Endogenous Entry in First-Price Auctions by P a i X u B . A . , Dongbe i Univers i ty of F inance and Economics , 1999 M . A . , Univers i ty of N e w B r u n s w i c k , 2002 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Doctor of Phi losophy i n T h e Facu l ty of Graduate Studies (Economics) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A (Vancouver) Ju ly , 2008 © P a i X u 2008 Abstract T h i s thesis studies the first-price auct ion models w i t h endogenous entry. In the first chapter, we propose using the nearest neighbor est imat ion tech- nique to est imate the entry cost i n the auct ion models. W e s tudy the large sample propert ies of the proposed est imator to establish sets of conditions sufficient for i ts consistency a n d asymptot ic normal i ty . To give an example, we further a p p l y the proposed method to estimate the cost of par t i c ipa - t i o n i n the M i c h i g a n Highway Procurement A u c t i o n s . O u r s tudy rejects the n u l l hypothesis of zero par t i c ipat i on costs. Based on our est imat ion result, we infer how the o p t i m a l auct ion outcomes can be real ized by using the regular po l i cy tools . W e demonstrate the improvement that the M i c h i g a n government cou ld have made on payments i f the o p t i m a l auctions h a d been employed. I n the second chapter, we propose app ly ing the s imulated non-l inear least squares ( S N L L S ) method of Laffont, Ossard a n d V u o n g (1995, L O V hereafter) [28] to estimate the d i s t r ibut i on of b idders ' pr ivate values i n the es t imat ion of a first-price auct ion mode l w i t h endogenous entry. U n l i k e the es t imat ion prob lem of L O V , however, the va luat ion d i s t r i b u t i o n of bidders is t runcated i n our prob lem, due to the presence of the entry cost. T h i s further causes the absence of continuous dif ferentiabi l i ty i n our s tat is t i ca l est ima- t i o n object ive funct ion , which was required i n L O V ' s large sample analysis. Therefore , we provide a separate analysis to s tudy the large sample behavior of the S N L L S est imator i n such setup. In the t h i r d chapter , we develop a nonparametr i c method that allows one to d i scr iminate among alternat ive models of entry i n first-price auctions. ABSTRACT Three models of entry are considered: L e v i n a n d S m i t h (1994) [31], Samuel - son (1985) [53], a n d a new model i n which the in format ion received at the entry stage is imperfect ly correlated w i t h valuations. W e show that these models impose different restrict ions on the quanti les of active bidders ' v a l - uations, and develop nonparametr ic tests of these restr ict ions. W e perform the tests us ing a dataset of highway procurement auctions i n O k l a h o m a . Depend ing on the project size, we f ind somewhat more support for the new model . Table of Contents A b s t r a c t i i Table of Contents iv List of Tables v i i List of F igures v i i i Acknowledgements i x D e d i c a t i o n x 1 N o n p a r a m e t r i c E s t i m a t i o n of the E n t r y C o s t in F i r s t - P r i c e A u c t i o n s 1 1.1 Introduct i on 1 1.2 Methodo l ogy 6 1.2.1 E s t i m a t o r of 0 ( x ) 7 1.2.2 A P l u g - i n E s t i m a t o r for a funct ion of <f>{x) 12 1.2.3 S i m u l a t i o n Results 13 1.3 A n A p p l i c a t i o n to the F i r s t - P r i c e A u c t i o n M o d e l 16 1.3.1 T h e A u c t i o n M o d e l 17 1.3.2 A p p l y i n g the E s t i m a t i o n Methodo l ogy 18 1.3.3 E m p i r i c a l A p p l i c a t i o n 20 1.4 M o r e E x a m p l e s on App l i ca t i ons 29 1.5 C o n c l u s i o n a n d Discussion 31 1.6 Tables a n d Figures 34 2 E s t i m a t i n g the F i r s t - P r i c e A u c t i o n M o d e l w i t h E n t r y : A P a r a m e t r i c A p p r o a c h 44 2.1 Introduct i on 44 2.2 Methodo logy 47 2.2.1 T h e first-price auct ion model w i t h entry 47 2.2.2 T h e s t ruc tura l econometric model 49 2.2.3 S imula ted nonlinear least squares est imator 51 2.2.4 A s y m p t o t i c P r o p e r t y of ^ 53 2.3 Conc lus i on and extensions 58 3 W h a t M o d e l for E n t r y i n F i r s t - P r i c e Auct ions? A N o n p a r a - metr ic A p p r o a c h 60 3.1 Introduct i on 60 3.2 Three models of entry and their testable restr ict ions 66 3.2.1 T h e L S and S models of entry 66 3.2.2 T h e aflShated model of entry ( A M E ) 68 3.3 Nonparametr i c identi f icat ion 71 3.4 Econometr i c implementat ion 76 3.4.1 Hypotheses 76 3.4.2 T h e d a t a generating process 79 3.4.3 E s t i m a t i o n of quantiles 80 3.4.4 C o m p a r i s o n w i t h the est imat ion method of H a i l e , H o n g a n d S h u m (2003) 84 3.4.5 Tests 85 3.5 M o n t e - C a r l o experiment 87 3.6 E m p i r i c a l appl i cat ion 90 3.7 C o n c l u d i n g remarks 96 3.8 Tables a n d Figures 99 B i b l i o g r a p h y 109 A p p e n d i c e s A Proofs for chapter 1 115 A . l Deta i l s of es t imat ion method 115 A . 2 Deta i l s of Identi f ication Issues 121 B Proofs for chapter 2 124 C Proofs for chapter 3 143 C l Deta i l s of the entry models 143 C.2 Deta i l s of the es t imat ion method 147 D Statement of C o - A u t h o r s h i p 163 List of Tables 1.1 S i m u l a t i o n Results on Coverage 35 1.2 S i m u l a t i o n Results , L o g n o r m a l ( 4 , l ) , k=T^/^ 36 1.3 B i d A n a l y s i s 37 1.4 E s t i m a t i o n Results on E n t r y Costs 38 1.5 P o l i c y Tools Towards O p t i m a l A u c t i o n s 39 3.1 Size of A M E Test 99 3.2 Size-corrected Power of the A M E Test 100 3.3 Descr ip t i on of Variables 101 3.4 L o g i t a n d O L S Regressions 102 3.5 E s t i m a t e d P r o b a b i l i t y n{N/x) of TV C o n d i t i o n a l on Pro jec t Size X 103 3.6 E s t i m a t e d P r o b a b i l i t y of B i d d i n g P{N, x) 104 3.7 E s t i m a t e d Med ians of Costs 105 3.8 E s t i m a t e d Transformed Medians of Costs 106 3.9 Test Resul ts 107 List of Figures 1.1 R a t e of P a r t i c i p a t i o n 40 1.2 P a r t i c i p a t i o n Costs 41 1.3 J / ( l - F ) for JV = 9 42 1.4 J / ( l -F)îoT N=12 43 3.1 Sample Frequencies of project sizes 108 Acknowledgements I w o u l d l ike to express my grat i tude to a l l those who gave me the pos- s ib i l i ty to complete this thesis. Especial ly , I a m deeply indebted to my supervisor Pro f . Sh in i ch i Sakata whose help, s t i m u l a t i n g suggestions and encouragement helped me i n a l l the t ime of research for a n d w r i t i n g of this thesis. I have furthermore to thank the members on my dissertat ion committee. Profs . M i c h a e l Peters , V a d i m M a r m e r , U n j y Song for their he lp , support , interest and valuable hints. Sincere thanks are also extended to my ofïicemates, colleagues, faculty and staff i n the department for their supports. L a s t , but not the least, I would like to give my special thanks to my family whose love enabled me to complete my study. Dedication To my parents. Chapter 1 Estimation of the Truncated Density Function at Its Unknown Truncation Point with Apphcation to Estimation of the Entry Cost in First-Price Auctions 1.1 Introduction I n a n economic mode l w i t h uncertainty, an e q u i l i b r i u m behavior often predicts the existence of a threshold i n a d i s t r ibut i on which represents the possible types of an i n d i v i d u a l . T h e threshold entails the point at which ind iv idua l s swi t ch from one choice to another and thus causes a t runcat ion of the d i s t r i b u t i o n . W e usual ly have no knowledge on the locat ion of this t runcat i on . However, the value of the probabi l i ty density funct ion at the threshold is often the interest of a researcher, par t i cu lar ly to discuss desirable economic pol ic ies. T o i l lustrate the s i tuat ion , we consider an independent private-value f irst- price auc t i on mode l w i t h endogenous entry. In order to part i c ipate in an auct ion , a b idder often bears various costs to prepare a b i d , such as the costs of t rave l ing to the auct ion site, and the bidders ' cognitive efforts i n the b i d d i n g process. Because these costs do not occur unless she enters the auct ion , she part ic ipates i n the auct ion if and only i f she expects profits from the auc t i on to be large enough to compensate the costs. T h e d i s t r ibut i on of bids is therefore t runcated by the cutoff introduced by the par t i c ipat ion costs. T h e m a r g i n a l part i c ipant of the auct ion right at the cutoff point must have the same expected profit as the amount of the entry costs. Therefore, the es t imat ion of the par t i c ipat i on costs amounts to evaluat ing an expected profit funct ion at this cutoff. Moreover, knowing the level of par t i c ipat i on costs is essential to design auctions for m a x i m i z i n g the seller's revenue. Ce - l ik a n d Y i l a n k a y a (2006) [8] show that the implementat ion of the op t imal auctions amounts to enforcing the o p t i m a l cutoff rules i n par t i c ipa t i on to a l l the bidders. For any given part i c ipat ion cost, the seller's prob lem is then reduced to appropr ia te ly choosing reserve price a n d / o r entry fee. In this chapter , we suggest the combinat ion of the one-sided nearest neigh- bor est imator [hereafter, N N estimator] and the extreme order statistics to estimate the t runcated univariate probab i l i ty density funct ion at its u n - k n o w n t r u n c a t i o n po ints . W e prove that the proposed est imator is consis- tent a n d asympto t i ca l l y normal ly d is t r ibuted under m i l d condit ions . W e also extend the method to cover the case i n w h i c h the parameter of inter- est is a smooth funct ion of the unknown density at the t runcat i on point a n d some other attr ibutes of the d i s t r ibut ion as is the case i n the auct ion example discussed earlier. It is possible to replace the one-sided N N estimators w i t h the one-sided version of the Rosenblat t -Paxzen class of b a n d w i d t h est imators [hereafter, b a n d w i d t h estimators] . T h e b a n d w i d t h est imators received a lot of attent ion from econometricians as a way of est imat ing density functions. In par t i cu lar , when the b a n d w i d t h est imator is appl ied i n boundary regions, the estimate is not necessarily consistent. T h i s inconsistency prob lem elicits an extensive l i terature on the correct ion of the boundary effect. N o t designed for the est imat ion of a density funct ion at the boundary po int , most of the exist ing methods focus on how to modi fy the weighting scheme i n such a way that boundary region can be automatical ly detected and adjusted. A c c o r d i n g to C h e n g et a l (1997) [10], some of the methods are quite diff icult to work w i t h . Moreover , the issue of the l i m i t i n g d i s t r ibut i on for these bias-corrected est imators is left unsett led. Bear in m i n d , however, that the one-sided a n d two-sided b a n d w i d t h estimators have quite different properties i n certain aspects. W e compare our one-sided N N estimator w i t h some typ i ca l bias-corrected b a n d w i d t h methods i n some Monte C a r l o experiments. I n the experiments , the one-sided N N estimate always has a smaller mean squared error than other b a n d w i d t h est imators . T h e stable performance of the one-sided N N est imator may reflect the local adaptive nature of the N N method , though we n a t u r a l l y expect that the results can vary, depending on the populat i on d i s t r i b u t i o n . To demonstrate usefulness of the proposed est imat ion method , we ap- p ly the m e t h o d to estimate the par t i c ipat ion costs i n M i c h i g a n Highway Procurement A u c t i o n , assuming that the auct ion is an independent private- value first-price auc t i on w i t h endogenous entry. O u r s tudy rejects the nu l l hypothesis that the par t i c ipat i on costs are zero at any reasonable significance levels. W i t h the est imated part i c ipat ion cost levels, we further investigate how to implement an o p t i m a l auct ion by employ ing only regular po l i cy tools, such as the reserve prices a n d / o r entry fees. W e find that , to m i n i m i z e the highway construct ion a n d maintenance costs, the government should subsi- dize par t i c ipants i f the reserve price is set at 110% of engineer's estimates. W e e x p l i c i t l y calculate how much the expected payment by the government wou ld change between the current auctions and the o p t i m a l auctions. O u r result suggests tha t the M i c h i g a n government cou ld have saved u p to 10- 1 5 % b y set t ing u p the auctions opt imal ly . N o t e that these numbers are not subject to the so-called extrapolat ion prob lem, despite that we have no in f o rmat i on on the valuations of the potent ia l bidders who d i d not part i c - ipate i n the auct ions , because the o p t i m a l po l i cy makes the threshold for p a r t i c i p a t i o n higher t h a n it ac tual ly was. T h u s , our work provides the first empir i ca l insights on how to use pol icy tools to a t t a i n o p t i m a l outcomes i n an auct ion w i t h endogenous entry. T h e above-mentioned appl i cat ion in our paper is re lated to a number of works i n the e m p i r i c a l auct ion l i terature. E m p i r i c a l studies on endoge- nous entry have recently received much attent ion from economists. B a j a r i and H o r t a s c u (2003) [5] empir i ca l ly investigate common-value eBay auctions through a s t r u c t u r a l approach. T h o u g h ident i fy ing p a r t i c i p a t i o n costs is not their m a i n interest, they recognize the average cost of b i d d i n g to be of a significant amount . T h i s reinforces the existence of costly prepara- t i on i n b i d d i n g a n d implies that understanding b idders ' entry decisions is information-reveal ing. Athey , L e v i n and Seira (2004) [4] s tudy entry and b i d d i n g patterns i n U . S . Forest Service t imber auct ions . T h r o u g h parame- ter iz ing the s t r u c t u r a l equations, they compare sealed-bid and open auctions to provide an assessment of b idder competitiveness. L i a n d Zheng (2005) [34] propose a semi-parametr ic Bayes ian method to j o i n t l y estimate entry and b i d d i n g models using a ful ly s t ruc tura l approach. B y us ing the highway mowing auct ion d a t a from Texas, they document e m p i r i c a l evidence of en- t r y costs i n s u b m i t t i n g bids. M o s t recently, L i (2005)[32] considers a s t ruc tura l mode l w i t h bo th entry process and b i n d i n g reserve prices i n first-price auct ions . H e proposes an M S M est imator , j o in t l y using observed bids a n d the number of ac tual b i d - ders, t o estimate the parameters i n the d is tr ibut ions of pr ivate values and the number of active bidders. T h e model is more general by considering b o t h entry costs and b ind ing reserve prices. For t rac tab i l i ty , however, L i assumes that the number of active bidders is k n o w n to bidders at the b i d - d ing stage of the game, which is on the contrary to the real i ty i n some of the interesting appl icat ions. O u r empir i ca l app l i cat ion is dist inguished from the ex is t ing empir i ca l auc- t i on l i terature b o t h i n est imat ion strategies a n d i n a u c t i o n models . F i r s t , our approach is s t r u c t u r a l and ful ly nonparametric . T h i s saves us from wor- r y i n g about the effects of misspecifications. Second, our mode l assumes the entry decisions are made ex post, that is , bidders decide whether to par - t i c ipate i n the auct ion after knowing their valuations. T o the best of our knowledge, th is work is the first empir ica l s tudy on such an auct ion model . I n terms of mode l ing costly entry, the works by Athey , L e v i n and Seira (2004)[4], L i a n d Zheng (2005)[34] and L i (2005)[32] are various versions of the L e v i n a n d S m i t h (1994)[31] model , i n which bidders make costly entry decisions ex ante (i.e., before knowing their own valuations) . T h e entry cost s tudied i n such models is sometimes referred to as "pre -part i c ipat ion invest- ment" i n the l i terature . A s the name suggests, this " investment" is more closely re lated to acquir ing information. O n the contrary, the entry cost i n our mode l occurs ex post (i.e., after bidders know their own valuations) . O u r mode l is a var iant of the Sameulson (1985) [53] entry mode l , which reflects that prepar ing the bids is a costly process.^ T h e different interpretations of costly entry process result in diff'erent equ i l i b r ium characterizations. '^ O u r Sameulson m o d e l has a pure strategy equ i l ib r ium by ho ld ing a cutoff on the va luat ion d i s t r i b u t i o n . Par t i c ipants i n auctions are allowed to have differ- ent (positive) expected profits from jo in ing the auct ion i n accordance w i t h their o w n valuat ions . O n the other hand , a mode l by L e v i n a n d S m i t h (1994) [31] suggests a m i x e d strategy equ i l ib r ium i n entry decisions due to the fact tha t no bidders have any pr ior knowledge on the va luat ion . T h i s feature of e q u i l i b r i u m restricts zero expected gains from entry to a l l bidders, w h i c h sometimes can hard ly be supported by appl icat ions . In general, one n a t u r a l l y expects the bidders to have some in format ion on the auctioned object when m a k i n g the par t i c ipat i on decision. A s i n our empir i ca l a p p l i - cat ion , v i z . the M i c h i g a n Highway Procurement A u c t i o n s , the bidders a l l have access to the detai led descriptions on the projects under auct ion . T h i s i n t u r n justif ies our choice of using Samuelson's entry mode l i n this chapter. ^Stegeman (1996) [55] provides a discussion of the distinction between the information acquisition ( " investment" ) and the costly bidding preparation. ^There m a y exist multiple equilibria if the equil ibrium strategies are asymmetric (see, for example, T a n and Y i l a n k a y a (2006)[57].) Throughout this chapter, we restrict our analysis to the unique symmetric equil ibrium. T h e rest of th is chapter is organized as follows. I n the next section, we propose a m e t h o d for est imat ing the value of a t runcated funct ion (pdf) at the t r u n c a t i o n po int and establish its large sample propert ies . W e also conduct M o n t e C a r l o experiments to study its f inite sample behavior. In Section 1.3, we app ly the proposed est imat ion methodology to est imat ing the entry cost incurred i n j o in ing the M i c h i g a n H i g h w a y Procurement A u c - tions. Moreover , we investigate how to implement the o p t i m a l auctions i n this section. Sect ion 1.4 i l lustrates some other examples i n w h i c h the pro- posed es t imat ion method can be appl ied and the last section concludes. T h e mathemat i ca l proofs are collected i n the appendix . 1.2 Methodology W e assume: A s s u m p t i o n 1.1 (Data Generating Process) (i) The data are a realization of an independently and identically dis- tributed (i.i.d.) univariate stochastic process {Xt}ten- (ii) The probability distribution of X\ has a support bounded below and a probability density function (pdf) <p that is positive and right-continuous at X, the lower bound of the support of X\. W e discuss the app l i cab i l i ty of assumptions i n the context of the f irst- price auct ion , since we are going to apply the proposed es t imat ion method to M i c h i g a n Highway Procurement Auct i ons i n this chapter. It is usual ly assumed that the bidders ' valuations are randomly d r a w n from an i.i.d. univar iate process for independent private-value auctions. T o ensure the existence a n d uniqueness of the b i d d i n g equ i l i b r ium, i t is further assumed that the va luat ion d is tr ibut ions are continuous and w i t h bounded supports . Because the e q u i l i b r i u m b idd ing is a s tr i c t ly monotonie t ransformat ion of va luat ion , the e q u i l i b r i u m bids are also i.i.d and cont inuously d i s t r ibuted . D u e to the endogenous entry, the equ ihbr ium may entai l a cutoff of b ids such that the bidder whose va luat ion is below the threshold level wou ld not submi t a b i d , whi le , on the contrary, the bidders w i t h his valuations above the cutoff w i l l a c tua l ly b i d . T h i s further implies that the bids are observed w i t h t runcat i on . T h i s t runcat i on point , i n t u r n , forms a lower bound for the observed (truncated) b i d d i s t r ibut ion . 1.2.1 E s t i m a t o r o f 0 ( x ) To est imate a t runcated univariate probabi l i ty density funct ion (pdf) at its u n k n o w n t r u n c a t i o n po int , we first look at two separate issues: (i) es t imat ion of the p d f at a given po int of t runcat ion ; and (ii) est imat ion of the t runcat i on po int . E s t i m a t o r of (f>{x) i n the case w i t h known x W e propose us ing the nearest neighbor est imat ion technique to estimate (f>{x). A s a way of es t imat ing densities, this est imat ion method , which is usual ly referred to as the fc-NN estimator i n the density es t imat ion l i terature . k is the number of closest observations taken from the sample to estimate the density at the po int of interest. T o s impl i fy the notat ion , we use to denote the t-th. order stat ist ic i n the sample {Xi,..., XT}- D u e to the t r u n c a t i o n , our observations are only f rom one side of the t r u n c a t i o n po int . T o accommodate this feature of our es t imat ion problem, we need to mod i fy the N N estimator slightly. T h e resul t ing est imator is the one-sided N N - e s t i m a t o r 0 at x: ^^^^ = w h - ^'-'^ T h e one-sided N N est imator (1.1) differs from the regular N N est imator in the denominator where regular N N estimator has 2|Xç^^ - x\ a n d i m p l i c i t l y requires s y m m e t r y on the defining interval for distance measure. W e impose restr ict ions o n k for the asymptot ics of our proposed est imator . I n what follows, we add subscript T to to stress its dependence on the sample size T a n d assume: A s s u m p t i o n 1.2 Let {kxjreN be a sequence of positive integers satisfying kr/T —> 0 as T" —> oo and one of the following: (i) kr —>• oo as T oo; (ii) fcr/loglogToo as T —> oo. A s s u m p t i o n 1.2 imposes the restrictions on the divergence rate of kr- T h e weak consistency of 4>ix) requires only A s s u m p t i o n 1.2(i), w h i c h is satisfied, for instance, i f kr — T" for some 0 < a < 1. A s s u m p t i o n 1.2(ii) is the weakest sufficient condi t ion on kx that guarantees s trong consistency of 4){x) (see, Kie fer (1972)[25]). T h e fol lowing assumpt ion is needed for the (}){x) to have the asymptot ic normal i ty property. A s s u m p t i o n 1.3 4> is right difjerentiable at x. Moreover, kr = o{T'^l^). T h e pd f (the R a d o n - N i k o d y m derivative) of a d i s t r i b u t i o n is not unique. A s s u m p t i o n 1.3 specifies that we attempt to estimate the one that is r ight - continuous at X, among a l l possible pdf ' s and requires that the p d f be right differentiable. T h e fo l lowing proposit ions state the consistency a n d asymptot i c normal i ty of the one-sided N N estimator. Let -^d denote the convergence i n d i s t r i b u - t ion . P r o p o s i t i o n 1.1 Suppose Assumptions LI and L2(i) hold. Then (p{x) —> 0 ( s ) as T oo in probability. If Assumptions LI and L2(ii) hold, then 4>{s.) "~* 0 ( 2 ) «5 T —> 0 0 a.s. P r o p o s i t i o n 1.2 Suppose Assumptions L1-L3 hold. Then R e m a r k . T h e N N method can be extended to a more general form by a d m i t t i n g weight ing functions, i.e., kernel functions i n the b a n d w i d t h estimators can also be appl ied to N N estimates. T h e asymptot ic properties of the N N est imator are not affected even if the kernel weight ing functions is employed. (See M o o r e a n d Yacke l (1976, 1977)[40][39] and Stone (1977)[56] among others.) T o ease the exposit ion, this chapter sticks to the uni form kernel weight ing functions as i n (1.1). However, a l l the results established i n the chapter r emain va l id when using other kernel weighting functions. R e m a r k . T h e N N method can also be extended to estimate a condi - t i ona l density funct ion . For example, suppose the econometric ian observes an i.i.d. b ivar iate stochastic process {Xt, Zt}t€N- Fur ther assume Zt follows a discrete d i s t r i b u t i o n a n d condi t ional on Zt, Xt is continuously d is t r ibuted w i t h (fi. T h e n , an est imator of condit ional density at the boundary x can be defined as 0{x\Z = z) := where X = {Xt : Zt — z'it}. If Zt follows a continuous d i s t r ibut i on , a kernel smooth ing technique can be appl ied to redefine the est imator . C o m p a r i s o n between the N N and b a n d w i d t h estimators T h e ex is t ing l i terature has been tack l ing w i t h the prob lem of density es- t i m a t i o n near boundaries using the b a n d w i d t h estimates, be ing silent on a p p l y i n g the N N technique for a whi le . W e provide some insights that demonstrate w h y the N N estimator can be a better choice t h a n the band - w i d t h est imators i n this subsection. We first broadly investigate the s i m i - larit ies between N N a n d b a n d w i d t h estimates. T h r o u g h this , we show that N N est imator is p romis ing i n the context of po int es t imat ion of density. W h e n a density funct ion as a whole is of interest, several disadvantages of N N density estimates el ic i ted cr i t ic isms from stat ist ic ians on us ing the N N es t imat ion technique, (see, S i l verman (1986)[54] and Stone et. a l . (1977)[56] among others) . Consequently, the N N estimator became less a n d less pop- ular for density est imat ion , after receiving attentions i n the late 70's. Nev - ertheless, as S i l v e r m a n (1986)[54] says: " . . .There is noth ing to choose between b a n d w i d t h a n d N N esti - mates w h e n est imat ing a density at a po int . E v e r y value k of the smooth ing parameter i n N N estimate w i l l give an estimate ident i ca l to that obtained w i t h a certain value of b a n d w i d t h i n the kerne l -bandwidth estimate. . ." Hard ie (1990) [18] further shows that the o p t i m a l choice of k entails a M S E convergence rate of n ~ ^ / ^ , which is the same order as for the o p t i m a l band - w i d t h estimates i n general. Under m i l d condit ions, b o t h N N a n d b a n d w i d t h estimators are consistent and asymptot ica l ly normal ly d i s t r ibuted . There - fore, i n the context of evaluating a density funct ion at a po in t , there is not much difference between N N and b a n d w i d t h estimates a n d the b a n d w i d t h estimate is not necessarily superior t h a n N N estimates. T h u s they share the same basic properties a n d efficiency. T h o u g h there is always a matching b a n d w i d t h est imator for a given N N est imator , M o o r e a n d Yacke l (1976) [40] have shown that the N N estimator is more efficient t h a n the matching b a n d w i d t h est imator at the points x where (p{x) is s m a l l . T h e difference i n efficiency of these two estimators comes from the difference of their smoothing parameters . T h i s argument reveals tha t the b a n d w i d t h estimator related to the N N est imator may be inferior sometimes. W e next b r i n g the comparison to the context of density es t imat ion at boundaries . M o s t of the exist ing l i terature focuses on how to manipulate the weight ing scheme to improve the performance of the b a n d w i d t h estimators near the boundaries . A m o n g these available proposals , the boundary-kernel methods (Jones 1993[24], M u l l e r 1991 [41]) involve on ly kernel modif ications and make no at tempts to estimate the populat i on density value i n correcting the bias. Therefore, these methods are always associated w i t h large variance. T h e pseudodata methods (CowHng and H a l l 1996) [11] require the knowledge of b o u n d a r y locat ion . Moreover, implement ing these proposed methods, i n general, is not an easy task. W h i l e examin ing the convergence property intensively, the exist ing l i terature leave the asymptot i c d i s t r i b u t i o n issue unexplored . T h e only exception among them, to the best knowledge of ours, is H a l l a n d P a r k (2002) [20]. T h e y propose to use t rans lat ion bootstrap to correct the bias at the boundaries. T h e l i m i t i n g d i s t r ibut i on of their method is on ly found to be of zero mean. O n the contrary, however, the N N m e t h o d at tempts to adapt the amount of smooth ing to the local density of d a t a observations. T h e adaptive nature of the N N method enables our N N est imator to outper form other methods i n terms of the mean squared error i n our s imula t i on study. O u r proposed est imator is easy to implement and shown to follow a normal d i s t r ibut ion i n the l i m i t , which allows us to construct confidence intervals a n d conduct testings i n appl icat ions. These features mot ivate us to choose the N N method over other candidates. It is fair to emphasize here that , to the prob lem of density est imat ion at boundaries , our proposed one-sided N N est imat ion method is just one of the solutions, w h i c h provides reasonably good performance i n s imula t i on s tud - ies. T h e compar ison w i t h some of the kernel-type bias correct ion estimators is brought up to front because of its popular i ty and close attent ion that had e l ic i ted f rom the l i terature . However, whether our one-sided N N estimator is always the on ly best among choices is beyond the scope of this work and , therefore, left open for the future research. E s t i m a t o r o f 0 (x) in the case w i t h u n k n o w n x In this subsect ion, we consider the est imat ion of a t runcated density func- t i o n at i ts u n k n o w n t runcat i on point . T h e t runcat i on po int i n the current mode l can be viewed as the lower bound of the t runcated d i s t r ibut i on . A n a t u r a l candidate to estimate the lower bound is the smallest order stat is - t ic i n the sample . W e denote such an est imator of ^ as X j - := X^^^y T h e fol lowing l e m m a shows that converges to x fast. L e m m a 1.3 Under Assumption 1-1, Xj- - x = Oa.s.