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Endogenous entry in first-price auctions Xu, Pai 2008

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Endogenous Entry in First-Price Auctions by Pai X u  B . A . , D o n g b e i U n i v e r s i t y of F i n a n c e a n d E c o n o m i c s ,  1999  M . A . , U n i v e r s i t y of N e w B r u n s w i c k , 2002  A THESIS S U B M I T T E D IN PARTIAL F U L F I L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF D o c t o r of P h i l o s o p h y in T h e F a c u l t y of G r a d u a t e Studies (Economics)  T H E UNIVERSITY OF BRITISH (Vancouver) J u l y , 2008 ©  P a i X u 2008  COLUMBIA  Abstract T h i s thesis studies the  first-price  a u c t i o n m o d e l s w i t h endogenous entry.  I n the first chapter, we propose using the nearest neighbor e s t i m a t i o n technique to e s t i m a t e the e n t r y cost i n the a u c t i o n models. W e s t u d y the large s a m p l e p r o p e r t i e s of the p r o p o s e d e s t i m a t o r t o establish sets of conditions sufficient for i t s consistency a n d a s y m p t o t i c n o r m a l i t y . T o give a n example, we f u r t h e r a p p l y the p r o p o s e d m e t h o d to estimate the cost of p a r t i c i p a t i o n i n t h e M i c h i g a n H i g h w a y P r o c u r e m e n t A u c t i o n s . O u r s t u d y rejects the n u l l h y p o t h e s i s of zero p a r t i c i p a t i o n costs. B a s e d o n o u r e s t i m a t i o n result, we infer h o w the o p t i m a l a u c t i o n outcomes c a n be r e a l i z e d by u s i n g the regular p o l i c y tools.  W e demonstrate the i m p r o v e m e n t t h a t the M i c h i g a n  government c o u l d have m a d e on p a y m e n t s i f the o p t i m a l a u c t i o n s h a d been employed.  I n the second chapter, we propose a p p l y i n g the s i m u l a t e d non-linear least squares ( S N L L S ) m e t h o d of Laffont, O s s a r d a n d V u o n g (1995, L O V hereafter) [28] t o estimate the d i s t r i b u t i o n of b i d d e r s ' p r i v a t e values i n the e s t i m a t i o n of a  first-price  a u c t i o n m o d e l w i t h endogenous entry. U n l i k e the  e s t i m a t i o n p r o b l e m of L O V , however, the v a l u a t i o n d i s t r i b u t i o n of bidders is t r u n c a t e d i n o u r p r o b l e m , due to the presence of the e n t r y cost. T h i s further causes the absence of continuous differentiability i n o u r s t a t i s t i c a l e s t i m a t i o n o b j e c t i v e f u n c t i o n , w h i c h was r e q u i r e d i n L O V ' s large s a m p l e analysis. T h e r e f o r e , we p r o v i d e a separate analysis to s t u d y the large s a m p l e b e h a v i o r of t h e S N L L S e s t i m a t o r i n such setup.  I n the t h i r d c h a p t e r , we develop a n o n p a r a m e t r i c m e t h o d t h a t allows one to d i s c r i m i n a t e a m o n g a l t e r n a t i v e models of e n t r y i n  first-price  auctions.  ABSTRACT  T h r e e models of e n t r y are considered: L e v i n a n d S m i t h (1994) [31], S a m u e l son (1985) [53], a n d a new m o d e l i n w h i c h the i n f o r m a t i o n received at t h e entry stage is i m p e r f e c t l y correlated w i t h v a l u a t i o n s . W e show t h a t these models i m p o s e different r e s t r i c t i o n s o n t h e quantiles of active b i d d e r s ' v a l u a t i o n s , a n d develop n o n p a r a m e t r i c tests of these r e s t r i c t i o n s . W e p e r f o r m the tests u s i n g a dataset of highway p r o c u r e m e n t a u c t i o n s i n O k l a h o m a . D e p e n d i n g o n t h e project size, we f i n d s o m e w h a t m o r e s u p p o r t for t h e new model.  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  viii  Acknowledgements  x  Dedication 1  ix  N o n p a r a m e t r i c E s t i m a t i o n of t h e E n t r y C o s t i n F i r s t - P r i c e Auctions  1  1.1  Introduction  1  1.2  Methodology  6  1.3  1.2.1  E s t i m a t o r of 0 ( x )  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 of <f>{x)  12  1.2.3  Simulation Results  13  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  7  16  1.3.1  The Auction Model  17  1.3.2  A p p l y i n g the E s t i m a t i o n M e t h o d o l o g y  18  1.3.3  Empirical Application  20  1.4  M o r e Examples on Applications  29  1.5  Conclusion and Discussion  31  1.6  Tables and 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 : Parametric Approach  44  2.1  Introduction  44  2.2  Methodology  2.3 3  A  47  2.2.1  The  first-price  auction model w i t h entry  2.2.2  T h e s t r u c t u r a l econometric m o d e l  47 49  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  51  2.2.4  A s y m p t o t i c P r o p e r t y of ^  53  C o n c l u s i o n a n d extensions  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 A u c t i o n s ? A  58 Nonpara-  metric A p p r o a c h  60  3.1  Introduction  60  3.2  T h r e e m o d e l s of entry a n d their testable r e s t r i c t i o n s  66  3.2.1  T h e L S a n d S models of e n t r y  66  3.2.2  T h e aflShated m o d e l of e n t r y ( A M E )  68  3.3  Nonparametric identification  71  3.4  Econometric implementation  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 e s t i m a t i o n m e t h o d of H a i l e , H o n g  3.4.5  a n d S h u m (2003)  84  Tests  85  3.5  M o n t e - C a r l o experiment  87  3.6  Empirical application  90  3.7  Concluding remarks  96  3.8  Tables a n d F i g u r e s  99  Bibliography  109  Appendices A  P r o o f s for c h a p t e r 1  115  A.l  D e t a i l s of e s t i m a t i o n m e t h o d  115  A.2  D e t a i l s of I d e n t i f i c a t i o n Issues  121  B  P r o o f s for c h a p t e r 2  124  C  P r o o f s for c h a p t e r 3  143  C l  D e t a i l s of the e n t r y models  143  C.2  D e t a i l s of the e s t i m a t i o n m e t h o d  147  D  S t a t e m e n t 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 R e s u l t s on Coverage  35  1.2  Simulation Results, Lognormal(4,l),  1.3  B i d Analysis  k=T^/^  36 37  1.4  E s t i m a t i o n R e s u l t s o n E n t r y Costs  38  1.5  Policy Tools Towards O p t i m a l Auctions  39  3.1  Size of A M E Test  99  3.2  S i z e - c o r r e c t e d P o w e r of the A M E Test  100  3.3  D e s c r i p t i o n of V a r i a b l e s  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 o n P r o j e c 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,  3.7  E s t i m a t e d M e d i a n s of Costs  105  3.8  E s t i m a t e d T r a n s f o r m e d M e d i a n s of C o s t s  106  3.9  Test R e s u l t s  107  x)  104  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  Participation Costs  41  1.3  J / ( l - F ) for JV = 9  42  1.4  J / ( l -F)îoT  43  3.1  S a m p l e Frequencies of p r o j e c t sizes  N=12  108  Acknowledgements I w o u l d like to express m y g r a t i t u d e to a l l those w h o gave m e the poss i b i l i t y to c o m p l e t e t h i s thesis.  E s p e c i a l l y , I a m deeply i n d e b t e d to m y  s u p e r v i s o r P r o f . S h i n i c h i S a k a t a whose help, s t i m u l a t i n g suggestions a n d encouragement h e l p e d me i n a l l the t i m e of research for a n d w r i t i n g of t h i s thesis. I have f u r t h e r m o r e to t h a n k the members o n m y d i s s e r t a t i o n  committee.  P r o f s . M i c h a e l P e t e r s , V a d i m M a r m e r , U n j y Song for t h e i r h e l p , s u p p o r t , interest a n d v a l u a b l e hints. Sincere t h a n k s are also extended to m y ofïicemates, colleagues, faculty a n d staff i n the d e p a r t m e n t for their supports. L a s t , b u t not the least, I w o u l d like to give m y special t h a n k s to m y f a m i l y whose love e n a b l e d me to complete m y 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 e c o n o m i c m o d e l w i t h uncertainty, an e q u i l i b r i u m b e h a v i o r  often  p r e d i c t s t h e existence of a t h r e s h o l d i n a d i s t r i b u t i o n w h i c h represents the possible types of a n i n d i v i d u a l . T h e t h r e s h o l d entails the p o i n t at w h i c h i n d i v i d u a l s s w i t c h f r o m one choice to another a n d t h u s causes a t r u n c a t i o n of t h e d i s t r i b u t i o n . W e u s u a l l y have no knowledge o n the l o c a t i o n of this truncation.  H o w e v e r , the value of the p r o b a b i l i t y d e n s i t y f u n c t i o n at the  t h r e s h o l d is often the interest of a researcher, p a r t i c u l a r l y to discuss desirable e c o n o m i c policies.  T o i l l u s t r a t e the s i t u a t i o n , we consider an independent p r i v a t e - v a l u e firstprice a u c t i o n m o d e l w i t h endogenous entry. I n order to p a r t i c i p a t e i n an a u c t i o n , a b i d d e r often bears various costs to prepare a b i d , s u c h as the costs of t r a v e l i n g to the a u c t i o n site, a n d the b i d d e r s ' cognitive efforts i n  the b i d d i n g process. Because these costs do not o c c u r unless she enters the a u c t i o n , she p a r t i c i p a t e s i n the a u c t i o n if a n d o n l y if she expects profits from the a u c t i o n to be large enough to compensate the costs.  The distribution  of bids is therefore t r u n c a t e d b y the cutoff i n t r o d u c e d b y the p a r t i c i p a t i o n costs. T h e m a r g i n a l p a r t i c i p a n t of the a u c t i o n r i g h t at the cutoff p o i n t must have t h e same expected profit as the a m o u n t of the e n t r y costs.  Therefore,  the e s t i m a t i o n of the p a r t i c i p a t i o n costs a m o u n t s to e v a l u a t i n g a n expected profit f u n c t i o n at t h i s cutoff.  