the simulated winning bid distribution using different bandwidths:
Visually, we can see that the simulated winning bid distributions fit the sample winning bid distributions very well for all bandwidths. This is complemented by the statistics in Table 5, which contains the bias (mean of simulated winning bids minus mean of sample winning bids), the percentage bias (bias divided by mean of sample winning bids), and the IMSE which is defined by
where $F_{B}(\cdot)$ is the CDF of the sample winning bids and $\widehat{F}_{B}(\cdot)$ is the CDF of the simulated winning bids. Table 5 shows that the percentage bias and the IMSE are all small, which suggest the fit of winning bid distribution is very good.
We can also construct the model implied distribution of the reserve price. We use $\widehat{V}(\alpha |X)$, $\widehat{\theta}$, and the observed ($N,X,W$) to calculate the seller's expected utility for $\alpha = 0.01,0.02,...,0.99$, then find the optimal screening level $\alpha_{R }$ as the one that maximizes the seller's expected utility. Finally, we use $\widehat{V}(\alpha_{R }|X)$ as the estimated optimal reserve price. Similar to the description above, we compare the CDF of observed reserve price and the CDF of estimated reserve price. The results are shown in the Figure 5.
Table 6 gives the bias, percentage bias, and IMSE of the estimated reserve price distribution, where the definition of bias, percentage bias, and IMSE is similar to that for the winning bid.
The figures and table presented above suggest our model generates the reserve price that fits the data well. Importantly, the results are qualitatively the same and are stable for all bandwidths considered.
As a simple counterfactual exercise, we construct CDFs of the reserve price distribution for a risk-neutral seller, which were constructed analogously as those in Figure 5. We provide these in Figure 6.
We see that the distribution of the observed reserve price is almost first order stochastic dominated by the distribution of the counterfactual reserve price when the seller is risk neutral, which implies the risk-neutral seller's reserve price tend to be higher than the risk-averse counterpart. To provide some quantitative comparisons, we calculate the average of amount increase and the percentage increase in the counterfactual reserve price for the whole sample and for three subsamples around each quartile of the apartment size: small group (with area between $20$th and $30$th percentiles), medium group (with area between $45$th and $55$th percentiles), and large group (with area between $70$th and $80$th percentiles). The results are summarized in Table 7.
Overall, the seller's reserve price would increase by 13% to 15% under the risk neutrality assumption, confirming that the seller's optimal reserve price is decreasing in the degree of risk aversion.
In this paper, we propose a framework to identify and estimate the seller’s risk-aversion parameter in ascending auctions. The model is semiparametric, with risk preferences assumed to come from a parametric utility family, while bidder valuations satisfy a linear quantile specification. We show that the risk-aversion parameter is identified under mild conditions that are commonly assumed in the theoretical literature. We then propose a two-step semiparametric estimator for it. This procedure uses the AQR approach of GG22 to estimate bidder valuation quantiles from winning bids in the first stage. We establish that the estimator is consistent and asymptotically normal under standard regularity conditions. Our estimator performs well in a simulation study. We then apply our methodology to real estate auction data in Brazil and find statistical evidence that sellers are risk-averse.
Our study makes a useful contribution to empirical auction research, as optimal auction design can depend crucially on whether the seller is risk averse or risk neutral. Moreover, any supply-side welfare analysis requires correct specification of the seller’s risk preference. Currently, empirical studies assume seller risk neutrality. Learning about seller-side primitives is, however, necessarily more demanding in terms of data requirements than learning about bidder-side primitives, although for the latter---when bid data alone are sometimes sufficient---identification and estimation still depend on the auction format, assumptions about bidders’ information structure, and data availability (e.g., all bids, bids below the reserve, or winning bids). Our application, which uses data from two-stage auction settings, is intended to serve as an illustration of the proposed methodology. More natural settings for its application include auctions with secret or hidden reserve prices, which are common in practice and for which the relevant data are observed by auction platforms and policymakers. While such data are less widely available to researchers, they have been used, for instance, in the first-price auction context (see elyakime1994first and andreyanov2022secret).
It should be noted that seller risk aversion is not the only mechanism that can rationalize low observed reserve prices. Alternative explanations include endogenous entry (levin1994equilibrium), affiliated types (levin1996optimal), and risk-averse bidders with interdependent values (hu2019low). We do not attempt to distinguish between these possibilities. To our knowledge, no existing empirical work has done so, making this an interesting direction for future studies.
Since bidders' behavior in a second-price auction is strategically equivalent to that of an ascending auction under IPV, the estimation strategy in this paper applies to second-price auctions. Our general approach to model and estimate the seller’s risk aversion through the revenue maximization condition can be applied to other auction settings. Beyond considering an alternative auction format, the IPV assumption may be relaxed. Various extensions from this framework have been proposed in the empirical auction literature, such as unobserved heterogeneity (krasnokutskaya2011identification, hu2013identification, and luo2023identification), endogenous entry (marmer2013model, gentry2014identification, chen2025identification), and interdependent values (gimenes2020nonparametric). Even within the narrower domain of AQR applications, which have thus far been applied to ascending and first-price auctions, some practical aspects will benefit from further studies. For instance, while convergence rates for AQR estimators and their corresponding optimal bandwidths have been derived, neither our paper nor GG22 provide practical guidance on bandwidth selection, and both rely on resampling methods for inference without formal justification. Despite encouraging finite-sample performance in Monte Carlo experiments and stable estimates with real data across bandwidth choices, further studies on these aspects will be useful for applied research.