{T B y ca l l ing the fast convergence property "super-consistency" , D o n a l d and Paarsch (1996) [15] has established a result s imi lar to above l e m m a . T h e i r approach however relies on the condit ion that the r a n d o m variable X d is - crete. Instead, we reach the same result w i t h continuous r a n d o m variables. W e now combine X j - a n d the one-sided N N est imator to est imate (i){x). T h e fo l lowing proposit ions demonstrate that i t is indeed the case, that is, the combined est imator is consistent and has a n o r m a l d i s t r i b u t i o n i n l i m i t . P r o p o s i t i o n 1.4 Suppose Assumptions 1.1 and 1.2(i) hold. Then^{X_rp) —» <f>{x) as T oo in probability. If, in addition. Assumption 1.2(ii) holds, then ^{2LT) ~* 'Pis.) 05 r —> 00 a.s. P r o p o s i t i o n 1.5 Suppose Assumptions 1.1-1.3 hold. Then ( A ; T ) ^ {mr) - Hs.)} A ^ [ O , 0 ( s ) ' ] . 1.2.2 A P l u g - i n E s t i m a t o r for a f u n c t i o n o f ^(x) W e consider how to estimate a parameter that is w r i t t e n as a funct ion of 4)(x) a n d other u n k n o w n parameters possibly dependent on x. T h e parame- ter thus can be w r i t t e n as an extended model ip{x) := i]j{(j){x),'d{x)), where ^{x) is another parameter . We assume: A s s u m p t i o n 1.4 (Extended model): (i) Let 1? : M —> M be a continuous function; (ii) Let t/) : —> R 6e a continuous function. A s s u m p t i o n 1.4 imposes the continuity of functions. I n our first-price auct ion mode l w i t h endogenous entry, we are interested i n knowing the value of b idder ' s expected payoff function at the t runcat i on po int , which is real ized by bidders ' par t i c ipat ion decisions. Besides the density (</)), the cumulat ive d i s t r i b u t i o n of submit ted bids is also part of the expected payoff funct ion (ip). To handle such s i tuations, we assume the existence of an est imator of ?? converging faster than ((> and propose a p lug - in estimator for the extended model . A s s u m p t i o n 1.5 (Faster convergence ofd ): d{X_j'), the estimator of ^{x), satisfies that •d{2Ç.T) " '̂ (s) = op{kj,^). In our auct ion mode l , '&{X_x) is an empir i ca l cumulat ive d i s t r ibut i on func- t i o n evaluated at Xsp. T h i s est imator converges at the rate T~^l'^, wh i ch is faster t h a n any other non-parametr ic estimators. Nevertheless, i n appl i ca - t ions, i9 is not l i m i t e d to continuous functions of X's cumulat ive d i s t r ibut ion . T h e fo l lowing theorems are derived by app ly ing the S l u t s k y theorem, and the de l ta m e t h o d to the previous proposit ions. T h e o r e m 1.6 Suppose Assumptions 1.1, 1.2(i), I.4, and 1.5 hold. Then, ifiKr) '4'{4>{K.T)^'^{K.T)) ~* fiî) as T 00 in probability. T h e o r e m 1.7 Suppose Assumptions 1.1-1.5 hold. Moreover, ip is contin- uously differentiable with respect to (f){x) and t9(x) at an open neighbor- hood of X. Then, (fcr)^ {'p{2LT) '~ v f e ) } asymptotically distributed with N\<d,il)i{(j){x))'^<j){x)'^], where ipi is the partial derivative of tp with respect to its first argument. 1.2.3 Simulation Results W e conduct M o n t e C a r l o s imulations to assess the finite sample behavior of the proposed est imators i n this section. W e first i l lustrate the s imulat ion method , a n d then remark on the s imulat ion results. T h e results are reported i n the tables i n appendix . Simulation Method In each experiment, we r a n d o m l y draw n numbers f rom the LogNormal(4,1) d i s t r ibut ion . T h e n , we choose q-th percentile as the cutoff value for the experiment, where q is 5, 10, or 25. T h e draws w i t h their values greater than the cutoff are selected to form a sample for est imat ion . For each s imulat ion , we repeat 10,000 t imes. Coverage W e first check whether the constructed 9 0 % a n d 9 5 % confi- dence intervals provide r ight coverage. T h e results are reported i n Table 1.1 i n appendix . W e experiment w i t h three different choices of the smoothing parameter k, T*/^, T^'^ and T^/^ j ^ v s jg chosen because i t is the M S E - b e s t rate for regular (two-sided) N N estimators. T^/^ a n d T^/^ are chosen w i t h a purpose to see how the coverage responds to the changes i n the rates. W e consider three sample sizes 200, 500, 1000 for the coverage study. T h e results i n Table 1.1 indicate that • T^l^, w h i c h violates A s s u m p t i o n 1.3, does not have r ight coverage. • T h e smal ler the smoothing parameter, as c o m p a r i n g T^/^ w i t h T^/^, the better the coverage that the estimator provides. • T h e slow rate of convergence, as the b u i l t - i n feature of the non-parametr ic estimates, requires quite a large sample to provide better coverage. In the s imulat ions , we need a sample size, at least, of 1000 for reasonable coverage. • Under-re ject ion may be a concern i n appl icat ions . Other Boundary Correction Methods There is an extensive l i terature on how to correct boundary effect when kernel est imators are i n use. A n i m p o r t a n t question is how wel l they compare to our suggested estimator. A n ideal approach to this issue would be careful analysis of a l l the ear- lier proposal , w i t h a detai led comparison of their properties. However, as commented by C h e n g et. a l . (1989) [10] who encountered a s imi lar s i tua - t i on w h e n propos ing their local po lynomia l fitting est imator , it wou ld be a tedious task because "there are so m a n y proposals and a number of them are quite compHcated, thus difficult to work w i t h bo th numerica l ly and analyt i ca l ly . . . " For the purpose of comparison, we select some wel l -known and representative methods to compare w i t h our one-sided N N estimators: • T h e pseudodata method (hereafter, Pseudodata ; see C o w l i n g and H a l l 1996[11]); • T h e boundary kernel method (hereafter. K e r n e l ; see, for example, Jones 1993[24], M u l l e r 1991[41] among others); • T h e loca l l inear f i t t ing (hereafter, L L F ; see C h e n g et a l . 1997[10]). For the mat ter of fairness, we choose the smoothing parameters for these boundary correct ion methods i n the way such that they either are M S E - best, or make the result ing bias at s imi lar level as one-sided N N estimates. Specifically, the rates of T~^^^ ate used for L L F a n d P s e u d o d a t a methods, a n d T~^/^ is used for the boundary kernel method . Fo l l owing the rule of t h u m b (S i lverman (1986)[54]), we use 1.06 for the constant i n bandwidths est imators. T h e number of random draws (n) we take are 100, 200, 500. T o satisfy the A s s u m p t i o n 1.3, we set smoothing parameters k = T^/^ for the N N est imators throughout this subsection of the s imula t i on study. Simulation for density estimation Table 1.2 reports the results when D G P is assumed as LogNormal{4,1). (i) F o r N N estimates, consistent w i t h the intu i t ions , the signs of bias are posit ive i n any case. Moreover, M S E is decreasing i n the sample size a n d increasing i n the cutoff po int . Over a l l , N N outperforms the other methods i n the s imulated cases. (ii) T h e bias of P s e u d o d a t a methods increase w i t h sample size, which further indicates tha t i t is an inconsistent est imator . T h e pseudo- d a t a m e t h o d is regarded as an improvement from reflection meth - ods, as i t is c la imed to be "considerably more adapt ive" (Cowl ing and H a l l , 1996) [11]. However, the generic pseudodata method requires the knowledge of t runcat i on po int , so that the sample order statistics can be used to interpolate the pseudodata. W h e n the boundary point is not k n o w n , the method essentially extrapolates the pseudodata, which causes the inconsistency prob lem as we have jus t seen from the s imu- la t i on . (iii) T h e kernel used i n s imulat ion is a one-sided kernel since we know exact ly how to correct bias at the boundary. T h e s imula t i on suggests that even though we can control bias to a desired level , the result ing variance is way too h igh compared to the N N est imator . T h i s result is consistent w i t h the findings in the l i terature - "Approaches invo lv ing on ly kernel modif icat ions wi thout regard to t rue density are always associated w i t h larger variance." (Zhang et. a l . , 1999)[63] T h e result reveals a real and pract i ca l phenomenon that the boundary kernel -related methods usual ly focus on gett ing the bias as one wants i t w i t h pay ing the price of increasing variance. It has gradual ly been real ized by researchers that this variance in f lat ion is impor tant . (iv) L L F , as a special case of the boundary kernel m e t h o d , is often thought of by some as a s imple, hard-to-beat default approach. However, the s imula t i on suggests that , i t is not as good as one-sided N N i n terms of M S B . It is fair to ment ion a l i m i t a t i o n of our s imulat ion study. T h o u g h our one- sided N N est imator outperforms other methods i n terms of M S E as reported i n Table 1.2, we make no attempts to c la im that our est imator is the only best among a l l other alternatives. M o r e thorough s tudy is beyond the scope of this work. 1.3 A n Application to the First-Price Auction Model W e now app ly the proposed est imation method i n the context of a first- price auct ion . W e set up a first-price auct ion model w i t h a par t i c ipat ion cost a n d discuss some identi f ication issues. W e then compare the auct ion mode l w i t h the extended model i n the methodology section to see how the p lug - in est imator can be appl ied i n this case. A t the end, we estimate the levels of par t i c ipa t i on costs i n the M i c h i g a n Highway Procurement Auc t i ons , w i t h w h i c h we further infer how the government should set the pol icy tools (reserve prices a n d / o r entry fee) for o p t i m a l auct ion outcomes. 1.3.1 The Auction Model W e consider a f irst-price sealed-bid auct ion of a single indiv is ib le good. W i t h i n the symmetr i c independent private-value ( I P V ) framework, each po- tent ia l r i sk neutra l part i c ipant i G {1,2, . . . , TV} knows her own value Vi for the object, but only knows the d i s t r ibut ion of the values to the other potent ia l bidders. It is assumed that the values to ind iv iduals are independently drawn from the absolutely continuous d i s t r ibut i on F{v) w i t h support {v,v] C M+. B idders submi t their bids s imultaneously and the object goes to the highest b idder . T h e winner pays her b i d to the seller, prov ided that the b i d is no less t h a n the reserve price r , which is assumed to be zero i n this chapter w i t h o u t loss of generality. O u r analysis deviates from the s tandard I P V framework by a l lowing for the presence of a c ommon par t i c ipat i on cost, K, w i t h w h i c h each bidder has to pay to j o i n the auct ion . G i v e n her pr ivate value, the bidder decides whether or not to submit a b i d (paying K) and becomes an ac tua l bidder. A l l po tent ia l bidders make this decision simultaneously. Therefore, they make their p a r t i c i p a t i o n decisions wi thout knowing how m a n y competitors they are ac tua l ly go ing to face. W e w i l l focus on the unique symmetr i c Bayes ian N a s h e q u i l i b r i u m ( M i l - grom, 2004) [37], i n which each potent ia l par t i c ipant jo ins the auct ion if her value is no less t h a n , Vp, the cut-off point (common to a l l b idders) , other- wise chooses not to part ic ipate . T h e cut-off point is such that the part i c ipant w i t h value Vp is indifferent i n entering the auct ion or not : so, Vp should solve the equal i ty VpF{vp)^~^ - K = 0. Throughout the chapter , we assume that the entry cost is moderate such that Vp G iv,v). T h i s assumpt ion effectively rules out the uninterest ing case of the entry cost is so large that there is no entry. T h e expected profit for the i t h bidder is then given by, ^iivi, y, {bj)j^i) = {v^ - v)[F{max{vp, b-\y)))f-' - K, (1.2) where y is the bidder i ' s b i d given Vi and a l l other parameters . is the inverse b i d d i n g strategy for bidders. T h u s , m a x i m i z i n g 11̂ w i t h respect to y yields the e q u i l i b r i u m b idd ing strategy: Kv) = ^—jèl [ uf{u)F{uf'^du if V > V, (1.3) A potent ia l b idder j part ic ipates i f and only i f Vj > Vp. In general , the observables i n our current model are the bids , the number of po tent ia l bidders and the number of ac tual bidders, whi le the pr ivate va l - ues a n d their d is tr ibut ions are not observed. T h e identi f icat ion prob lem here amounts to a discussion on whether the observed variables can uniquely de- termine b o t h the latent d i s t r ibut i on F and the par t i c ipat i on cost K. W e skip the discussion on the identi f icat ion issues i n our m a i n text . M o r e discussion on this identi f icat ion prob lem is discussed i n the appendix . 1.3.2 A p p l y i n g t h e E s t i m a t i o n M e t h o d o l o g y W e consider L homogeneous auctions w i t h the same number of po tent ia l bidders N, from w h i c h our d a t a are observed, n ; is the number of bidders who ac tua l ly submit a b i d i n the Ith auct ion , where I = 1,2, . . . L . W e define the sample of observed bids as J5, where B := {bu : i G {1, ...ni}, I € {l,...L}}. T h e t o t a l sample size is denoted by T j . B y the construct ion of the model , the cut-off value, Vp (wi th its o p t i m a l b i d d i n g bp, w h i c h is zero when the reserve price is zero.) makes the expected gain f rom entry is equal to the level of par t i c ipat i on cost, K. T h i s suggests that eva luat ing the expected profit function at Vp should give an estimate for K. T h a t is, K = 7r{bp,Vp)^{vp-bp)[F{vp)f-\ (1.4) W e could estimate K by replacing Vp, bp, and F w i t h the ir estimates. N e v - ertheless, the properties of such est imator are not k n o w n wel l ; i n part i cu lar , i t is diff icult to derive the l i m i t i n g d i s t r ibut ion of the est imator (see, for ex- ample , G P V ) . W e below derive an alternative expression for K, w h i c h leads to a convenient est imat ion method. Let b* denote the observed equ i l i b r ium b i d of the i t h ac tua l bidder, i € { l , . . . , n ; } , i n the Ith auct ion. G * and g* are the d i s t r i b u t i o n and den- sity functions of b*, respectively. T h u s , G* = Pr{b{v) < b*\v > Vp) — [F{v) - F{vp)\l[l - F{vp)\. Di f ferentiat ing w i t h respect to b* gives g*{b*) = WJv) i-F(vp)- ensure that g*{bp) is positive as i n A s s u m p t i o n l . l ( i i ) , f{v) has to be nonzero i n [v,v]. W e assume so for the rest of this chapter. It follows from (A . 24), by some elementary algebra, that S u b s t i t u t i n g (1.5) into (1.4), together w i t h the fact that G*{bp) = 0, we get 1 ^ f a ) " ^ (16 ) N-l[l-F{vp)]9*{bpy ^ • > F{vp) equals to one minus the expected par t i c ipa t i on rate , denoted by p. T h i s enables us to rewrite (1.6) as It is not h a r d to see that M and g* i n the current app l i ca t i on are ̂ {x) a n d (f) i n the preceding section, respectively. W e can app ly the proposed p lug - in est imator to estimate K . T h e rate of par t i c ipa t i on p, can be consistently est imated by the sample analogue p := j; J^iLiij^)• T h i s impl ies M can be est imated by M = ^ ^ ^ r i ) - - A p p l y i n g the one-sided N N est imator to g*{bp), we have k{n)/n where fc(T(,) is a sequence of positive integers sat is fying A s s u m p t i o n s 1.2 and 1.3. Therefore, our est imator for par t i c ipat ion costs K is defined as T h e fo l lowing corollaries state the asymptot ic properties of K according to the theorems 1.6 a n d 1.7 i n the preceding section. C o r o l l a r y 1.8 K converges to K almost surely. C o r o l l a r y 1.9 Assume the first derivative of g* exists and hounded around hp. Then, {k{n))\{K-K) -^a N[Q, ( g . ^ ' ) ) 2 ] - (1-10) 1.3.3 E m p i r i c a l A p p l i c a t i o n I n this subsection, we apply the est imator of K defined i n (1.9) to the M i c h i g a n Highway Procurement A u c t i o n data . W e first introduce the data set a n d form a subsample for est imation. T h e n , we estimate the par t i c ipa - t i o n costs for the auctions and present the estimates of the costs i n terms of b o t h absolute value ( in dollars) and relative value (to project size). F i - nal ly , we approx imate the levels of op t imal cutofl[s i n valuat ions , w i t h which we empir i ca l l y address the questions regarding implement ing the op t imal auctions. D a t a W e have d a t a for the highway procurement auctions he ld by the M i c h i g a n Depar tment of Transpor ta t i on between J a n u a r y 2001 a n d December 2002. T h e d a t a set consists of a to ta l of 1,538 projects. For each project , the dataset offers the l e t t ing date, the expected complet ion date , the locat ion , the tasks invo lved , the identities of a l l the bidders , a l l b ids , the engineer's estimate of the costs for complet ing the project , and a l ist of plan-holders for a l l projects i n the d a t a set. Letting Process T h e Department of Transpor ta t i on ( D o T ) announces a project to be let a n d the inv i ta t i on to submit bids begins. T h e length of this advert is ing per i od ranges from 4 to 10 weeks. T h e announcement of a project comes w i t h a brief descript ion of the project i n c l u d i n g the locat ion and complet ion t ime . Po tent ia l bidders who are interested i n the project may collect a deta i led b i d proposal from the D o T . Based on the proposal , bidders may submi t a b i d , w h i c h has to be 48 hours pr ior to the l e t t ing date. B idders do not know who else also submits a b i d . For each b i d , D o T checks that the b i d d i n g f i rm is among the firms that are quali f ied to do business w i t h D o T . T h e n , on the let t ing date, the project is awarded to the lowest b idder , prov ided that i t is below the reserve price. Engineer Estimates and Reserve Prices T h e D o T provides cost esti - mates for each project . These estimates are based on the engineers' assess- ment of the work needed to fulf i l l the task and in format ion extracted from the s imi lar projects let before. U S Federal law requires that the w i n n i n g b i d should be no greater than 110% of the engineers' estimates. However, a state is s t i l l al lowed to let the project w i t h a price higher than this threshold, i f the state can just i fy this act ion i n w r i t i n g . In this case, the engineer's est imate for th is project w i l l be revised a n d examined to see i f any possible mistake h a d been made. In the part i cu lar auctions examined here, the 110% l i m i t is quite frequently overridden. It is na tura l to expect that the potent ia l b idders of the auctions give l i t t l e considerat ion to the 110% l i m i t . T h e assumpt ion of no reserve price is thus not far f rom the reality i n our example. P r o m the same perspective, we consulted w i t h professionals i n the pro- curement business about the feasibil ity for a government to implement a b i n d i n g reserve price. E x p e r t s i n the industry informed us of the great difficulties, i f there were no any other compensation i n effect. A group of factors may be used by bidders to argue for inaccuracies i n engineer's es- t imates. Therefore , we are interested i n knowing how much we need to compensate part i c ipants to ensure an o p t i m a l auct ion outcome, i f we set a b ind ing reserve price at 110% of the engineers' estimates. T h i s motivates the scenario to be investigated i n the paper for the empir i ca l impl icat ions . Government's Maximum Willingness to Pay It has been recently noted i n the l i terature tha t the first (low-)price auctions w i thout b i n d i n g reserve prices, w h i c h is the case of procurement auctions considered here, does not have a unique finite Bayes ian -Nash equ i l ibr ium. To see this i n a more clear fashion, consider any bidder. If she knows that there is no b i n d i n g reserve price, a n d there is a non-zero probabi l i ty for her to be the single actual b idder , no mat ter how smal l this probabi l i ty is, her o p t i m a l strategy is always to b i d inf inity . To rat ional ize the b idders ' behavior, we use a s imi lar argument as L i and Zheng (2005) [34]. W e assume that a l l auct ion part i c ipants have a common belief that government has a maximum-willingness-to-pay, the upper bound of va luat ion d i s t r i b u t i o n , denoted as v. If the w i n n i n g b i d is above this value w i l l be rejected for sure. Moreover , v is not observed by econometricians.^ * O n e may argue that this line of justification amounts to assuming a reserve price. We, however, do not use this terminology is because reserve prices a n d entry costs may together complicate identification issue. Bidder Heterogeneity T h e m a i n sources of heterogeneity among b i d - ders i n the market are size a n d location.^ L o c a t i o n reflects the bidder 's cost for the company of mov ing equipment, materials and labor to the work site. Size of the f i rm entails the scale of economy. I n our analysis , we consider the projects invo lv ing on ly non-fringe bidders a n d from either M i c h i g a n or neighbor states. A quali f ied f i rm is considered as non-fringe i f i t part i c ipated more t h a n once i n the sample per iod . Project Type T h e choice of the project types is mot ivated by the objective that the auct ion environment of private value needs to be ensured. Ideally, we cou ld have focused on only one type of projects. However, since our empir i ca l method is nonparametr ic , we can not afford to be l i m i t e d to a very s m a l l sample. O u r entire sample has up to 30 different type codes. N o t a single type can provide a sample w i t h a reasonable size. Therefore, as a compromise, we include only the auctions that involve p a v i n g or grading to form the subsample. T h e l i terature has documented some evidence that paving- type road jobs seem to have more private-value components (c f . H o n g and S h u m (2002) [22]). Therefore, our control led subsample is composed of the observations only from the auctions for p a v i n g / g r a d i n g type projects a n d involves only non - fringe firms from M i c h i g a n or neighbor states. Unobserved Project Heterogeneity K r a s n o k u t s k a y a (2003) [26] uses the sample of highway maintenance projects let by M i c h i g a n D o T between February 1997 a n d December 2003. Her results suggest that " fa i l ing to account for unobserved auct ion heterogeneity may lead to overest imating uncerta inty that bidders face when s u b m i t t i n g their b i d s " . W e conduct several pre l iminary regressions to check whether the unob- served project heterogeneity prob lem exists i n the sample we use. T h e re- •"The specialization of firm may also matter in terms of bidder heterogeneity. However, we do not have enough information in our data set to handle this. suits are reported i n Table 1.3. A l l regressions indicate that the Engineer 's Es t imate is the on ly variable has a stat ist ical ly significant impact on the b i d level. T h i s i n t u r n justifies our idea of normal i z ing the bids by the engineer's estimates to homogenize the auctions. O L S analysis produces equal to 0.9780 w h i c h indicates that variables inc luded i n the regression capture factors affecting b i d level quite well . W h e n the between-effect mode l (regression on group means) is used to exp la in the var iat ion i n the b ids , B? remains roughly same at 0.9779 w h i c h indicates that there is no substant ia l amount of inter -auct ion var ia t i on unexplained by the variables available to the researchers. T h e random-effect regression provides a s imi lar result to the first two sets of regressions: the unobserved auct ion heterogeneity may not present i n our control led subsample. Est imates of part ic ipat ion costs A potent ia l b idder i n this appl i cat ion is defined as a quali f ied firm who re- quested for an official b i d d i n g proposal before the b i d d i n g starts . W e group the auctions by the number of potent ia l bidders. For example , G r o u p 5 means the auct ions w i t h 5 potent ia l bidders. W i t h i n each group, a l l the auctions are regarded as homogeneous, while the under ly ing value d i s t r i b u - tions (and, thus the par t i c ipa t i on costs) across groups are taken as different.^ A l l bids are normal i zed by the engineer's estimates. F i g u r e 1.1 plots the est imated par t i c ipat i on rates for groups. I n the dataset, we observe that the number of potent ia l bidders ranges from two to twenty-two.^ W h e n there are only two potent ia l bidders , the par t i c ipa t i on rate is one, so that we cannot estimate the entry cost i n our approach. W e ^ A n alternative estimation approach may assume the entry cost is same across the auction groups a n d pool all the auctions together to estimate the entry cost jointly. How- ever, this approach employs the variation in the number of potential bidders. Therefore, it implici t ly assumes such variation is exogenous and there were no unobserved hetero- geneity across the groups of auctions. We decide not to impose these further assumptions by estimating the model separately across number of potential bidders. ^We do not observe any auction with 19 potential bidders in the date set. e l iminate the auct ions w i t h 16 to 22 potent ia l bidders f rom our analysis, because they offer less t h a n 100 observations. Based on these observations, we choose G r o u p s 3-15 for the next steps. C o l u m n 3 i n Table 1.4 reports the according numbers for F igure 1.1. T h e rates of par t i c ipa t i on monotonica l ly decrease from .8 to .5 for the chosen groups. T h e p a r t i c i p a t i o n costs are est imated for each group according to K pro- posed i n (1.9). M o d i f i c a t i o n on K is needed for the empir i ca l appl i cat ion . In our theory framework, we follow the convention i n l i terature to consider an auct ion where the highest b i d wins. However, our empir i ca l app l i cat ion is procurement auctions i n which the lowest bids w i n . It should be noted that , i n highway procurement auctions, v is the construct ion cost by the f i rm and the b idder submits a b i d only i f her construct ion cost is less t h a n the cutoff value Vp. E s t i m a t i o n results on the par t i c ipat i on costs are l is ted i n C o l u m n 4 of Table 1.4 a n d p lo t ted i n F igure 1.2. T h e est imated par t i c ipa t i on cost varies from 2 % to .001% of the engineer's estimates. T h e dotted lines i n the figure are 9 5 % confidence intervals for the estimates. T h e results show that the par t i c ipa t i on costs are signif icantly different from zero for a l l groups. T h i s es t imat ion result should be interpreted w i t h care. It indicates that the entry cost is proport iona l to the engineer's estimates. It is compatible w i t h the constancy of the entry cost i n the auct ion theory by the fol lowing argument. I n pract ice , the larger auct ioned projects ( w i t h larger engineer's estimates) can be more costly on b i d submission. T h i s may reflect the fact tha t larger projects require more work on complet ing b i d d i n g plans (forms) a n d more efforts involved i n the b i d d i n g process. Therefore, the constant entry cost entails the fixed proport ion to engineer's estimates i n procurement auctions. I n absolute terms, we investigate the levels of p a r t i c i p a t i o n costs i n an average-size projects i n each group. T h e numbers are l is ted i n Table 1.4 co lumn 5. W e f ind that the costs i n dollar values decrease d ra ma t i ca l l y over the range. T h e costs can be as h igh as $13,416 when N = 3 , then drop down to $3,721 when N = 5 . W h e n N > 8 , the costs f luctuate between $1,000 a n d $2,000. T h e dec l in ing pat tern of entry cost w i t h the groups i n b o t h rela- t ive a n d absolute terms is consistent w i t h the fact that the average project sizes decrease w i t h the groups. A n ins t i tu i t i ona l reason may be that the auctions w i t h less potent ia l bidders are less complex i n nature , require less for p a r t i c i p a t i o n a n d therefore at tract more bidders. Counter factual experiments : implementing o p t i m a l auctions W i t h the help of knowing the level of par t i c ipat ion cost, is there anyth ing that a government can do to improve its s i tuat ion? W e empir i ca l ly address this question by ca l cu lat ing the o p t i m a l level of cutoffs i n valuat ions , which accordingly suggests the po l i cy tools (entry fee/subsidy, or reserve prices) a government should undertake for o p t i m a l auct ion outcomes. A s counterfac- t u a l experiments , we compute the savings a government could have made from the o p t i m a l auctions. T h e op t ima l i ty analyzed i n this subsection is from revenue perspective, i.e., i t refers to m i n i m i z i n g the government's ex- pected payment on the auct ioned project. Optimal cut-off O p t i m a l auctions have been studied from theoretical perspective. (See, for example, M y e r s o n (1981)[42] among others.) In par- t i cu lar , C e l i k a n d Y i l a n k a y a (2006) [8] consider the o p t i m a l auctions w i t h par t i c ipa t i on costs. T h e y show that the cutoff i n valuations for an op t imal symmetr i c auc t i on , v*, should solve the fol lowing equation: {v ~ J{v*mi - Fiv*)f-'= K, (1.11) where J{v) =^ v -\- is the v i r t u a l va luat ion funct ion , fo l lowing the nota- t ions i n the l i terature . E q u a t i o n (1.11) can be interpreted as follows. Suppose a l l bidders at the cutoff V*. In t roduc ing a sl ight decrease of the cutoff to one bidder w i l l increase the gross ga in to the seller by {v - J{v*)){l - F{v*))^~^ (gaining V — J{v*), the v i r t u a l va luat ion , when a l l the other va luat ions are greater t h a n V*, i .e., w i t h probabi l i ty (1 - F{v*))'^~^), whi le p a y i n g the bidder K, the m a r g i n a l cost of induc ing the par t i c ipat ion . T h e value of v, recovered through (1.5), is used to create a pseudo sam- ple of valuat ions . W e estimate F by an empir i ca l C D F a n d / by kernel- b a n d w i d t h est imator , respectively. Fo l l owing the e m p i r i c a l auct ion l i tera - ture, we use a tr iweight function for the kernel a n d 2.978 x 1.06 x T~^^^ x a for b a n d w i d t h where T is the sample size and a is the s tandard deviat ion . 