M o r e o v e r , k n o w i n g the level of p a r t i c i p a t i o n  costs is essential to design auctions for m a x i m i z i n g t h e seller's revenue. C e l i k a n d Y i l a n k a y a (2006) [8] show t h a t the i m p l e m e n t a t i o n of the o p t i m a l auctions a m o u n t s t o enforcing the o p t i m a l cutoff rules i n p a r t i c i p a t i o n to a l l the b i d d e r s . F o r a n y given p a r t i c i p a t i o n cost, the seller's p r o b l e m is t h e n reduced to a p p r o p r i a t e l y choosing reserve price a n d / o r e n t r y fee.  I n t h i s c h a p t e r , we suggest the c o m b i n a t i o n of the one-sided nearest n e i g h b o r e s t i m a t o r [hereafter, N N estimator] a n d the e x t r e m e order statistics to estimate t h e t r u n c a t e d u n i v a r i a t e p r o b a b i l i t y density f u n c t i o n at its u n k n o w n t r u n c a t i o n p o i n t s . W e prove t h a t the p r o p o s e d e s t i m a t o r is consistent a n d a s y m p t o t i c a l l y n o r m a l l y d i s t r i b u t e d u n d e r m i l d c o n d i t i o n s .  We  also e x t e n d the m e t h o d to cover the case i n w h i c h the p a r a m e t e r of i n t e r est is a s m o o t h f u n c t i o n of the u n k n o w n density at the t r u n c a t i o n p o i n t a n d some other a t t r i b u t e s of the d i s t r i b u t i o n as is the case i n the a u c t i o n e x a m p l e discussed earlier.  It is possible t o replace the one-sided N N e s t i m a t o r s w i t h the one-sided version of t h e R o s e n b l a t t - P a x z e n class of b a n d w i d t h e s t i m a t o r s  [hereafter,  b a n d w i d t h estimators]. T h e b a n d w i d t h e s t i m a t o r s received a lot of a t t e n t i o n f r o m e c o n o m e t r i c i a n s as a w a y of e s t i m a t i n g density f u n c t i o n s . I n p a r t i c u l a r , w h e n the b a n d w i d t h e s t i m a t o r is a p p l i e d i n b o u n d a r y regions, the estimate is not necessarily consistent. T h i s inconsistency p r o b l e m elicits a n extensive l i t e r a t u r e o n the c o r r e c t i o n of the b o u n d a r y effect.  N o t designed for the  e s t i m a t i o n of a density f u n c t i o n at the b o u n d a r y p o i n t , most of the e x i s t i n g m e t h o d s focus o n how t o m o d i f y the w e i g h t i n g scheme i n s u c h a w a y t h a t  b o u n d a r y r e g i o n c a n be a u t o m a t i c a l l y detected a n d a d j u s t e d .  According  to C h e n g et a l (1997) [10], some of the m e t h o d s are q u i t e difficult to w o r k w i t h . M o r e o v e r , t h e issue of the l i m i t i n g d i s t r i b u t i o n for these bias-corrected e s t i m a t o r s is left u n s e t t l e d . B e a r i n m i n d , however, t h a t the one-sided a n d t w o - s i d e d b a n d w i d t h estimators have quite different properties i n c e r t a i n aspects. W e c o m p a r e o u r one-sided N N e s t i m a t o r w i t h some t y p i c a l bias-corrected b a n d w i d t h m e t h o d s i n some M o n t e C a r l o e x p e r i m e n t s . I n t h e e x p e r i m e n t s , the o n e - s i d e d N N estimate always has a s m a l l e r m e a n s q u a r e d error t h a n other b a n d w i d t h e s t i m a t o r s . T h e stable performance of the one-sided N N e s t i m a t o r m a y reflect the l o c a l adaptive n a t u r e of the N N m e t h o d , t h o u g h we n a t u r a l l y e x p e c t t h a t the results can vary, d e p e n d i n g o n the p o p u l a t i o n distribution.  T o d e m o n s t r a t e usefulness of the proposed e s t i m a t i o n m e t h o d , we a p p l y the m e t h o d to estimate the p a r t i c i p a t i o n costs i n M i c h i g a n H i g h w a y P r o c u r e m e n t A u c t i o n , a s s u m i n g t h a t the a u c t i o n is a n independent p r i v a t e value  first-price  a u c t i o n w i t h endogenous entry. O u r s t u d y rejects the n u l l  h y p o t h e s i s t h a t the p a r t i c i p a t i o n costs are zero at any reasonable significance levels. W i t h t h e e s t i m a t e d p a r t i c i p a t i o n cost levels, we further investigate how t o i m p l e m e n t a n o p t i m a l a u c t i o n b y e m p l o y i n g o n l y regular p o l i c y tools, s u c h as t h e reserve prices a n d / o r entry fees. W e find t h a t , to m i n i m i z e the h i g h w a y c o n s t r u c t i o n a n d maintenance costs, the government s h o u l d s u b s i dize p a r t i c i p a n t s 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 c a l c u l a t e how m u c h the expected p a y m e n t b y the government w o u l d change between the current auctions a n d the o p t i m a l a u c t i o n s . O u r result suggests t h a t the M i c h i g a n government c o u l d have saved u p to 101 5 % b y s e t t i n g u p the auctions o p t i m a l l y .  N o t e t h a t these n u m b e r s are  not s u b j e c t t o t h e so-called e x t r a p o l a t i o n p r o b l e m , despite t h a t we have no i n f o r m a t i o n o n the v a l u a t i o n s of the p o t e n t i a l bidders w h o d i d not p a r t i c i p a t e i n t h e a u c t i o n s , because the o p t i m a l p o l i c y makes the t h r e s h o l d for p a r t i c i p a t i o n higher t h a n it a c t u a l l y was. T h u s , our w o r k provides the first  e m p i r i c a l insights o n how to use p o l i c y tools to a t t a i n o p t i m a l outcomes i n a n a u c t i o n w i t h endogenous entry. T h e above-mentioned  a p p l i c a t i o n i n our p a p e r is r e l a t e d to a n u m b e r  of w o r k s i n the e m p i r i c a l a u c t i o n l i t e r a t u r e . E m p i r i c a l studies o n endogenous e n t r y have r e c e n t l y received m u c h a t t e n t i o n f r o m economists.  Bajari  a n d H o r t a s c u (2003) [5] e m p i r i c a l l y investigate c o m m o n - v a l u e e B a y auctions through a structural approach.  T h o u g h i d e n t i f y i n g p a r t i c i p a t i o n costs is  not t h e i r m a i n interest, they recognize the average cost of b i d d i n g t o be of a significant a m o u n t .  T h i s reinforces the existence of costly p r e p a r a -  t i o n i n b i d d i n g a n d i m p l i e s t h a t u n d e r s t a n d i n g b i d d e r s ' e n t r y decisions is information-revealing.  A t h e y , L e v i n a n d S e i r a (2004) [4] s t u d y e n t r y a n d  b i d d i n g p a t t e r n s i n U . S . Forest Service t i m b e r a u c t i o n s . T h r o u g h p a r a m e t e r i z i n g t h e s t r u c t u r a l equations, they compare s e a l e d - b i d a n d o p e n auctions to p r o v i d e a n assessment of b i d d e r competitiveness. L i a n d Z h e n g (2005) [34] propose a s e m i - p a r a m e t r i c B a y e s i a n m e t h o d to j o i n t l y e s t i m a t e e n t r y a n d bidding models using a fully structural approach.  B y u s i n g the highway  m o w i n g a u c t i o n d a t a f r o m T e x a s , t h e y d o c u m e n t e m p i r i c a l evidence of ent 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 r u c t u r a l m o d e l w i t h b o t h e n t r y process a n d b i n d i n g reserve prices i n first-price a u c t i o n s .  H e proposes a n  M S M e s t i m a t o r , j o i n t l y using observed bids a n d the n u m b e r of a c t u a l b i d ders, t o e s t i m a t e t h e p a r a m e t e r s i n the d i s t r i b u t i o n s of p r i v a t e values a n d the n u m b e r of active bidders.  T h e m o d e l is m o r e general b y c o n s i d e r i n g  b o t h e n t r y costs a n d b i n d i n g reserve prices.  F o r t r a c t a b i l i t y , however, L i  assumes t h a t the n u m b e r of active bidders is k n o w n to b i d d e r s at the b i d d i n g stage of the game, w h i c h is o n the c o n t r a r y to t h e r e a l i t y i n some of the interesting applications.  O u r e m p i r i c a l a p p l i c a t i o n is d i s t i n g u i s h e d from the e x i s t i n g e m p i r i c a l a u c t i o n l i t e r a t u r e b o t h i n e s t i m a t i o n strategies a n d i n a u c t i o n m o d e l s .  First,  o u r a p p r o a c h is s t r u c t u r a l a n d fully n o n p a r a m e t r i c . T h i s saves us f r o m wor-  r y i n g a b o u t t h e effects of misspecifications. Second, o u r m o d e l assumes t h e e n t r y decisions are m a d e ex post, t h a t i s , bidders decide w h e t h e r to p a r t i c i p a t e i n t h e a u c t i o n after k n o w i n g t h e i r v a l u a t i o n s . T o t h e best of o u r k n o w l e d g e , t h i s w o r k is t h e first e m p i r i c a l s t u d y o n s u c h a n a u c t i o n m o d e l . I n t e r m s of m o d e l i n g costly entry, t h e works b y A t h e y , L e v i n a n d S e i r a (2004)[4], L i a n d Z h e n g (2005)[34] a n d L i (2005)[32] are various versions o f the L e v i n a n d S m i t h (1994)[31] m o d e l , i n w h i c h bidders m a k e costly e n t r y decisions ex ante (i.e., before k n o w i n g t h e i r o w n v a l u a t i o n s ) . T h e e n t r y cost s t u d i e d i n s u c h m o d e l s is sometimes referred t o as " p r e - p a r t i c i p a t i o n investm e n t " i n t h e l i t e r a t u r e . A s t h e name suggests, t h i s " i n v e s t m e n t " is more closely r e l a t e d t o a c q u i r i n g i n f o r m a t i o n . O n t h e c o n t r a r y , t h e e n t r y cost i n o u r m o d e l occurs ex post (i.e., after bidders k n o w t h e i r o w n v a l u a t i o n s ) . O u r m o d e l is a v a r i a n t o f t h e S a m e u l s o n (1985) [53] e n t r y m o d e l , w h i c h reflects t h a t p r e p a r i n g t h e bids is a costly process.^ T h e different i n t e r p r e t a t i o n s o f costly e n t r y process result i n diff'erent e q u i l i b r i u m characterizations.'^ O u r S a m e u l s o n m o d e l has a pure strategy e q u i l i b r i u m b y h o l d i n g a cutoff o n t h e v a l u a t i o n d i s t r i b u t i o n . P a r t i c i p a n t s i n auctions are allowed t o have different (positive) e x p e c t e d profits f r o m j o i n i n g t h e a u c t i o n i n accordance w i t h their o w n v a l u a t i o n s .  O n t h e other h a n d , a m o d e l b y L e v i n a n d S m i t h  (1994) [31] suggests a m i x e d strategy e q u i l i b r i u m i n e n t r y decisions due t o the fact t h a t n o b i d d e r s have a n y p r i o r knowledge o n t h e v a l u a t i o n . T h i s feature of e q u i l i b r i u m restricts zero expected gains f r o m e n t r y t o a l l bidders, w h i c h sometimes c a n h a r d l y be s u p p o r t e d b y a p p l i c a t i o n s . I n general, one n a t u r a l l y e x p e c t s t h e bidders t o have some i n f o r m a t i o n o n t h e a u c t i o n e d o b j e c t w h e n m a k i n g t h e p a r t i c i p a t i o n decision. A s i n o u r e m p i r i c a l a p p l i c a t i o n , v i z . t h e M i c h i g a n H i g h w a y P r o c u r e m e n t A u c t i o n s , t h e bidders a l l have access t o t h e d e t a i l e d descriptions o n t h e projects u n d e r a u c t i o n . T h i s i n t u r n justifies o u r choice o f using Samuelson's e n t r y m o d e l i n this chapter. ^Stegeman (1996) [55] provides a discussion of the d i s t i n c t i o n between t h e information a c q u i s i t i o n ( " i n v e s t m e n t " ) a n d the costly b i d d i n g p r e p a r a t i o n . ^ T h e r e m a y exist m u l t i p l e e q u i l i b r i a i f the e q u i l i b r i u m strategies are a s y m m e t r i c (see, for e x a m p l e ,  T a n a n d Y i l a n k a y a (2006)[57].)  analysis t o t h e u n i q u e s y m m e t r i c e q u i l i b r i u m .  T h r o u g h o u t this chapter, we restrict o u r  T h e rest of t h i s chapter is o r g a n i z e d as follows. I n t h e n e x t section, we propose a m e t h o d for e s t i m a t i n g the value of a t r u n c a t e d f u n c t i o n  (pdf)  at the t r u n c a t i o n p o i n t a n d establish its large s a m p l e p r o p e r t i e s . W e also c o n d u c t M o n t e C a r l o experiments t o s t u d y its finite s a m p l e b e h a v i o r .  In  Section 1.3, we a p p l y the proposed e s t i m a t i o n m e t h o d o l o g y to e s t i m a t i n g the e n t r y cost i n c u r r e d i n j o i n i n g the M i c h i g a n H i g h w a y P r o c u r e m e n t A u c tions. M o r e o v e r , we investigate h o w to i m p l e m e n t the o p t i m a l auctions i n this section. S e c t i o n 1.4 i l l u s t r a t e s some other e x a m p l e s i n w h i c h the p r o posed e s t i m a t i o n m e t h o d c a n be a p p l i e d a n d the last s e c t i o n concludes. T h e m a t h e m a t i c a l proofs are collected i n the a p p e n d i x .  1.2  Methodology  W e assume: A s s u m p t i o n 1.1 (i)  The  (Data  Generating  data are a realization  tributed  (i.i.d.)  (ii) The probability probability  univariate distribution  density function  Process)  of an independently stochastic  process  and identically {Xt}ten-  of X\ has a support (pdf) <p that is positive  at X, the lower bound of the support of  dis-  bounded below and a and  right-continuous  X\.  W e discuss the a p p l i c a b i l i t y of a s s u m p t i o n s i n t h e c o n t e x t of the firstprice a u c t i o n , since we are going to a p p l y the p r o p o s e d e s t i m a t i o n m e t h o d t o M i c h i g a n H i g h w a y P r o c u r e m e n t A u c t i o n s i n t h i s c h a p t e r . It is u s u a l l y a s s u m e d t h a t t h e b i d d e r s ' v a l u a t i o n s are r a n d o m l y d r a w n f r o m a n u n i v a r i a t e process for independent private-value a u c t i o n s .  i.i.d.  T o ensure the  existence a n d uniqueness of the b i d d i n g e q u i l i b r i u m , i t is f u r t h e r assumed t h a t the v a l u a t i o n d i s t r i b u t i o n s are continuous a n d w i t h b o u n d e d s u p p o r t s . B e c a u s e the e q u i l i b r i u m b i d d i n g is a s t r i c t l y m o n o t o n i e t r a n s f o r m a t i o n of v a l u a t i o n , the e q u i l i b r i u m bids are also i.i.d a n d c o n t i n u o u s l y d i s t r i b u t e d .  D u e t o t h e endogenous entry, the e q u i h b r i u m m a y e n t a i l a cutoff of b i d s s u c h t h a t the b i d d e r whose v a l u a t i o n is below the t h r e s h o l d level w o u l d not s u b m i t a b i d , w h i l e , o n the contrary, the bidders w i t h his v a l u a t i o n s above the cutoff w i l l a c t u a l l y b i d . T h i s further i m p l i e s t h a t the b i d s are observed w i t h t r u n c a t i o n . T h i s t r u n c a t i o n p o i n t , i n t u r n , forms a lower b o u n d for the observed ( t r u n c a t e d ) b i d d i s t r i b u t i o n .  1.2.1  E s t i m a t o r of 0(x)  T o e s t i m a t e a t r u n c a t e d u n i v a r i a t e p r o b a b i l i t y density f u n c t i o n (pdf) at its u n k n o w n t r u n c a t i o n p o i n t , we first look at two separate issues: (i) e s t i m a t i o n of the p d f at a g i v e n p o i n t of t r u n c a t i o n ; a n d (ii) e s t i m a t i o n of the t r u n c a t i o n point. E s t i m a t o r of (f>{x) i n the case w i t h k n o w n x W e p r o p o s e u s i n g the nearest neighbor e s t i m a t i o n technique to estimate (f>{x). A s a way of e s t i m a t i n g densities, this e s t i m a t i o n m e t h o d , w h i c h is u s u a l l y referred to as the fc-NN estimator i n the density e s t i m a t i o n l i t e r a t u r e . k is the n u m b e r of closest observations t a k e n f r o m t h e s a m p l e to estimate the d e n s i t y at t h e p o i n t of interest. T o s i m p l i f y the n o t a t i o n , we use denote t h e t-th. order s t a t i s t i c i n the sample {Xi,...,  to  XT}-  D u e t o the t r u n c a t i o n , our observations are o n l y f r o m one side of the t r u n c a t i o n p o i n t . T o a c c o m m o d a t e this feature of our e s t i m a t i o n p r o b l e m , we need t o m o d i f y the N N e s t i m a t o r slightly. T h e r e s u l t i n g e s t i m a t o r 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 e s t i m a t o r (1.1) differs f r o m the regular N N e s t i m a t o r i n the d e n o m i n a t o r where regular N N e s t i m a t o r 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 o n the defining i n t e r v a l for distance measure. W e impose r e s t r i c t i o n s o n k for t h e a s y m p t o t i c s of our p r o p o s e d e s t i m a t o r .  In what  follows, we a d d s u b s c r i p t T to  to stress its dependence o n the sample size  T a n d assume: A s s u m p t i o n 1.2 kr/T  —> 0  Let {kxjreN  T" —> oo and one of the  as  (i) kr —>• oo as T (ii)  be a sequence of positive  fcr/loglogToo  integers  satisfying  following:  oo; as T —> oo.  A s s u m p t i o n 1.2 imposes the restrictions o n the divergence r a t e of kr-  The  weak consistency of 4>ix) requires o n l y A s s u m p t i o n 1.2(i), w h i c h is satisfied, for i n s t a n c e , 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 c o n d i t i o n o n kx t h a t guarantees s t r o n g consistency of 4){x)  (see, K i e f e r (1972)[25]). T h e following a s s u m p t i o n is needed for the  (}){x) to have the a s y m p t o t i c n o r m a l i t y p r o p e r t y . 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 p d 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 t h a t we a t t e m p t to e s t i m a t e the one t h a t is r i g h t c o n t i n u o u s at X, a m o n g a l l possible p d f ' s a n d requires t h a t the p d f be right differentiable.  T h e f o l l o w i n g p r o p o s i t i o n s state the consistency a n d a s y m p t o t i c n o r m a l i t y of the one-sided N N e s t i m a t o r . L e t -^d denote the convergence i n d i s t r i b u tion. P r o p o s i t i o n 1.1 0 ( s ) as T  Suppose Assumptions  oo in probability.  LI  and L2(i)  If Assumptions  hold.  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  Then (p{x) —> hold,  then  Remark.  T h e N N m e t h o d c a n be extended to a more general  form  b y a d m i t t i n g w e i g h t i n g functions, i.e., kernel functions i n the b a n d w i d t h e s t i m a t o r s c a n also be a p p l i e d to N N estimates. T h e a s y m p t o t i c properties of the N N e s t i m a t o r are not affected even if the k e r n e l w e i g h t i n g functions is employed. (See M o o r e a n d Y a c k e l (1976, 1977)[40][39] a n d Stone (1977)[56] a m o n g others.)  T o ease the e x p o s i t i o n , t h i s chapter sticks to the u n i f o r m  kernel w e i g h t i n g functions as i n (1.1). However, a l l the results established i n t h e c h a p t e r r e m a i n v a l i d w h e n using other kernel w e i g h t i n g functions.  Remark.  T h e N N m e t h o d can also be extended to estimate a c o n d i -  t i o n a l d e n s i t y f u n c t i o n . F o r e x a m p l e , suppose the e c o n o m e t r i c i a n observes a n i.i.d. b i v a r i a t e stochastic process {Xt,  Zt}t€N- F u r t h e r assume Zt follows  a discrete d i s t r i b u t i o n a n d c o n d i t i o n a l o n Zt, Xt is c o n t i n u o u s l y d i s t r i b u t e d w i t h (fi. T h e n , a n e s t i m a t o r of c o n d i t i o n a l density  at the b o u n d a r y x can  be defined as 0{x\Z  where X = {Xt  : Zt — z'it}.  = z)  :=  If Zt follows a continuous d i s t r i b u t i o n , a k e r n e l  s m o o t h i n g t e c h n i q u e c a n be a p p l i e d t o redefine the e s t i m a t o r . 