2.978 is needed because of the triweight kernel. (Examples can be found i n G P V a n d L i , Perr igne and V u o n g (2000)[35].) W e use the m a x i m u m bids i n the sample as the estimates for v. W e then numer ica l ly solve (1-11) from the pseudo sample a n d derived an estimate for v*. T h e estimates are reported i n the second c o l u m n of Table 1.5. For a l l the auctions i n the sample, the o p t i m a l cutoff values vary mostly between 0.9 a n d 1.1. Restricting entry T h e fol lowing propos i t ion shows that i n our endoge- nous entry mode l , the cutoff i n the symmetr i c e q u i l i b r i u m Vp a n d the cutoff i n an o p t i m a l auct ion v* are comparable. P r o p o s i t i o n 1.10 In a first-price IPV auction model with participation costs, zero entry fee and zero reserve price, for any given number of po- tential bidders, the seller maximizes the expected revenue, in the symmetric equilibrium, by discouraging entry. T h e preceding propos i t ion indicates that , i n our current first ( low-)bid auct ion , v* < Vp. Intuit ively , a decrease i n the e q u i l i b r i u m cutoff (vp) make the auctioneer better-off by extract ing from al l bidders who have sufficiently favorable values that they decide to enter and by further screening least favorable va luat ion bidders. T h i s theoretical i m p l i c a t i o n provides a device for us to check the reUabil i ty of computed cutoffs. W e report the m a x i m u m of recovered valuat ions (after t r i m m i n g near boundaries) i n C o l u m n 3 of Tab le 1.5, w h i c h clearly shows that v* < Vp. T h i s result may be more s t r ik ing than it appears. W e can observe d a t a only from the t runcated density rather than the ful l range of d i s t r ibut i on . T h i s fact makes one reasonably expect that , i n nonparametr ic framework, the pred i c t i on power of counterfactuals that econometricians may conduct is l i m i t e d . W e are able to provide an example that , even w i t h the case of t r u n c a t e d density, do ing such reliable counterfactuals is possible. It is because the o p t i m a l po l i cy (of restr ic t ing entry) uses in format ion from the observed range on the (truncated) d i s t r ibut ion that enables us to do so. Optimal policy tools W e next investigate how to implement the o p t i m a l auctions by us ing only regular pol icy tools, the reserve price (r) a n d entry fee (<5). These po l i cy parameters should be chosen such that the fol lowing equal i ty holds : (r - v*)il - F{v*)f-^ = K + S, (1.12) where K + Ô can be interpreted as effective par t i c ipa t i on costs. Note that 6 can be negative, i m p l y i n g an entry subsidy. Obviously , there are many pairs of r a n d S can sustain the equahty (1.12) and any such a pa i r should entai l a way to implement o p t i m a l auctions. W e w i l l focus on the two scenarios to show how it works : (I) how m u c h is r, when S = 0; (II) how m u c h is S, when r = 1.1. Scenario I entails the s i tuat i on of o p t i m a l reserve prices only. In Scenario I I , the government wou ld rather set an o p t i m a l level of entry subsidy/ fee whi le enforcing the reserve prices at 110% of engineer's estimates. T h e reserve price that should be imposed i n Scenario I is l isted i n C o l u m n 6 of Table 1.5. T h e levels of reserve prices may be as h igh as 3.2, but most ly vary f rom 1.1 to 1.3. T o further show the o p t i m a l po l i cy entails discouraging entry, we report the m a x i m u m of bids for each group i n the table. T h e est imated Ô, o p t i m a l fee, i n Scenario II are reported i n C o l u m n 4 of Table 1.5. E x c e p t for N = 1 4 , the ô's are a l l negative, w h i c h entai l par t i c ipa t i on subsidies for the auctions i n the sample. For G r o u p N = 1 4 , Table 1.5 shows that reserve prices at 110% of engineer's estimates does not restrict entry enough. Together this poUcy, an entry fee is needed towards the op t imal auct ion outcome. T h e amount of the subsidies decreases along the increase of number of po tent ia l bidders. Revenue improvement A t the end, we calculate the expected payment {EP) by the government i n Scenario II through the fo l lowing equation: r' EP = N J{v){l-F{v)f~^f{v)dv-5NF{v*). (1.13) Jo T h e first t e r m i n (1.13) is the expected b u y i n g price a n d the second te rm is the expected revenue /payment for entry. B o t h the ca lculated expected pay- ment a n d the ac tua l payment i n current auctions are prov ided i n Table 1.5. T h e difference between two payments indicates the amount a government wou ld have saved if he h a d implemented the o p t i m a l auct ions specified as Scenario II . O u r results indicate that , for a l l the auct ions i n the sample, the government w o u l d have p a i d less i f the o p t i m a l auctions were implemented. For some of the auctions, the improvement can be up to 10-15%. 1.4 More Examples on Applications T h i s section l ists some other situations where the proposed est imation methodology can be appl ied . For the first several scenarios, we extend the auct ion framework elaborated above to other economic fields, because an auct ion is jus t a mechanism for transactions. W e then provide another example where no strategic structure is involved. T h r o u g h this , we at tempt to a p p l y our proposed est imation method for a broader range of appl icat ions i n empir i ca l studies. Menu Costs Suppose there are N f irms conduct ing p r o d u c t i o n i n the economy. I n one per iod , each f i rm is subject to a r a n d o m shock e which follows a d i s t r ibut i on . If the real ization of e is below a threshold Cp, the f i rm does not need to proceed w i t h any significant change on its "menu" . Otherwise , the f i rm has to pay K for " repr int ing menus" . Therefore, at the end, we observe n firms actual ly make the changes of their menus. How much change the f i rm w i l l take is determined by the real izat ion of e. It is in format ive for a government to know this K for po l i cy -mak ing purposes. Efficiency Wages Consider a monopsonist employer that wants to pro- vide incentives for workers to make efforts to acquire a specific s k i l l i n their labor market . Suppose a l l potent ia l workers are w i t h s imi lar backgrounds, say w i t h at least a h igh school education. W o r k i n g experience i n a related area a n d a higher degree can promote the worker 's product iv i ty . T h i s labor market p a r t i c i p a t i o n prob lem is identical to our f irst-price auct ion model w i t h an entry cost. W i t h our est imat ion framework, we can estimate how much "effective investment" a potent ia l worker w i l l have to incur to j o in the market . Moreover , we can help the employer to identi fy the o p t i m a l incentive they w i l l provide to maximize its benefits. T h i s can provide em- p i r i c a l evidence of "efficiency wages", i.e. how to compensate workers for investment i n h u m a n cap i ta l . Jump at a discontinuity point Consider a social program that a govern- ment conducts . O n l y households w i t h an annual income lower t h a n $30,000 are qual i f ied. N o w the government is interested i n knowing whether the threshold is correct ly imposed i n the fol lowing senses: (1) For the house- holds w i t h an income close to $30,000, the program is not he lp ing them to a higher u t i l i t y t h a n the one reached by the houesehold whose income is above $30,000 but close. Because otherwise, the program w i l l encourage certain people p u t t i n g less effort into the labor market . (2) T h e program is prov id - ing a id to the households w i t h a low income so that their l i v i n g standards are not far off f rom the people outside the program. These purposes entai l to evaluate the behavior of households from b o t h sides of the threshold . T h e difference then estimates the " j u m p " at the d iscont inuity point . In such a s i tuat i on , our proposed est imat ion method applies. W e also note tha t , i n this case, our est imat ion strategy improves the accu- racy of estimates i n the exist ing methods. Current ly , e m p i r i c a l economists use a parametr i c approach to estimate b o t h groups ( w i t h i n a n d outside the program) . T h e y then evaluate the two est imated functions at $30,000. O u r approach does not lose accuracy i n estimates i n order to fit the entire func- tions. O u r interest is to know the difference between two points at the boundaries anyway. 1.5 Conclusion and Discussion In this chapter , we propose using the combinat ion of one-sided nearest neighbor est imator w i t h extreme order stat ist ic i n the sample to estimate t runcated univar iate probab i l i ty density functions (pdf) at the i r u n k n o w n t r u n c a t i o n po ints . I n practice, the parameter of interest is often a smooth funct ion of the u n k n o w n p d f and some other at tr ibutes of the d i s t r ibut i on at the t r u n c a t i o n po int . W e also suggest a p l u g - i n est imator i n w h i c h such parameter is es t imated by first est imating the p d f a n d some other relevant attr ibutes of the d i s t r ibut i on at the t runcat i on po int , and then plugging the est imated values into the smooth function. W e s tudy the large sample properties of the p l u g - i n estimator to establish sets of condit ions sufficient for i ts consistency and asymptot ic normality . F u r t h e r , we app ly the proposed method to estimate the p a r t i c i p a t i o n costs incurred to the bidders i n order to j o i n the M i c h i g a n Highway Procurement A u c t i o n s . O u r empir i ca l result suggests that the p a r t i c i p a t i o n costs are s tat i s t i ca l ly significant f rom zero. Based on this est imate, we infer how the o p t i m a l auct ion outcomes can be implemented by using the regular po l i cy tools. W e exp l i c i t l y show the improvement the M i c h i g a n government could have made on payments , i f the entry subsidies were prov ided whi le enforcing the current reserve price policy. O u r est imation method is easy to implement a n d not l i m i t e d to empir i ca l auct ion works. W e have l isted some examples where the m e t h o d is applicable. W i t h more successful appl i cat ions of game theoret ical tools to other economic topics, we are confident to foresee more employment of th is est imat ion method to address questions that arise i n other economic fields. In our e m p i r i c a l appl icat ions, we only considered homogeneous auctions i n th is chapter . Heterogeneous auctions have been manipu la ted through contro l l ing some covariates i n the l i terature. (See, for example, G P V . ) I n - formally, we can reconcile a heterogeneous case to our framework; i f the covariates on ly affect the locations of F rather t h a n its shapes, one can a l - ways normal ize the bids to control the shifts of F by covariates so that the heterogeneous case is transformed into a homogeneous case. T h i s so lut ion, however, is heurist ic . W e expect a more rigorous treatment to this problem i n the future research. In terms of b i d d i n g strategies, we focus on the unique symmetr i c equihb- r i u m i n this chapter. However, along the line of o p t i m a l auctions, Ce l ik and Y i l a n k a y a (2006) [8] point out that there are s i tuat ions where the equi l ibr ia w i t h asymmetr i c cutoffs may further improve the expected revenue (or pay- ment) . T h e authors provide sufficient conditions for the existence of such equi l ibr ia : the behavior of T{v) := YZ^^- We report the plots of T(v) for G r o u p s N = 9 a n d N = 1 2 as Figures 1.3 and 1.4. T h e plots indicate that for auctions w i t h nine potent ia l bidders, T{v) increases globally, which i n t u r n may suggest the existence of equ i l ibr ia for o p t i m a l auctions w i t h asymmet- ric cutoffs. (See P r o p o s i t i o n 3 i n CeUk a n d Y i l a n k a y a (2006)[8] for details . ) . O n the contrary, though F igure 1.4 does not show any global monotonic i ty of T{v) for N = 1 2 , i t is increasing between 1.1 a n d 1.2, where the o p t i m a l cutoff V* for symmetr i c equ i l i b r ium locates for th is group. T h i s may also i m p l y the equ i l i b r ia for o p t i m a l asymmetr ic auctions through a sufficient condi t ion on the loca l behavior of T{v) at v*. (See P r o p o s i t i o n 2 i n Ce l ik a n d Y i l a n k a y a (2006) [8] for details.) T h i s is just a p re l imina ry treatment on checking the existence of equi l ibr ia w i t h asymmetr ic cutoffs. T h e interest of our work is of es t imat ing entry cost and showing its appl icat ions . T h e focus of this chapter is, therefore, l im i ted to the wor ld of symmetr i c equ i l i b r ium. M o r e r igorous empir i ca l examinat ion on the existence a n d implementab i l - i ty of equ i l ibr ia w i t h asymmetr ic cutoffs may be w o r t h exp lor ing i n future researches. 1.6 Tables and Figures Table 1.1: S i m u l a t i o n Results o n Coverage 5% 10% 25% 200 500 1000 200 500 1000 200 500 1000 90% Interval 95% Interval 0.9027 0.9906 0.8224 0.9442 0.6964 0.8617 0.9142 0.9921 0.8664 0.9616 0.8031 0.911 0.9205 0.9976 0.8318 0.9554 0.6954 0.8546 7-3/5 90% Interval 95% Interval 0.9995 0.9999 0.9862 0.9998 0.9311 0.9912 0.9976 0.999 0.982 0.9979 0.9247 0.9912 0.9975 0.9985 0.9821 0.9993 0.8964 0.9902 T ' / 2 90% Interval 0.9872 0.9514 0.9142 0.9852 0.9499 0.9195 0.9866 0.9276 0.8675 95% Interval 0.9999 0.9998 0.9821 0.999 0.9983 0.9847 0.9998 0.999 0.9735 Table 1.2: S i m u l a t i o n Results , L o g n o r m a l ( 4 , l ) , k ^ T ^ / ^ 5% 10% 25% 100 200 500 100 200 500 100 200 500 N N R M S E Bias S T D 0.004693 0.003705 0.002880 0.003877 0.003241 0.002128 0.003034 0.002660 0.001459 0.004360 0.002847 0.003302 0.003447 0.002427 0.002449 0.002744 0.002143 0.001714 0.005496 0.003690 0.004074 0.004641 0.00354 0.003003 0.004038 0.003429 0.002133 L L F R M S E Bias S T D 0.031893 0.027133 0.016761 0.020386 0.015935 0.012715 0.011775 0.008007 0.008634 0.034401 0.028858 0.018726 0.022065 0.016948 0.014128 0.01329 0.008892 0.009876 0.041942 0.035646 0.022101 0.027865 0.022264 0.016756 0.017046 0.012554 0.011531 Kernel R M S E Bias S T D 0.053803 0.048792 0.022674 0.035848 0.031056 0.017906 0.021449 0.017108 0.012938 0.057082 0.051156 0.025327 0.038254 0.032618 0.019985 0.023641 0.018302 0.014964 0.067057 0.060352 0.029228 0.045892 0.03946 0.023431 0.0286 0.023008 0.016989 Pseudodata R M S E Bias S T D 0.012061 0.008673 0.008381 0.007062 0.003074 0.006358 0.004407 -0.000889 0.004317 0.01274 0.008639 0.009363 0.007557 0.002684 0.007064 0.005117 -0.001343 0.004938 0.016394 0.012U 0.011050 0.009978 0.005419 0.008378 0.005793 0.000564 0.005766 Table 1.3: B i d Ana lys i s Variable O L S Between Effects Random Effects Log(Estimate) 0.9843* 0.9939* 0.9933* Log(Back!og) (.0119) (.0164) (.0157) -0.0066 -0.0145 -0.0012 (.0120) (.0202) (.0091) Number of Days -0.0001 -0.0001 -.00001 Distance (.0000) (.0001) (.0001) 0.0191 0.0168 .0239 Out-of-State F i r m (dummy) (.0109) (.0185) (.0093) -0.0336 -0.0449 -0.0110 (.0434) (.0768) (.0324) Letting Date (dummy) -0,0001 -0.0000 -0.0000 Constant (.0001) (.0001) (.0001) 2.2313 1.3474 1.1930 (2.6427) (3.4205) (3.3130) 0.9780 0.9779 0.9779 Dependent Variable: Log(Bid) Number of observations: 171 Number of Potential Bidders: 5 Projects: Paving and Grading Bidders: non-fringe firms * Statistically significant at 5% level. Table 1.4: E s t i m a t i o n Results on E n t r y Costs pot* nob" K* 3 257 .79 .02391 13416 4 354 .77 .01318 11557 5 473 .73 .00341 3721 6 503 .72 .00126 2916 7 468 .68 .00156 3446 8 410 .64 .00055 2007 9 356 .62 .00013 1326 10 119 .51 .00059 1871 11 228 .53 .00017 1621 12 249 .52 .00009 1226 13 215 .52 .00003 1043 14 145 .47 .00004 1201 15 191 .49 .00001 1042 * Number of potential bidders. Sample size. § Rate of participation. t Estimated levels of participation costs(in engineer's estimate). * Estimated levels of participation costs (in Dollars). Table 1.5: P o l i c y Tools Towards O p t i m a l A u c t i o n s pot v' Max{ÏÏ} fee*(i) Max{b} r* A P E P 3 0.9448 0.9514 -0.01496 1.8447 1.3707 0.9413 0.8395 4 1.0458 1.0468 -0.01463 2.9872 2.2567 0.9503 0.9007 5 1.0625 1.0692 -0.00369 2.3447 1.6640 0.9166 0.8676 6 1.1104 1.1306 -0.00152 2.7703 1.8178 0.9325 0.8699 7 1.0867 1.0877 -0.00202 3.9949 3.2271 0.9287 0.8933 8 1.0223 1.0233 -0.00064 2.6707 2.0821 0.8799 0.8102 9 1.0706 1.0875 -0.00013 1.7973 1.2658 0.9132 0.8918 10 1.0744 1.0940 -0.00057 1.9711 1.2857 0.9473 0.8163 11 1.1016 1.1435 -0.00019 1.9684 1.3002 0.9422 0.8280 12 1.0959 1.1019 -0.00010 1.8021 1.3074 0.9319 0.9105 13 1.0787 1.1214 -0.00001 1.5400 1.1252 0.9129 0.7585 14 1.0396 1.1018 0.00002 1.4922 1.0767 0.8901 0.8669 15 1.1108 1.1492 -0.00001 1.5305 1.1527 0.9259 0.8205 (i) pot: Number of potential bidders. (ii) v': Optimal cutoffs, calculating according to equation (1.11). (iii) Max{v}: Maximum of recovered valuations, i;(after trimming near boundaries). (iv) fee*((5): Entry fee/subsidy in the counterfactual experiment, where the reserve price is set as 1.1. (v) Max{b}: Maximum of observed bids. (vi) r*: Optimal reserve price, when no entry fee or subsidy is in effect. (vii) A P (Actual Payment): Average actual payments (buying prices from data). (viii) E P (Expected Payment): Expected payment in the counterfactual experiment, where the reserve price is 1.1 and entry fee/subsidy is in effect. Figure 1.1: Rate of P a r t i c i p a t i o n Rate of Participation * rate of participation (p) 90% confidence band 7 9 11 13 15 17 19 21 Number of Potential Bidders Figure 1.2: P a r t i c i p a t i o n Costs Participation Costs 0.031 1 1 1 , •S 0.025 - Number of Potential Bidders F i g u r e 1.4: J / ( l - F) for N = 12 J/(1-F) for N=12 0.4 0.6 0.8 1 Valuations (relative to engineer's estimates) 1.2 1.4 Chapter 2 Estimating the First-Price Auction Model with Entry: A Parametric Approach 2.1 Introduction A fact i n the rea l -wor ld auctions is that not a l l the eligible bidders ac- t u a l l y submit a b i d , even though they have i n i t i a l l y shown their interest on the auct ioned object. T h i s observation suggests that i t may be costly p a r t i c i p a t i n g the auct ion . M o s t recently, several empir i ca l works have at tempted to s tudy the auc- t i o n models w i t h endogenous entry. Athey , L e v i n and Seira (2004) [4] s tudy the par t i c ipa t i on p a t t e r n i n the t imber auctions w i t h costly entry. K r a s - nokutskaya a n d Se im (2006) [27] consider two types of bidders dec iding to enter the auctions by introduc ing b i d preference programs. L i a n d Zheng (2005) [34] propose a semi-parametr ic Bayes ian method to j o in t l y estimate entry a n d b i d d i n g . B a j a r i , H o n g and R y a n (2004) [6] use a l ikehhood-based es t imat ion approach i n the presence of mult ip le equi l ibr ia . A l l of these works estimate variants of an entry model based on L e v i n and S m i t h (1994) [31] [hereafter, L S ] , i n w h i c h a l l the potent ia l bidders have no in format ion on their va luat ion of the auct ioned object when deciding whether or not to enter. B y i n c u r r i n g a cost, they then can become informed on the va lua- t ions a n d submi t a b i d . T h e equ i l ib r ium of the mode l predicts that a l l the potent ia l b idders randomize their decisions on entry so that their expected payoffs f rom entry become zero. In m a n y appl i cat ions , fiowever, tfie equ i l i b r ium condit ions for L S mode l are h a r d to just i fy . For example, consider the highway procurement auctions i n U S , w h i c h are the appl icat ions studied i n those recent empir i ca l works. T h e D e p a r t m e n t of Transpor ta t i on ( D O T ) i n the state usual ly announces the project under auct ion a few weeks before the le t t ing date. T h e D O T also describes the jobs and t ime frame involved i n the project . T h o u g h actual part i c ipants i n the auct ion obta in more in format ion later, the descript ion provided at this stage s t i l l contains valuable in format ion for t y p i c a l potent ia l bidders, who are experienced construct ion companies. Therefore, i t does not seem reasonable to assume that potent ia l bidders have no in format ion on the projects u p o n their decision on part i c ipat ion . T h i s chapter considers an alternative entry mode l i n auctions. O u r model is based on a theoretical work by Samuelson (1985) [53], i n w h i c h the b i d - ders already know their valuations when m a k i n g the entry decisions. T h e endogenous entry is due to a cost incurred i n b i d submiss ion. In equi l ib - r i u m , the b idder does not submit a b i d i f and only i f his expected profit from p a r t i c i p a t i o n is not large enough to cover the b i d d i n g cost. Therefore, the zero expected profit from part i c ipat ion i n the L S mode l is discarded i n our framework. C h a p t e r 1 adopts the same entry model to s tudy M i c h i g a n H i g h w a y pro- curement auct ions . There we propose a non-parametr i c est imate for the entry cost, w i t h w h i c h we further suggest how to implement o p t i m a l auc- tions. W i t h i n the nonparametr ic framework, certain types of counterfactual experiments are impossible , because we have no way to nonparametr i ca l ly recover the d i s t r i b u t i o n function below the cut-off po int . For this reason, we sometimes employ a parametr ic model to investigate the consequences of po l i cy changes i n an auct ion. I n this chapter, we develop a method to estimate the auct ion model w i t h endogenous entry based on a p a r a m e t r i - ca l ly specified value d i s t r ibut ion . W i t h the est imated parametr i c auct ion model , we can conduct various counterfactual experiments invo lv ing the en- t ire value d i s t r i b u t i o n . T h i s , for example, lets us investigate the effectiveness of asymmetr i c o p t i m a l mechanism (Cel ik and Y i l a n k a y a (2006) [8]). A challenge i n est imat ing an auct ion model is computat ion . W e need to calculate mul t i - f o ld integral each t ime we evaluate the o p t i m a l b idd ing funct ion , for example . A remedy for this type of problems is the use of a s imulat ion-based method . I n a seminal work by Laffont , Ossard a n d V u o n g (1995) [28] [hereafter, L O V ] , a s imulation-based est imator is proposed to esti - mate one of the simplest theoretical auct ion models, the f irst-price sealed b i d independent private-value auctions (without entry) . I n terms of es t imat ion m e t h o d , th is chapter can be viewed as an extension of L O V . However, there is an impor tant difference between L O V and our work. I n L O V ' s mode l , the d i s t r ibut i on of the observed bids is t runcated by a constant (and observed) reserve price, which does not depend on the parameters to est imate. O n the contrary, the t runcat ion point of the b i d d i s t r i b u t i o n in our mode l depends on the parameters i n est imat ion , because the entry decision depends on the private-value d i s t r ibut ion . It turns out that this feature of our p r o b l e m complicates the large sample analysis of the s imulated least squares m e t h o d , m a k i n g the est imat ion objective funct ion non-smooth i n the parameters . Fo l l owing P o l l a r d (1985) [51], we derive a set of sufficient condit ions for the asymptot i c properties of our est imator . Therefore, our work compl iments to the l i terature by showing the app l i cab i l i t y of L O V ' s framework to the context of endogenous entry. T h e rest of th is chapter is organized as follows. W e first describe our auct ion mode l w i t h endogenous entry and develop a s t r u c t u r a l econometric mode l based on i t . W e then propose a method to estimate the econometric mode l a n d s tudy the asymptot ic properties of the est imator . In the last section we conclude the chapter w i t h some discussion on possible extensions of th is work. T h e mathemat i ca l proofs are collected i n the appendix . 2.2 Methodology 2.2.1 T h e first-price a u c t i o n m o d e l w i t h e n t r y W e consider a f irst-price sealed-bid auct ion of a single ind iv i s ib le good. However, dev ia t ing from the s tandard I P V framework, we exp l i c i t l y model an entry stage pr ior to the b idd ing . T h e b idd ing process evolves as follows. F i r s t , w i t h i n the independent private-value ( I P V ) framework, each poten- t i a l r isk neutra l part i c ipant i € {1,2, . . . , A''} knows her own value Vi for the object, but only knows the d i s t r ibut i on of the values to the other poten- t i a l b idders . It is assumed that the values to ind iv idua l s are independently d r a w n from the absolutely continuous d i s t r ibut i on F{v) ( w i t h its p d f fy) w i t h support [v,v] C K + . In the second stage, there presents a common p a r t i c i p a t i o n cost, K, w i t h w h i c h each b idder has to pay to part ic ipate in the auct ion . G i v e n her p r i - vate value, the bidder decides whether or not to submit a b i d (paying K) and becomes an ac tua l bidder. A l l potent ia l bidders make this decision s i m u l t a - neously. T h e y therefore make their par t i c ipat i on decisions w i t h o u t knowing how m a n y compet i tors they are actual ly going to face. F i n a l l y , b idders submit their bids s imultaneously and the object goes to the highest b idder . T h e winner pays her b i d to the seller, prov ided that the b i d is no less t h a n the reserve price r , which is assumed to be zero i n this paper w i t h o u t loss of generality. W e focus on the unique symmetr ic Bayes ian N a s h e q u i l i b r i u m of the auc- t i o n game, (see Milgrom(2004)[37]). In equ i l i b r ium, each po tent ia l par t i c - ipant jo ins the auct ion if her value is no less t h a n , Vp, the cut-off point ( common to a l l bidders) , otherwise chooses not to part i c ipate . T h e cut-off po int Vp is such that i t is indifferent to the part i c ipant w i t h value Vp whether or not to enter the auct ion . T h u s , Vp should solve the equat ion VpF{vpf-' - K - 0. (2.1) T h r o u g h o u t the chapter, we assume that the entry cost is moderate so that Vp is always an interior point i n [v,v]. T o derive the e q u i l i b r i u m b idd ing strategy for bidder i, define i 's ex- pected profits ( I i i ) from par t i c ipat i on as follows, Uiivi^y, {bj)j^i) = {vi - y)[F{max{vp,b-\y)))f-^ - AC, where y is the bidder i ' s b i d given Vi and a l l other parameters. b~^ is the inverse b i d d i n g strategy for bidders. M a x i m i z a t i o n of 11̂ w i t h respect to y yields the e q u i h b r i u m b i d d i n g strategy: b = eiv, N, F) = J^^'J:^ £ uf{u)F{uf-^du if v > Vp (2.2) A potent ia l b idder j part ic ipates i f a n d only i f Vj > Vp. A n important property is that 6 as a funct ion of v, is s t r i c t ly increasing on [vp,v]. Hence, the e q u i l i b r i u m is reveal ing, provided that Vi > Vp. D u e to the s tr i c t monotonic i ty of b idd ing strategy (2.2), the winner of the auct ion must be the bidder w i t h the highest pr ivate value, prov ided that her value is no less t h a n Vp. T h u s , the w i n n i n g b i d b^ can be w r i t t e n as: 6 - = e ( « ( i ) , i V , F ) , (2.3) where ^(i) denotes the largest order statist ic among vi, ...v^. If < Vp, i t represents a case that no part ic ipants submits a b i d and , therefore, w i n n i n g bids are undefined. T h e we l l -known Revenue Equivalence Theorem implies that b"" = e{v, N, F) = E[max(^;(2), F- i (p (Ar) ) )| i ; ( i ) = v, N, F] (2.4) (see, for example , M i l g r o m and Weber (1982)[38], M i l g r o m (2004)[37]). v^2) denotes the second largest order statist ic among vi, ...VN, is the inverse funct ion of F, a n d p{N) = F{vp\N) represents the rate of par t i c ipat ion condi t iona l on N. A n intu i t ive explanat ion to equation (2.4) is that , condi - t i ona l on w i n n i n g , the bidder would just b i d the expected level of the second highest pr ivate values. T h e difference between her b i d a n d the pr ivate value is n o r m a l l y ca l led the in format ion rent for the bidder. T a k i n g the expectat ion w i t h respect to v, (2.4) gives B[b^\N,F] = E[max{vç2),F'\piN))}]. T h i s cond i t i ona l expectat ion can be viewed as an integral w i t h respect to the density i n the fol lowing way: E{b'^\N,F) = j^max{u(2),F-\p{N))}U{ui)...U{uN)dui...duN (2.5) where U(2)is the second largest order statist ic among u i , . . .ujy- (2.5) provides us a way to s imulate the moment of w i n n i n g bids i n our es t imat ion method . 2.2.2 T h e s t r u c t u r a l e c o n o m e t r i c m o d e l W e now specify an econometric model based on the theoret ical auct ion mode l w i t h endogenous entry discussed above. W e here impose a s impl i fy ing assumpt ion that the number of potent ia l bidders n is the same across a l l auctions under study. W h i l e this assumption is not essential i n order for the es t imat ion m e t h o d described below to work, i t allows us to avoid excessive compl i ca t i on i n our large sample analysis. A l s o , the constancy of the number of po tent ia l bidders is often reasonable i n practice. For example , pr ior to the highway procurement auctions, D O T requires the qual i f i cat ion check for the construct ion firms to be eligible for any par t i cu lar types of jobs at the beg inning of each year. T h e n , when let t ing process starts , only those eligible bidders can get involved. T h i s quali f ication stage effectively restricts the set of potent ia l b idders for the year. L e t Vu denote the value of the auctioned object for the i t h potent ia l bidder i n auct ion t. O u r auct ion model requires that va be independently and iden- t i ca l l y d i s t r ibuted across potent ia l bidders. W e here add the independence of Vit across auctions. N a m e l y : A s s u m p t i o n 2.1 {vu : i = 2,... ,n, t £ N} is an i.i.d. array of ran- dom variables absolutely continuously distributed with a pdf fviv;ôo) that is strictly positive and continuous on [v,v], where OQ lives in O , a compact, non-empty subset ofW. T h e w i n n i n g b i d is given by (2.4), as long as at least one potent ia l bidder ac tua l ly part ic ipates i n the auct ion. However, the w i n n i n g b i d is not defined when the auct ioned object is unsold , that is, when < Vp^t- T o provide technical convenience for the discont inuity point at v^iyt — '^p,t, we extend (2.4) to M i n the fol lowing way. br = E [max {t ; (2 ) ,F- i ( /5)}|T ; ( i ) = v]l{v^,^ > F-\p)} +F-\p)l{v<2) < F-\p)}, (2.6) where l { . } is notat ion for the indicator funct ion w h i c h takes the value of one i f the logical condi t ion inside i t is satisfied, and zero elsewhere. L e t H{-\n) denote the cumulat ive d i s t r ibut i on of 6^ condi t iona l on n for n > 1. It satisfies that ^ ^ ' 1 - F ( u p ) " I n this equality, H{-\n) can be est imated by us ing the observations of w i n n i n g bids for each n > 1, and F{vp), wh i ch is the probab i l i ty of par t i c ipa t i on i n the auc t i on , can be est imated i f the number of ac tua l bidders is observed for each auc t i on along w i t h the number of po tent ia l part i c ipants . W h e n es t imat ing a parametr ic probabi l i ty mode l l ike ours, the m a x i m u m l ike l ihood ( M L ) est imat ion method is usual ly taken as the first choice, as the M L m e t h o d provides a more efficient estimates t h a n any other alternatives. Unfor tunate ly , M L est imation involves two major difficulties i n auct ion ap- pl icat ions , w h i c h makes i t hard to implement . F i r s t , i t has been shown that the support of the b ids ' d i s t r ibut ion depends on the parameter 9, wh i ch violates the regular i ty assumptions under ly ing the fami l iar nice properties of M L est imat ion . D o n a l d and Paarsch (1993) [14] address the issue i n de- t a i l . In p a r t i c u l a r , they show that the M L est imator has a nonstandard asymptot i c d i s t r i b u t i o n . T h e second difficulty, which is not l imi ted to the M L m e t h o d , lies i n com- p u t a t i o n . I n general , the equ i l ib r ium b i d d i n g funct ion (2.2) is h ighly non l in - ear i n pr ivate values. E v a l u a t i o n of the b i d d i n g funct ion (2.2), w h i c h has no closed-form expression, therefore requires intensive c o m p u t a t i o n . T h i s fea- ture makes i t nearly imprac t i ca l to estimate the mode l by the M L method or any other methods that require exact evaluat ion of the inverse b idd ing funct ion . To overcome the above-mentioned problems, we propose us ing L O V ' s s imulat ion-based est imation method , which does not require the exact com- p u t a t i o n of either the equi l ibr ium b idd ing strategy (2.2) or the moments of the w i n n i n g b i d . 2.2.3 S i m u l a t e d n o n l i n e a r least squares e s t i m a t o r W e now describe our s imulated nonlinear least squares es t imat ion method, w h i c h is based on the first moment of w i n n i n g bids. L e t l{9) denote the expression obta ined by replacing fy w i t h fy{-,0) i n the r i g h t - h a n d side of (2.5). T h e usual N L L S estimator should minimizes an object ive funct ion QNLLS ^ ( l / T ) ^ ^ ^ i [ 6 J " - l{9)]^ w i t h respect to 9. G i v e n that l{9) is diff icult to evaluate, we replace l{9) w i t h an unbiased s imulator X{9), i.e., a s imulator such that E [ X ( e ) ] = l{9). There are several ways to simulate the condi t iona l mean of w i n n i n g bids. Fo l l owing L O V , we use importance sampl ing to s imulate the integrals in (2.5). A s noted i n L O V , this s imulat ion technique can reduce the sam- p l i n g var iab ih ty of the r a n d o m draws a n d hence improve the precis ion of s imulat ion-based estimators. L e t 3 be a k n o w n continuous density w i t h a support V g at least conta in ing the support of fviv). T h e n , (2.5) can be rewri t ten as 1(6) = f m a x { w ( 2 ) , F - i ( p , ^ ) } JVg fv{ui;6)...Uun\e) g{ui)...g{un) g{ui)...g{un)dui...dun For each t = 1, . . . , T , we draw S independent samples, each of size n . T h e n , for every t, lt{d) can be est imated by the sample mean: X ( « " , . . . , « ^ * , ^ ) = i5^X(««*,^), where u * ' (s = 1 , 2 , . . . , S ) is an i.i.d. n x 1 r a n d o m vectors, whose compo- nents are independently and ident ical ly d i s t r ibuted w i t h density g, and Xiu-,e) = maxK^) ,F -^ (pO, , ) ] / -K^g- - /^^^ (2.7) T w o features regarding the s imulat ion are wor th not ing . F i r s t , E [ X ( u ^ ' , u^^,6)] = l{9) by the construct ion. Second, a l l the r a n d o m draws u are independent of 6 a n d are drawn before est imat ion . T h u s , for any given 6, the variables X ( u * ' , ^ ) , (therefore, X ( u ^ * , ...u^*,6)), are independent of bf. N o w , we are ready to define our s imulated nonlinear least square est ima- tor , 6, w h i c h should min imize the objective funct ion , for any fixed number of s imulat ions S: 1 ^ _ - s { s ^ ) ^ ~ ( ^ - ^ ^ T h e second t e r m on the right h a n d side of (2.8) is the bias correct ion term. W e show i n the next section that incorporat ing such a s imple adjustment t e r m enables QT un i f o rmly converges to Q{0) = E[6J" - 1(0)]^ a.s. 2.2.4 A s y m p t o t i c P r o p e r t y of 6 There exists a significant difference between our objective funct ion (2.8) a n d the one used i n L O V . Due to the part i cu lar t r u n c a t i o n feature induced b y the entry cost i n our entry model , 6 appears i n the " m a x " operator i n (2.7). Hence, our objective funct ion defined i n (2.8) is not twice continuously differentiable i n 6. Therefore, a separate invest igat ion on the asymptot i c properties of 6 is needed. I n this subsection, we derive a set of sufficient condit ions that ensures 6 is b o t h consistent a n d asympto t i ca l l y normal ly d i s t r ibuted . To ease the expos i t ion , we use the fol lowing notations. /(, is the density funct ion of the d i s t r i b u t i o n for observed bids whose support is denoted as V b . |{.|| denotes the E u c l i d e a n norm. Moreover , 1 ^ — W e omi t the subscript and superscript in the proofs whenever there is no confusion. Consistency W e assume the fol lowing for the the desired consistency property. A s s u m p t i o n 2.2 (Consistency:) (i) ^0 uniquely minimizes Q over 9 . (ii) 7^ = sup0çe,^Glt!,ïï] < oo. (iii) g = sup„gVg 5(w) < oo. (iv) g = inf„evg 9{u) > 0. (v) S is a fixed natural number no less than 2 and independent oft. A s Q is continuous on the compact parameter set 6 , Q at ta ins its m i n - i m u m on O . A s s u m p t i o n 2.2(i) requires there be only one min imizer . A s - s u m p t i o n 2.2(ii) states that is bounded. T h i s assumption is generally assumed i n the auct ion setup, as otherwise the b i d d i n g strategy (2.2) w i l l not be well -defined. A s s u m p t i o n 2.2(hi)-(iv) is not restrict ive i n the sense that g is chosen by the researcher. T h e restr ict ion can ho ld w i thout much dif f iculty i f g is specified i n most of the common d i s t r i b u t i o n family. T h e last i t em i n assumpt ion 2.2 reinforces that our asymptot i c analysis is for a fixed number of s imulat ions as T —» oo. T h e fo l lowing l e m m a shows that our sample objective funct ion QT is un i f o rmly consistent to Q, w h i c h is m i n i m i z e d at OQ. T h e l e m m a is used to prove the consistency result . L e m m a 2.1 (Uniform Consistency:) Given Assumptions 2.1, 2.2(ii)-(v), s u p e g e l Q T ( ^ ) - Q ( ^ ) H O . T h e fo l lowing propos i t ion states the consistency property of 6. P r o p o s i t i o n 2.2 Given Assumptions 2.1 and 2.2, 9 is a strongly consistent estimator of OQ as T ^ OO. A s y m p t o t i c N o r m a l i t y T o derive the asymptot i c normal i ty result , we first introduce the fol lowing notat ion . L e t /io = Vi (^o ) , (TQ = Var[6]. L e t Xso = X{u^,6o) a n d Yso = VgXiu^Oo). W e further impose the following assumption for the asymptot i c d i s t r i b u - t ion . A s s u m p t i o n 2.3 (Asymptotic Normality:) (i) ^0 is an interior point of Q. (ii) { w i , . . . , u „ } are a random sample from the distribution with density g that is continuous. (iii) m{0^e,ve[v,v] f{v,0) = > 0. (iv) fv{v,.) : Q i-^R is Lipschitz uniformly in v e [v,v]. (v) For all v e [v,v], fy{v,.) : O R is twice continuously differentiable and the Henssian matrix V'^fviv,6) satisfies that supy^^^g^Q \\v^fy{v,e)\\<oo. (vi) For all v e [v,v], F{v;.) : © i—» R is twice differentiable and the Henssian matrix V'^F{v,6) satisfies that sup„g[j,_^] ^ I ^ Q ||V^F( i i , )̂|| < oo. (vii) For all v £ [v,v] and O&Q, VeF{v,e) ^ 0. (vii i ) C = /XO/ZQ is nonsingular. A s s u m p t i o n 2.3(iii) asks fy to be bounded away from zero. In the auc- t i o n sett ing , i t is usual ly assumed so. Otherwise the b i d d i n g strategy (2.2) is not s t r i c t l y increasing, which further hurts b o t h the uniqueness of b i d - d i n g e q u i l i b r i u m a n d the identi f ication of s t ruc tura l analysis . Assumpt ions 2.3(iv)-(vii) impose condit ions on fy and F over the parameter set 0 . These restrict ions can be satisfied i f fy is d iosen from ttie commonly used d i s t r i - but ions . Espec ia l ly , i n most of empir i ca l auct ion works that use parametr ic approaches, the pr ivate values are usual ly specified to follow n o r m a l or log- n o r m a l d i s t r ibut ions . T h e n these assumptions ho ld . Before in t roduc ing the asymptot ic d i s t r ibut i on result , we present the fol- lowing l e m m a . L e m m a 2.3 Given Assumptions 2.1, 2.2, and 2.3, Xgo and Yso have finite second moments. Together w i t h A s s u m p t i o n 2.3(vi i i ) , the above l e m m a ensures that the co- variance m a t r i x i n the proposi t ion 2.4 is well-defined. W e now present the fol lowing proposi t ion , w h i c h states the asymptot ic n o r m a l i t y for 6. P r o p o s i t i o n 2.4 Given Assumptions 2.1, 2.2 and 2.3, T,~'^/'^VT(6-Oo) is asymptotically distributed with N{0,ïp) as T oo, where E = C~^AC~^, where A == fr^(|var[no] + MMo) + |var[X,o]/̂ oMo + ; ç ^ ^ ^ [ V a r [ X , o ] V a r [ n o ] + C o v ( X , o , n o ) C o v ( X , o , ^o)']. O u r proof for P r o p o s i t i o n 2.4 makes a connection i n the l i terature . P o l - l a r d (1985) [51] relaxes the dif ferentiabil ity assumptions on the stat ist ica l objective funct ion . T h i s effectively breaks down the asymptot i c normal i ty for the extreme estimators under wel l -known proofs. H e describes some new techniques to prove the central l i m i t theorems. W e benefit f rom his work by using the theorem 2 i n his paper for the desired asymptot i c result . H o w - ever, w h e n checking the stochastic dif ferentiabil ity condi t ion required by his theorem, two methods suggested i n his work, bracket ing a n d combinator ia l methods are h a r d to apply i n our context, i f at a l l . To avoid this diff i - culty, we instead employ another approach mentioned i n Pakes and P o l l a r d (1989) [46], by ver i fy ing the E u c l i d e a n property of a class of functions. In terms of efficiency, i t is possible to compare our S N L L S est imator to the usual N L L S est imator , w h i c h has to compute l{8). T h e asymptot ic variance-covariance m a t r i x of N L L S estimator wou ld be C ~ M C " ^ where A = (To/ioA'o- Therefore, our S N L L S estimator is less efficient t h a n N L L S est imator by a t e r m of order (l/S). T h i s efficiency loss is due to the s imula - t i on involved i n our est imat ion . If, however, the number of s imula t i on draws S can be increased to infinity, then our S N L L S est imator 6 becomes asymp- tot i ca l ly efficient as the usual N L L S estimator. Moreover , as we ignore the auct ion heterogeneity i n analysis, the N L L S est imator is asympto t i ca l l y ef- ficient i n the class of estimators that uses moment res tr i c t ion Efè*"] = IÇOQ). For the purpose of conduct ing inferences on ̂ o, i t is an essential question on how to consistently estimate the variance-covariance m a t r i x , E . T h i s m a t r i x invloves /XQ, CTQ, a n d the variances and covariances of Xgo and YSQ. A l l the variance a n d covariance terms can be easily est imated , but /XQ- A s /xo = V/ (^o ) , i t cannot be expl i c i t ly determined for the same reason as l{6o) cannot . W e follow L O V ' s proposal to s imulate /UQ th rough Ygo, w h i c h is also consistent for a fixed number of s imulations as T goes to infinity. W e define _ _ 1 ^ _ _ e = Y{e)Y{êy - j ^ ^ ^ ^ ( n w - Y{e)){Ys{e) - Y{ê)y Note that the way we construct C is exactly same as we define QT- It then follows that C7 —» C for any fixed S. Denote A = A A ' , where _ _ 1 ^ _ Â = {(6 - xiê))Y{ê) + gj^rrY) - ^(oWsiê)]}- T h e n , for any fixed S, our est imator S := C ~ ^ A C ~ ^ is consistent for S , as shown i n L O V . 2.3 Conclusion and extensions A seminal work by L O V provides a new parametr ic est imat ion strategy for ana lyz ing auct ion data . T h e y propose a s imulated N L L S est imator to approx imate the bidders private-value d istr ibut ions . T h e i r work greatly broaden the class of d is tr ibut ions that empir i ca l researchers can handle when analyz ing auct ion d a t a sets. T h i s chapter can be viewed as an extension to L O V ' s work. W e apply their method to a f irst-price auct ion mode l w i t h endogenous entry. T h e part i cu lar entry p a t t e r n i n such models makes L O V ' s objective funct ion no longer twice cont inuously differentiable. T h e loss of smoothness on the stat ist ica l objective demands another investigation on the asymptot i c properties for L O V ' s est imators . W e derive a set of sufficient condit ions for the consistency a n d asymptot i c normal i ty to r emain va l id . There are several lines along w h i c h this work can be extended. F i r s t , i n some appl icat ions , the econometricians can observe more t h a n w i n n i n g bids. If the other observed bids are assumed to be of e q u i l i b r i u m b idd ing , they surely are in format ive on the private-value d i s t r i b u t i o n as wel l . However, w i t h i n the framework of endogenous entry, inc lud ing these bids i n the esti - m a t i o n w i l l introduce other compl icat ion i n asymptot i c analysis . Because the bids are no longer completely independently d i s t r ibuted , the develop- ment of the proposed est imat ion strategy for other types of auctions seems appeal ing . T h i s chapter assumes the auctions are homogeneous. I n empir i ca l work, this assumpt ion can typ i ca l l y be justi f ied by contro l l ing the auct ion het- erogeneity through one random variable . T h e n a l l the observed bids can be normal i zed so that a l l auctions are taken as homogeneous. However, i n most of the appl i cat ions , f inding such a one-dimensional variable to homog- enize the auctions may be a h a r d mission, i f possible at a l l . T h u s t a k i n g care of b o t h the observed and unobserved auct ion heterogeneity i n empir i ca l analysis can be an interesting direct ion to pursue i n the future. If the auct ion heterogeneity is exphcit ly considered, the es t imat ion method of N L L S may not be asymptot i ca l ly efficient i n general. One possible way to improve efficiency, as proposed i n C h a m b e r l a i n (1987) [9], is to construct an est imator exp lo i t ing the orthogonal ity condit ions by E [ W [ 6 - i(^o))] = 0. W can be a vector of auction-specific variables. S imi lar ly , Pakes and P o l l a r d (1989)[46] consider the method of s imulated moments i n their appl ications. However, how one can carefully choose W to ga in efficiency is an unsett led question. T h e proposed es t imat ion method involves s imula t ing the moments of the observed bids . It is a v i t a l role that the number of r a n d o m draws S plays i n the s imula t i on . Specifically, S appears i n the variance-covariance m a t r i x . Therefore, i t is not a t r i v i a l question how the choice of S is going to affect the es t imat ion accuracy i n the finite sample. W e leave this for the future research. Chapter 3 What Model for Entry in First-Price Auctions? A Nonparametric Approach 3.1 Introduction Sealed tenders are a widespread mechanism for procur ing goods and ser- vices i n the U n i t e d States. T h i s is a large a n d impor tant market , and unders tanding its workings is a topic of general interest. A robust and wel l -documented feature of many real -world auctions is that not a l l b i d - ders who are el igible to submit a b i d choose to do so, suggesting that entry into the auc t i on may be costly. I n this chapter, we develop nonparametr ic approaches that w i l l al low the empir i ca l researcher to d iscr iminate among different models of entry. M o s t of the empir i ca l auctions l i terature to date is based on the theoretical work of L e v i n a n d S m i t h (1994) [31] ( L S hereafter). I n their mode l , potent ia l b idders are i n i t i a l l y uninformed about their valuations of the good, but may become in formed a n d submit a b i d at a cost. In e q u i l i b r i u m , the potent ia l entrants randomize their entry decisions a n d earn zero expected profit . Several e m p i r i c a l papers, most of them recent, est imated variants of this mode l . B a j a r i a n d Hor tacsu (2003) [5] have studied entry and b i d d i n g i n eBay auct ions , w i t h i n a common value framework. A Bayes ian est imat ion method is implemented using a dataset of mint a n d proof sets of U S coins. T h e magni tude of the entry cost is est imated, and expected seller revenues axe s imulated under different reserve prices. Athey , L e v i n a n d Seira (2004) [4] estimate a mode l of b i d d i n g i n t imber auctions w i t h costly entry. T h e entry cost is assumed to be private in format ion of the potent ia l bidders, who sort into the poo l of entrants based on their draws of the entry cost. L i a n d Zheng (2005) [34] study entry and b idd ing for lawn mowing con- tracts us ing the L S model . To our knowledge, it is the first paper i n the l i terature that uti l izes the number of planholders as a measure of potent ia l c ompet i t i on i n highway procurement. L i and Zheng (2005) [34] propose and implement a Bayes ian est imat ion method a n d use their s t r u c t u r a l estimates to investigate the effect of restr ict ing potent ia l compet i t i on on the expected revenue. I n add i t i on , L i (2005) [32] develops a general parametr ic approach for auctions w i t h entry. Krasnokutskaya and Se im (2006) [27] s tudy b i d pref- erence programs a n d bidder par t i c ipat ion using C a l i f o r n i a da ta . T h e i r paper also uses the L S mode l , and as i n Athey , L e v i n a n d Seira (2004) [4], the fo- cus is on asymmetr i c equi l ibr ia . B a j a r i , H o n g a n d R y a n (2004) [6] propose a parametr ic l ikel ihood-based est imat ion strategy i n the presence of m u l t i - ple equ i l ib r ia , a n d app ly it to highway procurement auctions, using the L S mode l . A n a l ternat ive mode l of entry was developed i n Samuelson (1985) [53] (S hereafter). In this mode l , bidders make their entry decisions after they have learned their valuat ions . T h e entry cost is interpreted solely as the cost of prepar ing a b i d , a n d bidders choose to enter i f their valuations exceed a certain cutoff. T h e set of entrants is therefore a selected sample, biased towards bidders w i t h higher valuations. T o the best knowledge of ours, chapter 1 of th is thesis is the only work app ly ing Samuelson model to d a t a so far. B o t h L S a n d S models are styl ized to capture the amount of in forma- t i o n available to bidders at the entry stage: no in format ion is available i n L S , whi le the in format ion is perfect i n S. These polar assumptions lead to drast i ca l ly differing po l i cy impl icat ions . One of the most impor tant and wel l -studied po l i cy instruments i n auctions is the reserve price. In a seminal paper, R i l e y a n d Samuelson (1981) [52] show that , when the entry costs are n u l l , the o p t i m a l po l i cy for the seller is to set the reserve price above the level tha t he w o u l d be w i l l i n g to accept. Moreover , the o p t i m a l reserve price does not depend on the number of potent ia l bidders N . In the S mode l , while the o p t i m a l reserve price is also above the seller's wil l ingness to accept, it increases w i t h A''. L S , on the other h a n d , reach a s t r i k i n g conclusion that i t is o p t i m a l to set the reserve price at the m a x i m a l wil l ingness to accept level. G i v e n that po l i cy impl icat ions are so different, i t is impor tant to be able to d iscr iminate between these models empir ical ly . W e b u i l d on the insight i n H a i l e , H o n g a n d S h u m (2003) [19] (HSS hereafter) a n d propose to use exogenous var ia t i on i n A/' as a basis for such a test. L e t F{v) denote the cumulat ive d i s t r i b u t i o n funct ion ( C D F ) of valuations, and let F*{v\N) de- note the C D F for those potent ia l bidders that have s u b m i t t e d a b id . T h e C D F F*{v\N) is a c ruc ia l parameter whose behavior across N allows us to d i scr iminate among the alternative models of entry. Fo l l owing the approach of Guerre , Perr igne a n d V u o n g (2000)[17] ( G P V hereafter), we show that this d i s t r i b u t i o n can be nonparametr ica l ly identif ied i n b o t h models i f the number of po tent ia l bidders and a l l bids i n each auct ion are observed. W e show that , whi le F*(i;|7V) does not depend on TV i n the L S mode l , it does i n the S mode l . T h e i n t u i t i o n here is s imp ly that , i n the S mode l , the va lua- t ions of active bidders are t runcated by the entry cutoffs v*{N) that depend on TV, but a l l share the same parent d i s t r ibut i on across TV. T h i s imposes a res tr i c t ion on F*{v\N) across TV. It is not too h a r d to show that this res tr i c t ion impl ies a stochastic dominance order ing for F*{v\N): F*iv\N) > F*{v\N') for N' > TV. (3.1) I n other words, as TV becomes larger, the d i s t r i b u t i o n become more t i l t ed towards bidders w i t h higher valuations. T h i s is of course an intu i t ive i m p l i - ca t i on of selective entry. It is also t r i v i a l l y satisfied by the L S model , w i t h equal ity signs for a l l N. I n this chapter , we also propose a generalized mode l that allows for selec- t ive entry but dispenses w i t h the stark assumption that po tent ia l bidders perfectly know their valuations at the entry stage as i n S, thus sharing w i t h the L S mode l a costly valuat ion discovery stage. It f ormal ly nests the L S model . T h i s mode l , w h i c h we refer to as an affi l iated mode l of entry ( A M E hereafter), is as follows. A t the entry stage, the po tent ia l b idders each ob- serve a pr ivate s ignal correlated w i t h their yet u n k n o w n va luat i on of the good. Based on this private s ignal , a bidder may learn the va luat ion upon incurr ing an entry cost k. T h e bidder who entered w i l l on ly b i d i f the v a l - uat i on exceeds the reserve price. T h e signals may be in format ive about the valuat ions , however unlike i n the S model , they are not perfectly infor- mat ive . B o t h L S a n d S models can be viewed as its l i m i t cases: the L S mode l corresponds to uninformative signals, whi le the S mode l corresponds to perfectly informative signals. M o d e l s s imi lar to A M E have been looked at i n the l i terature . Hendr i cks , P i n k s e a n d Por te r (2003) [21] estimate a model of b i d d i n g for off-shore o i l . T h e y sketch a mode l of entry that is i n some respects s imi lar to ours, but w i t h a common-value component. T h e focus of their paper is however not on entry but on test ing an equ i l ib r ium model of b idd ing . T h e mode l is also out l ined i n the conc luding section of Ye (2007) [62]. To implement the tests, we follow the approach of G P V and show that the d i s t r i b u t i o n of entrants ' valuations can be nonparametr i ca l ly identif ied from the d a t a i f N a n d a l l bids i n each auct ion are observed. T h i s enables us to develop a nonparametr i c quantile-based test of selective entry i n the spir i t of H a i l e , H o n g a n d S h u m (2003) [19]. A l t h o u g h our approach shares w i t h Hai le , H o n g a n d S h u m (2003) [19] the basic idea that exogenous var iat ion i n the number of b idders can be used for test ing the in format ion environment of the game, there is a number of i m p o r t a n t differences. Hai le , H o n g and S h u m (2003) [19] consider a different model i n w h i c h bidders ' valuations may have a c ommon component. T h e y propose a test for c ommon values based on the var iat ion i n the number of actual b idders , whi le we test for selective entry using the var iat ion i n the number of potential bidders. O u r approach is also different i n the implementat ion i n that we use a direct quanti le es t imat ion method . T h e method is easy to implement , does not require the c omputat i on of pseudo values of G P V , and also allows arb i t rary form of dependence on covaxiates. T h i s last feature is p a r t i c u l a r l y important since the method of covariate control i n Ha i l e , H o n g and S h u m (2003) [19] is not appl icable i n the sett ing w i t h entry considered i n this chapter. W e make a number of observations about the identi f icat ion of model p r i m - itives i n the L S a n d S models. A s tandard reference for identi f icat ion in auctions is A t h e y and Hai le (2002)[2]. However, they do not address iden- t i f i cat ion i n models w i t h endogenous entry. These observations are discussed i n Sect ion 3.3. I n part i cu lar , i f the reserve price is b i n d i n g a n d there is no var ia t i on i n the number of potent ia l bidders, the entry cost i n the L S model is not identi f ied. T h e reason is that da ta allow an equivalent interpretat ion as be ing generated i n a model w i t h zero entry cost, a n d nonpart i c ipat i on is s i m p l y expla ined by the fact that some bidders draw valuations below the reserve price . Note that this exp lanat ion was or ig inal ly put forward i n Paarsch (1997) [44]. If there is var iat ion i n the number of po tent ia l bidders, then the entry cost may be identified. W e observe that a sufficient condi t ion for ident i f i cat ion is that the pat tern of the probab i l i ty of s u b m i t t i n g a b i d has a flat i n i t i a l segment followed by a decreasing segment, piN) = ...=piN,)<...<p(N). O n the flat segment, we are certain that bidders enter w i t h probab i l i ty 1 and nonpar t i c ipa t i on is due to the t runcat ing effect of the reserve price ^See also A t h e y and Haile (2005)[3]. only, a n d therefore are able to identify F{r) = 1 -p{N). O n the decreasing segment, we are certain that bidders are indifferent between entering or not , a n d are able to identi fy the entry cost from the indifference condi t ion given the knowledge of F{r). However, the estimate of the entry cost may be sensitive to model mis - speci f ication. W e show that , i f the data are generated according to a model w i t h selective entry (either S or our model) , but the researcher uses the L S mode l , the est imated entry cost w i l l be upward biased. Moreover , we show b y the way of an example that the bias may be severe. T h e i n t u i t i o n for this result is the fo l lowing. W h e n the entry cost is est imated i n the L S model , it is assumed that each potent ia l entrant is indifferent between entering or not , so that the entry cost is equal to the expected profit of a bidder who w i l l draw, so to speak, an average valuat ion upon entry. I n the models w i t h selectivity, the S and A M E , the entry cost is equal to the expected profit of a marg ina l b idder . W h e n the signals are posit ively correlated w i t h valuations, the va luat ion that a marg ina l bidder w i l l draw may p laus ib ly be less than the average va luat ion . I n our e m p i r i c a l appl i cat ion , we use a dataset of auct ions conducted by the O k l a h o m a Department of Transpor ta t i on ( O D O T ) . In add i t i on to a l l w i n n i n g a n d losing bids and certain project characteristics, we also observe the number of f irms that obtained construct ion plans, a variable that can serve as a reasonable proxy for the number of po tent ia l b idders . W e argue that , because the qual i f icat ion process essentially selects bidders based on w o r k i n g c a p i t a l requirements, the number of planholders may be assumed to be exogenous. T h e empir i ca l results are somewhat m i x e d , but we do have a number of f indings. F i r s t , the S model is robust ly rejected. Second, there is some support for the L S model , but somewhat more support for the A M E model . 