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 e x i s t i n g l i t e r a t u r e has been t a c k l i n g w i t h the p r o b l e m of density est i m a t i o n near b o u n d a r i e s u s i n g the b a n d w i d t h estimates, b e i n g silent on a p p l y i n g the N N technique for a w h i l e .  W e p r o v i d e some insights t h a t  d e m o n s t r a t e w h y the N N e s t i m a t o r can be a better choice t h a n the b a n d w i d t h e s t i m a t o r s i n t h i s subsection. W e first b r o a d l y investigate the s i m i larities between N N a n d b a n d w i d t h estimates. T h r o u g h t h i s , we show t h a t N N e s t i m a t o r is p r o m i s i n g i n the context of p o i n t e s t i m a t i o n of density.  W h e n a d e n s i t y f u n c t i o n as a whole is of interest, several disadvantages of N N d e n s i t y estimates e l i c i t e d c r i t i c i s m s f r o m s t a t i s t i c i a n s on u s i n g the N N e s t i m a t i o n t e c h n i q u e , (see, S i l v e r m a n (1986)[54] a n d Stone et. a l . (1977)[56]  a m o n g others). C o n s e q u e n t l y , the N N e s t i m a t o r b e c a m e less a n d less p o p u l a r for d e n s i t y e s t i m a t i o n , after receiving a t t e n t i o n s i n t h e late 70's. N e v ertheless, as S i l v e r m a n (1986)[54] says: " . . . T h e r e is n o t h i n g to choose between b a n d w i d t h a n d N N estimates w h e n e s t i m a t i n g a density at a p o i n t .  E v e r y value k of  t h e s m o o t h i n g p a r a m e t e r i n N N e s t i m a t e w i l l give a n e s t i m a t e i d e n t i c a l to t h a t o b t a i n e d w i t h a c e r t a i n value of b a n d w i d t h i n the k e r n e l - b a n d w i d t h estimate..." H a r d i e (1990) [18] further shows t h a t the o p t i m a l choice of k entails a M S E convergence r a t e of n ~ ^ / ^ , w h i c h is the same order as for t h e o p t i m a l b a n d w i d t h estimates i n general. U n d e r m i l d c o n d i t i o n s , b o t h N N a n d b a n d w i d t h e s t i m a t o r s are consistent a n d a s y m p t o t i c a l l y n o r m a l l y d i s t r i b u t e d . T h e r e fore, i n t h e c o n t e x t of e v a l u a t i n g a density f u n c t i o n at a p o i n t , there is not m u c h difference between N N a n d 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 t h e y share the same basic p r o p e r t i e s a n d efficiency.  T h o u g h there is always a m a t c h i n g b a n d w i d t h e s t i m a t o r for a given N N e s t i m a t o r , M o o r e a n d Y a c k e l (1976) [40] have s h o w n t h a t the N N e s t i m a t o r is more efficient t h a n the m a t c h i n g b a n d w i d t h e s t i m a t o r 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 f r o m the difference of t h e i r s m o o t h i n g p a r a m e t e r s .  T h i s argument  reveals t h a t t h e b a n d w i d t h e s t i m a t o r related to the N N e s t i m a t o r m a y be inferior sometimes.  W e next b r i n g the c o m p a r i s o n to the context of d e n s i t y e s t i m a t i o n at b o u n d a r i e s . M o s t of the e x i s t i n g l i t e r a t u r e focuses o n h o w to m a n i p u l a t e the w e i g h t i n g scheme to i m p r o v e the performance of the b a n d w i d t h e s t i m a t o r s near the b o u n d a r i e s . A m o n g these available p r o p o s a l s , the b o u n d a r y - k e r n e l m e t h o d s (Jones 1993[24], M u l l e r 1991 [41]) involve o n l y k e r n e l m o d i f i c a t i o n s a n d m a k e n o a t t e m p t s to estimate the p o p u l a t i o n d e n s i t y value i n c o r r e c t i n g the bias. T h e r e f o r e , these methods are always associated w i t h large variance.  T h e p s e u d o d a t a m e t h o d s ( C o w H n g a n d H a l l 1996) [11] require the knowledge of b o u n d a r y l o c a t i o n . M o r e o v e r , i m p l e m e n t i n g these proposed m e t h o d s , i n general, is not a n easy task.  W h i l e e x a m i n i n g the convergence p r o p e r t y  intensively, the e x i s t i n g l i t e r a t u r e leave the a s y m p t o t i c d i s t r i b u t i o n issue u n e x p l o r e d . T h e o n l y e x c e p t i o n a m o n g t h e m , t o 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 r a n s l a t i o n b o o t s t r a p  to correct t h e bias at the b o u n d a r i e s .  T h e l i m i t i n g d i s t r i b u t i o n of their  m e t h o d is o n l y f o u n d to be of zero m e a n .  O n the contrary, however, the  N N m e t h o d a t t e m p t s t o a d a p t the a m o u n t of s m o o t h i n g to the l o c a l density of d a t a observations.  T h e a d a p t i v e n a t u r e of the N N m e t h o d enables our  N N e s t i m a t o r to o u t p e r f o r m other m e t h o d s i n t e r m s of the m e a n squared error i n o u r s i m u l a t i o n study. O u r proposed e s t i m a t o r is easy to i m p l e m e n t a n d s h o w n t o follow a n o r m a l d i s t r i b u t i o n i n the l i m i t , w h i c h allows us to c o n s t r u c t confidence intervals a n d conduct testings i n a p p l i c a t i o n s .  These  features m o t i v a t e us to choose the N N m e t h o d over other c a n d i d a t e s .  It is fair t o emphasize here t h a t , to the p r o b l e m of density e s t i m a t i o n at b o u n d a r i e s , o u r p r o p o s e d one-sided N N e s t i m a t i o n m e t h o d is j u s t one of the s o l u t i o n s , w h i c h provides reasonably g o o d performance i n s i m u l a t i o n s t u d ies. T h e c o m p a r i s o n w i t h some of the k e r n e l - t y p e bias c o r r e c t i o n e s t i m a t o r s is b r o u g h t u p t o front because of its p o p u l a r i t y a n d close a t t e n t i o n t h a t h a d e l i c i t e d f r o m t h e l i t e r a t u r e . However, whether o u r one-sided N N e s t i m a t o r is always the o n l y best a m o n g choices is b e y o n d the scope of t h i s w o r k a n d , therefore, left o p e n for the future research.  E s t i m a t o r o f 0 ( x ) i n t h e case w i t h u n k n o w n x In t h i s s u b s e c t i o n , we consider the e s t i m a t i o n of a t r u n c a t e d density funct i o n at i t s u n k n o w n t r u n c a t i o n p o i n t . T h e t r u n c a t i o n p o i n t i n t h e current m o d e l c a n be v i e w e d as the lower b o u n d of the t r u n c a t e d d i s t r i b u t i o n . A n a t u r a l c a n d i d a t e to e s t i m a t e the lower b o u n d is the smallest order s t a t i s tic i n t h e s a m p l e . W e denote such a n e s t i m a t o r of ^ as X j - := X^^^y T h e f o l l o w i n g l e m m a shows t h a t  converges to x fast.  L e m m a 1.3  Under Assumption  1-1, Xj- - x = Oa.s.{T  B y c a l l i n g t h e fast convergence p r o p e r t y " s u p e r - c o n s i s t e n c y " , D o n a l d a n d P a a r s c h (1996) [15] has established a result s i m i l a r t o a b o v e l e m m a . T h e i r a p p r o a c h however relies o n t h e c o n d i t i o n t h a t t h e r a n d o m v a r i a b l e X d i s crete. I n s t e a d , we r e a c h t h e same result w i t h c o n t i n u o u s r a n d o m variables.  W e n o w c o m b i n e X j - a n d t h e one-sided N N e s t i m a t o r t o e s t i m a t e (i){x). T h e f o l l o w i n g p r o p o s i t i o n s demonstrate t h a t i t is i n d e e d t h e case, t h a t is, the c o m b i n e d e s t i m a t o r is consistent a n d 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 <f>{x) as T  oo in probability.  1.1 and 1.2(i) hold. Then^{X_rp)  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  (A;T)^  1.2.2  {mr)  1.1-1.3 hold.  - Hs.)}  Then  A^[O,0(s)'].  A P l u g - i n E s t i m a t o r f o r a f u n c t i o n o f ^(x)  W e consider h o w t o e s t i m a t e a p a r a m e t e r t h a t is w r i t t e n as a f u n c t i o n of 4)(x) a n d o t h e r u n k n o w n parameters p o s s i b l y dependent o n x. T h e p a r a m e ter thus c a n be w r i t t e n as a n extended m o d e l ip{x) : = i]j{(j){x),'d{x)),  where  ^{x) is a n o t h e r p a r a m e t e r . W e assume: A s s u m p t i o n 1.4 (Extended  model):  (i) Let 1? : M —> M be a continuous (ii) Let t/) :  —> R 6e a continuous  function; function.  A s s u m p t i o n 1.4 imposes t h e c o n t i n u i t y of f u n c t i o n s . a u c t i o n m o d e l w i t h endogenous  I n our first-price  entry, we are i n t e r e s t e d i n k n o w i n g t h e  value of b i d d e r ' s expected payoff f u n c t i o n at t h e t r u n c a t i o n p o i n t , w h i c h  is r e a l i z e d b y b i d d e r s ' p a r t i c i p a t i o n decisions. Besides the d e n s i t y (</)), the c u m u l a t i v e d i s t r i b u t i o n of s u b m i t t e d bids  is also p a r t of the expected  payoff f u n c t i o n (ip). T o h a n d l e such s i t u a t i o n s , we assume t h e existence of a n e s t i m a t o r o f ?? converging faster t h a n ((> a n d propose a p l u g - i n e s t i m a t o r for t h e e x t e n d e d m o d e l . A s s u m p t i o n 1.5 satisfies  (Faster convergence  that •d{2Ç.T) " '^(s) =  ofd ): d{X_j'),  the estimator  of ^{x),  op{kj,^).  I n o u r a u c t i o n m o d e l , '&{X_x) is a n e m p i r i c a l c u m u l a t i v e d i s t r i b u t i o n funct i o n e v a l u a t e d at Xsp. T h i s e s t i m a t o r converges at the rate T~^l'^, w h i c h is faster t h a n a n y o t h e r n o n - p a r a m e t r i c e s t i m a t o r s . Nevertheless, i n a p p l i c a t i o n s , i9 is n o t l i m i t e d t o continuous functions of X's c u m u l a t i v e d i s t r i b u t i o n . T h e f o l l o w i n g t h e o r e m s are d e r i v e d b y a p p l y i n g t h e S l u t s k y t h e o r e m , a n d the d e l t a m e t h o d t o t h e previous p r o p o s i t i o n s . T h e o r e m 1.6 Suppose Assumptions '4'{4>{K.T)^'^{K.T))  ifiKr)  T h e o r e m 1.7 Suppose uously  Then,  Assumptions  as T  00 in  1.1-1.5 hold.  (fcr)^ {'p{2LT) '~ v f e ) }  first  1.2.3  Then,  probability. Moreover,  asymptotically  N\<d,il)i{(j){x))'^<j){x)'^], where ipi is the partial its  I.4, and 1.5 hold.  ip is  with respect to (f){x) and t9(x) at an open  differentiable  hood of X.  ~* fiî)  1.1, 1.2(i),  derivative  contin-  neighbor-  distributed  with  of tp with respect to  argument.  Simulation Results  W e c o n d u c t M o n t e C a r l o s i m u l a t i o n s t o assess t h e finite s a m p l e b e h a v i o r of the p r o p o s e d e s t i m a t o r s i n t h i s section. W e first i l l u s t r a t e t h e s i m u l a t i o n m e t h o d , a n d t h e n r e m a r k o n the s i m u l a t i o n results. T h e results are r e p o r t e d i n the tables i n a p p e n d i x .  Simulation  Method  f r o m t h e LogNormal(4,1)  I n each e x p e r i m e n t , we r a n d o m l y d r a w n n u m b e r s d i s t r i b u t i o n . T h e n , we choose q-th percentile as  the cutoff value for t h e e x p e r i m e n t , where q is 5, 10, or 25. T h e draws  w i t h t h e i r values greater t h a n the cutoff are selected to f o r m a sample for e s t i m a t i o n . F o r each s i m u l a t i o n , we repeat 10,000 t i m e s . Coverage  W e first check whether the c o n s t r u c t e d 9 0 % a n d 9 5 % confi-  dence intervals p r o v i d e r i g h t coverage. T h e results are r e p o r t e d i n T a b l e 1.1 i n a p p e n d i x . W e e x p e r i m e n t w i t h three different choices of the s m o o t h i n g p a r a m e t e r k, T*/^, T^'^ a n d T^/^ j ^ v s jg chosen because i t is the M S E - b e s t rate for r e g u l a r (two-sided) N N estimators. T^/^ a n d T ^ / ^ are chosen w i t h a purpose to see h o w the coverage responds to the changes i n t h e rates. W e consider three s a m p l e sizes 200, 500, 1000 for the coverage study. T h e results i n T a b l e 1.1 i n d i c a t e t h a t • T^l^, w h i c h violates A s s u m p t i o n 1.3, does not have r i g h t coverage. • T h e s m a l l e r the s m o o t h i n g p a r a m e t e r , as c o m p a r i n g T^/^ w i t h T^/^, the b e t t e r the coverage t h a t the e s t i m a t o r p r o v i d e s . • T h e slow rate of convergence, as the b u i l t - i n feature of the n o n - p a r a m e t r i c estimates, requires quite a large sample to p r o v i d e b e t t e r coverage. I n the s i m u l a t i o n s , we need a sample size, at least, of 1000 for reasonable coverage. • U n d e r - r e j e c t i o n m a y be a concern i n a p p l i c a t i o n s . Other Boundary  Correction  Methods  T h e r e is a n extensive l i t e r a t u r e  o n how t o correct b o u n d a r y effect w h e n k e r n e l e s t i m a t o r s are i n use. i m p o r t a n t q u e s t i o n is how w e l l they compare to our suggested  An  estimator.  A n i d e a l a p p r o a c h to this issue w o u l d be careful a n a l y s i s of a l l the earlier p r o p o s a l , w i t h a d e t a i l e d c o m p a r i s o n of t h e i r properties. c o m m e n t e d b y C h e n g et.  However, as  a l . (1989) [10] w h o encountered a s i m i l a r s i t u a -  t i o n w h e n p r o p o s i n g t h e i r l o c a l p o l y n o m i a l fitting e s t i m a t o r , it w o u l d be a tedious task because "there are so m a n y proposals a n d a n u m b e r of t h e m are quite c o m p H c a t e d , t h u s difficult to w o r k w i t h b o t h n u m e r i c a l l y a n d analytically..."  F o r the p u r p o s e of c o m p a r i s o n , we select some w e l l - k n o w n a n d representative m e t h o d s to c o m p a r e w i t h our one-sided N N e s t i m a t o r s : • T h e p s e u d o d a t a m e t h o d (hereafter, P s e u d o d a t a ; see C o w l i n g a n d H a l l 1996[11]); • T h e b o u n d a r y kernel m e t h o d (hereafter. K e r n e l ; see, for e x a m p l e , Jones 1993[24], M u l l e r 1991[41] a m o n g others); • T h e l o c a l linear f i t t i n g (hereafter, L L F ; see C h e n g et a l . 1997[10]). F o r t h e m a t t e r of fairness, we choose the s m o o t h i n g p a r a m e t e r s for these b o u n d a r y c o r r e c t i o n m e t h o d s i n the way s u c h t h a t they either are M S E best, or m a k e the r e s u l t i n g bias at s i m i l a r 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 m e t h o d s , a n d T~^/^ is used for the b o u n d a r y k e r n e l m e t h o d .  F o l l o w i n g the rule of  t h u m b ( S i l v e r m a n (1986)[54]), we use 1.06 for the constant i n b a n d w i d t h s e s t i m a t o r s . T h e n u m b e r of r a n d o m 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 s m o o t h i n g p a r a m e t e r s k = T^/^ for the N N e s t i m a t o r s t h r o u g h o u t this subsection of the s i m u l a t i o n study.  Simulation  for  density estimation  D G P is a s s u m e d as  T a b l e 1.2 r e p o r t s the results w h e n  LogNormal{4,1).  (i) F o r N N estimates, consistent w i t h the i n t u i t i o n s , the signs of bias are p o s i t i v e i n a n y case. M o r e o v e r , M S E is decreasing i n the sample size a n d i n c r e a s i n g i n the cutoff p o i n t . O v e r a l l , N N o u t p e r f o r m s the other m e t h o d s i n the s i m u l a t e d cases. (ii) T h e bias of P s e u d o d a t a m e t h o d s increase w i t h s a m p l e size, w h i c h f u r t h e r i n d i c a t e s t h a t i t is a n inconsistent e s t i m a t o r .  T h e pseudo-  d a t a m e t h o d is regarded as a n i m p r o v e m e n t f r o m reflection m e t h ods, as i t is c l a i m e d to be " c o n s i d e r a b l y more a d a p t i v e " ( C o w l i n g a n d H a l l , 1996) [11]. However, the generic p s e u d o d a t a m e t h o d requires the knowledge of t r u n c a t i o n p o i n t , so t h a t the s a m p l e order s t a t i s t i c s c a n be used to i n t e r p o l a t e the p s e u d o d a t a . W h e n the b o u n d a r y p o i n t is  not k n o w n , the m e t h o d essentially e x t r a p o l a t e s t h e p s e u d o d a t a , w h i c h causes the inconsistency p r o b l e m as we have j u s t seen f r o m the s i m u lation. (iii) T h e k e r n e l used i n s i m u l a t i o n is a one-sided k e r n e l since we k n o w e x a c t l y how to correct bias at the b o u n d a r y . T h e s i m u l a t i o n suggests t h a t even t h o u g h we c a n c o n t r o l bias to a desired level, the r e s u l t i n g variance is w a y too h i g h c o m p a r e d to the N N e s t i m a t o r . T h i s result is consistent w i t h the findings i n the l i t e r a t u r e - " A p p r o a c h e s i n v o l v i n g o n l y k e r n e l m o d i f i c a t i o n s w i t h o u t r e g a r d t o t r u e d e n s i t y are always associated w i t h larger variance." ( Z h a n g et. a l . , 1999)[63] T h e result reveals a r e a l a n d p r a c t i c a l p h e n o m e n o n t h a t the b o u n d a r y k e r n e l - r e l a t e d m e t h o d s u s u a l l y focus o n g e t t i n g t h e b i a s as one wants it w i t h p a y i n g t h e price of increasing variance. It has g r a d u a l l y been r e a l i z e d b y researchers t h a t this variance i n f l a t i o n is i m p o r t a n t . (iv) L L F , as a s p e c i a l case of the b o u n d a r y k e r n e l m e t h o d , is often thought of b y some as a s i m p l e , h a r d - t o - b e a t default a p p r o a c h . However, the s i m u l a t i o n suggests t h a t , i t is not as g o o d as one-sided N N i n terms of M S B . It is fair to m e n t i o n a l i m i t a t i o n of our s i m u l a t i o n s t u d y . T h o u g h our onesided N N e s t i m a t o r outperforms other methods i n t e r m s of M S E as reported i n T a b l e 1.2, we m a k e no a t t e m p t s to c l a i m t h a t o u r e s t i m a t o r is the only best a m o n g a l l other alternatives. M o r e t h o r o u g h s t u d y is b e y o n d the scope of t h i s w o r k .  1.3  A n Application to the First-Price Auction Model  W e now a p p l y the p r o p o s e d e s t i m a t i o n m e t h o d i n the c o n t e x t of a price auction.  W e set u p a  first-price  first-  auction model w i t h a participation  cost a n d discuss some i d e n t i f i c a t i o n issues. W e t h e n c o m p a r e the a u c t i o n m o d e l w i t h the e x t e n d e d m o d e l i n the m e t h o d o l o g y section t o see how the  p l u g - i n e s t i m a t o r c a n be a p p l i e d i n this case. A t the e n d , we estimate the levels of p a r t i c i p a t i o n costs i n the M i c h i g a n H i g h w a y P r o c u r e m e n t A u c t i o n s , w i t h w h i c h we f u r t h e r infer how the government s h o u l d set the p o l i c y tools (reserve prices a n d / o r e n t r y fee) for o p t i m a l a u c t i o n outcomes.  1.3.1  The Auction Model  W e consider a first-price sealed-bid a u c t i o n of a single i n d i v i s i b l e g o o d . W i t h i n the s y m m e t r i c independent private-value ( I P V ) framework, each p o t e n t i a l r i s k n e u t r a l p a r t i c i p a n t i G { 1 , 2 , . . . , TV} k n o w s her o w n value Vi for the object, b u t o n l y k n o w s the d i s t r i b u t i o n of the values to the other p o t e n t i a l b i d d e r s . It is a s s u m e d t h a t the values to i n d i v i d u a l s are i n d e p e n d e n t l y d r a w n f r o m t h e a b s o l u t e l y continuous d i s t r i b u t i o n F{v)  w i t h s u p p o r t {v,v]  C M+.  