3.2 Three models of entry and their testable restrictions 3.2.1 The LS and S models of entry T h e L S a n d S models share a common structure. There is an entry stage i n w h i c h N potent ia l bidders contemplate entry into the auct ion . A t the auct ion stage, a b i d d i n g game transpires among those bidders that have entered. T h e auct ion is f irst -price sealed b i d , possibly w i t h a reserve price r. O n l y the bidders w i t h valuations above the reserve price actual ly submit bids. W e cal l t h e m actual bidders. W e assume the Independent P r i v a t e Values ( I P V ) environment. T h e bidders ' valuations are d is tr ibuted according to the C D F F{-) that has support [v,v], a corresponding density /(•) posit ive on the support . E n t r y is costly; only the bidders that have incurred the entry cost k can b i d i n the auct ion . T h e two models differ i n the in format ion available at the entry stage. T h e L S mode l assumes that no in format ion is available. U p o n i n c u r r i n g the entry cost, the bidders learn their valuations a n d proceed to the b i d d i n g stage. O n l y the entrants w i t h v > r submit a b id . L e v i n and S m i t h characterize a symmetr i c perfect -Bayesian equ i l i b r ium of this game i n w h i c h bidders submi t a b i d w i t h probab i l i ty p G [0,1]. T h e e q u i l i b r i u m value of p as a funct ion of N is denoted as p{N). T h e equ i l ib r ium is character ized i n the fol lowing propos i t i on . W e assume that the reserve price is b i n d i n g , but the result carries over w i t h minor changes to the case when i t is not b ind ing . Proposition 3.1 (Levin and Smith, 1994[31j; Milgrom, 2004[37]) A symmetric equilibrium is characterized by the probability of submitting a bidp and bidding strategy B{v). The ex-ante equilibrium profit from bidding is equal to n ( p , N) - j\l - F{v)){l - p + pF*{v)f~^dv. (3.2) The equilibrium distribution of active bidders valuations is given by F*{v) = {{Fiv) - F ( r ) ) / ( 1 - F ( r ) ) ) and does not depend on N. Denote the equilibrium probability p as a function of N asp{N). IfU{l,N) > 0, then p{N) = 1, and ifn{0,N) < k, then p{N) = 0. Otherwise p{N) € (0,1) is determined from the zero expected profit equation U{p{N),N) = 0. (3.3) There is a quahf icat ion to be added to the above propos i t i on , as wel l as to s imi lar results for other models. Throughout the chapter , we assume away the uninterest ing case of the entry cost so large that there is no entry, p{N) = 0. T h e equ i l i b r ium b idd ing strategy B{v) is e x p l i c i t l y derived i n L S . T h e L S mode l has the fol lowing impl icat ions (we w i l l show later i n this chapter tha t these impl i cat ions are testable). F i r s t , since the profit function i n (3.2) is decreasing i n the r i va l b idd ing probab i l i ty p as wel l as i n the number of po tent ia l rivals N, we can see that the e q u i l i b r i u m probab i l i t y of s u b m i t t i n g a b i d is at least non-increasing, p{N) > p{N') y N <N', (3.4) w i t h s tr i c t inequal i ty i f N' is sufficiently large. Second, the d i s t r i b u t i o n of entrants valuat ions coincides w i t h the d i s t r ibut i on of po tent ia l bidders va l - uations. T h e C D F of valuations condit ional on entry F*{v) has the support [r, v] a n d is independent of N, F*{v\N) = F*{v\N') V N,N'. (3.5) In the S mode l , the potent ia l bidders know their valuat ions already at the entry stage. I n any symmetr ic equ i l ibr ium, a bidder whose va luat ion is at the lower end of the support , i» = u, is unable to w i n w i t h a posit ive probabi l i ty , a n d w i l l not enter. Samuelson shows that there is a cutoff v*{N) such that a b idder s t r i c t l y prefers to enter i f a n d only i f v > v*{N), so that the equ iUbr ium probab ihty of entry isp(7V) = l-F{v*{N)). Note that , since v*{N) > r , i n the S model this is the same as the probab i l i ty of s u b m i t t i n g a b i d . T h e e q u i l i b r i u m is formally characterized i n the fol lowing proposit ion. P r o p o s i t i o n 3.2 (Samuelson, 1985[53])The bidding stage has a unique symmetric equilibrium, in which the bidding strategy B(v) is an increasing and continuous function. The profit at the bidding stage of the marginal entrant with valuation v*{N) is given by {v*{N) - r ) ( l - p{N))^~^, where p{N) = 1 - F{v*{N)) is the probability of bidding. The cutoff v*{N) is determined by the requirement that bidder with valuation v* (N) makes zero expected profit: k = {v*{N) - r ) ( l - p{N))^-\ (3.6) There is always entry with probability less than 1, i.e. v*{N) e {r,v — k). T h e S mode l shares w i t h the L S mode l restr ic t ion (3.4) that b idd ing probabi l i t ies are non-increasing (they must ac tua l ly be s t r i c t l y decreasing i n the S model ) . B u t it implies a different restr ic t ion for the d i s t r i b u t i o n of active b idders valuat ions F*{v\N). For v > v*{N), 1 - F{v*{N)) Fiv)-{l-p{N)) piN) (3.7) where we used the fact that the entry probab i l i ty p{N) is equal to 1 - F{v*{N)). Since the d i s t r ibut i on F does not depend on A'', a m a n i p u l a t i o n of (3.7) leads to the fol lowing restr ict ion of the S model : p{N)F*iv\N) + 1 - p{N) = piN')F*{v\N') + 1 - p{N') V N, N'. (3.8) 3.2.2 T h e af f i l iated m o d e l o f e n t r y ( A M E ) A m o d e l of selective entry proposed i n this chapter occupies a middle ground between S a n d L S . Specifically, i t shares w i t h S the assumption that in f o rmat i on about the va luat ion is available at the b i d d i n g stage, but dispenses w i t h the stark assumption that this in format ion is perfect. T h e game begins w i t h the entry stage in w h i c h TV potent ia l r i sk -neutra l bidders ob ta in p r e l i m i n a r y estimates (signals) Si of their true values Vi; i t is assumed that this in fo rmat ion is available to them for free. U p o n observing Si, a bidder may expend an entry cost k, wh i ch results i n observing the true value Vi a n d entering the auct ion. O n l y the bidders that have learned Vi are eligible to submi t a b i d i n the auct ion. Moreover , only those w i t h valuations at or above the reserve price r submit a b id . W e assume that the pairs {Vi,Si) are ident ical ly a n d independently dis - t r i b u t e d across potent ia l bidders i = 1, ...,7V and are d r a w n from d i s t r i b u - t i o n F{v, s) w i t h support [v,v] x [0,1] and density f{v, s). For convenience, we assume that the marg ina l d i s t r ibut ion of the signals is un i f o rm on [0,1]. Since the in format iona l content of signals is preserved under a monotone t rans format ion , th is assumption is w i thout loss of generality. T h e entry stage is followed by the b idd ing stage. A c t i v e bidders draw their values Vi, a n d then simultaneously and independently submit sealed bids. A c t i v e bidders do not know the number of active bidders, on ly the number of po tent ia l bidders TV. T h e good is awarded to the highest b idder who pays its b i d . W e assume that the signals are informative a n d that higher signals are "good news" . Formal ly , we assume aff i l iation, i n the sense of M i l g r o m a n d Weber (1982) [38]. A s s u m p t i o n 3.1 For each bidder i, the variables {Vi,Si) are affiliated: for any z — {v,s) and z' = {v',s'), / ( m a x { 2 , ^; ' } ) / (min{z, z'}) = f{z)f{z'). Note that b o t h the L S and S models can be viewed as l i m i t cases of the A M E . T h e L S model is formally nested since i t corresponds to signals being independent of the valuations; this wou ld effectively pur i fy the m i x e d - strategy e q u i l i b r i u m . T h e S model corresponds to the other extreme, namely the signals a n d valuat ions being perfectly correlated. B u t because we assume existence of a j o int density of valuations a n d signals f{v, s), the S model is not nested. ^ A symmetr i c equ i l i b r ium of the A M E model can be characterized i n a manner s imi lar to the L S model . T h i s is done i n the propos i t ion below, whose proof is i n the A p p e n d i x (the proof also contains a f ormula for the b i d d i n g strategy) . Once again, we assume that the reserve price is b ind ing , but the result carries over w i t h minor changes to the case when it is not b i n d i n g . P r o p o s i t i o n 3.3 A symmetric equilibrium is characterized by a signal cut- off s such that only those potential bidders with Si>s choose to enter. The equilibrium probability of submitting a bid is P{s)^Pr{Si>s,V^>r}, (3.9) and the distribution of active bidders valuations is F*{v\s) = Pr{Vi < v\Si >s,Vi> r}. For any bidder i with signal Si = s, the equilibrium profit from is equal to n{s,s,N) = j\l - F{v\s)){l - P{s) + P{s)F*{v\s)f-^dv - k. (3.10) / / n ( 0 , 0 , TV) > 0, then s = 0, and all potential bidders always enter. If n ( 0 , 0 , TV) < 0, then s = 1 and there is no entry. Otherwise, the bidder with signal s is indifferent between entering or not so that s is determined from n{s,s,N) = 0. (3.11) Define the cutoffs a function of N as s{N). Ifs(N) G (0 ,1) , then s(TV') G (0,1) also for N' > TV, and s(N') > s{N). * A n interesting question that we do not address is whether the equi l ibrium of the S model can be supported as a limit point of our class of models. If the reserve price is non-b inding , i.e. r < v, some modi f icat ions are necessary. In the A p p e n d i x , we show that the entry equat ion (3.11) becomes = (1 - Pis))^-\v -r)+ Hi - Fiv\s)){l - P{s) + Pis)F*{v\s)f-^dv, Jv (3.12) where the presence of the first t e rm reflects the fact tha t the bidder w i t h type V makes profit by b idd ing the reserve price r a n d w i n n i n g the auct ion on ly when no one else enters. T h e b idd ing strategy also needs to be modif ied accordingly. Since the cutoff s{N) is non-decreasing i n TV, the p r o b a b i l i t y of b idd ing p(TV) = P{s{N)) is also a non-decreasing funct ion of TV. T h i s is the res t r i c t i on (3.4) that we have seen before. It is shared w i t h other models of entry considered i n th is chapter. B u t the restr ict ion on active b idders ' C D F F*(z;|TV) is different from either L S or S. To derive this condi t ion , note that F*{v\N) is equal to Pr{Vi < v\Si > s(TV), Vi > r}. T h e assumpt ion 3.1 impl ies that Pr{Vi < v\Si > s,Vi > r} is non-decreasing i n s (Theorem 23 i n M i l g r o m a n d Weber(1982)[38]). Since the cutoff s(TV) is non-decreasing i n TV, the F*(t;|TV)'s are t i l t e d towards bidders w i t h higher valuat ions : F*{v\N) > F*iv\N') V TV < AT' (3.13) Note that this restr i c t ion is impl i ed by restrictions (3.5) of the L S mode l as wel l as restr i c t ion (3.8) of the S model , but is clearly weaker. 3.3 Nonparametric identification In order to be able to test the restrict ions derived i n the previous section w i t h o u t m a k i n g parametr ic assumptions, it is necessary to show that the required quantit ies are nonparametr ica l ly identif ied. T h i s section is devoted to ident i f i cat ion of the models of entry considered i n this chapter. It is assumed that the econometrician can observe a l l the bids a n d there- fore also the number of active bidders n. A n impor tant add i t i ona l in forma- t i o n that is assumed to be also available is the number of po tent ia l bidders N. I n other words, we assume that the d a t a generating process identifies p{N) a n d G*{-\N) where p(N) = E[n\N]/N is the probab i l i ty of submi t t ing a b i d a n d G*{b\N) is the d i s t r ibut i on of entrants ' b ids condi t iona l on N. W e now show that i n a l l models considered i n this chapter, the d i s t r ibut i on F*{v\N) can be recovered from the first-order condit ions . O u r identi f ica- t i on strategy follows G P V . Consider first-order e q u i l i b r i u m condit ions of the b i d d i n g game. A bidder w i t h value v who submits a b i d b has a probab i l i ty of w i n n i n g over a given r iva l equal to 1 - p{N) + p{N)G*{b\N). Since there are i V — 1 ident i ca l r ivals , it follows by independence that the probab i l i ty of w i n n i n g is (1 - p{N) +p{N)G*{b\N))^-'^, and the expected profit is n(b,v) = ib-v){l~piN)+p{N)G*{b\N)f-\ W r i t i n g out the f irst-order condi t ion , i.e. t a k i n g the derivative of Û{b, v) w i t h respect to b a n d sett ing i t equal to 0, gives the inverse b i d d i n g strategy . . l-piN)+p{N)G*{b\N) mN) = b+ . (3.14) T h e inverse b i d d i n g strategy (̂•|A'̂ ) is identif ied f rom the observables, a n d i ts inverse, the b i d d i n g strategy B{v\N), is also identi f ied. T h e n the d i s t r i b u t i o n of active bidders ' valuations F*{v\N) is identi f ied according to F*{v\N) = G*{B{v)\N). It is interest ing to note that , i f there is no var iat ion i n A^, the L S a n d A M E models are observat ional ly equivalent. T h i s is because, even i f the true d a t a generating process corresponds to the A M E model , the d i s t r ibut i on F*{v\N) can be interpreted as the d i s t r ibut i on of valuations Vi condi t iona l on Vi > r that w o u l d arise i f the true model was L S . If the reserve price is non-b ind ing , r < v, then the S model is also observational ly equivalent. If, on the other h a n d , the reserve price is b ind ing , r e {v,v), then the S model is not observat ional ly equivalent since the lower bound of the support of F*{-\N), identi f ied as ^{b\N), must be greater than r i n the S mode l , but is equal to r i n b o t h L S and A M E models. C o n t i n u i n g to assume that N is fixed, a further interest ing question i f the mode l pr imi t ives are identif ied. In the A M E model , the pr imi t ives are the entry cost k a n d the d i s t r ibut i on F{v\s). Ne i ther is nonparametr i ca l ly iden- tif ied. T h e reason is that the d a t a generating process only reveals the d i s t r i - b u t i o n of b idders ' valuations, i.e. F*{v\N) = Pr{Vi < v\Si > s{N), Vi > r}, but not F{v\s). T h e knowledge of F{v\s) wou ld also be needed to identify the entry cost according to (3.11). T h e L S mode l is ful ly identif ied i f r < u a n d p{N) 6 (0,1) . T h e d i s t r i - b u t i o n F*{v) is equal to the d i s t r ibut i on of valuations of potent ia l bidders, a n d since Pr{Si > s} = p{N) e (0,1) , the entry cost is identi f ied from the indifference condi t ion k = {l-piN))^-\v-r) + Hi - F*{v)){l-p{N)+p{N)F*{v)f-'dv. Jv (3.15) Ii r < V a n d p{N) = 1, then we can only conclude that the reserve price is bounded from above by the expected profit that appears i n the r ight -hand side of (3.15). S i m i l a r l y , i f the reserve price is b ind ing , r G {v,v), the entry cost is not identi f ied. T h i s is because now k = {l-Fir))J\l-F\v)){l-piN)+p{N)F*iv)f-'di^.l6) p{N) = {l-F{r))Pr{Si = s{N)}, (3.17) a n d the m o d e l is observationally equivalent to the one w i t h = 0 so that Pf{Si > s{N)} — 1 and the probabi l i ty of b i d d i n g is equal to the probab i l i ty of d rawing Vi>r. In the S mode l , the d i s t r ibut ion of potent ia l b idders ' valuations is t r u n - cated (at v*{N) > v) even if r < v. Since the probab i l i ty of b i d d i n g is now equal to 1 - F{v*{N)), and v*{N) is identified as ^{b\N), i t follows that we can ident i fy F{v), but only for v > v*{N). O n the other hand , the entry cost is identi f ied, k = {v*{N) - r ) ( l - p{N))^-\ (3.18) Note that for the identi f ication of the S model , whether or not the reserve price is b i n d i n g plays no role. N o w assume that the number of potent ia l bidders N G N,N_ + I, ...jN where T£ < N. In other words, there is var iat ion i n N. It is easy to show that the var ia t i on i n N does not lead to the identi f icat ion of the pr imit ives of the A M E model . In the L S model , this var iat ion can sometimes (but not always) lead to identi f icat ion of the entry cost even i f the reserve price is b i n d i n g . Cons ider the fol lowing general pa t te rn for the probabi l i t ies of b i d d i n g p{N): p{N) = ...^piN.)<...<p(N) where < M. W e allow for b o t h Nt = N and A^, = N. I n the A p p e n d i x , we prove the fol lowing proposit ion. P r o p o s i t i o n 3.4 In the LS model with a binding reserve price, r G {v,v), k is identified if and only if G {N_ + 1 , N — 1}. T h e i n t u i t i o n for th is result is as follows. W h e n TV belongs to the fiat segment. A'" < 7V», we are certain that bidders enter w i t h probab i l i t y 1 and nonpar t i c ipa t i on is due to the t runcat ing effect of the reserve price only, a n d therefore are able to identify F{r) = 1 — p(N_). W h e n A'' belongs to the decreasing segment, TV > A^*, we are certain that bidders are indifferent between enter ing or not , and are able to identi fy the entry cost from the indifference cond i t i on given the knowledge of F{r), according to (3.16). A l s o note that i n the S model , the only imp l i ca t i on of the var ia t i on i n N is that we can identify the d i s t r ibut i on F ( - ) for a l l v = v*{N). O u r final remark i n this section concerns the bias i n the e s t imat i on of the entry cost. Suppose that the reserve price is n o n - b i n d i n g a n d p{N) < 1, so that the entry cost is identified i n the L S mode l . However , the data are generated according to A M E , w i t h str ict aff i l iation i n the sense that F{v\s) < F{v\s'), V U G [V,V],S' > s. (3.19) For s impl i c i ty , assume that the number of po tent ia l b idders N is fixed (this is not c ruc ia l ) , a n d that the researcher estimates the en t ry cost wrongly assuming that the d a t a are generated according to the L S mode l , i.e. as i f the cost was determined by equation (3.15). T h e difference from the correct ly specified mode l is that the researcher uses the w r o n g expression 1 — F*{v) instead of the correct one 1 — F{v\s{N)), because the true cost is given by (3.12). B u t w i t h the str ict affi l iation assumption (3.19), F*{v\N) > F{v\s{N)), so that i n the case of misspecif ication, true k is smal ler than the one produced b y (3.15). T h e i n t u i t i o n for the existence of this bias is as follows. T h e entry cost i n the L S mode l is equal to the equ i l ibr ium expected profit of the average potent ia l b idder , whi le i t is equal to the expected profit of the marg ina l potent ia l b idder ( w i t h signal s{N)) i n our model . Because the signals are pos i t ive ly re lated to the valuations, the marg ina l b idder m a y p laus ib ly have an entry cost on average smaller than the average bidder , so the entry cost w o u l d be overestimated. T h e same bias is present also when the d a t a are generated according to the S mode l , so that the true entry cost is given by equat ion (3.18). F r o m (3.15), the est imated entry cost i n the misspecified L S model is greater t h a n the true k. T h e fol lowing example helps i l lustrate that the bias may be very severe. E x a m p l e . Suppose that the valuations are un i f o rmly d i s t r i b u t e d on [0,1] a n d the reserve price is r = 0. T h e entry cost is k £ (0,1) a n d the true model is S. T h e cutoff v*{N) is determined by (3.18), a n d v*(N) = k^l^. T h u s the d i s t r i b u t i o n of active bidders ' valuations F*{v\N) is a t runcated uni form [0,1], w i t h a t runcat ion point given by v*{N). T h e probab i l i ty of s u b m i t t i n g a b i d is p{N) = 1 — v*{N). T h e researcher misspecifies the mode l as L S a n d estimates the entry cost according to (3.15), subst i tut ing F4v\N) = {v - v*{N))/{l - v*{N)) and v = v''{N). A f t e r evaluat ing the integral i n (3.15), one obtains on the right hand side ~^l-v*iN)\ N N+1 J' instead of k, where the second s u m m a n d is the bias t e rm. Observe that the bias can be substant ia l when TV is smal l , even i f true cost is negligible. Since limk-^QV*{N) = 0, the bias becomes 1/(TV(TV + 1)) when A; - » 0. 3.4 Econometric implementation In w h a t follows, we allow for auctions heterogeneity by in t roduc ing the vector of auct ion specific covariates x. W e assume now that the d i s t r ibut i on of valuat ions can change from auct ion to auct ion depending on the value of X a n d is denoted by F{v\x). S imi lar ly , the d i s t r i b u t i o n of valuations cond i t i ona l on entry is now denoted as F*{v\N,x), a n d the probab i l i ty of s u b m i t t i n g a b i d as p{N,x). T h e mode l selection is also cond i t i ona l on x, i.e. different models may be true for different values of x. 3.4.1 Hypotheses T h e previous section shows that the d is t r ibut ions of valuat ions condi t ional on b i d d i n g are identif ied for the three alternat ive models, a n d therefore i n pr inc ip le , m o d e l selection tests can be formulated i n terms oî F*{v\N,x) or equivalently, i n terms of quantiles of this d i s t r i b u t i o n , as suggested i n Hai le , H o n g a n d S h u m (2003) [19]. Define Q*(r|TV,a;) = F * - i ( r | T V , x ) to be the r - t h quanti le of the d i s t r ibut i on of entrants ' valuations. Assume that N varies between the lower bound N a n d the upper bound TV. In terms of the quanti les , the testable restr ict ion of the L S model is HLS--Q*iT\N,x) = ... = Q*{T\N,x), V r e [0,1], whi le the A M E mode l implies the restr ict ion HAME : Q*{T\N,X) =< ... < Q*{T\N,X), V T G [0,1]. (3.20) T h e testable restr i c t ion (3.8) of the Samuelson mode l can also be expressed using the quanti les funct ion Q*{-\N,x) as follows. F i r s t , by the def init ion a n d since F has a compact support , for any r € [0,1], F{Q{T\X)\X) = r . N e x t , for those quantiles of F ( - ) that correspond to valuations above the cutoffs v*{N), i.e. for r > 1 -p{N,x), equation (3.7) implies w h i c h i n t u r n impl ies that Define a funct ion air N x ) - - - i ' - P i ^ ^ ^ ) ) (^ir,N,x)- ^^^^^^ T h e quanti les i n the le f t -hand side of (3.21) do not depend on TV because they correspond to the d is t r ibut ion of potential b idders ' valuat ions , and we then have that Q*(û:(r, TV,x)|TV, 2;) must be constant across TV's for a l l T > 1 - p ( T V , x ) : Hs : Q * ( a ( T , i V , a ; ) | T V , x ) = ... = Q*(a(r,TV,a;)|TV,a;), V r > 1 - p ( T V , i ) . T h e restr i c t ion i n Hs is l imi ted to a part i cu lar range of r ' s . A s imi lar res tr i c t ion , however w i t h r G [0,1], can be obtained from (3.8) directly. Define a funct ion Note tha t , since p(N,x) < p{N,x),0 < /3{T,N,X) < 1 for a l l r G [0,1], and therefore can be interpreted as a legit imate t rans format ion of the quantile order r . ^ T h e condi t ion i n (3.8) imphes that for a l l TV, F*{v\N, x) = PiF*iv\N, x), TV, x), (3.22) and , by the same argument as before, we obta in the fol lowing restr ic t ion i n terms of the transformed quantiles: H's : Q*{P{T,N,X)\N,X) = . . . = Q*{(3{T,N,X)\N,X), V r G [0,1]. P r o m the prac t i ca l po int of v iew, test ing Hs is s imi lar to test ing H'g; how- ever, the last one does not require t runcat i on of r ' s . Therefore, we focus only on H'g. N o t e also that because /3(r, TV, x) is decreasing i n TV, the restrictions under Hs a n d H'g are consistent w i t h the restr ic t ion of A M E (3.20) on the quantiles Q*{T\N,X) w i thout the transformation (3, but are stronger. In this chapter , we consider independent test ing of His, H A M E , and H'g against the ir corresponding unrestricted alternatives. I n add i t i on , we also consider test ing whether the entry probabi l i t ies p(TV, x) are non-increasing i n TV. T h e n u l l hypothesis , for a given value of x, is Hp:l> p{N, x)>...> p{N, x) > 0, a n d i t is also tested against its corresponding unrestr icted alternative . T h e last test is of independent interest. T h e fact tha t the equ i l i b r ium probabi l i t ies of s u b m i t t i n g a b i d decline i n the number of po tent ia l bidders ^Whi le any other fixed value of N can be used in the place of in the definition of 0, the choice N = N ensures that (3 takes on values in the zero-one interval. is p robab ly a common feature of many other models of entry. Whenever a model w i t h cost ly entry is brought to expla in why some potent ia l bidders do not b i d , one must confront an alternative exp lanat ion . Namely , fol lowing Paarsch (1997) [44], even i f there is no entry cost, non -par t i c ipa t i on may s t i l l be expla ined by the fact that some bidders draw their valuat ions below the reserve price. B u t i n that case, the probab i l i ty of b i d d i n g is equal to PriVi < r} a n d therefore does not depend on the number of potent ia l bidders. V i e w e d this way, the above hypothesis Hp is a testable restr ic t ion of cost ly versus costless entry. 3.4.2 T h e d a t a g e n e r a t i n g process W e assume that a sample of L auctions is available, a n d index auct ions by / = 1,.., L. E a c h auct ion is characterized by the vector of covariates x ; € X. We assume that the covaxiates xi are drawn independently for each auct ion from a d i s t r i b u t i o n w i t h density (p{-). C o n d i t i o n a l on x / , the number of potent ia l b idders , Ni, is drawn independently from the d i s t r i b u t i o n 7r(7V|x/); it is assumed that 7r(-|x;) has support A/" = {N,N}}° T h e entry cost k{x) is assumed to be a determinist ic function of x. There is a b i n d i n g reserve price r/ a n d it is observable. C o n d i t i o n a l on x ; = x and TV; = TV, the valuations Vu of potent ia l bidders i = 1 , T V ; are d r a w n independently from a d i s t r i b u t i o n w i t h density / ( - [x) that does not depend on TV. T h e support of Vu is [v{x),v{x)], where i = 1,...,TV;. These valuations are unobservable. T h e central to our approach is the assumpt ion that the number of potent ia l bidders TV is exogenous cond i t i ona l on x ; = x . T h i s assumption allows us to use the var ia t i on i n TV for the purpose of test ing. I n Section 3.6, we exp la in w h y this assumption is plausible i n the context of our empir i ca l app l i cat ion . A s s u m p t i o n 3.2 Vu and Ni are independent conditional onxi. ^°FoT simplicity, we assume that the support does not depend on x, but the results continue to hold even without this assumption. T h e b i d bu corresponding to the valuat ion Vu is generated according to the b i d d i n g strategy bu = B{Vu\Ni,xi). T h e decisions to submit a b i d , yu G {0,1}, are generated according to the cutoff strategy yu = 1 if Su > s{Ni,xi) and Vu > ri, where the signals are uni formly d is t r ibuted . Su ~ t^[0,1]. T h e b i d d i n g strategy B a n d the cutoff funct ion s depend on the model 's pr imi t ives / a n d k th rough the equ ihbr ium conditions of each model ; neither B nor s is available i n closed form. T h e number of ac tua l bidders is given by ni = J Z i ^ i yu- N o t e that ni is posit ively correlated w i t h valuations i f the mode l is not L S . O u r formal assumptions guarantee that the d i s t r i b u t i o n of bids has a density g*{b\N, x). Furthermore , the fol lowing l e m m a s imi lar to Propos i t i on 1 of G P V holds. L e m m a 3.5 Under Assumption C.l(f), for all N e M and x e X, the dis- tribution of bids has the compact support [b{N,x),b(N,x)], with b{N,x) = r if there is a binding reserve price, andg*{-\N, •) has at least R-\-l continuous partial derivatives on its interior. Furthermore, g*{b\N,x) is bounded away from zero. A fu l l l ist of technical econometric assumption on the d a t a generating process needed for our results is given i n the A p p e n d i x i n A s s u m p t i o n C . l . 3.4.3 E s t i m a t i o n o f q u a n t i l e s In this section, we present our nonparametr ic es t imat ion method for Q*{T\N,X). O u r es t imat ion method is based on the fact t h a t , since the b i d d i n g strategies are increasing, the quantiles of valuat ions Q*{T\N, X) and bids q*iT\N,x) = G*-\T\N,X) = mi{b:G*{b\N,x)>T} are l inked through the (inverse) b idd ing strategy, Q*{T\N,x) = aq*ir\N,x)\N,x). Since b o t h ^(.[TV, x ) and q*{T\N,x) can be est imated nonparametr ica l ly , we consider a n a t u r a l p l u g - i n estimator Q*iT\N, x) = ê(q*(T|7V, x)\N, x). (3.23) T h e nonparametr i c estimators for Ç and q* axe constructed as follows. R e c a l l i n g that the inverse b i d d i n g strategy Ç(-|Af, x) is given by i;WJ^,x)-0+ ^N-l)p{N,x)9*{b\N,x) ' our est imator ^ ( - l i V , x ) is obtained by replacing p{N,x), G*{b\N,x), and g*ib\N, x) w i t h nonparametr ic estimators, p{N, x ) , G*{b\N, x ) , and g*{b\N, x ) . T h e cond i t i ona l quanti le q*{T\N,x) is est imated by invert ing the nonpara- metr ic est imator for the C D F G(6|A'',x): q*{T\N,x) = inf{6 : G(fc|7V,x) > r } . T h e t rans format ion /3(T, N, X) can be s imi lar ly est imated by using the est imators p{N, x ) , a n d the transformed quantiles est imated as Q*{/3{T,N,x)\N,x). O u r nonparametr i c estimators for t l ie required input functions g*{b\N, x), G*{b\N, x), a n d p{N, x) are based on the kernel method . Specif ically, we use the fo l lowing est imators : where h is the b a n d w i d t h parameter , iiT is a kernel funct ion satisfying A s - sumpt ion C.2 i n the A p p e n d i x , and n/ = l ^ S i Vu ^he number of actual bidders i n auct ion I. Since the probab i l i ty of observing N condi t ional on x a n d the probab i l i t y of s u b m i t t i n g a b i d condi t iona l on N a n d x can be w r i t t e n as n{N\x) = E[l{Ni = N}\x] a n d p{N,x) = E[n\N,x]/N, their estimators are s tandard nonparametr ic regression estimators. In P r o p o s i t i o n C.2 i n the A p p e n d i x we show that the est imator of b i d sub- mission probab i l i t y p{N, x) is asymptot i ca l ly n o r m a l and derive its asymp- totic variance Moreover , we show that the estimators p{N, x) are a sympto t i ca l l y indepen- dent for any d is t inct N,N' e{N,...,N} a n d x,x' e A ' t . T h e condi t iona l b i d densities a n d distr ibut ions are est imated by a kernel method , w i t h an adjustment needed to account for a r a n d o m number of observations w i t h i n each auct ion . W e estimate first the expected number of b i d observations that correspond to TV-bidder auctions i n the sample w i t h covariates x as p{N,x) TriN\x) E f = i n L i ^ ( ^ ) T.Lnil{Ni = N}nUK{^-M^) ê{N, x) = p{N, x)9{N\x)NL. T h e proposed est imators of g* a n d G* are ^^^'^'''^ - h<iê{N,x)ip{x) ' where (p{x) is the s tandard mult ivar iate kernel density est imator . T h e esti - mators g* a n d G* are essentially s tandard nonparametr ic cond i t i ona l density and C D F est imators w i t h the number of bids observations Ylf=i E i ^ i Vu l{Ni = N} replaced by its est imated expected value ê{N,x). I n the A p p e n d i x , we prove that the estimators Q*{r\N, x) a n d Q*{P{T, N, X) \N, x) are asympto t i ca l l y normal . Specifically, we prove t h a t , under certain technica l but s tandard econometric assumptions, VLh'i+^Q*{T\N,x) - Q*{T\N,X)) is a sympto t i ca l l y n o r m a l w i t h mean zero and variance / /• - \'^+' ( l - p ( 7 V , x ) ( l - r ) ) 2 VQ{N,r,x)= (^J K{ufdi^ (N - l)2Np3{N,x)g*^{q*{T\N,x)\N,x)Tr{N\x)ip{x) A consistent est imator VQ{N,T,X) can be obta ined by replac ing p{N,x), q*{T\N,x) a n d other u n k n o w n functions by their est imators . W e also show that VLf^{Q*0{r, N, x)\N,x) - Q*(P{T, N,X)\N,X)) converges i n d i s t r i b u t i o n to a n o r m a l random variable w i t h mean zero and variance VQ{N, /3(r, N, x ) , x ) ) . Moreover , for any d ist inct A'', N' £ {T£, . . . , ÎV} , T,T' e T , a n d x , x ' £ X\ the estimators Q*{T\N,X) are asymptot i ca l ly independent, as are the estimators Q*{^{T,N,x)\N,x). These results are contained i n P r o p o s i t i o n C.5 i n the A p p e n d i x . 3.4.4 Comparison with the estimation method of Haile, Hong and Shum (2003) It is interest ing to compare our estimator Q*{T\N,X) w i t h that of H H S , who present their method i n a model different from ours i n that they allow for c o m m o n value effects. T h e i r method is semiparametr ic a n d i n the present context, reduces to the fol lowing. One begins w i t h removing the effect of covariates on valuations by per- forming a p r e l i m i n a r y regression. T h e m a i n assumption i n H H S is that the valuations depend on covariates addit ively, w i t h the mean va luat ion spec- ified by some funct ion r{xi;d) that depends on x a n d a f inite-dimensional parameter 9: Vii=rixi;e)+eii. (3.24) T h e error t e r m en is mean-zero a n d d i s t r ibuted independent ly of xi w i t h C D F Fe(-) . It is also assumed that the reserve price has the same addit ive form, ri = ro + r{xi;9). One can always write the b i d d i n g strategy as the va luat ion minus the m a r k u p , B{vii\Ni,xi) = vu - m{vu,xi,Ni). H S S show that i n their sett ing, the m a r k u p Tn{vu,xi, Ni) does not depend on xi: m{vu,xi,Ni) = mo{£u,Ni). It is then easy to show that the b ids regression takes the addit ive form bu = a{Ni) + r{xi;9) + e'u, (3.25) where a(Ni) = E[moieu,Ni)\Ni] and ê , = eu - moiu,Ni) + a(A^()- T h e parameter can then be est imated by any nonlinear regression m e t h o d , and the bids "homogenized" according to bu = bu — T{xi;9). N e x t , the inverse b i d d i n g strategy ^{b\N) is est imated nonparametr ica l ly , i n essentially the same way as i n our method, a n d a pseudo sample of bids (cf. G P V ) is formed according to vu = C{bu\Ni), t r i m m e d appropr iate ly to avoid b ias ing boundary effects. Las t ly , the quantiles Q*{T\N) are est imated as sample quanti les of the t r i m m e d sample. If desired, the cond i t i ona l quantiles can be est imated as Q*{T\N,X) = T{x;9) + Q*{T\N). T h i s "homogenizat ion" is not feasible i n our sett ing, since even i f one as- sumes the addi t ive form (3.24), the m a r k u p m(uj ; , xi,Ni) i n general depends on xi- T h i s is because, unlike in the models considered i n H S S , i n our case the equat ion for the b i d d i n g strategy (C.5) also contains the entry prob - ab i l i ty p{N,x), an u n k n o w n funct ion of x. Therefore the regression (3.25) produces inconsistent estimates of 9. O u r method effectively changes the or- der of steps of the H S S estimator. U n l i k e H H S , our treatment of covariates is fu l ly nonparametr i c . W e first nonparametr ica l ly estimate the quantiles of bids q(T\N,x) a n d then insert them into the est imator ^{-{NjX) to o b t a i n our cond i t i ona l quanti le est imator Q*{T\N,X) = ^{q{T\N,x)\N, x). 3.4.5 T e s t s I n v iew of the results of subsection 3.4.3, quanti le restrict ions derived from the L S , S, a n d A M E models can be tested i n a s tandard manner as equal i ty or inequal i ty constraints . W e implement the tests us ing a finite set of r ' s from (0,1) in terva l , T = { n , r 2 , T j t } . T h e tests of H L S a n d H'g are based on the corresponding statist ics T'^^lx) a n d T^{x) that measure deviations of the est imated quanti les Q*{T\N,X) and Q*0{T,N,X)\N,X) f rom their B y the results of P r o p o s i t i o n C .5 , Q*{T\N,X) and Q*(P{T,N,X)\N,X) are approx imate ly n o r m a l i n the large samples and independent across N; there- fore, the L S mode l is rejected at level a for the auctions w i t h covariates' values X whenever T^^{x) > xf#j^-,i)k,i-a' ^^^^^ ^\#M-\)k,i-<x denotes ' ^Recal l that Q'{T\N,X) is the same for all N under HLS, and Q'{P{T, N,x)\N,x) is the same for all under H's. means: 11 the 1 — a quant i le of the chi-squaxe d i s t r ibut i on w i t h degrees of freedom {#J\f ~l)k, where #A/ ' denotes the number of elements i n M. S imi lar ly , one rejects H'g i f T^{x) > X(^jv_i)A: i-a Note that due to asymptot i c normal i ty a n d independence of the quanti le estimators, T^^{x) a n d T^{x) can be also viewed as l ike l ihood rat io ( L R ) statist ics . W e now t u r n to test ing of the A M E model . In this case, the distance or L R stat ist ic is T^^^ix)^ m i n Lh'^^' V (Q^WAT, x) - , ; . , . ) ^ VK.<-<m.r^rer ^ ^ ^ ^ ^ V Q ( 7 V , T , X ) and one rejects the n u l l of A M E i n favor of the general a l ternat ive when T^^^{x) takes on large values. T h e A M E mode l does not determine uniquely the n u l l d i s t r i b u t i o n of the T^^^{x) s tat is t i c ; however, we show i n Propos i t i on C.6 i n the A p p e n d i x that the p r o b a b i l i t y of type I error is m a x i m i z e d when a l l inequalities are replaced w i t h the equalities, i.e. the same restrict ions as i n the L S model . T h i s propos i t i on also shows that , under the restr i c t ion of the L S model , the s tat is t i c T^^^{x) is asymptot i ca l ly equivalent to the r a n d o m variable defined by T^M'^{x)^y m i n (3.27) where Zr ~ A''(0, I#Af-i) and independent across r ' s , Q{T, X) = RVQ{T, X ) R ' , VQ{T, X) = diag(VQ{N, r , x),VQ{N, r , x)), and R is the ( # A / ' - 1) x (#A^) differencing m a t r i x 1 - 1 0 ... 0 \ R= ' ' « ^ 0 0 ... 1 - 1 y Consequently, a test that rejects HAME when T'^^^{x) > c{x)AME,i-a, where c{x)AME,i-a is the 1 - a quanti le of the d i s t r i b u t i o n of T ^ ^ ^ ( a ; ) , has asymptot i c size a. T h e d i s t r i b u t i o n of T'^^^(a;) depends on asymptot ic variances of the quanti le est imators ; however, the c r i t i ca l values can be s imulated as follows. F i r s t , f rom the est imated asymptot ic variances VQ{N_,T,X), ...,VQ{N,T,X) construct the matrices VQ{T,X) a n d CI{T,X) for T i , . . . , T f c . Second, for m = 1 , M , generate independent iV(0 , I # A / ' - I ) vectors Zri,m, Zri^m, a n d com- pute f^^^{x) as defined i n (?? ) , but w i t h Zr replaced by Zr,m, and ft replaced w i t h Q. T h e s imulated cr i t i ca l value for a test w i t h asymptot i c size a, denoted by c{x)AME,i-a, is then computed as the 1 — a sample quanti le of {f^^^ix) : m = 1,...,M}. One should reject the n u l l of A M E when T^^^ix) > c{x)AME,l-a- Test ing whether the entry probabi l i t ies are non-increasing i n A'', the Hp hypothesis , can be performed s imi lar ly to test ing H A M E - Define T^ix)= m i n m > - > v ^ ^ VpiN,x) One should reject Hp whenever TP{x) > c{x)p^i-a, where c(x)pj-a is the 1 - a quanti le of TP(x) = rnm^y2^^^^^^ \\Z - fif, fip(i) = RVp{x)R\ Vp{x) is a d iagonal m a t r i x w i t h the m a i n diagonal elements Vp{N,, x ) , V p { N , i ) , a n d Z ~ A / ' ( 0 , S u c h a test has asymptot ic size a a n d is consistent. A s i n the case of T^'^^[x), the c r i t i ca l values for the T P ( X ) test can be s imulated fo l lowing the steps described above. 3.5 Monte-Carlo experiment I n th is section we present a M o n t e - C a r l o s tudy of the s m a l l sample proper- ties of the tests. I n par t i cu lar , we are interested how the choice of quantiles T affects size a n d power of the tests. In our s imulat ions , we focus on test ing the A M E mode l w i t h o u t covariates x. W e s imulate the r a n d o m signals S a n d valuat ions V us ing the Gauss ian copula . L e t {Zi,Z-i) be bivariate normal w i t h zero means, variances equal to one, a n d the correlat ion coefficient p. L e t $ denote the s tandard normal C D F . A pa i r ( 5 , y ) is generated as 5 = «>(Zi) , a n d V = $(^2). Nonzero values of p correspond to the case of informative signals a n d selective entry; whi le /9 = 0 corresponds to the case of the L S mode l . T h e details of the computat ion of the d is tr ibut ions F{v\S') and F*{v\N) that are needed i n order to solve for the e q u i l i b r i u m of the auct ion are as follows. F i r s t , recal l that ^ N{pzi,l - p^), and , consequently, the cond i t i ona l d i s t r i b u t i o n of V given 5 is given by F{v\S) = P{V<v\S) = P{Z2<^-\v)\^-\S)) \ N e x t , note that the marg ina l d i s t r ibut i on of 5" is un i f o rm o n the [0,1] inter- v a l , a n d F*{v\N) = F{v\S>-s{N)) 1 - s h{N) / where the cutoff s ignal 's{N^ can be found, given the value of TV, as a so lut ion to equat ion (3.11). Last ly , for S > s{N), the bids are computed according to the b i d d i n g strategy (C.5) . I n our s imulat ions , we set L = 250, A/" = {2 ,3 ,4 ,5 } , 7r(TV) — 1/4 for a l l TV e A/ ' , a n d k — 0.17. T h e number of M o n t e C a r l o repl icat ions is 1,000; i n each rep l i cat ion , the c r i t i ca l values for the T^'^^ test are obta ined using 999 repl icat ions . W e use the triweight kernel funct ion K{u) = ( 35 /32 ) ( l - u'^)^l{|u| < 1} for nonparametr ic kernel est imat ion . T o reflect the fact that the number of active bidders varies from auct ion to auct ion depending on the number of po tent ia l bidders i n the auct ion N a n d p{N), we decided to use a b a n d w i d t h that depends on N. Specifically, we used h = {LNp{N))~'^/^. Table 3.2 reports the results of size s imulat ions for p = 0,0.5 , a n d the fol lowing sets of quanti les: {0.5}, {0.3,0.5,0.7}, and {0.3,0.4,0.5,0.6,0.7} . W h i l e the asymptot i c approx imat ion works reasonable wel l for a s m a l l n u m - ber quanti les , the finite sample size properties of the test deteriorate when the number of quanti les used to construct the test increases. For example , the T^M^ test over rejects the n u l l when p = 0 a n d r e {0 .3,0.4,0.5,0.6,0.7} : for the n o m i n a l size of 10%, 5%, and 1% the s imulated rejection rates are approx imate ly 16%, 10% and 4% respectively. Note also that the rejection rates for p = 0.5 are smaller than for p = 0.5 and below the nomina l re- jec t ion rates. T h i s reflects the fact that the probab i l i ty of type I error is m a x i m i z e d at p = 0. Tab le 3.3 reports the size corrected power results (the c r i t i c a l values are computed from the s imulated d i s t r ibut i on of the test s tat ist ic under the nul l ) . T o address the power issue, i t is necessary to come up w i t h an alter- nat ive . Ideally, this w o u l d be achieved by considering a s t r u c t u r a l model . Absent a s t r u c t u r a l model , however, we are allowed to consider any configu- ra t i on of b i d d i n g quantiles, i n part i cu lar we may reverse their order, m a k i n g decreasing as opposed to increasing i n A''. T o do this i n the simplest fashion possible, we m u l t i p l y each quanti le by minus one and then add a constant to a l l quanti les to assure that they are posit ive. Tab le 3.3 shows that the power increases w i t h the distance f rom the n u l l . T h e power also increases when we use quantiles 0.3 a n d 0.7 i n add i t i on to the med ian . However, we also observe that i n some cases the size corrected power decreases w i t h the number of quantiles, when the number of quantiles used to construct the test is large. For example, i n the case of {0.3,0.5,0.7} quanti les , p = 0.9, and the n o m i n a l size of 5%, the s imulated rejection rate is about 20%; however, when we use i n add i t i on the quanti les 0.4 and 0.6, the rejection rate is on ly about 18%. In practice, given samples of moderate size, a rule of t h u m b would be to use 3 fixed quanti les i n order to m a i n t a i n good size a n d reasonable power. 3.6 Empirical application O u r dataset consists of 547 auctions for surface pav ing and grading con- tracts let by O k l a h o m a Department of Transpor ta t i on ( O D O T ) d u r i n g the per iod of January , 2002 to December, 2005. ^̂ ^̂ T h e available d a t a items include a l l b ids , the engineer's estimate, the t ime length of the contract ( in days) , the number of items i n the proposal and the length of the road . T h e O D O T implements a po l i cy under which a l l bids over 7% of the engineer's est imate are t y p i c a l l y rejected, so there is a b i n d i n g reserve price. In reality, we do observe b ids above the reserve price (a l though extremely few w i n n i n g bids were above the reserve price) . W e treat these bids as non-serious. T h e y are only used i n the est imat ion of b idd ing probabi l i t ies , but are otherwise e l iminated f rom the sample of bids. Important ly , we observe the l ist of eligible bidders (planholders) for each auct ion . In the vocabulary of this chapter, these eligible bidders are the potent ia l bidders. T h e l ist of planholders is publ ished on the O D O T website pr ior to b idd ing . A f i rm becomes a planholder through the fo l lowing process. A l l projects to be auct ioned are advertised by the O D O T 4 to 10 weeks prior to the le t t ing date. These advertisements include the engineer's estimate, a brief s u m m a r y of the project , locat ion of the work and the type of the work involved. B u t the advertisements lack detailed schedules of work items which are on ly revealed i n construct ion plans. ' ^ T h e data were obtained from the O D O T website, h t t p : / / w w w . o k l a d o t . s t a t e . o k . u s / . ^ ' O u r choice of surface paving and grading contracts is motivated by the fact that H o n g and S h u m (2002) [22], in their study of highway procurement auctions in New Jer- sey, find little support for common values for this type of contracts. See also De Silva, Kankanamge , a n d K o s m o p o u l o u (2007)[13l. T h i s is important because in this chapter, we assume independent private values (costs). Interested firms can then submit a request for plans and b i d d i n g proposals, the documents that contain the specifics of the project ( in part i cu lar , the items schedule). A n important feature of the qual i f i cat ion process is that only eligible firms are allowed access to these documents. A firm is deemed eligible i f i t satisfies certain qual i f icat ion requirements. T h e goal of the qual i f i cat ion process is to ensure that the w i n n i n g firm w i l l have sufficient expertise a n d capacity to undertake the project . W h i l e the expertise part is t y p i c a l l y determined at the pre-qual i f icat ion stage ( in most cases, once per year) , the capacity part is project-specific. A n i m p o r t a n t requirement is that the prospective bidder is not qualif ied for the aggregate amount of work that exceeds 2.5 times its current work ing cap i ta l . G i v e n that the bidders know the sizes of a l l projects to be let but not the project specifics, i t is plausible that the decision of a firm to request the p l a n for a part i cu lar project is p r i m a r i l y determined by the project size as wel l as the sizes of other projects for w h i c h i t is pre-quali f ied, i n re lat ion to the available capacity of the firm. T h e capac i ty may be determined by a number of factors, such as for example the amount of resources commit ted to other projects , inc lud ing but not l i m i t e d to those previously contracted w i t h O D O T . Before t u r n i n g t o our nonparametr ic tests, we investigate the importance of various observable covariates on b i d levels and the decisions to submit a b i d w i t h the help of usual O L S and logit regressions. T h e variables used i n the regressions are described i n Table 3.4. T h e results of the entry logit regression a n d O L S b idd ing regression are presented i n Table 3.5. In the O L S regression, the dependent variable is log{bid), where b i d is the amount of b i d i n mi l l i ons of dollars. T h e size of the project has a strongly posit ive effect on the bids. Clear ly , i t is the most impor tant variable i n the O L S regression. U s i n g i t alone produced of about 0.79, so the impact of the other variables is m u c h smaller. I n the order of importance , the next variable is the number of potent ia l bidders N; i f i t is inc luded i n the regression, R"^ " T h i s requirement is explicitly stated in the O D O T rule O A C 730:25. ' ^ T h e covariates are basically the same as in other papers on procurement auctions (e.g. B a j a r i and Ye (2003) [7]; Pesendorfer and Jofre-Bonet (2003) [48]; Krasnokutskaya (2003)[26]; Krasnokutskaya and Seim (2006)[27l; L i and Zheng (2005)[34l). increases to about 0.94. W e also mention that the project size has a negative (but not s ta t i s t i ca l ly significant) effect on the probab i l i t y of s u b m i t t i n g a b i d . T h e effect of the number of potent ia l bidders is s tat i s t i ca l ly significant i n a l l regressions. H a v i n g more potent ia l bidders reduces the b i d submission rate: increasing A'' by 1 reduces the odds of s u b m i t t i n g a b i d by about 4%. H a v i n g more bidders also results i n lower bids. T h i s is of course consistent w i t h the models considered i n this chapter. T h e logit regression also shows that project size has a negative effect on the probab i l i ty of b idd ing . T h i s effect, however, is not stat ist ical ly significant. T h e complex i ty of the project is captured by the number of i tems i n the construct ion p l a n . T h i s is the variable Nitems. Table 3.4 shows that there is a substant ia l var ia t i on i n Nitems; the mean is 72 a n d the s tandard dev i - at ion is 71 items. One might expect that the cost of prepar ing a b i d is an increasing funct ion of Nitems, even contro l l ing for the size of the project. T h e conjectured effect of Nitems is therefore to reduce the probab i l i ty of s u b m i t t i n g a b i d . T h e estimate of the Nitems coefficient i n the logit regres- s ion confirms this conjecture. However, the effect is quite smal l : increasing Nitems b y one s tandard deviat ion , i.e. adding about 70 pay i tems, reduces the odds by on ly about 1%. Inc luded i n b o t h regressions are d u m m y variables for 20 firms that appear on the planholders l ist most frequently. T h e other firms are treated as fringe f irms. Observe that even though not a l l firms enter at the same rate and b i d s imi lar ly , the empir i ca l evidence of asymmetries is s trong only for out- of-state f irms (the f irms w i t h headquarters outside the state of Oklahoma) that enter less frequently a n d also b i d less, and for the fol lowing three firms: B r o c e C o n s t r u c t i o n , Glover C o n s t r u c t i o n and Becco Contractors . Since our mode l assumes b idder symmetry , we decided to exclude the auctions i n w h i c h either out-of-state firms or these three f irms were on the planholders l i s t . I n the prac t i ca l implementat ion of our estimators, we are confronted w i t h the usual bias a n d variance trade-off. Inc luding more variables w i l l reduce the bias, but at the same t ime w i l l increase the sample var iab i l i ty of our est imators. G i v e n the pre l iminary regression results, we on ly condi t ion on the project size. F i g u r e 3.1 shows the empir i ca l frequencies (i.e. the histogram) of project sizes. T h e pat tern is h ighly skewed towards smaller projects: the average project size is about $3.6 m i l l i o n (from Table 3.4), but the projects for which there are at least 10 auctions i n the dataset have sizes not exceeding $2.5 m i l l i o n . Since we need a moderate number of observations to implement our nonparametr i c tests, we have chosen the set of project sizes X to be the equal p a r t i t i o n of the interval [0,2.5], i.e. i n mi l l i ons of dol lars, X = {0 .5 ,1 ,1 .5 ,2 ,2 .5} . T h e projects of larger size may be, other things equal , more attract ive to the bidder . For example , th is would be so i f there are economies of scale. There is some evidence of this i n the data . Table 3.6 shows esti - mated cond i t i ona l probabi l i t ies 7r(7V|a;), where as before N is the number of po tent ia l bidders and x is the size of the project as measured by the en- gineer's est imate. One can see that the est imated mean E[N\x] increases w i t h X from E{N\x = 2]= 4.69 to E[N\x = 2.5] = 7.97, and the differences E[N\x] - £;[iV|a;'], x > x', are s tat is t i ca l ly significant for a l l a; G X . Tab le 3.6 also shows that there s t i l l remains a substant ia l var ia t i on i n the number of potent ia l bidders even after contro l l ing for project size. T h i s var ia t i on w i l l impor tant for our tests. E q u a l l y impor tant is that the var iat ion i n the number of planholders is l ikely to be exogenous, uncorrelated w i t h unobservable project characteristics since the latter on ly become available i n the plans. A n o t h e r pract i ca l issue is that , as is wel l known , nonparametr i c estimators suffer from substant ia l loss of precision when sample size is very smal l . W h e n we t r i ed to inc lude a l l auctions, the estimates of the quanti les Q{T\N, X) were h ighly errat ic . T h e prob lem is that , because the d a t a is sparse, for some {N,x) pairs the est imated probabi l i ty 7r(A''|a;) is very close to 0. To make our est imators stable, we decided to exclude those pairs {N,x) where the number of observations that TV; = TV a n d xi G [x - h,x + h] is less t h a n 15. T h e w o r k i n g sample u l t imate ly consisted of 258 auctions, a n d a l l the results discussed below were obtained using this smaller sample. T h e tests are performed condit ional on project size x G X. W e first test the pred i c t i on shared by a l l models considered i n this chapter, namely that b i d submiss ion probabi l i t ies p{N,x) are dec l ining i n the number of potent ia l b idders TV for each x. Refer to Table 3.7, where the est imated b i d d i n g probabi l i t ies p(N, x) as wel l as the results of the tests are reported. T h e average rate of b i d submission is about 62%. E x c e p t for re lat ively large projects , x = $2.5 m i l . , there is a strongly dec l in ing p a t t e r n over TV. For example , for moderate ly sized projects, x = $1.5 m i l . , the probabi l i ty of b i d d i n g p(N, x) falls from a relatively large value of 0.56 when there are 3 potent ia l bidders , to about half of that , 0.28 when there are 9 potent ia l bidders. F o r other values of x G X\{2.5}, the p a t t e r n is less pronounced, but the f o rmal tests of the monotonic i ty restrict ions s t i l l do not reject the n u l l at the conventional 5% significance level. W e now t u r n to the tests of the models: L S , S, a n d A M E . F i r s t note that , because the procurement auctions are l ow-b id , the n u l l hypothesis tha t corresponds to the A M E model must be changed accordingly, i.e. the quanti les must decreasing rather than increasing. A l s o , the inverse strategy is now c ( 6 i T v , . ) = 6 - ^-p(^^-)o*m,x) {N-l)p{N,x)g*ib\N,x)' a n d the asympto t i c variances of the quantiles are ( l - p ( T V , x ) r ) 2 VQ{N,T,X) = (^j K{ufdv^ {N - \YNp^{N,x)g*^{q'{T\N,x)\N,x)TT{N\x)^(xy ° T h e b a n d w i d t h was chosen according to the same rule as in Section 3.5. but a l l other aspects of est imat ion are unchanged. T h e results of the tests are presented i n Table 3.9. Cons ider first the results for just one quanti le , the median (Tables 3.7 a n d 3.8 report the esti - mated median a n d transformed median costs respectively a n d their s tandard errors) . T h e A M E mode l is rejected for relatively smal l projects, x = $0.5 a n d a; = $1 m i l . , but i t is not rejected for larger projects. Nei ther L S models are rejected for most project sizes. Since the findings are somewhat counter in tu i t ive ; but recal l that our M o n t e - C a r l o studies have shown that the power of the tests can be increased substant ia l ly i f we increase the number of quan - tiles f rom one to three. W h e n 0.3, 0.5 a n d 0.7 quanti les are s imultaneously considered, the support for the S model disappears completely at 5% signif- icance for a l l project sizes. T h e largest p-value of the test is 0.02. T h e L S mode l fairs better , but i t too is rejected for a l l projects but the largest ones, w i t h X = $2.5 m i l . T h e A M E model , on the other h a n d , is now rejected also for x = $1 m i l . , but s imi lar ly to the one quanti le case, is not rejected for project sizes x = $2.0 a n d x = $2.5 m i l . Increasing the number of quantiles from three to five (we have chosen the quanti les 0.3, 0.4, 0.5, 0.6 and 0.7) leads to the same results: the S mode l is robust ly rejected, the L S mode l is rejected for a l l but the largest projects. T h e A M E mode l is not rejected for the two largest project sizes we consid- ered. T h e reason that a l l models are rejected for the s m a l l project can be as follows. It is possible that for smal l projects, f irms coordinate their de- cisions to request plans and become potent ia l bidders, whi le these decisions are made independently for larger projects. C o n f i r m i n g this hypothesis em- p i r i c a l l y is l ike ly to be difficult i n view of potent ia l ly confounding effects of unobserved project heterogeneity. T h e fact that the S model is always rejected is probab ly not very surpr is - ing . R e c a l l that the entry cost i n the S model is solely the cost of prepar ing a b i d rather t h a n the jo int cost that also includes in format ion acquis i t ion. E v e n for s m a l l projects, firms may face uncertainty about the exact level of their construct ion costs that can only be resolved through cost ly in forma- t ion acquis i t ion , so i t is on ly natura l that this is conf irmed empir ica l ly . T h e empir i ca l evidence regarding the L S model can also be expla ined intui t ive ly . It is p lausible that project complexity increases w i t h project size. O u r i n - ab i l i ty to reject the L S mode l for large projects may be due to the fact that for these projects , the in format ion received by the bidders before the plans are available is re lat ive ly less precise. 3.7 Concluding remarks I n th is chapter , we have proposed nonparametr ic tests to d iscr iminate among al ternat ive models of entry in first-price auctions. T h e models con- sidered are: (a) the L e v i n and S m i t h (1994) [31] model w i t h randomized entry strategies, (b) the Samuelson (1985) [53] mode l that assumes that b i d - ders are perfect ly informed about their valuations at the entry stage, and select into the p o o l of entrants based on this in format ion , a n d (c) a new mode l that allows for selective entry but i n a less stark form t h a n Samuel - son (1985) [53]. Specif ically, our model assumes that bidders receive signals that are in format ive about their valuations a n d make their entry decisions based on these signals. I n the e m p i r i c a l app l i ca t i on , we have tested the restrict ions of each model against the unrestr i c ted alternatives using a dataset of h ighway procurement auct ions f rom the O k l a h o m a Department of T r a n s p o r t a t i o n ( O D O T ) . W e have found strong evidence for selective entry according to our mode l . W h i l e these f indings are encouraging and the test ing framework can be used i n other appl i cat ions , th is research could be extended i n a number of directions. One i m p o r t a n t extension would be developing more powerful tests. One cou ld improve power by considering tests based on a c o n t i n u u m of quan - ti les rather t h a n a f inite set. H S S have pursued this approach, developing a K o l m o g o r o v - S m i r n o v type test. A s we have already discussed, their es- t i m a t i o n m e t h o d cannot be direct ly transferred to our sett ing . T h i s would be an i m p o r t a n t extension of our approach left for future work. S i m i l a r hy - potheses are considered i n the recent l i terature on tests of stochastic d o m i - nance a n d monotonic i ty (see, for example, Lee, L i n t o n , W h a n g , Suntory, for Economics , a n d Disc ip l ines (2006)[29]); however, their approach cannot be appl ied d i rec t ly i n our case, since private valuations are unobservable, and our stat ist ics are based on kernel density estimators. O u r test ing framework is quite general and the empir i ca l f indings are by a n d large in tu i t i ve . B u t there is also an important l i m i t a t i o n that the future research should address. I n auct ion datasets, one t y p i c a l l y finds that the var iat ion i n bids is only par t ia l l y explained by their var ia t i on w i t h i n auc- t ions. T h e between-auction var iat ion is typ i ca l ly present. It is also observed i n our dataset. T h i s pat tern can be explained w i t h i n the I P V framework only by unobserved project heterogeneity. Recently, K r a s n o k u t s k a y a (2003) [26] has developed a s t ruc tura l est imat ion method that can be appl ied even i n the presence of unobserved heterogeneity. She assumes that the heterogene- i ty enters into the specif ication of valuations as a m u l t i p l y i n g factor. Her method relies on the fact that the same mult ip l i cat ive structure carries over to the bids. Unfortunate ly , this is not true for the models w i t h entry consid- ered i n this chapter , for reasons largely s imi lar to those described i n Section 3.4.4. • A l te rnat ive ly , one can expla in between-auction var ia t i on us ing a model w i t h afïiUated private values ( A P V ) , as i n L i , Perr igne a n d V u o n g (2002) [33]. Note however that the theoretical models of entry that we b u i l d on are a l l w i t h i n the I P V sett ing. It is known that the A P V mode l can lead to qua l - i ta t ive ly different predict ions (Pinkse a n d T a n (2005) [49]), so i t is an open question if i t can lead to testable impl i cat ions s imi lar to those considered here. W e leave this for future research. A n o t h e r extension would be to allow bidder asymmetries (e.g., a recent w o r k i n g paper by Krasnokutskaya a n d Se im (2006) [27]). T h e obvious dif- f i culty here wou ld the necessity to deal w i t h mul t ip le equ i l ib r ia . B a j a r i , H o n g a n d R y a n (2004) [6] obta in a number of identi f icat ion results i n this d irect ion a n d estimate a parametr ic model w i t h mul t ip le equ i l i b r ia for h igh - way procurement auctions. F i n a l l y , incorporat ing d y n a m i c features as i n Pesendorfer a n d Jofre -Bonet (2003) [48] is also left for future research. 