B i d d e r s s u b m i t t h e i r b i d s s i m u l t a n e o u s l y a n d the o b j e c t goes to the highest b i d d e r . T h e w i n n e r pays her b i d to the seller, p r o v i d e d t h a t the b i d is no less t h a n the reserve p r i c e r , w h i c h is assumed to be zero i n t h i s chapter w i t h o u t loss of generality.  O u r a n a l y s i s deviates f r o m the s t a n d a r d I P V f r a m e w o r k b y a l l o w i n g for the presence of a c o m m o n 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 bidder has t o p a y to j o i n the a u c t i o n . G i v e n her p r i v a t e value, the b i d d e r decides w h e t h e r or not t o s u b m i t a b i d (paying K) a n d becomes a n a c t u a l b i d d e r . A l l p o t e n t i a l b i d d e r s m a k e t h i s decision s i m u l t a n e o u s l y . Therefore, t h e y make their p a r t i c i p a t i o n decisions w i t h o u t k n o w i n g how m a n y c o m p e t i t o r s t h e y are a c t u a l l y g o i n g to face.  W e w i l l focus o n the u n i q u e s y m m e t r i c B a y e s i a n N a s h e q u i l i b r i u m ( M i l g r o m , 2004) [37], i n w h i c h each p o t e n t i a l p a r t i c i p a n t j o i n s the a u c t i o n if her value is n o less t h a n , Vp, the cut-off p o i n t ( c o m m o n to a l l b i d d e r s ) , otherwise chooses not to p a r t i c i p a t e . T h e cut-off p o i n t is s u c h t h a t the p a r t i c i p a n t w i t h v a l u e Vp is indifferent i n entering the a u c t i o n or n o t : so, Vp s h o u l d solve the e q u a l i t y VpF{vp)^~^  - K = 0. T h r o u g h o u t the c h a p t e r , we assume t h a t  the e n t r y cost is m o d e r a t e such t h a t Vp G iv,v).  T h i s a s s u m p t i o n effectively  rules out the u n i n t e r e s t i n g case of the e n t r y cost is so large t h a t there is no entry. T h e e x p e c t e d profit for the i t h bidder is t h e n given by, ^iivi,  y, {bj)j^i)  = {v^ - v)[F{max{vp,  b-\y)))f-'  - K,  where y is t h e b i d d e r i ' s b i d given Vi a n d a l l other p a r a m e t e r s .  (1.2) 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 11^ w i t h respect to y yields the e q u i l i b r i u m b i d d i n g strategy:  Kv)  = ^—jèl  [  uf{u)F{uf'^du  if  V > V,  (1.3)  A p o t e n t i a l b i d d e r j p a r t i c i p a t e s if a n d o n l y i f Vj > Vp. I n general, the observables i n our current m o d e l are the b i d s , t h e n u m b e r of p o t e n t i a l b i d d e r s a n d the n u m b e r of a c t u a l bidders, w h i l e the p r i v a t e v a l ues a n d t h e i r d i s t r i b u t i o n s are not observed. T h e i d e n t i f i c a t i o n p r o b l e m here a m o u n t s to a discussion on whether the observed variables c a n u n i q u e l y det e r m i n e b o t h the latent d i s t r i b u t i o n F a n d the p a r t i c i p a t i o n cost K. W e s k i p the discussion o n the i d e n t i f i c a t i o n issues i n o u r m a i n t e x t . M o r e discussion on t h i s i d e n t i f i c a t i o n p r o b l e m is discussed i n the a p p e n d i x .  1.3.2  A p p l y i n g the Estimation  Methodology  W e consider L homogeneous auctions w i t h the same n u m b e r of p o t e n t i a l b i d d e r s N, from w h i c h our d a t a are observed, n ; is the n u m b e r of bidders w h o a c t u a l l y s u b m i t a b i d i n the Ith a u c t i o n , where I =  1,2, . . . L .  We  define the s a m p l e 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 t h e c o n s t r u c t i o n of the m o d e l , the cut-off value, Vp ( w i t h its o p t i m a l b i d d i n g bp, w h i c h is zero w h e n the reserve price is zero.) makes the e x p e c t e d g a i n f r o m e n t r y is equal to the level of p a r t i c i p a t i o n cost, K. T h i s suggests  t h a t e v a l u a t i n g the expected profit f u n c t i o n at Vp s h o u l d give a n estimate for K. T h a t i s , (1.4)  K = 7r{bp,Vp)^{vp-bp)[F{vp)f-\  W e c o u l d e s t i m a t e K b y r e p l a c i n g Vp, bp, a n d F w i t h t h e i r estimates. N e v ertheless, t h e p r o p e r t i e s of such e s t i m a t o r are n o t k n o w n w e l l ; i n p a r t i c u l a r , it is difficult t o derive t h e l i m i t i n g d i s t r i b u t i o n of t h e e s t i m a t o r (see, for exa m p l e , G P V ) . W e below derive a n a l t e r n a t i v e expression for K, w h i c h leads to a convenient e s t i m a t i o n m e t h o d . L e t b* d e n o t e t h e observed e q u i l i b r i u m b i d of t h e i t h a c t u a l b i d d e r , i € { l , . . . , n ; } , i n t h e Ith a u c t i o n . G * a n d g* are the d i s t r i b u t i o n a n d d e n sity f u n c t i o n s of b*, respectively. [F{v) - F{vp)\l[l WJv)  i-F(vp)-  T h u s , G* = Pr{b{v)  < b*\v > Vp) —  - F{vp)\. D i f f e r e n t i a t i n g w i t h respect t o b* gives g*{b*) = ensure t h a t g*{bp) is positive as i n A s s u m p t i o n l . l ( i i ) ,  f{v)  has t o be nonzero i n [v,v]. W e assume so for the rest o f t h i s chapter. It follows f r o m ( A . 24), b y some elementary a l g e b r a , t h a t  S u b s t i t u t i n g (1.5) i n t o (1.4), together w i t h the fact t h a t G*{bp) = 0, we get  1  ^fa)"^  N-l[l-F{vp)]9*{bpy  F{vp)  (16) ^ • >  equals t o one m i n u s the expected p a r t i c i p a t i o n r a t e , d e n o t e d b y p.  T h i s enables us t o r e w r i t e (1.6) as  It is n o t h a r d t o see t h a t M a n d g* i n t h e c u r r e n t a p p l i c a t i o n are ^{x) a n d (f) i n t h e p r e c e d i n g section, respectively. W e c a n a p p l y t h e p r o p o s e d p l u g - i n e s t i m a t o r t o e s t i m a t e K . T h e rate of p a r t i c i p a t i o n p, c a n be consistently  e s t i m a t e d b y the sample analogue p := be e s t i m a t e d b y M  = ^^^ri)--  j; J^iLiij^)•  T h i s implies M  can  A p p l y i n g the one-sided N N e s t i m a t o r to  g*{bp), we have k{n)/n  where fc(T(,) is a sequence of positive integers s a t i s f y i n g A s s u m p t i o n s 1.2 a n d 1.3. T h e r e f o r e , o u r e s t i m a t o r for p a r t i c i p a t i o n costs K is defined as  T h e f o l l o w i n g corollaries state the a s y m p t o t i c p r o p e r t i e s of K a c c o r d i n g 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  C o r o l l a r y 1.9  Assume  hp.  the first derivative  surely. of g* exists and hounded  around  Then, {k{n))\{K-K)  1.3.3  -^a N[Q,  (g.^'))2]-  (1-10)  Empirical Application  I n t h i s s u b s e c t i o n , we a p p l y the e s t i m a t o r of K defined i n (1.9) to the M i c h i g a n H i g h w a y P r o c u r e m e n t A u c t i o n d a t a . W e first i n t r o d u c e the d a t a set a n d f o r m a s u b s a m p l e for e s t i m a t i o n . T h e n , we e s t i m a t e the p a r t i c i p a t i o n costs for the a u c t i o n s a n d present the estimates of the costs i n terms of b o t h absolute value (in dollars) a n d relative value (to p r o j e c t size). F i n a l l y , we a p p r o x i m a t e the levels of o p t i m a l cutofl[s i n v a l u a t i o n s , w i t h w h i c h we e m p i r i c a l l y address the questions r e g a r d i n g i m p l e m e n t i n g the o p t i m a l auctions.  Data W e have d a t a for the h i g h w a y p r o c u r e m e n t auctions h e l d b y the M i c h i g a n D e p a r t m e n t of T r a n s p o r t a t i o n between J a n u a r y 2001 a n d D e c e m b e r 2002. T h e d a t a set consists of a t o t a l of 1,538 projects.  F o r each project, the  dataset offers the l e t t i n g date, the expected c o m p l e t i o n d a t e , the l o c a t i o n , the tasks i n v o l v e d , the identities of a l l the b i d d e r s , a l l b i d s , t h e engineer's estimate of the costs for c o m p l e t i n g the project, a n d a list of p l a n - h o l d e r s for a l l p r o j e c t s i n t h e d a t a set. Letting  Process  T h e D e p a r t m e n t of T r a n s p o r t a t i o n ( D o T )  announces  a project to be let a n d the i n v i t a t i o n to s u b m i t bids begins. T h e l e n g t h of this a d v e r t i s i n g p e r i o d ranges from 4 to 10 weeks. T h e a n n o u n c e m e n t of a project comes w i t h a brief d e s c r i p t i o n of the project i n c l u d i n g the l o c a t i o n a n d c o m p l e t i o n t i m e . P o t e n t i a l bidders w h o are interested i n the project m a y collect a d e t a i l e d b i d p r o p o s a l f r o m the D o T . B a s e d o n t h e p r o p o s a l , b i d d e r s m a y s u b m i t a b i d , w h i c h has t o be 48 hours p r i o r to the l e t t i n g date. B i d d e r s do not k n o w w h o else also s u b m i t s a b i d . F o r each b i d , D o T checks t h a t the b i d d i n g f i r m is a m o n g the firms t h a t are q u a l i f i e d to do business w i t h D o T . T h e n , o n the l e t t i n g date, the project is a w a r d e d to the lowest b i d d e r , p r o v i d e d t h a t i t is below the reserve price.  Engineer  Estimates  and Reserve  Prices  T h e D o T p r o v i d e s cost esti-  mates for each p r o j e c t . T h e s e estimates are based o n the engineers' assessment of t h e w o r k needed to fulfill the task a n d i n f o r m a t i o n e x t r a c t e d from the s i m i l a r p r o j e c t s let before. U S F e d e r a l l a w requires t h a t the w i n n i n g b i d s h o u l d be no greater t h a n 1 1 0 % of the engineers' estimates. However, a state is s t i l l a l l o w e d to let the project w i t h a p r i c e higher t h a n this t h r e s h o l d , i f the state c a n j u s t i f y t h i s a c t i o n i n w r i t i n g . I n t h i s case, the engineer's e s t i m a t e for t h i s p r o j e c t w i l l be revised a n d e x a m i n e d t o see i f any possible m i s t a k e h a d been m a d e . I n the p a r t i c u l a r a u c t i o n s e x a m i n e d here, the 1 1 0 % l i m i t is q u i t e frequently o v e r r i d d e n . It is n a t u r a l to expect t h a t the p o t e n t i a l b i d d e r s of the a u c t i o n s  give l i t t l e c o n s i d e r a t i o n to the 110% l i m i t .  