3.8 Tables and Figures Table 3.1: Size of A M E Test Nominal size Q u a n t i l e s 0.5 0.3, 0.5, 0.7 0.3, 0.4, 0.5, 0.6, 0.7 rho=0 0.10 0.0760 0.1520 0.1580 0.05 0.0310 0.0730 0.1030 0.01 0.0070 0.0200 0.0360 rho=0.5 0.10 0.0420 0.0660 0.0640 0.05 0.0240 0.0350 0.0310 0.01 0.0010 0.0130 0.0100 Tab le 3.2: Size-corrected Power of the A M E Test Nominal size Quanti les 0.5 0.3, 0.5, 0.7 0.3, 0.4, 0.5, 0.6, 0.7 rho=0.5 0.10 0.1622 0.2800 0.2683 0.05 0.0941 0.2030 0.1812 0.01 0.0210 0.0630 0.0651 rho=0.9 0.10 0.2693 0.3624 0.4124 0.05 0.1842 0.2853 0.2983 0.01 0.0661 0.1231 0.1552 Table 3.3: Descr ip t i on of Var iables V a r i a b l e D e s c r i p t i o n M e a n S t d . D e v . M i n M a x EngEst The engineer's estimate for the project, in mil. dollars 3.647 4.488 0.066 24.800 B i d Bid divided by the engineer's estimate 1.067 0.173 0.385 2.106 Nitems Number of pay items in the project ad 71.736 70.704 1.000 363.000 Ndays Number of business days to complete the project 195.995 142.370 10.000 681.000 Length Length of the road (in miles) 4.699 4.794 0.000 36.630 Distance Distance in miles from the headquarters of the firm of the bidding firm to the project site 344.237 382.469 0.000 1702.016 Backlog The total amount of unfinished work on a given day and normalized by the bidder-specific maxi- mum, the value is between 0 and 1 0.219 0.297 0.000 1.000 Npotential Number of planholders 8.299 4.244 2.000 26.000 Nfictual Number of actual bidders 3.451 1.428 0.000 7.000 Out-of-state dummy =1 if the firm has headquarters outside 0.1358936 0.342929 0.000 1.000 the state of Oklahoma Table 3.4: Log i t and O L S Regressions L o g i t O L S V a r i a b l e Coeff. s.e. Coeflf. s.e. Intercept 3.565 1.148 0.497 0.123 Log(EngEst) -0.01 0.034 0.970 0.010 Npotential -0.043 0.019 -0.008 0.002 Length 0.009 0.012 0.001 0.001 Ndays 0.000 0.001 0.000 0.000 Nitems -0.0002 0.000 0.0004 0.000 Distance 0.000 0.000 0.000 0.000 Backlog 0.137 0.170 0.019 0.019 Out-of-state -0.359 0.166 -0.034 0.020 Fringe firm 0.250 0.175 0.027 0.034 APAC-OKLAHOMA, INC. 0.303 0.497 0.013 0.025 T H E CUMMINS CONST. CO., INC. 0.266 0.385 0.005 0.023 HASKELL LEMON CONST. CO. -0.206 0.351 0.028 0.023 BROCE CONSTRUCTION CO., INC. -1.803 0.313 0.052 0.029 WESTERN PLAINS CONSTRUCTION COMPANY -1.338 0.756 -0.011 0.031 BELLCO MATERIALS, INC. -0.672 0.411 -0.086 0.277 OVERLAND CORPORATION -0.726 0.399 0.057 0.029 GLOVER CONST. CO., INC. -0.919 0.395 0.061 0.003 T Si G CONSTRUCTION, INC, -0.974 0.992 -0.036 0.028 TIGER INDUSTRIAL TRANS. SYS., INC -0.061 0.479 -0.004 0.029 HORIZON CONST. CO., INC. -0.181 0.473 0.009 0.033 CORNELL CONST. CO., INC. -0.481 0.450 0.007 0.030 SEWBLL BROTHERS, INC. -0.075 0.472 -0.018 0.034 BECCO CONTRACTORS, INC. -1.818 0.876 -0.068 0.033 EVANS & ASSOC. CONST. CO., INC. -0.228 0.582 -0.044 0.036 SHERWOOD CONST. CO., INC. -0.561 0.416 0.006 0.035 VANTAGE PAVING, INC. -1.619 0.962 0.030 0.047 ALLEN CONTRACTING, INC. 0.245 0.493 0.014 0.035 DUIT CONSTRUCTION CO., INC. 1.275 0.707 0.027 0.037 MUSKOGEE BRIDGE CO., INC. -1.122 0.809 0.006 0.037 Observations 4485 1860 Log-Iikelihood/fl2 -1543.750 0.983 Significemt coefficients (at 5% level) are marked in bold. The dependent variables were: for the logit regression, the indicator variable equal to 1 if the bid is submitted; for the O L S regression, the amount of bid in $ mil. Table 3.5: E s t i m a t e d P r o b a b i l i t y 7r{N/x) of A'' C o n d i t i o n a l on Pro j e c t Size x P r o j e c t size ($ mi l . ) x=0.5 x = l x=1.5 x=2 x=2.5 N n{N/x) s.e. •K(N/X) s.e. 7r(Ar/x) s.e. 7r(iV/x) s.e. s.e. 2 0.07 0.02 0.04 0.02 0.02 0.02 0.01 0.02 0.00 0.00 3 0.26 0.04 0.21 0.04 0.14 0.04 0.07 0.04 0.01 0.02 4 0.22 0.04 0.22 0.04 0.16 0.04 0.10 0.05 0.12 0.06 5 0.15 0.03 0.17 0.04 0.18 0.05 0.13 0.05 0.06 0.04 6 0.17 0.03 0.15 0.03 0.19 0.05 0.26 0.07 0.18 0.07 7 0.02 0.01 0.04 0.02 0.05 0.03 0.11 0.05 0.20 0.07 8 0.05 0.02 0.06 0.02 0.08 0.03 0.11 0.05 0.12 0.06 9 0.02 0.01 0.04 0.02 0.07 0.03 0.08 0.04 0.07 0.05 10 0.01 0.01 0.03 0.02 0.06 0.03 0.06 0.04 0.04 0.03 11 0.01 0.01 0.01 0.01 0.02 0.02 0.04 0.03 0.08 0.05 12 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.02 0.03 13 0.00 0.00 0.02 0.02 0.04 0.03 14 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.03 0.03 15 0.01 0.02 0.02 0.04 0.03 E{N/x) 4.69 0.02 5.23 0.02 5.95 0.03 6.84 0.04 7.97 0.04 Table 3.6: E s t i m a t e d P r o b a b i l i t y of B i d d i n g P{N, x) P r o j e c t size ($ mi l . ) x=0.5 x = l x=1.5 x=2 x=2.5 N P{N,x) s.e. P{N,x) s.e. P(iV,x) s.e. P{N,x) s.e. P{N,x) s.e. 2 3 0.49 0.05 0.53 0.06 0.56 0.09 0.47 0.17 4 0.45 0.05 0.45 0.05 0.45 0.07 0.56 0.12 5 0.39 0.05 0.40 0.05 0.38 0.06 0.33 0.09 6 7 0.37 0.04 0.36 0.05 0.34 0.05 0.30 0.06 0.27 0.08 8 0.30 0.06 0.31 0.06 0.27 0.07 9 0.28 0.07 0.35 0.09 0.45 0.11 10 11 0.21 0.08 Test Statistic 0.00 0.00 0.01 4.73 19.74 p-value 0.99 1.00 1.00 0.17 0.00 Table 3.7: E s t i m a t e d M e d i a n s of Costs P r o j e c t size ($ mi l . ) N x=0.5 x = l x=1.5 x=2 x=2.5 Mediein s.e. Median s.e. Median s.e. Median s.e. Median s.e. 2 3 0.79 0.15 0.78 0.18 0.83 0.22 0.84 0.33 4 0.89 0.07 0.89 0.08 0.90 0.11 0.94 0.10 5 0.90 0.08 0.89 0.09 0.84 0.18 0.69 0.45 6 7 0.90 0.06 0.90 0.06 0.88 0.07 0.87 0.08 0.84 0.13 / 8 0.90 0.07 0.92 0.07 0.93 0.09 9 0.76 0.22 0.82 0.08 0.80 0.03 10 11 0.81 0.11 Tab le 3.8: E s t i m a t e d Transformed M e d i a n s of Costs P r o j e c t size ($ mi l . ) x=0.5 x = l x=1.5 x=2 x=2.5 N Trans. Median s.e. Trans. Median s.e. Trans. Median s.e. Trans. Median s.e. Treins. s.e. Median 2 3 0.85 0.14 0.85 0.16 0.86 0.20 0.84 0.33 4 0.92 0.07 0.91 0.08 0.93 0.11 0.95 0.10 5 0.91 0.07 0.92 0.08 0.89 0.14 0.69 0.45 6 7 0.90 0.06 0.91 0.06 0.90 0.07 0.87 0.08 0.85 0.13 ( 8 0.90 0.07 0.92 0.07 0.93 0.09 9 0.76 0.22 0.82 0.08 0.84 0.07 10 11 0.81 0.11 Table 3.9: Test Results P r o j e c t Size ($ mi l . ) A M E s t a t i s t i c A M E p-value L S Stat is t ic L S p -value s s t a t i s t i c S p -value Quantiles ; 0.5 0.5 9.56 0.02 9.56 0.05 8.35 0.08 1.0 9.21 0.02 9.21 0.06 5.85 0.21 1.5 6.84 0.11 12.69 0.03 12.24 0.03 2.0 4.68 0.19 17.04 0.00 20.03 0.00 2.5 0.04 0.67 1.26 0.53 3.95 0.14 Quantiles : 0.3,0.5,0.7 0.5 30.45 0.00 30.47 0.00 41.77 0.00 1.0 31.77 0.00 31.77 0.00 41.57 0.00 1.5 20.44 0.02 32.10 0.01 46.46 0.00 2.0 12.89 0.12 44.32 0.00 62.33 0.00 2.5 1.02 0.73 3.82 0.70 15.65 0.02 Quantiles : 0.3,0.4,0.5,0.6,0.7 0.5 50.55 0.00 50.57 0.00 65.13 0.00 1.0 54.21 0.00 54.21 0.00 61.78 0.00 1.5 37.96 0.00 57.69 0.00 64.57 0.00 2.0 19.49 0.14 72.63 0.00 104.08 0.00 2.5 1.61 0.85 5.50 0.86 24.11 0.01 Bibliography B i b l i o g r a p h y [1] A n d r e w s , D . 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T h e measure of Sr is denoted dr, wh i ch s imply equals r i n the univariate case. Therefore, the weak consistency i n the propos i t i on entails: 4>{x) = limP{Sr)/dr, ( A . l ) r—>0 i.e., for any a r b i t r a r y e > 0, there exists a R such that iî r < R, \PiSr)/dr - m\ < e. (A.2) Fur ther denote r^^ := -x. It is easy to show that PiSr^^ ) — Ukj. has a beta d i s t r i b u t i o n w i t h parameters kr a n d T — k-r + l (Theorem 8.7.1, p.236, W i l k s (1962)[61]). W e first show that Ukj-ldr^^ —> 0 ( x ) i n probabi l i ty . A n app l i ca t i on of Tchebychev inequal i ty yields Uk^. —» 0 i n probabi l i ty . However, this can only happen when dr^^ 0 i n probabi l i ty , i.e., rfe^ 0 i n probabi l i ty . Le t i î be as defined i n (A.2) . There exists a t such that for T > i a n d any a r b i t r a r y > 0, P{rk^ <R}>l-ri- (A.3) T h i s suffices to i m p l y that UkT/dr,^^ —> (p{x) i n probabi l i ty . For the desired result , we are left to show that {T/kT}Ukj. —> 1 i n probab i l - ity. T h i s can be shown by using the fact that Ukj. has a beta d i s t r ibut i on along w i t h an app l i ca t i on of the Tchebychev inequality. Strong Consistency : W e follow the line of proof for the a.s. consistency of a regular N N estimate by M o o r e a n d Yacke l (1976)[40]. Wagner (1973)[59] provides another proof for the strong consistency through a different ap- proach. H i s approach can also be modif ied for our one-sided N N estimates. B y not i c ing that Ukj. can be viewed as sample kr/T-tile f rom T i i d uni form r a n d o m variable on the uni t interval , i t can be shown that under A s s u m p t i o n 1.2(ii) '-^-.1 a.. (A.4) [cf., Kie fer (1973)[25], M o o r e and Yacke l (1976)[40]] T h i s further implies that Ukj, —» 0 a.s. W e first c l a i m that -^0 = i n f { r : Ur > 0} a.s. (A.5) It is clear tha t r^^, > 0 a.s. for each T. If for any a r b i t r a r y e > 0, rkj, > e for a sequence of T at a sample point w, then C/fê > > 0 for these T at Lj. Since Ukj, —> 0 a.s., this can occur only on a set of ui hav ing zero probabi l i t ies . W e note that Ur = P{x-x<r}= I (t>{x)dx. (A.6) A p p l y i n g the mean value theorem for integrals, there exists dx sat is fying inf (t){x) <dT< sup ^{x) (A.7) such that Ukr=dTdr.. (A.8) W i t h (A .4 ) , (A.8) implies that ${x)/dT 1 a.s. (A.5) together w i t h the cont inui ty of at x ensures that dr —> (p{x) a.s. Therefore, the desired results. • P r o o f of P r o p o s i t i o n 1.2. F i r s t , we note that the right differen- t i a b i l i t y of 0 at X al lows us to wri te k]/^[(f){xt) — 0 (x ) ] = k]r^'^[4>'{x){xt - x) + oQxt -x|) ] when xt — x = o{kT/T). (A.9) T h e n , kr = o(T'^/^) i n the A s s u m p t i o n 1.3 further impUes that kll^[<Pixt) - <P{^)] ^ 0. (A . 10) B y the cont inui ty of 0 at x , as i n the proof of the previous proposi t ion , we can wr i te k It 4>i£.) = -Tj—<P{xt) where xt~ x< rfc„. ( A . l l ) UkT Moreover , we have shown that rkj. —> 0 and 1 i n probabi l i ty . - ^(x)) = k ^ i ' - ^ - 1) + k^ipl - 1 ) ^ . (A.12) <p(x) Ukr <P{x) Ukr Therefore, (A.10) impl ies that for the desired result , we only need to show 4 ' ' ( ^ - l ) - N ( 0 , l ) . (A.13) Since Ukj, is the / jT - th order statist ic of T i i d u n i f o r m r a n d o m variables UuU2,...,UTon[Q, 1], P „ ( a ) = P [ 4 / 2 ( ^ _ i ) < a ] 1 -I- akj. ' = P[BT<kT], where BT is the number OÎUI,...,UT fa l l ing below TT^ = — ^ ' ' l i / 2 a n d has the b i n o m i a l d i s t r i b u t i o n w i t h (TJTTT) . B y assumption, Trr —» 0 and TTTT oo, so that BT is asympto t i ca l l y normal . W r i t i n g p^(a) = p f T - T n T ^ kr-T^T^ ^^^^^ CTT O'T where CTT = [ T 7 r r ( l - TTr)]^''^, and not ic ing that kj. - TTTT O'T a. (A.15) W e o b t a i n PT{O) —> ^ (a ) , $ being the standard n o r m a l d i s t r ibut i on . T h i s completes the proof. • P r o o f o f P r o p o s i t i o n 1.4. I f e ) - 4>{x)\ < I f e ) - + - m \ (A.16) Loftsgaarden a n d Quensenberry (1965) [36] show that the second t e r m on the r ight h a n d side of (A.16) converges to zero i n probab i l i t y under Assumpt ions 1.1, 1.2(i) a n d 1.2(ii). If, i n add i t i on . A s s u m p t i o n 1.2 (iii) holds, Wagner (1973) [59] establishes the almost sure consistency for the second t e r m on the r ight h a n d side of (A.16) . Therefore, for the desired result to ho ld , i t suffices to show the first t e r m on the right h a n d side of (A.16) to converge. To this end , we rewrite the first term on the right h a n d side of (A . 16) kr/T kr/T kr/T ^^^^^ kr/T = 0,(fcj ') . (A.17) where x < x < Xsr- T h e second equaUty follows from the mean value theo- rem a n d the second last equal i ty follows from the fact — x = Oa.s.{T~^) shown i n the l e m m a . T h i s completes the proof. • P r o o f of P r o p o s i t i o n 1.5. 4>a.T) - 0 ( 2 ) = ( ? ( ^ T ) - ^(2)) + ( 0 ( £ ) - 4>{^) (A.18) I n l i m i t , the behavior of the t e r m i n the first bracket on the r ight h a n d side of (A.18) is dominated by the terms in the second bracket as shown i n the proof of previous propos i t ion . {kT)2{<p{x) — (j){x)) is shown to follow a normal d i s t r i b u t i o n w i t h zero mean a n d variance of (j){x)'^ by P r o p o s i t i o n 1.2. T h e desired result therefore follows by the asymptot i c equivalence l emma. • P r o o f of P r o p o s i t i o n 1.10. For the sake of consistency, we prove the propos i t i on i n the context of f i rst (high) -b id auct ion . C e l i k a n d Y i l a n k a y a (2006) [8] show that for any (symmetric ) o p t i m a l auc- t i on , the o p t i m a l cutoff v* should solve the fol lowing: l^*~^-J^]F(^T''=^- (A.19) W h i l e , as shown i n previous section, the equ i l i b r ium should entai l the cutoff, v^, as so lv ing the fol lowing: v'F{vT~^ = «• (A.20) For the desired result , we need to show v* > v^. I n negation, we suppose v* < v^. T h e n it must be the case that F ( u * ) " ~ ^ < T h i s further implies < [v* - ^ T p p ] < v*, wh i ch contradicts to the suppos i t ion . • P r o o f o f L e m m a 1.3. W e first show the a.s. convergence of X j . to X. Note that Xj^^ > X^^^^ for every T > 1. T h i s means that , for every sample po int u i n the under ly ing probabi l i ty space, X ^ j ( w ) > X^^^(a ; ) . Therefore, {Xj^^(a;),T' > l } i s a non-increasing real sequence, so i t has a l i m i t , denoted by X ( a ; ) . I n other words, the sequence of r a n d o m variables {X^^yT > 1} converges a.s. to the random variable X , a n d X < X j ^ ^ for T > I. So i t w i l l suffice to prove that X = £ a.s. T o this end, for any a r b i t r a r y e, ProhiXf^) - x > e ) = Prob{Xf^^ >x + e) = [l-F{x + e)f -> 0 as T oo, (A.21) where F is X ' s C D F . S o , X ^ j —> i i n probabi l i ty . W e have just shown that X^^j —> X a.s., w h i c h also implies X j^^ —> X i n probabi l i ty . Since no sequence can have d ist inct l i m i t s , X = ^ a.s. Therefore, Xj^ ~^ £ N o w we are left to f ind the convergence rate. Denote QF as the quanti le funct ion of F a n d Yi = FiX'l'^^). T h e n X j ^ j = Q F ( I ^ I ) - B y mean value theorem, we have Xf,) -x = QF{Y,) - QF{Q) = Q'F{Y)YU ( A . 2 2 ) where the second equaUty is by the mean value theorem a n d Y is from some neighborhood of 0. Since Yi is the smallest of T independent observations from the un i f o rm d i s t r i b u t i o n on the unit interval , " ^ " {T+l)...{T + k) • ^^-^^^ Therefore, i t follows that the mean squared error of Yi is of order T ~ ^ , i.e., E F j ^ = 0 ( T - 2 ) . T h i s further implies that Yi = 0{T-'^). Since Q'p{Y) — Q'p{0) —> 0 from the fact that (p (therefore, Q'p) is continuous i n a neighborhood of 0. P u t t i n g a l l together, we have Xj^ - x = 0(̂ 1) = o(T~^). T h i s completes the proof. • A.2 Details of Identification Issues G P V investigate some identi f icat ion issues i n the f irst-price auctions w i t h endogenous entry. T h e y consider a b ind ing reserve price i n first-price auc- t i o n (Theorem 4 , p . 5 4 8 ) . In that case, they assume only the number of ac tua l bidders is observed, besides the bids. However, we employ a different framework to set up the entry process. Therefore, modi f i cat ion is needed to a p p l y G P V ' s identi f icat ion results to our entry model . L e t P denote the set of a l l absolutely continuous d is t r ibut ions w i t h an interva l support i n R""". L e t G be the jo int d i s t r ibut i on of ( 6 1 , 6 „ ) , where n is the number of ac tua l bidders. G and g denote the d i s t r i b u t i o n of 6j and its density funct ion , respectively. Define Vi:=^{h,,G,N) = hi + j ^ ^ if bi>bp. ( A . 2 4 ) (A.24) can be views as the inverse b idd ing funct ion. T h e fol lowing proposi - tions provide the results for identi f ication of F on [vp, v] a n d the par t i c ipat ion cost K, respectively. P r o p o s i t i o n A . l (GPV, 2000) Assume in the endogenous participation setting, both the number of potential bidders, N, and the number of actual bidders, n, are observed. Let G * belong to the set P " with support [0,6]". There exists a distribution of bidders' private values F S P such that G is the distribution of the observed equilibrium bids in a first-price sealed-hid auction with independent private values and endogenous participation if and only if (1) G* (6 i , . . . , 6 „ ) = n r = i G * ( 6 i ) ; (2) The inverse bidding function (A.24)is strictly increasing on [0, b] and its inverse is differentiable on [vp,v] = [^{bp,G*,N),^(b,G*,N)]. Moreover, when F exists, it is unique on the support of [vp,v] and it sat- isfies F{v) = [1 - F{vp)]G*{Ç-^{v,G*,N)) for all v e [vp,v]. In addition, ^{•,G*,N) is the quasi-inverse of the equilibrium strategy in the sense that ^{b, G*,N) = b-\b, F, N) for all b € [0, b]. P r o o f for P r o p o s i t i o n A . l . Denote u) as the event of par t i c ipa - t i o n . T h e o r e m 1 i n G P V identifies condit ional d i s t r i b u t i o n , F{-\u)) f rom the observations. Therefore, i t suffices to show that there is a unique m a p p i n g from the cond i t i ona l d i s t r ibut i on , F{-\uj), to the uncondi t i ona l d i s t r ibut i on , F. Since we can observe n for each auct ion , the number of ac tua l bidders, a n d TV, the number of potent ia l bidders, i t is true that F = F{-\w) • Pr{w) for any given auct ion . T h e rat io , n/7V gives the probab i l i ty of par t i c ipat i on for the given auct ion . Therefore, Pr{ui) is now observed. Therefore, such a unique m a p p i n g exists. For the second part of the proposi t ion , i t is needed to show that (A.24) is i n fact an inverse b idd ing function. To this end, take the first-order derivative of ( 1 . 3 ) w i t h respect to v and rearrange terms, the fol lowing is derived l = ( . . - M ( i V - l ) | | g ^ ^f v.>v, ( A . 2 5 ) T h i s is ident i ca l to the equation ( 2 ) given i n G P V ( 2 0 0 0 ) [17] . T h e n their discussions about the inverse of b i d d i n g funct ion follow. T h u s , the desired results are derived by employing Theorem 1 i n G P V . B P r o p o s i t i o n A . 2 Given the result of Proposition A.l, the participation cost, K, is identified if and only if F is strictly increasing in a right neigh- borhood ofvp. P r o o f for P r o p o s i t i o n A . 2 B y the construct ion of the model , P r ( w ) = Pr[ivi - bi){Fimax{vp,b'\bi)})f-^ -K>0]. ( A . 2 6 ) Define _ Vi • [F(vp)]'^ 1 Vi<Vp Differentiate q{vi) w i t h respect to vf. F ( u p ) ^ * Vi <Vp T h u s , q{vi) is increasing i n Vi and str i c t ly increasing i n the r ight neighbor- hood of Vp, so is the cdf of qivi). Moreover , Pr{uj) is observable for each auct ion . Therefore the desired result f o l l ows .• Appendix B Proofs for chapter 2. P r o o f for L e m m a 2.1. W e first show that \QT - E (9 t ) l - » 0 un i f o rmly on 0 as T —» oo. T o this end, we apply Jennr ich ' s uni form law of large numbers (Jennr i ch (1969) [23], T h e o r e m 2). Specif ical ly, we need to show there exists a funct ion D such that \q{b,u,9)\ < D{b,u) for a l l 9 GQ a n d a l l b a n d u i n the supports of F a n d G. Moreover , E [ J D ( 6 , U ) ] = / D{b,u)dF{b)dG{u) < oo. N o t i n g that the second t e r m i n (2.9) is always non-negative, we have \qib,u,9)\ < \{b-Xiu,9)f\ < 2b'^ + 2X{u,9f / S \ 2 < 2b'^ + p^:=D{b,u). S g{u) F u r t h e r , we have E [ Z ) ( 6 , u ) ] = E [ 2 6 2 + | ! ^ ] = 2 j e{vf U{v;9)dv + | ï ï ( / J " < 0 0 , where the last inequal i ty is guaranteed by the finiteness of / a n d b i d d i n g strategy e. N e x t , we work on the expression for E ( ç f ) . Note that 'E,[qt{9)] is the same for every t € {1 ,2 , ...T} due to the property of i.i.d. i n our context . W e therefore suppress the subscript t i n the analysis. _ 1 ^ _ E (g ) = E[b - Xie)f - _ ^^ElY^iXM - X{9))'] = E[b-i{9)f + -E[i{e)-xie)]'^ +2E[6 - - X{e)] - ^VarlXsid)] = E[6 - 1(6)]' + ^Var[Xs{e)] - ~Var[Xs{e)] = nb-i{e)f = Qie), where the second equal i ty follows from the unbiased es t imat ion of Xs{6). T h e t h i r d equal i ty follows from the fact that b a n d s imula t i on draws are independent. Therefore, the desired result holds, i.e., S U P I Q T - Q W I I - ^ 0 . Bee P r o o f for L e m m a 2.3. For the desired result , i t is equivalent to showing each component of A is f inite. To this end, it is enough to show CTQ, Var [Xso ] , Var[yso] and /io are finite. Note that the finiteness of Coiv{XsQ,YsQ) can be imp l i ed if b o t h V a r f X s o ] < oo a n d VarfYso] < oo. W e first show CTQ < oo. R e c a l l that b is continuously d i s t r ibuted w i t h o n the support V b - fb is f inite and V b is a closed interval defined on R + . These facts ensure that CTQ < oo. N e x t , we consider Var [Xso] . R e c a l l (2.7), the def ini t ion of X^o- W e observe the fo l lowing: 1^. ̂0 = m a x [ . ^ , ) , F - n / , ^ ) ] - ^ ^ ^ " ^ ' ^ ^ - ^ ' ' ^ " " ' ^ ^ 5 ( u f ) . . . ^ « ) 5 " where u is the upper bound of V g . Moreover, g the density funct ion of u is finite. Therefore, the second moment of Xgo exists. S imi lar ly , for Var [Fso ] , we notice where V% := sup„g[j,^^]_ege l|V/^|| and V F := sup„g[„_5]_e6e l|VF||. T h e existence of such upper bounds are impl i ed by the assumptions. A g a i n , as u is cont inuously d i s t r ibuted w i t h g, wh i ch is finite. These facts together indicate tha t Var[yso] < oo- Las t l y , we show /io < oo. To this end, we note that KOo) = / ••• / m a x { u ? 2 ) , F ^(p,eo)fviui;do)---fv{un;Oo)dui...dun, i i n o i i < \\ui^)ym/g{u)\\ + \\vmF-\e)-vF{e)\\ w h i c h can be rewr i t t en as iieo) = Loieo) + Li{eo) + ... + Lsieo), where LoiOo) = ... F-\p-M Ju Ju fv{ui;Ôo)...fv{un;9o)dui...dun JF-^(P,0O) JU JU fv{ui;9o):. L2{9o) = / / ... / JP'-^ipfio) JF-^(pfio) Ju Ju i;9o).. fviun;9o)dui...dun F-^pfio) rF-^pfio) fviui;9o)...fviun;9o)dui...dun Lsi9o) = r ... r u^2) Jp-Hpfio) JF-Hpfio) fv{ui;9o)...fv{un;9o)dui...dun. Note that Lo(^o), •••,Ls{9o) are a l l differentiable w i t h respect to 9o. T h e n the der ivat ive of l{9o) is finite, provided that L's a n d their the derivatives are finite. F u r t h e r note that F~^, f^, U(2), V / t , a n d V F are a l l finite. These facts together are enough to i m p l y that /XQ < oo. • P r o o f for P r o p o s i t i o n 2.2. T h e propos i t ion follows d irect ly from the theorem due to W h i t e (1994) [60] [Theorem 3.4 on p.28]. • P r o o f for P r o p o s i t i o n 2.4. T o ease the expos i t ion , we s i m - pl i fy the notat ions i n the fol lowing way. Xs{9) denotes X{u^,n,9). X{9) is Xt{v},...,u^,n,9). T h e subscript 0 denotes the r a n d o m variables are evalu- ated at ^0- T h e subscript t is omit ted . T h e arguments i n the functions may be suppressed whenever there involves no confusion. Let Wsi0) = max[u%.,F-^p;e)]. T h e n , q{e) = u>s{e)fyie) -, 2 1 a n d = î ^ ^ ^ ^ ^ , = 1 E l l Xs{e). Denote n 2 « = 1 _ J "(2) " f 2 ) V / ( ^ ) / f l ( « ) a n d Y = {1 /S) J2Li Ys{0). Note y , (0) = {dXs{e)/de)' wherever X , is differentiable. T h e n , we further denote V 9 ( 6 , u, 0) = -2{{b - x{e))Y{e) + ^ ^ ^ ^ f;}{Xs{9) - x{e))Ym]- Fo l l owing d i rec t ly from T h e o r e m 2 i n P o l l a r d (1985)[51], the proof of the propos i t i on amounts to checking the fol lowing regular i ty condit ions are satisfied: (i) E[q(0)] has a nonsingular second derivative J at 9Q. (ii) E A = 0 a n d E ( A A ' ) < oo, where A ( 6 , u ) = Vg(6 ,u ,6lo ) . (iii) Stochast ic di f ferentiabi l i ty condit ion as defined i n the equat ion (4) of P o l l a r d (1985) holds for r (6 ,u,6l ) , where . ( M , ^ ) 4 ^ ^ ^ ^ ' ^ - ^ ^ ' ° ^ - ^ ' - ' ° ^ ' ^ ^ ( B . l ) ' ' \ 0 ^ = ^0 F i r s t , we show the stochastic dif ferentiabil ity cond i t i on holds for r{.,9). To this end , we first establish a result that 'E[r{.,d)'^] 0 as 9 OQ. Note that where Hi = [b-X{9)f-[b-X{9o)f + 2{9-9oy{b-Xo)Yo H2 = [Xsie)-Xi9)f-[Xsi9o)-Xi9o)]'' + 2i9-9on{Xso-Xo)Yso] Dependent on the real ized values of (6, u), there exists a real e > 0 such that q is differentiable on the e - o p e n b a l l i n 9 centered at ^o- T h e existence of such a b a l l is ensured by the fact that 9o is non-differentiable w i t h probab i l i ty zero. T h e n , the mean value theorem implies that there exists 9 i n the ba l l such that where the convergence happens because of the continuous dif ferentiabil ity of / „ i n the assumpt ion and the fact that (9 — 9o)/\\9 — 9o\\ is bounded by a uni t b a l l . S i m i l a r l y , i t can be shown that 0. Therefore, we have established that r{.,9) —+ 0 as 9 —> ^o- Fur ther note it is also true that r{.,9f^0as9^9o. N o w we show that there exists a funct ion W such that \r{9)\ < W for a l l 6 e@ a n d E [ W j < oo. W e notice the following IXsiOo) - Xs{e)\ 1 \\e - eoW \\e - 9o\\ Ws{9)Ue)/g{u) - wsieo)fM/giu) ^ •i{uL.>F-\e)}ufMe)/9iu) '̂(2) +i{u%, < F-Ho)}F-\e)ue)/g{u) < 2 ~l{ul2^>F-\6o)}ul2)Meo)/g{u) - l { i i ^ 2 ) < F-'(9o)}F-\eo)Meo)/9{u)\ \F-H6)M9)/g{u) - F-\eo)fM/g{u)\ +2 l l ^ - ^ o l l l » ( 2 ) / . W / g ( " ) - » ( 2 ) ^ ( ^ o ) / g M l l ^ - ^ o l l where 1{.} is no ta t i on for the indicator funct ion w h i c h takes the value of one i f the logical condi t ion inside i t is satisfied, a n d zero elsewhere. B u t \F-^{e)U{e)/9{u) - F-\eQ)U{e(>)/g{u)\/\\e - 04 is n o t h i n g else t h a n the slope of the funct ion F"^{6)fy{6)/g{u) w i t h respect to 6. T h e assumed uni form L i p s c h i t z property on ensures that there must exist W\ such that \F-'{9)U{e)/g{u) - F-\eo)fvi9o)/9{u) Simi lar ly , there must also exist W2 such that < Wi. ul^^Ue)l9{u)-ul^^fMl9{u)\ < H'2- Therefore, we have \Xs{eQ) - Xs{6)\ <2{Wi + W2). N e x t , we consider \Xs{0) + Xs{Oo)\ for any 6» G 9 . \Xs{e) + XM\ = \ws{e)ue)/g{u) + wM/M/giu)] < 2\mBx{u%yF-\9))Me)/g{u)\ 9W where û denotes the upper bound of Vg. / „ is the max; imum of / „ , whose existence is again i m p h e d by the assumption of un i f o rm L ipsch i t z property on / „ . T h e last inequal i ty follows from the fact that Vg contains the set [u,v]- W e wr i te r{d) as the following 1 1 s r{.,e) = \\o-eo\\ SiS-1) ^ i C 2 - ( ^ - e o ) ' A ] s = l where Ki = [b-X{6)f-[b-Xo]^ K2 = [Xso-Xo?-[x,ie)-xie)] |2 Note that 1^11 1 -A[2b-Xo-x{e)][Xo-x{e)]\ \\9-eo\\ we-eoV = pZ-ô^\\^^(^o - x(e)) - ( X o - x ( ^ ) ) ( X o + x{e))\ 2\b\ ^ |X,o - Xs{6)\ - S ^i P-^o\\ s s s s=l s=l s=l a n d 1^21 1 Mx,o + Xs{e))-{Xo + x{e))] \\e-eo\\ \\6-eo\\' x[{Xso-Xsie))-iXo-x{em P - U |(X«o + XsimXsO - X,{Q)) - ( X , o + X , ( ^ ) ) ( X o - x(e)) - ( X o + X ( e ) ) ( X , o - X , ( ^ ) ) + ( X o + X ( 0 ) ) ( X o - Xm\ - l^«o + X , ( e ) | +|X.o + X , ( ^ ) | +1^0 + + i x o + x ( . ) | F ° - S 52 s=l s=l s=l Fol lowing the facts that fy is s t r i c t ly positive and uni formly L ipsch i t z and that (6 - Oo)/\\0 - 6o\\ is bounded by a unit b a l l , there must exist such that \\0-ôo\\ - ' N o t i n g the fact that r is continuous i n K\,K2 a n d A , we therefore have shown that there exists a funct ion W := W2, M 3̂i Ŵ 4) such that \r{OY\ < for a l l 6» G G and E[W'^] < 00. Together w i t h the result that r{.,9y —> 0 as 9 —» ^0, the dominated convergence theorem implies that E [ r ( . , e ) 2 ] - . 0 as 9 ^ 9o. (B.2) L e t i? be a class of functions R = {r( . , 9) : 9 G © } . For the desired stochastic di f ferentiabi l i ty condi t ion , we are left to show that r{.,9) has the stochastic equicont inuity property, that is for r G i? a n d each e > 0, there always exists a, Ô > 0 such that l i m sup P r o 6 { sup |t>rr| > e} < e, (B.3) p{r)<S where p{r) = ( E r ^ ) ! / ^ and vrr = T~'^/^ Tj^iin-Er). T o this end, we next show that R has the E u c l i d e a n property mentioned i n Pakes a n d P o l l a r d (1989) [46]. N o t e that for any given {b,u) G V b x V g ^ , the po ints of 9 where q is non-differentiable form a closed set wi thout interiors . © is separated by such a closed set into finite numbers of subsets. T h e resul t ing parts contain points such that q is always differentiable. Use i G {1,2... , 2'̂ } to index these sets. W e first show that for 9 G 6 ' , the Hessian of q, V^q{b,u,9) is bounded by a p X p m a t r i x (f){b, u). Note that , for any given (6, u) a n d 9 G q{9) = S ~ n 2 s=l S r ~ 5(5 - 1) ^ [ Ws{m:i9) 9{u) -, 2 where ûJs takes on the value either u^^^^ or F~^{p;9), a n d no longer involves the m a x operat ion . Therefore, q is twice differentiable everywhere on 0 \ provided that Fy a n d fy are differentiable. W e then get a p x p m a t r i x for V^q{9) = {b-Yi9)yY{9) + ib-X{9))VY{e)-{- 5 ( 5 - 1 ) J2 [(n(^) - y(^)) 'n (e) + iXsi9) - Xi9))VYsie)]iBA) s=l where VYs{9) = dYs{9)/d9 and V y ( ^ ) = l / 5 X ^ f ^ i Vys (6 l ) . Denote the j - t h element i n the vector 9. fyj = dfy{9)/d9j. Fj = dF{9)/d9j. Ysj is the j-th element i n Ys- T h e bounded Hess ian of / „ implies that the gradient of fy is bounded by a constant vector as we l l . W e use to represent such a b o u n d i n g vector and its j - t h element is Zj. S imi lar ly , we define the b o u n d i n g vector for the gradient of F as a n d its j - t h element T o show that V^q is bounded, we notice the fol lowing: \Ysm < u'f, V] < < 9{u) fyj{9)F-Hp,9)-Fj{9) 9{u) 9{u) \fvj 1 + 9{u) + 1 J 9{u) 9{u) IF , D] •.= Mb,u) N o w consider the jA ; - th element of V l ^ ( & ) . Denote Z the bounding m a t r i x for V ^ / „ a n d F the bounding m a t r i x for V ^ F . W e use subscript jk for the jk-th element in the corresponding m a t r i x . < 9{u) + < + v 9{u) < +v \Zik\ + 9{u) 1 îv{e)9{u) z]\\Dl\ + 1 1 3k 9[u) 9{u) Djk\ ••= 4>i{b,u) Furthermore , 9W (B.4) suggests that V ^ g is continuous i n Yg, X g , V F ^ , w h i c h i n t u r n i m - plies there must exist a composed funct ion (j){b, u) := (j>{(j)\{b, u), 4>2{b, u), cj)2,{b, u)) such that |V2<7(6,u,6l)| < Since i is chosen a r b i t r a r i l y and (i){h,u) does not depend on i , the cont inuity of q over 0 impl ies the existence of (j) : JR-^+i R such that \\Vq{h,u,ei)-Vq(b,u,e2)\\ < 4>ib,u) Pi-O^l (6 ,« ,^1,^2) € Y b x Y ^ x O ^ . W e next establ ish the result that r is a L i p s c h i t z funct ion . P i c k 6 e 0\{0o}. T h e n we have that Vr{b,u,e) = Note that V g ( b , u , g ) - V g ( b , u , g o ) l l ^ - ^ o l l q{b, u, 9) - q{b, u,9o)- V'qjb, u, 9o)id - OQ) ^q{b,u,9)-Vqib,u,9o) \\o-eo\\ <4>{b,u). A l s o , note that for some 9 on the Hne segment j o in ing 9 a n d 9o q{b, u, 9) - u , 9o) - Vq{h, u, 9o){9 - 6o) = \Vq{b, u, 9){9 - ^o) - V 'g (6 , u, 9o){9 - 60) <\\Vq{b,U,9)-Vqib,U,9o)\\ \\9-9o\\ <(j>{b,u) \\9-e4 \\9-94 <4>{b,u) \\9-9of. (B.5) It follows that q{b, u, 9) - q{b, u, 9o) - Vq{b, u, 9Q){9 - OQ) \\e-9or T h u s , we have that ||Vr(6,«,^)||<20(fe,w). Because {b,u) G V b x V g ' ^ and 9 G 6\{6'o}, it follows that for each {b,u) G V b X V g ' ^ a n d 9Q G 9 , r ( 6 , u , ) : 6 ^ M is a L i p s c h i t z func- t i o n on 6\{^o} w i t h the L ipsch i t z constant set equal to 20(6 ,u ) . F u r t h e r , i t follows from (B.5) that for each 9 G 0\{^o} a n d each {b,u) G V b X V g - ^ that \r{h,u,9)-r{b,uM\ = \r{h,u,9)\ _ 1 = cj>{b,u)\\9-94<24>{b,u)\\9-94. q{b, u, 9) - q{b, u, 9o) - V ' g ( 6 , u, 9o) {9 - 9o) \ T h u s , for each (6, u) G V b x V g ' ^ , r{b, u , •) : 9 —» R is L i p s c h i t z w i t h the L i p s c h i t z constant set equal to 20(6, u) . T h e L e m m a 2.13 i n Pakes and P o l l a r d (1989)[46] impl ies that R is Euc l idean for the envelope n{b,u) := 2 y ^ s u p 0 ||^-^o||0(6, u) . Since 9 is compact, then supe || -̂̂ o|| < 0 0 . M o r e - over, the finiteness of 0(6, u) is impl i ed by the formulat ion of 0 i , 02 and 03. A l l these facts together ensure that ETZ < 0 0 , wh i ch further impl ies that (B.3) holds by the L e m m a 2.16 i n Pakes and P o l l a r d (1989)[46]. Fur thermore , (B.2) and (B.3) together i m p l y that s u p | i ; T r ( . , ^ ) | - ^ 0 (B.6) UT for each sequence of bal ls {UT} that center at 9o a n d shr ink down to 0. B y no t ing that (B.6) is a stronger condi t ion than ( B . l ) , we have shown that the remainder funct ion r{9) satisfies the stochastic di f ferentiabi l i ty condi t ion as desired. N e x t , we show the dif ferentiabil ity of E[ç] at 9o. O n l y i n the case where U(2) = F~^(p;9o), q is not differentiable as WS{9Q) is not. However, notice that E[ç] can be w r i t t e n as S folds of integrals, each of which represents the averaging over the r a n d o m variable u^^^y T h e each of such integrals can be separated into tak ing averaging over two sets: u^2) > F~^{P\9Q) a n d tt*2) < F~^{P;9Q), because the event U(2) — F~^{p;9o) happens w i t h probab i l i t y zero as u is continuously d is t r ibuted w i t h density g. Further note that a l l the terms i n E[g] that involves 9 are twice differentiable by assumptions . These facts together ensure the twice di f ferentiabi l i ty of E[g] at w h i c h is denoted as J. T h e nonsingular i ty of J is i n the assumption. N o w we show that E A = 0. E A = EWeq{9o) = VeEç(^o) = ^eQ{9o) = 0, where the last equal i ty follows from the ident i f iabi l i ty assumption . T h e second equal i ty follows from the interchangeabi l i ty of expectat ion and dif- ferent iat ion, w h i c h i n t u r n follows from the fact that di f ferentiation of Efg] w i t h respect to the 9 on the l imits of integrals is equal to zero. To complete the proof for the theorem, now it remains to derive the covariance m a t r i x , S . B y the theorem due to P o l l a r d (1985)[51], S = J - i E ( A A ' ) J - ^ W e first consider J . W e note that A = dq{6)/d9o. Di f ferent iat ing E A , we obta in J = - 2 ( J i + J2 + J3 + Ji), where J i = - E I F O F Q ] , _ J2 = E [ ( 6 - X o ) ^ ] , 1 ^ - = E [ - ^ ^ ^ r - ^ 5 ^ n o ( n o - y o ) ' ] , = E [ ^ | : ( X . o - X o ) ^ ] . N o w for any s £ {1 ,2, [Xsa^yso) are i.i.d. a n d independent of b. T h u s , us ing the fact that E[6] = E[X3o] = l{0(i), we get J i = - [ V a r ( F o ) + E F o E y [ ) ] = - ( i v a r ( y , o ) + E r , o E y ; o ) , J2 = E [ a ( 0 o ) - X o ) ^ ] = - - C o v ( X . o , ^ ) , J3 = - ^ V a r ( n o ) , Cance l ing terms a n d using the Ey^o = dEXso/dO = fiQ by inver t ing differ- ent iat ion and expectat ion. Therefore, we have J = 2non'o. W e now t u r n to compute E ( A A ' ) . Denot ing /xxo = '(^0) and /xyo = dl{9o)/9. W e have A = - 2 ( > l i - ^ 2 - ^ 3 + >l4), where Al = {b-fixo)Yo, A2 = (Xo - p.xo)lJ-Yo, g 1 ^ A4 = _ YliXsO - fJ-xo){Yso - Myo)- Because E A = 0, E ( A A ' ) = V a r [ A ] . Note that , for j = 2 ,3 ,4 , Cov{Ai,Aj) = E[(6 - fixo)YoA'j] = 0, a n d 1 ^ C o v [ ^ 2 , ^ 3 ] = g 2 ( g _ ^ ) m E [ ( ^ ( X , o - M x o ) ' s +2 ^ ( X , o - Mxo) (^to - f^xo))iJ2(^so - MKo)')] s<t s=l ^ fJ-YOmXsQ-fiXO?