T h e a s s u m p t i o n of no reserve  price is t h u s not far f r o m the reality i n our e x a m p l e . P r o m t h e same perspective, we consulted w i t h professionals i n the p r o c u r e m e n t business a b o u t the feasibility for a government to i m p l e m e n t a b i n d i n g reserve p r i c e .  E x p e r t s i n the i n d u s t r y i n f o r m e d us of the great  difficulties, if there were no any other c o m p e n s a t i o n i n effect.  A g r o u p of  factors m a y be u s e d b y b i d d e r s to argue for i n a c c u r a c i e s i n engineer's estimates.  T h e r e f o r e , we are interested i n k n o w i n g how m u c h we need to  compensate p a r t i c i p a n t s to ensure a n o p t i m a l a u c t i o n o u t c o m e , if we set a b i n d i n g reserve price at 110% of the engineers' estimates. T h i s motivates the scenario t o be i n v e s t i g a t e d i n the paper for the e m p i r i c a l i m p l i c a t i o n s .  Government's  Maximum  Willingness  to Pay  It has b e e n recently n o t e d  i n the l i t e r a t u r e t h a t the first (low-)price auctions w i t h o u t b i n d i n g reserve prices, w h i c h is the case of procurement a u c t i o n s considered here, does not have a u n i q u e finite B a y e s i a n - N a s h e q u i l i b r i u m . T o see t h i s i n a more clear fashion, consider a n y b i d d e r . If she knows t h a t there is no b i n d i n g reserve price, a n d there is a non-zero p r o b a b i l i t y for her to be the single a c t u a l b i d d e r , n o m a t t e r h o w s m a l l t h i s p r o b a b i l i t y is, her o p t i m a l strategy is always t o b i d infinity.  T o r a t i o n a l i z e the b i d d e r s ' b e h a v i o r , we use a s i m i l a r a r g u m e n t as L i a n d Z h e n g (2005) [34]. W e assume t h a t a l l a u c t i o n p a r t i c i p a n t s have a c o m m o n belief t h a t g o v e r n m e n t has a maximum-willingness-to-pay,  the u p p e r b o u n d  of v a l u a t i o n 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 t h i s value w i l l be rejected for sure. M o r e o v e r , v is not observed b y econometricians.^ * O n e m a y argue t h a t this line of justification a m o u n t s to a s s u m i n g a reserve price. W e , however, d o not use this t e r m i n o l o g y is because reserve prices a n d entry costs may together c o m p l i c a t e identification issue.  Bidder  Heterogeneity  T h e m a i n sources of heterogeneity a m o n g b i d -  ders i n the m a r k e t are size a n d location.^ L o c a t i o n reflects t h e b i d d e r ' s cost for t h e c o m p a n y of m o v i n g e q u i p m e n t , m a t e r i a l s a n d l a b o r to t h e w o r k site. Size of the f i r m entails the scale of economy. I n our a n a l y s i s , we consider the projects i n v o l v i n g o n l y non-fringe bidders a n d f r o m either M i c h i g a n or neighbor states. A qualified f i r m is considered as non-fringe i f i t p a r t i c i p a t e d more t h a n once i n the s a m p l e p e r i o d .  Project  Type  T h e choice of the project types is m o t i v a t e d b y the  objective t h a t the a u c t i o n e n v i r o n m e n t of p r i v a t e value needs to be ensured. Ideally, we c o u l d have focused on o n l y one t y p e of projects. However, since o u r e m p i r i c a l m e t h o d is n o n p a r a m e t r i c , we c a n not afford to be l i m i t e d to a very s m a l l s a m p l e . O u r entire s a m p l e has u p t o 30 different t y p e codes. N o t a single t y p e c a n p r o v i d e a sample w i t h a reasonable size. Therefore, as a c o m p r o m i s e , we i n c l u d e o n l y the auctions t h a t involve p a v i n g or g r a d i n g to f o r m the s u b s a m p l e . T h e l i t e r a t u r e has d o c u m e n t e d some evidence t h a t p a v i n g - t y p e r o a d jobs seem to have more p r i v a t e - v a l u e c o m p o n e n t s  (cf.  H o n g a n d S h u m (2002) [22]).  T h e r e f o r e , our c o n t r o l l e d s u b s a m p l e is c o m p o s e d of the observations o n l y f r o m t h e a u c t i o n s for p a v i n g / g r a d i n g t y p e projects a n d involves o n l y n o n fringe firms f r o m 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  s a m p l e of h i g h w a y m a i n t e n a n c e projects F e b r u a r y 1997 a n d D e c e m b e r 2003.  let b y M i c h i g a n D o T  between  H e r results suggest t h a t " f a i l i n g to  account for unobserved a u c t i o n heterogeneity m a y lead to o v e r e s t i m a t i n g u n c e r t a i n t y t h a t bidders face w h e n s u b m i t t i n g t h e i r b i d s " . W e c o n d u c t several p r e l i m i n a r y regressions t o check w h e t h e r the u n o b served p r o j e c t heterogeneity p r o b l e m exists i n the s a m p l e we use. T h e re•"The specialization of firm m a y also m a t t e r i n terms of bidder heterogeneity. we do not have enough information i n our d a t a set to handle this.  However,  suits are r e p o r t e d i n T a b l e 1.3. A l l regressions i n d i c a t e t h a t the E n g i n e e r ' s E s t i m a t e is t h e o n l y variable has a s t a t i s t i c a l l y significant i m p a c t o n the b i d level. T h i s i n t u r n justifies our i d e a of n o r m a l i z i n g the b i d s b y t h e engineer's estimates t o homogenize the auctions. O L S analysis produces  equal to 0.9780 w h i c h indicates t h a t variables  i n c l u d e d i n t h e regression c a p t u r e factors affecting b i d level quite well. W h e n the between-effect m o d e l (regression on group means) is used to e x p l a i n the v a r i a t i o n i n t h e b i d s , B? r e m a i n s r o u g h l y same at 0.9779 w h i c h indicates t h a t there is n o s u b s t a n t i a l a m o u n t of i n t e r - a u c t i o n v a r i a t i o n u n e x p l a i n e d b y t h e v a r i a b l e s available to the researchers. T h e random-effect  regression  provides a s i m i l a r result to the first two sets of regressions: the unobserved a u c t i o n heterogeneity m a y not present i n o u r c o n t r o l l e d s u b s a m p l e .  E s t i m a t e s of p a r t i c i p a t i o n costs A p o t e n t i a l b i d d e r i n t h i s a p p l i c a t i o n is defined as a qualified firm w h o requested for a n official b i d d i n g p r o p o s a l before the b i d d i n g s t a r t s . W e group the a u c t i o n s b y t h e n u m b e r of p o t e n t i a l bidders. means the a u c t i o n s w i t h 5 p o t e n t i a l bidders.  For example, G r o u p 5  W i t h i n each g r o u p , a l l the  auctions are r e g a r d e d as homogeneous, w h i l e the u n d e r l y i n g value d i s t r i b u tions ( a n d , t h u s the p a r t i c i p a t i o n costs) across groups are t a k e n as different.^ A l l bids are n o r m a l i z e d by the engineer's estimates.  F i g u r e 1.1 p l o t s the e s t i m a t e d p a r t i c i p a t i o n rates for groups.  I n the  dataset, we observe t h a t the n u m b e r of p o t e n t i a l bidders ranges f r o m two to twenty-two.^ W h e n there are o n l y two p o t e n t i a l b i d d e r s , the p a r t i c i p a t i o n rate is one, so t h a t we cannot estimate the e n t r y cost i n our a p p r o a c h . ^ A n alternative e s t i m a t i o n  We  a p p r o a c h m a y assume the entry cost is same across  a u c t i o n groups a n d p o o l all the auctions together to estimate the entry cost jointly. ever, this a p p r o a c h e m p l o y s the variation i n the n u m b e r of p o t e n t i a l bidders.  Therefore,  it i m p l i c i t l y assumes such variation is exogenous a n d there were no unobserved geneity across the g r o u p s of auctions. W e decide not to i m p o s e these further b y e s t i m a t i n g the m o d e l separately across number of potential bidders. ^We d o not observe any a u c t i o n w i t h 19 potential bidders in the date set.  the  Howhetero-  assumptions  e l i m i n a t e the a u c t i o n s w i t h 16 t o 22 p o t e n t i a l b i d d e r s f r o m our analysis, because t h e y offer less t h a n 100 observations. B a s e d o n 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 T a b l e 1.4 reports the a c c o r d i n g n u m b e r s for F i g u r e 1.1. T h e rates of p a r t i c i p a t i o n m o n o t o n i c a l l y decrease f r o m .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 e s t i m a t e d for each group a c c o r d i n g to K p r o posed i n (1.9). M o d i f i c a t i o n o n K is needed for the e m p i r i c a l a p p l i c a t i o n . I n our t h e o r y f r a m e w o r k , we follow the convention i n l i t e r a t u r e to consider a n a u c t i o n where the highest b i d w i n s . However, our e m p i r i c a l a p p l i c a t i o n is p r o c u r e m e n t a u c t i o n s i n w h i c h the lowest bids w i n . It s h o u l d be n o t e d t h a t , i n h i g h w a y p r o c u r e m e n t auctions, v is the c o n s t r u c t i o n cost b y the f i r m a n d the b i d d e r s u b m i t s a b i d o n l y if her c o n s t r u c t i o n cost is less t h a n t h e cutoff value Vp.  E s t i m a t i o n results o n the p a r t i c i p a t i o n costs are l i s t e d i n C o l u m n 4 of T a b l e 1.4 a n d p l o t t e d i n F i g u r e 1.2. T h e e s t i m a t e d p a r t i c i p a t i o n cost varies f r o m 2 % to . 0 0 1 % of the engineer's estimates. T h e d o t t e d lines i n the figure are 9 5 % confidence intervals for the estimates.  T h e results show t h a t the  p a r t i c i p a t i o n costs are s i g n i f i c a n t l y different f r o m zero for a l l groups.  