{YsO-fJ-YoY], S{S - 1) 1 ^ C o v [ ^ 2 , ^ 4 ] = g 2 ( g _ ;^) /^V-oE[(^(XsO - / ixo ) ) s 3 = 1 T h u s Cov[A2, A3 - A4] = 0. Hence V a r [ A ] = 4 (Var [A i ] + Varf / la] + Var[^3 - ^4]). N o w it remains to compute the variance matrices of Ai,A2 a n d {A3 — A4). Note that these vectors have zero expectations. Since <JQ = E[(6 — p-xo)'^]: we o b t a i n V a r [ A i ] = a ^ ( | v a r [ n o ] + / iyo /xVo) , Var[yl2] = ^Var[Xso]fiYO(J-'Yù- For the t h i r d variance covariance m a t r i x , we have V a r [ A 3 - A 4 ] = E l A s A y + £[^4^^] - E l A s A ^ ] - E [ A 4 A ^ ] . B u t , s s<t s=l +2Y,{Yso-^iYo){Yto-^iYo)')] s<t s s ^ ' s=\ s = l + 4 ( ^ ( X , o - /xxo)(Xto - / i xo ) E(^-o - /^yo)(Fto - /iyo)')] 1 = g 2 ( g , i ) 2 E [ E ( ^ ^ Q - ^^0)^(^0 - M y o ) ( n o - / i yo ) ' + "^{Xsù - /xxo)^(l^to - Myo)(i'«o - Myo) ' +4 E("''-^o ~ AiA:o)(^to - A'xo)('5^so - Myo)(yto - /^yo)'] = g ( g ^ 1 )2^1 (^^0 - /xxo)^(no - M y o ) ( n o - /iyo)'] + ^ ^ ^ ^ [ V a r X , o V a r n o + 2 C o v ( X , o n o ) C o v ( X , o n o ) ' ] , + ^{Xso - Mxo)(^to - lJ-xo){Yso - ^Yo){Yto - HYO)'] = g (g^_ i ) 2^[ (^«o - Mxo ) ' (no - Myo ) (no - Myo)'] ^~^^Cov{XsoYsu)Cow{XsM', 1 ^4] = g2(5 _ i)2 [̂(I](-̂ ^Q ~ iJ^xaWsQ - /xro) + l^C^so - /ixo)(yto - m))(E(̂ «o - Mxo ) (no - Myo)')] si^t s=i 1 ^ = S2 (g _ i)2 [̂(IZ(̂ «o ~ /̂ xo)(>".o - /^yo)) s= l = E [ A 4 A ^ ] . Therefore, we o b t a i n V a r [ ^ 3 - ^ 4 ] = E[^3^^] - E[^4A^] = JiJZY) [ V a r X , o V a r n o + Cov(X,oV;o)Cov(Xsono) ' l T h u s , V a r [ A ] = 4 { a g ( | v a r [ n o ] + HYOP'YO) + | v a r [ X s o ] / x y o M y o + g ^ g ^ _ ^ ^ [ V a r X , o V a r y , Q + C o v ( X , o y s o ) C o v ( X , o n o ) ' ] } = 4 { a g ( ^ V a r [ n o ] + MoMo) + | v a r [ X , o ] M o M o + ^ ^ ^ ^ [ V a r X , o V a r n o + C o v ( X , o n o ) C o v ( X , o n o ) ' ] } , where the last equal i ty follows by observing that / iyo = dl{6o)/6 = /io- • Appendix C Proofs for chapter 3 C . l Details of the entry models T h e fo l lowing l e m m a is used for the proof of P r o p o s i t i o n 3.3. L e m m a C . l The function Tl{s,s,N) is a non-decreasing function of s. Further, the function ]I{s,s,N) is a continuous and increasing function of s. For s G [0,1), ll{s,s,N) is a decreasing function of N, and constant in N ifs = 1. P r o o f for L e m m a C . l . Subs t i tu t ing P ( s ) a n d F*{v\s) from (C.2) a n d (C.3) respectively into (C.6) gives after some m a n i p u l a t i o n the fol lowing expression for \l{s,s,N): j\\ - F{v\s)){\ - Pr{Si >-s,Vi>r) + Pr{Si > s, S [r, V\)f-^dv. ( C . l ) Note that by the aff i l iation of Vi and Si, F{v\s) is non- increasing i n s (Theo- rem 23 i n M i l g r o m a n d Weber (1982)[38]), so 1 - F{v\s) is non-decreasing in s a n d consequently n ( s , s, N) is non-decreasing i n s. A l s o , the t e r m w i t h i n the second parentheses i n ( C . l ) is increasing i n s when v € {v,v): i^Pr{Si >s,Vi£ [r, V]} - Pr{Si >s,Vi> r}] = ^ (^j\nv\s) - F ( r | 5 ) ] ^ - F{r\s)di^ = F{r\s) - F(v\s) + F{r\s)] = 1 - F{v\s) > 0, because F{v\s) 6 (0,1) i f u S {v, v). It follows that II (s , s, N) is a continuous a n d increasing funct ion of s on [0,1). A l s o observe that for any s G [0,1), the t e r m w i t h i n the second parentheses 1 - Pr{S^ >s,Vi>r} + Pr{Si > 5, K € [r, V]} € [0,1), so that II(s, s, N) is a decreasing funct ion of N. • P r o o f for P r o p o s i t i o n 3.3. F i x bidder i a n d assume that his r ivals follow their equ i l i b r ium strategies represented by a cutoff s. F r o m bidder i ' t h v i ewpo int , condi t ional on entry he is p a r t i c i p a t i n g i n a f irst-price auct ion w i t h a r a n d o m number of bidders. Specifically, the number of his r ivals follows a b i n o m i a l d i s t r ibut ion w i t h parameters A'̂ a n d the probab i l i ty of b i d d i n g equal to P{s) = Pr{Si>s,Vi>r}. (C.2) (we suppress the dependence of s on iV for now), a n d the r ivals have iid d i s t r i b u t e d va luat ions according to F*{v\s) = Pr{Vi < v\Si >s,Vi> r}. (C.3) Note that i f s = 1, then P{s) = 0 and F*{v\s) is not defined. A s s u m i n g existence of a b i d d i n g equ i l ib r ium i n which a l l bidders use the same b idd ing strategy B : [v,v] —* R+, an increasing funct ion , a bidder wins against a given po tent ia l r i v a l either i f the r i va l does not b i d , or bids but his valuat ion cond i t i ona l on b i d d i n g is less than v. T h i s probab i l i ty is equal to 1 — P{s) + P{s)F*(v\s). B y independence, the probabi l i ty of w i n n i n g the auct ion for a b idder w i t h va luat i on v is (1 - Pis) + P{s)F*{v\s))'^-K (C.4) Standard envelope-theorem argument implies that z ' th profit condi t iona l on receiving the signal Si = s at the b idd ing stage is equal to J\l-Pis)+Pis)F*{v\s)f-'d^, A s s u m i n g provis ional ly that an equ i l ib r ium given by a cutoff s exists, the b i d d i n g strategy B{v) can be found from the alternative expression for this profit , {v - B{v)){l - Pis) + P{s)F*iv\s))''-\ which gives the e q u i l i b r i u m b i d d i n g strategy j;{l-P{s) + P{s)F*{v\s))^-'dd il-Pis)+Pis)F*iv\s))^ S t a n d a r d arguments (e.g. M i l g r o m (2004)[37]) i m p l y that B{-) is an increas- ing funct ion , a n d is indeed a best response. T h e expected profit at the entry stage n(s,s,N) = £ f{v\s) J\l-P{s) + P(s)F*{v\s))^-^éiidv-k = J\l - Fiv\s))il - P{s) + P{s)F*iv\s))^-'^dv - k, where the last line follows by integrat ion by parts. If the reserve price is b ind ing , then the bidder w i t h the lowest active type is the one w i t h the lowest type possible, i.e. v. U n l i k e i n the b i n d i n g reserve price case, this bidder now makes a posit ive profit a n d w i n n i n g the auct ion i n the event when no one else chooses to enter, i.e. w i t h probab i l i t y (1 - P{s))^~^{v - r). A p p h c a t i o n of the Envelope T h e o r e m now results i n U{s,s,N) = {l-P{s))'^-\v-r) + J\l - F ( i ;|s ) ) ( l - P ( s ) -t- P ( s )P* ( ï ï|s ) )^ - id ï ï - k, a n d the b i d d i n g strategy must be modif ied accordingly. T h e c ruc ia l quant i ty w i l l be the marg ina l bidder 's profit n ( s , s, TV), i.e. when bidder i has a s ignal equal to the equ i l i b r ium c u t o f i ' s . Note that Pi{s, s, N) defined even i f s = 1 (no rivals enter), i n w h i c h case i t does not depend on A^: n ( l , 1, N) = j \ l - F{v\s))éS - k. N e x t , note that i n v iew of L e m m a C . l , for a given N >2, either n ( 0 , 0 , TV) > 0, so that a n e q u i l i b r i u m w i t h cuto f f s = 0 exists, or 11(1,1, TV) < 0 so that an equ i l i b r ium w i t h cutoff s = 1 (no entry) exists, or an e q u i l i b r i u m such that a bidder w i t h s G (0,1) that solves the indifl?erence equat ion n ( s , s, TV) = 0 exists. T h i s i m p l i c i t l y defines the equ i l ib r ium cutoff s as a funct ion of TV, s(TV). Since n ( s , s , TV) is an increasing continuous funct ion of s and a de- creasing funct ion of TV, t a k i n g on the same value for a l l TV if s = 1, it follows that i f s(TV) G (0 ,1) , then also s(TV) G (0,1) for N' > TV, a n d s{N') > s(TV). • P r o o f for P r o p o s i t i o n 3.4. Consider the only i f part first. There are two cases to consider. F i r s t , suppose that p{N) < ... < p{N). Denote for future reference w i t h i n this proof T(TV) = J\l - F*{v))(l-p{N) +p{N)F*{v)f-^dv. W e know that for TV > TV, s (TV) G (0,1) and the marg ina l b idder is indiflï'er- ent between enter ing or not , fc = [1 - F ( r ) ] r ( A ^ ) V TV > TV, (C.6) For TV = TV, i t m a y be either that s(TV) = 0, so that bidders enter w i t h probab i l i ty 1, or s(TV) G (0,1) . T h e key observation is tha t the quant i ty 1 - F{r) is not identi f ied, so that (C.6) cannot be used to identi fy k. W e only know that w i t h str ict equal i ty on ly i f s ( iV) = 0, and the weak inequal i ty iis{N_) € (0,1). N o w , suppose that p{If) = ... = p(N). In this case, s{N) = 0 so that 1 - F{r) = p{N) V N eAf, ( in part i cu lar , 1 - F ( r ) is identif ied) . W e can only put an upper b o u n d on k: k < [l-F{r)]TiN) < [l-F{r)]T(N), where the last inequal i ty follows from the fact that [1 - F{r)]T{N) is the profit i f a potent ia l b idder when each r iva l enters w i t h probab i l i t y 1. W e now prove the " i f part . If H < < N, then by the same logic as i n case 2 above, 1 - F(r) is identif ied since whi le by the same logic as i n case 1 above, k is identi f ied f rom (C.6) since for AT = A''» + 1, bidders enter w i t h probabi l i ty s{N) £ (0,1) and therefore are indifferent between entering or not. • C.2 Details of the estimation method W e make the fol lowing assumptions concerning the d a t a generating pro- cess. A s s u m p t i o n C . l (a) {{Ni,xi) : I = 1,...,L} are i.i.d. (b) The marginal PDF of xi,ip , is strictly positive, continuous on its com- pact support X C R**, and admits at least R > 2 continuous and bounded partial derivatives on Interior (X). (c) The distribution of Ni conditional on xi, Tr{N\x), has support N = {N, ...,N} for allx€X,N> 2. (d) Vil o,nd Ni are independent conditional on xi. (^) {Vit '• i = ^,---,Ni;l = l,...,L} are i.i.d. conditional on {Ni,xi) (f) For ail X £ X, the density of valuations f{-\-) is strictly positive and bounded away from zero on its support, a compact interval [v{x),v{x)\ C R + , and admits at least R continuous and bounded partial derivatives its interior. (g) 7r(iV|-) admit at least R>2 continuous bounded derivatives on Interior (X) for all N eAf. (h) The entry probability conditional on {N,x), p{N,x), is strictly posi- tive for all N & M and x G X, and p{N, •) admits at least R > 2 continuous derivatives bounded away from zero on an open subset X+ e Interior{X) and all N eAf. A s s u m p t i o n C . l (a) is the usual iid assumption on the d a t a generating process for the covariates. Assumpt ions C . l ( b ) , (f), a n d smoothness of functions i n (g) and (h) are s tandard i n the nonparametr i c auctions l i ter - ature (see, for example , G P V ) . A s s u m p t i o n C . l ( c ) defines the support of the d i s t r i b u t i o n of TVj condi t iona l on the covariates. A s s u m p t i o n C . l ( d ) is one of the most impor tant assumptions; it asserts that i n the number of potent ia l b idders N is exogenous condi t ional on xi = x, w h i c h allows us to use the var ia t i on i n Â ^ for the purpose of test ing. In Sect ion 3.6, we exp la in w h y th is assumpt ion is plausible i n the context of our e m p i r i c a l apphcat ion . A s s u m p t i o n C . l ( e ) is the I P V assumption. For kernel es t imat ion , we use kernel functions K sat is fy ing the fol lowing s tandard assumpt ion (see, for example , Newey (1994)). A s s u m p t i o n C.2 The kernel K has at least R>2 continuous and bounded derivatives on R, compactly supported on[—l, 1] and is of order R: J K{u)du = I, J uŒ{u)du = 0 for j = 1 , R - l . T h e s t a n d a r d nonparametr ic regression arguments i m p l y that the est ima- tor of entry probabi l i t ies p{N, x) is asymptot i ca l ly n o r m a l as well (see, for example , P a g a n and U l l a h (1999)[45], Theorem 3.5, p . l l O ) : P r o p o s i t i o n C . 2 Suppose that x G A ' t . Assume that the bandwidth h sat- isfies as L oo; Lh^ —» oo and \/U?h^ —» 0. Then, under Assumptions 3 and 4, VLhM{p{N,x) —p{N,x)) is asymptotically normal with mean zero and variance Moreover, the estimators p{N,x) are asymptotically independent for any distinct N, N' G {TV, ...N} and x, x' G Ml Since the d i s t r i b u t i o n of values and, consequently, the d i s t r ibut i on of bids have compact supports , the est imator of the density g* is asymp- to t i ca l l y biased near the boundaries. O u r quanti le approach allows one to avoid the prob lem by considering only inner intervals of the supports . Specif ically, let [v{N,x),v{N,x)] denote the support of F*{v\N,x), a n d let A be some compact inner interval , A{N,x) = [vi(N,x),V2{N,x)] C \v{N,x),v{N,x)]. T h e quanti le orders corresponding to vi a n d V2 are given by Ti{N,x) = F*{vi{N,x)\N,x) for i = 1,2. Hence, we consider quanti le or- ders i n T(A' ' , x) = [TI{N, X), T2{N, X)]. N e x t , the corresponding inner interval of the support of G* is given by the values between the n a n d T2 quantiles: e ( 7V ,x ) = \bi{N,x),b2(N,x)], where bi{N,x) = q*{Ti{N,x)\N,x), i = 1,2. Simi lar ly , we define the interval of quantile orders for transformed quantiles: T'^iN,x) = [T^{N,x),T^iN,x)] such that r f ( 7 V , x ) = { inf T|/3(r, A^, x) > r i ( i V , x ) , r G [0,1]} and r2^(iV,x) = sup{r|/?(r, A^,x) < r 2 ( i V , x ) , r G [0,1]}. L e m m a C . 3 Under Assumptions C.l and C.2, for allx G Interior{X) and N€^f, (a) ^ ( x ) - ^ ( x ) = O p ( ( j ^ ) - i / 2 + (b) 7r(7V|x) - 7r(7V|x) = Op{{{^)-'/' + h^). (c) piN,x) - p{N,x) = Op{i{^)-'/^ + h^). (d) ^^PbmN,.),m.)] \G*{b\N,x) - G*{b\N,x)\ = O p ( ( | ^ ) - i / 2 + h^). (e) snp,^r(N,.) ITirlN, x) - q*{T\N, x)\ = Op{{{^r'l^ + h^). ( f ) sup, ,e ( ;v , . ) \rm,x) - 9*{b\N,x)\ = 0 , ( ( ^ ) - V 2 + ^f l ) . (g) snp,^riN,x) \Q*{r\N,x) - Q* ( r| iV ,x )| = O p ( ( M f l i ) - i / 2 + (h) sup , e [ o , i ] l /S ( r , 7V ,a ; ) - / ? ( r , iV , : r )| = 0 p ( ( j g ) - V 2 + / , f l ) . ( i ) sup,eT^( ;v ,x)IQ*( -3(r , iV,x)|iV,a ; ) -Q*( /3(r , iV,a ; )|7V,a ; )| - L e m m a B . 3 of Newey (1994)[43]. For par t (d), define a funct ion G*o{b,N,x) = Np{N,x)7r{N\x)G*{b\N,xMx), a n d its est imator as G*o{b,N,x) = -rjjr^^yuHNi = i V } l { 6 , , < b}K,h{xi - a;), P r o o f o f L e m m a C . 3 . P a r t s (a)-(c) of the l e m m a follow from L N, 1=1 i=l where (C.9) (C.8) N e x t , / JV, \ EGl{h,N,x) = E l{Ni = N)K,h{xi-x)Y,yul{hu<h} \ r=l / = NE[l{Ni = N}K,h{xi - x)yal{hu < b}) = NE{E{l{bu < h}\N,xuVii = l)yul{Ni = N}K,H{XI - x)) = NE[G\h\N,xi)p{N,x:)n{N\xi)K,n{xi-x)) = N J G*ib\N,u)p{N,u)7r{N\u)K,h{x - u)^{u)du = JG*o{b,N,x + hu)Kdi^)du. B y L e m m a 3.5, G*(6|7V, •) admits at least R+1 continuous derivatives. T h e n , as i n the proof of L e m m a B.2 of N e w y (1994[43]), A s s u m p t i o n s C . l ( b ) , (g) a n d (h) i m p l y that there exists a constant c > 0 such that |G5(6 , iV ,a ; ) -SG5(6 , iV ,x )| < c / i « (^j \K^i^)\\\u\\^du^ \\vec{D^GUb, N,xm, where || • || denotes the E u c l i d e a n norm, and D^GQ denotes the R - t h p a r t i a l derivative of G Q w i t h respect to a;. It follows then that ^^Pbem,-),mx)]\Goib,N,x) - EGUb,N,x)\ = 0{h\ N o w , we show that (C.IO) ^mmN,r)hN,x)]\Gl{b,N,x) - EGl{b,N,x)\ = M — j ( C . l l ) W e follow the approach of P o l l a r d (1984) [50]. Consider , for given N € ^f a n d X € Interior(X), a class of functions Z indexed by h a n d b, w i t h a representative funct ion ziib,N,x) = J2yiinNi = N}l{bu < b]h''K,h{xi - x). i=l B y the result i n P o l l a r d (1984) [50] (Prob lem 28), the class Z has po lyno- m i a l d i s c r i m i n a t i o n . Theorem 37 i n P o l l a r d (1984) [50](see also E x a m p l e 38) impl ies tha t for any sequences 5L, ai such that L5\a\jlogL —>• oo, Ezf{b)<ôl al^l" sup \-Y^zi{h,N,x)-Ezi{h,N,x)\^Q (C.12) b€[b(N,x),b(N,x)] ^ 1=1 almost surely. W e c l a i m that this implies / rud \ 1/2 — sup \GUb,N,x)-EGUb,N,x)\ \LogL,/ beUN,x),b{i^,x)] is bounded as L ^ oo almost surely. T h i s implies tha t sup \Gl{b,N,x)-EG*o{b,N,x)\ = Op (-—) 6e[È(Ar,x),b(iv,x)] \\i'Ogi^j j T h e proof is by contradic t ion . Suppose not. T h e n there exist a sequence 7 i —> oo a n d a subsequence of L such that along this subsequence sup | G S ( 6 , 7 V , x ) - F G S ( & , i V , 3 : ) | > 7 L 7 - 7 • (C.13) 6e[6(W,x),6(7V,x)] on a set of events fi' C w i t h a posit ive probab i l i ty measure. N o w if we let &\ = / i * * a n d = 1L^{J^Y^I'^-, then the def ini t ion of z implies that , a long the subsequence, on a set of events fi'. aL^'5L^sup^e[6(7v,x),6(iV,x)] I i E t i zi{b,N,x) - Ezi{b,N,x)\ \l0ghj i,6[b(iV,x),6(JV,i)] ^ /=i / r , d \ 1/2 \iogijj be[b{N,x)fi{N,x)] - [logLj ^"-yiogL) 1/2 = It - ^ 0 0 , where the inequahty follows by (C.13) , a contradic t ion to (C.12) . T h i s establishes ( C . l l ) , so that (C.IO) , ( C . U ) and the tr iangle inequal i ty together i m p l y that sup \Gl{b,N,x)-Gl{b,N,x)\ = O p { { ^ \ + h ^ \ (C.14) be\b(N,x),b{N,x)] Wog^J / To complete the proof, recal l that , f rom the definitions of GQ{b,N,x) and G*oib,N,x), G*ib\N,x) = G*{b\n,x) = so that by the mean-value theorem. G*o{b,N,x) p{N,xMN\xMxy G*o{b,N,x) p{N,x)Tf{N\x)(p{x)' \G*{b\N, x) - G*{b\N, x)\ < C{b, N, x) GUb,N,x)-G*a{b,N,x) p{N,x)-p{N,x) n{N\x) - 7r(iV|a;) 0{x) - ip{x) / (C.15) where C{b, N, x) is given by p{N, X)T!-{N, x)ip{x) ' Gl{b,N,x) G*o{b,N,x) Gl{b,N,x'^ ^ ' p{N,x) ' n{N,x) ' ^{N,x) J a n d | | ( G 0 - G ' O , p - p , ^ - 7 r , ^ - < ^ ) | | < ||(G" - G ^ p - p , 7 f - T T , ^ - f o r a l l {b,N,x). Fur ther , by A s s u m p t i o n C . l ( b ) , (c) a n d (h), a n d the results in parts (a)-(c) of the l emma, w i t h the probab i l i ty approaching one p,7f and axe bounded away from zero. T h e desired result follows from (C.14) , (C.15) a n d parts (a)-(c) of the lemma. For par t (e) of the l emma, since G*{-\N,x) is monotone by construct ion, P{q*{n{N,x)\N,x) <b{N,x)) = p(^m{{b:G'ib\N,x) >TI{N,X)} <b{N,x)^ = p(G'{b{N,x)\N,x)>n{N,x)) = 0(1), where the last equal i ty is by the result i n part (d). S imi lar ly , PiriT2{N,x)\N,x) > b{N,x)) = P{G{b{N,x)\N,x) < T2{N,X)) = 0(1). Hence, for a l l x € Interior{X) a n d N G M, w i t h the p r o b a b i l i t y approaching one, b{N,x) < q*{TiiN,x)\N,x) < q*{T2{N,x)\N,x) < b{N,x). Since the d i s t r i b u t i o n G*{b\N,x) is continuous in b, G*(q*{T\N,x)\N,x) = r , and, f o r r 6 T(N, x), we can wr i te the ident i ty G%r{r\N,x)\N,x)-G*{q*{T\N,x)\N,x) = G*{q'(T\N,x)\N,x)-T. (C.16) U s i n g L e m m a 21.1(ii) of van der Vaar t (1998)[58], a n d by the def init ion of G * , 0<G*{q*(T\N,x)\N,x)-T < ^ p{N,x)n{N\x)(p{x)NLh'i' a n d by the results i n (a)-(c), G*iq*iT\N,x)\N,x) = T + Oj, [{Lh'')-') (C.17) un i f o rmly over r . C o m b i n i n g (C.16) and (C.17) , a n d app ly ing the mean- value theorem to the left -hand side of (C.16) , we obta in rir\N,x)-rir\N,x) = Ç m ^ l ^ ' " ^ ' ( 5 * W ^ ^ - g*iq*{T\N,x)\N,x) + 0 p [{Lh'")-') , (C.18) where q* lies between q* a n d q* for a l l {T,N,X). N O W , by L e m m a 3.5, g*{b\N,x) is bounded away from zero, a n d the result i n part (e) follows from (C.18) a n d part (d) of the l emma. T o prove par t (f), by L e m m a 3.5, 5*(-|iV, •) admits at least iï-1-l continuous bounded p a r t i a l derivatives. Le t g*o{b,N,x)=piN,x)7:iN\xMx)g*{b\N,x), (C.19) roib, N, x) = p{N, x)niN\x)0{x)rm, x). (C.20) B y L e m m a B . 3 of N e w y (1994)[43], gQ{b,N,x) is uni formly consistent over beeiN,x): / / r ^ d + i \ - i / 2 \ sup \roib,N,x)-g'oib,N,x)\ = Op {^-^] + • (C.21) 6€e(N,x) V V y y B y the results i n parts (a)-(c), the estimators p{N,x), n{N\x) a n d 0{x) converge at the rate faster than that i n (C.21). T h e desired result follows by the same argument as i n the proof of part (d), equat ion (C.15) . N e x t , we prove part (g). B y L e m m a 3.5, g*{b\N,x) > Cg > 0. T h e n \Q*iT\N,x)-Q*iT\N,x)\ < \rir\N,x)-q*{r\N,x) , \r ( r iT\N,x)\N,x) - g'iq*{T\N,X)\N,X)\ +2^ +- p(N,x)r{q*{T\N,x)\N,x)c, \p{N,x)-piN,x)\ p{N,x)p{N,x)riq'{r\N,x)\N,x) - \^ p{N,x)g*{q'{T\N,x)\N,x)cJ x\q[T\n,x) - q{T\n,x)\ \r{q'{T\N,x)\N,x) - g*{q'{T\N,x)\N,x)\ p{N,x)g'{^{T\N,x)\N,x)cg \p{N,x)-p{N,x)\ ^p{N,x)p{N,x)g'{q*{T\N,x)\N,x)- ' Define an event EL{N,X) = {r{n{N,x)\N,x) > bxiN,x),rir2iN,x)\N,x) < b^iN^x)}, a n d let 0L = {^^T'^^^ + f'^^- the result i n par t (e), P{El{N,x)) = o ( l ) . Hence, i t follows from part (e) of the l e m m a the est imator ^(g*(T| A'', x) \N,x) is bounded away from zero w i t h the probab i l i ty approaching one. Consequently, i t follows by L e m m a 3.5 and part (e) of this l e m m a that the first s u m m a n d o n the r ight -hand side of (C.22) is Op(/3^^ un i f o rmly over r{N,x). N e x t , P (sup^^r^^^,^(iL\r{q*ir\N,x)\N,x)-g*{r{r\N,x)\N,x)\ > M ) < p( sup /3L\ririr\N,x)\N,x)-g'ir{T\N,x)\N,x)\>M,EUx) \ T € Ï ( J V , I ) / +PiEi{x)) < p( sup PL\g'{b\N,x)-g'ib\N,x)\>M]+o(\). (C.23) \bee(.N,x) ) T h e result of par t (g) follows from parts (c) a n d (f ) of the l e m m a a n d (C.23). For par t (h), by A s s u m p t i o n C . l ( h ) and part (c) of the l e m m a , P{T, N, X) — f3{r,N,x) for a l l T,N, a n d x. T h e result of part (h) follows since P is l inear i n T (see A n d r e w s (1992)[1]; also Theorems 21.9 a n d 21.10 on pp . 337-339 of D a v i d s o n (1994)[12]). Las t l y , we prove part (i). W e have SUPreT ' ' {Ar ,x ) q* ( ^ ( T , N, x)\N, x) - q*{(3{T, N, x)\N, x) = n k r , N, x)\N, x) - q*0iT, TV, x)\N, x) +q*0{T, N, x)\N,x) - q*{Pir, N, x)\N, x) < sup \q*{T\N,x)-q*{T\N,x)\+Op Ter(N,x) V iogL + h R (C.24) where the inequal i ty follows from part (h) of the l e m m a a n d L e m m a 3.5. T h e result of par t (i) follows from the def init ion of Q* i n (3.23) a n d (C.24). L e m m a C . 4 Let Q{N,x) be as in Lemma C.3. Suppose that Assumptions C.l and C.2 hold, and that the bandwidth h is such that Lh'^+^ —> oo, Vlhd+T^h^-^0. Then V l h ^ i r m , x) - g*ib\N. x)) 7V(0, Vgib, N, x)) for b € Q{N,x), X e Interior{X), and N e Af, where Vg{b,N,x) is given by 9' Vg{N,b,x)^ Np(N. ,xMN\xMx) \J J Furthermore, g*{b\Ni,x) andg*{b\N2,x) are asymptotically independent for allNi^N2, Ni,N2eAf. P r o o f f o r L e m m a C . 4 . Consider g^ib, n , x) a n d g^{b, n, x) defined i n (C.19) a n d (C.20) respectively. It follows from parts (a)-(c) of L e m m a C . 3 , -55(6,7V,x)) + Op(l). (C.25) Fur thermore , as i n L e m m a B 2 of N e w y (1994), Eg^ib, N, x) — g^ib, N, x) = 0{h^) un i formly i n 6 € Q{N,x) for a l l x € Interior{X) a n d N € J\f. T h u s , i t remains to establish asymptot ic normal i ty of wLhP^ig^(b,N,x) — Eroib,N,x)). Define (=1 i=l where is defined i n (C.9) . W i t h above definitions we have that VNLhd^iToib,N, x) - EToib,N,x)) = ^(WL^N - E W L , N ) . (C.28) T h e n , by the L i a p u n o v C L T (see, for example, C o r o l l a r y 11.2.1 on p. 427 of L e h m a n a n d R o m a n o (2005) [30]), ^ / N L { W L , N - EwL,N)/yjNLVar{wL,N) N{0,1), (C.29) prov ided that Ew^i ^ < oo, a n d for some S > 0, l i m --^E\wii^N - Ewii^N\^~^^ = 0- T h e last cond i t i on follows from the L iapunov ' s condi t ion (equation (11.12) on p. 427 of L e h m a n a n d R o m a n o (2005)[30]) a n d because WU^N are iid. N e x t , Ewu^!\f is given by ^I^E{p(N, xiMN\xi) ! K { ^ ) g*{u\N,xi)duKa (^) = Vh^+ï J Jp{N,x + hy)Tr{N\x + hy) K{u)g*{b + hu\N, x + hy)Kd{y)<p{x + hy)dudy 0. Fur ther , Ewfij^ is given by T;è^!!p{N,y)n{N\y)K^ {^) g*{u\N,y)Kl {^) cp{y)dudy = J JP(^^ ^ + hy)iT{N\x + hy) K\u)g*ib + hu\N, X + hy)KJiy)ipix + hy)dudy < 0 0 . Hence, NLVaiT{wL,N) ^ p{N,x)7ciN\x)g*ib\N, x)ip{x) (^JK^(u)du^ du. (C.30) N e x t , E\wii^N\^~^^ is bounded by wmrrm 1 1 l ' ^ ' 9*{u\N, y) \Kd{^)f^' ^{y)dudy = h i ^ J Jmu)\'+'9*{b + hu\N,x + hy)\Kd{y)\''+'<p{x + hy)dudy ^ Tm)72 ^ " P \K{u)f''^'^^^+'hupv{x) sup g*{b\N,x) '̂ ^ « € [ - 1 , 1 ] xex beB{N,x) h(d+l)S/2 • Last ly , ^ 0, (C.31) since Lh'^'^^ —> CXD by the assumption. T h e first result of the l emma follows now from (C.25) - (C.31) . N e x t , note tha t the asymptot ic covariance of W L . M a n d WL,iV2 involves a product of the two indicator functions, l{Ni = i V i } l { J V / = N2}, which is zero for a l l Ni ^ N^. T h e jo int asymptot ic n o r m a l i t y a n d asymptot ic independence oig*{b\Ni,x) a n d g{b\N2, x) follows then by the C r a m e r - W o l d device. • P r o p o s i t i o n C . 5 Suppose that T e (0,1) and x s Xl Assume that the bandwidth h satisfies as L -* 00 : LW^^^ —> 0 0 and y/Th^h^ 0. Then, under Assumptions C.l and C.2, VLh^{Q'{T\N,x)-Q'{T\N,X)) A f ( 0 , V g ( 7 V , r , x ) ) , '/Lh^{Q'0{r,N,x)\N,x)-Q'{P{T,N,x)\N,x)) ->d N{0,VQ{N, I3{T, N,x),x))), where and Vg{N, T, X) is defined in Lemma C.4. Moreover, for any distinct N, N' £ { i V , A ' " } , r , r ' e T , and x,x' € X\ the estimators Q*{T\N,X) are asymp- totically independent, as well as the estimators Q*{f3{T,N,x)\N,x). P r o o f for P r o p o s i t i o n C . 5 . F i r s t , by L e m m a C.3(c) , (e) and (f), a n d the mean-value theorem. (N-l)p{N,x)g*^{q*{T\N,x)\N,x) xiriq*iT\N,x))-g*iq*{r\N,x))) + Op ( ^ 7 = ) , where g* is a mean-value between g* and g* for b = q*{T\N,x). T h e result follows then by L e m m a C.4 . • P r o p o s i t i o n C.6 Let x € X^. Assume that Lh"^ -> 0 0 and VLh^h^ 0 as L 00, and Assumptions C.l and C.2 hold. Then s u p P / f ^ ^ ^ ( T ^ ^ ^ ( a ; ) > c ) = PH,S{T^''^{X) > c) (C.32) _^ P ( r ^ ^ ^ ( a ; ) > c), (C.33) where PHAME ^•'"''^ ^HC^S denotes probabilities under the inequality restric- tions of AME and equality restrictions of LS respectively, and T '^^^(a;) is defined in (3.27). P r o o f for P r o p o s i t i o n C.6 . T h e result i n (C.32) follows by L e m m a 8.2 of P e r l m a n (1969)[47], I n order to show (C.33) , consider first the case of Â; = 1. B y the results i n Chapter 21.3.3 of Gour i e roux and M o n f o r t (1995)[16], , T ' * ^ ^ ( x ) is asymptot i ca l ly equivalent to f ^ ^ ^ ( x ) = m i n L/ i ' '+ i||7-^||2, where VLÎ?+^^ A''(0, J ^ A T - I ) ; however. '-^^^{x) = m i n \ / L ^ 7 - / i m m n^/^(T,x)fi<o and the result follows by the Cont inuous M a p p i n g T h e o r e m . Extens ion to the case of fc > 1 is straightforward since there axe no cross r restrictions i n (3.26), a n d the quanti le estimators are asymptot i ca l ly independent across r . Appendix D Statement of Co-Authorship T h e t h i r d chapter i n this thesis reports the results of a j o int research w i t h Professors V a d i m M a r m e r a n d A r t y o m Shneyerov. T h e current version is a further development of a paper c irculated earlier under the t i t l e "Selective E n t r y i n F i r s t - P r i c e A u c t i o n s . " T h e author took central roles i n a l l stages of the research, i n c l u d i n g iden- t i f i cat ion of the research questions, development of the s ta t i s t i ca l methodo l - ogy, invest igat ion of the finite sample performance of the proposed stat is t i ca l methods through M o n t e C a r l o s imulat ions, acquis i t ion of the d a t a set used i n the e m p i r i c a l analysis , and design of the counter factual experiments .
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Endogenous entry in first-price auctions Xu, Pai 2008
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Title | Endogenous entry in first-price auctions |
Creator |
Xu, Pai |
Publisher | University of British Columbia |
Date Issued | 2008 |
Description | This thesis studies the first-price auction models with endogenous entry. In the first chapter, we propose using the nearest neighbor estimation technique to estimate the entry cost in the auction models. We study the large sample properties of the proposed estimator to establish sets of conditions sufficient for its consistency and asymptotic normality. To give an example, we further apply the proposed method to estimate the cost of participation in the Michigan Highway Procurement Auctions. Our study rejects the null hypothesis of zero participation costs. Based on our estimation result, we infer how the optimal auction outcomes can be realized by using the regular policy tools. We demonstrate the improvement that the Michigan government could have made on payments if the optimal auctions had been employed. In the second chapter, we propose applying the simulated non-linear least squares (SNLLS) method of Laffont, Ossard and Vuong (1995, LOV hereafter) [28] to estimate the distribution of bidders' private values in the estimation of a first-price auction model with endogenous entry. Unlike the estimation problem of LOV , however, the valuation distribution of bidders is truncated in our problem, due to the presence of the entry cost. This further causes the absence of continuous differentiability in our statistical estimation objective function, which was required in LOV's large sample analysis. Therefore, we provide a separate analysis to study the large sample behavior of the SNLLS estimator in such setup. In the third chapter, we develop a nonparametric method that allows one to discriminate among alternative models of entry in first-price auctions. Three models of entry are considered: Levin and Smith (1994) [31], Samuelson (1985) [53], and a new model in which the information received at the entry stage is imperfectly correlated with valuations. We show that these models impose different restrictions on the quantiles of active bidders' valuations, and develop nonparametric tests of these restrictions. We perform the tests using a dataset of highway procurement auctions in Oklahoma. Depending on the project size, we find somewhat more support for the new model. |
Extent | 5597558 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-03-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0067042 |
URI | http://hdl.handle.net/2429/5639 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
GraduationDate | 2008-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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