T h i s e s t i m a t i o n result s h o u l d be i n t e r p r e t e d w i t h care. It indicates t h a t the e n t r y cost is p r o p o r t i o n a l to the engineer's estimates. It is c o m p a t i b l e w i t h the c o n s t a n c y of the entry cost i n the a u c t i o n t h e o r y b y the following a r g u m e n t . I n p r a c t i c e , the larger a u c t i o n e d projects ( w i t h larger engineer's estimates) c a n be more costly o n b i d s u b m i s s i o n . T h i s m a y reflect t h e fact t h a t larger p r o j e c t s require more w o r k o n c o m p l e t i n g b i d d i n g p l a n s (forms) a n d m o r e efforts i n v o l v e d i n the b i d d i n g process.  T h e r e f o r e , the constant  e n t r y cost entails the fixed p r o p o r t i o n to engineer's estimates i n p r o c u r e m e n t auctions.  I n absolute t e r m s , we investigate the levels of p a r t i c i p a t i o n costs i n a n average-size projects i n each group.  T h e n u m b e r s are l i s t e d i n T a b l e 1.4  c o l u m n 5. W e find t h a t the costs i n dollar values decrease d r a m a t i c a l l y over the range. T h e costs c a n be as h i g h as $13,416 w h e n N = 3 , t h e n d r o p d o w n to $3,721 w h e n N = 5 . W h e n N > 8 , the costs fluctuate between $1,000 a n d $2,000.  T h e d e c l i n i n g p a t t e r n of e n t r y cost w i t h the groups i n b o t h r e l a -  tive a n d a b s o l u t e t e r m s is consistent w i t h the fact t h a t the average project sizes decrease w i t h the groups.  A n i n s t i t u i t i o n a l reason m a y be t h a t the  a u c t i o n s w i t h less p o t e n t i a l b i d d e r s are less c o m p l e x i n n a t u r e , require less for p a r t i c i p a t i o n a n d therefore a t t r a c t more b i d d e r s .  Counterfactual experiments:  implementing optimal auctions  W i t h the h e l p of k n o w i n g the level of p a r t i c i p a t i o n cost, is there a n y t h i n g t h a t a government c a n do to i m p r o v e its s i t u a t i o n ? W e e m p i r i c a l l y address t h i s q u e s t i o n b y c a l c u l a t i n g the o p t i m a l level of cutoffs i n v a l u a t i o n s , w h i c h a c c o r d i n g l y suggests the p o l i c y tools (entry fee/subsidy, or reserve prices) a government s h o u l d u n d e r t a k e for o p t i m a l a u c t i o n outcomes. A s counterfact u a l e x p e r i m e n t s , we c o m p u t e the savings a government c o u l d have made f r o m the o p t i m a l a u c t i o n s .  T h e o p t i m a l i t y a n a l y z e d i n t h i s subsection is  f r o m revenue p e r s p e c t i v e , i.e., i t refers to m i n i m i z i n g the government's exp e c t e d p a y m e n t o n the a u c t i o n e d project.  Optimal  O p t i m a l a u c t i o n s have been s t u d i e d f r o m t h e o r e t i c a l  cut-off  p e r s p e c t i v e . (See, for e x a m p l e , M y e r s o n (1981)[42] a m o n g others.) I n p a r t i c u l a r , 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 a u c t i o n s w i t h p a r t i c i p a t i o n costs. T h e y show t h a t the cutoff i n v a l u a t i o n s for a n o p t i m a l s y m m e t r i c a u c t i o n , v*, s h o u l d solve the following e q u a t i o n : {v ~ J{v*mi where J{v)  =^ v -\-  - Fiv*)f-'=  K,  (1.11)  is the v i r t u a l v a l u a t i o n f u n c t i o n , f o l l o w i n g the n o t a -  t i o n s i n the l i t e r a t u r e . E q u a t i o n (1.11) c a n be i n t e r p r e t e d as follows.  S u p p o s e a l l b i d d e r s at  the cutoff V*. I n t r o d u c i n g a slight decrease of the cutoff t o one b i d d e r w i l l  increase the gross g a i n to the seller b y {v - J{v*)){l V — J{v*),  (gaining  - F{v*))^~^  the v i r t u a l v a l u a t i o n , w h e n a l l the other v a l u a t i o n s are greater  t h a n V*, i.e., w i t h p r o b a b i l i t y (1 - F{v*))'^~^),  w h i l e p a y i n g the b i d d e r K,  the m a r g i n a l cost of i n d u c i n g the p a r t i c i p a t i o n . T h e value of v, recovered t h r o u g h (1.5), is used to create a pseudo s a m ple of v a l u a t i o n s . W e estimate F b y a n e m p i r i c a l C D F a n d / b a n d w i d t h e s t i m a t o r , respectively.  by kernel-  F o l l o w i n g the e m p i r i c a l a u c t i o n l i t e r a -  ture, we use a t r i w e i g h t f u n c t i o n for the k e r n e l a n d 2.978 x 1.06 x T~^^^ x a for b a n d w i d t h w h e r e T is the s a m p l e size a n d a is t h e s t a n d a r d d e v i a t i o n . 2.978 is needed because of the triweight kernel. ( E x a m p l e s c a n be found i n G P V a n d L i , P e r r i g n e a n d V u o n g (2000)[35].) W e use t h e m a x i m u m bids i n the s a m p l e as t h e estimates for v. W e t h e n n u m e r i c a l l y solve (1-11) from the pseudo s a m p l e a n d derived a n estimate for v*. T h e estimates are r e p o r t e d i n the second c o l u m n of T a b l e 1.5. F o r a l l the a u c t i o n s i n the s a m p l e , the o p t i m a l cutoff values v a r y m o s t l y between 0.9 a n d 1.1.  Restricting  entry  T h e following p r o p o s i t i o n shows t h a t i n our endoge-  nous e n t r y m o d e l , the cutoff i n the s y m m e t r i c e q u i l i b r i u m Vp a n d the cutoff i n a n o p t i m a l a u c t i o n v* are c o m p a r a b l e . P r o p o s i t i o n 1.10 costs, zero entry fee tential  bidders,  equilibrium,  In a  first-price  IPV  auction  and zero reserve price,  the seller maximizes  by discouraging  for  model with any given  the expected revenue,  participation  number  in the  of  po-  symmetric  entry.  T h e p r e c e d i n g p r o p o s i t i o n indicates t h a t , i n o u r c u r r e n t first ( l o w - ) b i d a u c t i o n , v* < Vp. I n t u i t i v e l y , a decrease i n the e q u i l i b r i u m cutoff (vp) make the auctioneer better-off by e x t r a c t i n g f r o m a l l bidders w h o have sufficiently favorable values t h a t t h e y decide to enter a n d b y f u r t h e r screening least favorable v a l u a t i o n bidders. T h i s t h e o r e t i c a l i m p l i c a t i o n provides a device for us to check the reUability of c o m p u t e d cutoffs. W e r e p o r t the m a x i m u m of recovered v a l u a t i o n s (after t r i m m i n g near b o u n d a r i e s ) i n C o l u m n 3 of T a b l e 1.5, w h i c h clearly shows t h a t v* < Vp.  T h i s result m a y be more s t r i k i n g t h a n it appears. W e c a n observe d a t a o n l y f r o m the t r u n c a t e d density r a t h e r t h a n the full range of d i s t r i b u t i o n . T h i s fact makes one reasonably expect t h a t , i n n o n p a r a m e t r i c framework, the p r e d i c t i o n power of counterfactuals t h a t econometricians m a y c o n d u c t is l i m i t e d .  W e are able to provide a n example t h a t , even w i t h the case  of t r u n c a t e d density, d o i n g s u c h reliable counterfactuals is possible.  It is  because the o p t i m a l p o l i c y (of r e s t r i c t i n g entry) uses i n f o r m a t i o n f r o m the observed range o n the (truncated) d i s t r i b u t i o n t h a t enables us t o do so. Optimal  policy tools  W e next investigate how to i m p l e m e n t the o p t i m a l  auctions b y u s i n g o n l y regular p o l i c y tools, the reserve price (r) a n d entry fee (<5). T h e s e p o l i c y p a r a m e t e r s s h o u l d be chosen such t h a t the following equality holds: (r - v*)il  - F{v*)f-^  = K + S,  (1.12)  where K + Ô c a n be i n t e r p r e t e d as effective p a r t i c i p a t i o n costs. N o t e t h a t 6 c a n be negative, i m p l y i n g a n e n t r y subsidy. O b v i o u s l y , there are m a n y pairs of r a n d S c a n s u s t a i n the e q u a h t y (1.12) a n d any such a p a i r s h o u l d e n t a i l a w a y to i m p l e m e n t o p t i m a l auctions. W e w i l l focus o n the t w o scenarios to show h o w i t w o r k s : (I) how m u c h is r, w h e n S = 0; (II) h o w m u c h is S, w h e n r = 1.1. Scenario I entails the s i t u a t i o n of o p t i m a l reserve prices only. I n Scenario II, the government w o u l d r a t h e r set a n o p t i m a l level of e n t r y s u b s i d y / f e e w h i l e e n f o r c i n g t h e reserve prices at 1 1 0 % of engineer's estimates. T h e reserve p r i c e t h a t s h o u l d be i m p o s e d i n Scenario I is l i s t e d i n C o l u m n 6 of T a b l e 1.5. T h e levels of reserve prices m a y be as h i g h as 3.2, b u t m o s t l y v a r y f r o m 1.1 to 1.3. T o further show the o p t i m a l p o l i c y entails d i s c o u r a g i n g entry, we r e p o r t the m a x i m u m of bids for each group i n the table.  The  e s t i m a t e d Ô, o p t i m a l fee, i n Scenario I I are r e p o r t e d i n C o l u m n 4 of T a b l e 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 e n t a i l p a r t i c i p a t i o n  subsidies for the a u c t i o n s i n the sample. F o r G r o u p N = 1 4 , T a b l e 1.5 shows  t h a t reserve prices at 1 1 0 % of engineer's estimates does not r e s t r i c t entry enough.  T o g e t h e r t h i s poUcy, a n entry fee is needed t o w a r d s the o p t i m a l  a u c t i o n o u t c o m e . T h e a m o u n t of the subsidies decreases a l o n g the increase of n u m b e r of p o t e n t i a l bidders.  Revenue {EP)  improvement  A t the e n d , we c a l c u l a t e the e x p e c t e d p a y m e n t  b y the government i n Scenario II t h r o u g h the f o l l o w i n g e q u a t i o n :  EP  = N  r' Jo  J{v){l-F{v)f~^f{v)dv-5NF{v*).  (1.13)  T h e first t e r m i n (1.13) is the expected b u y i n g p r i c e a n d t h e second t e r m is the e x p e c t e d r e v e n u e / p a y m e n t for entry. B o t h the c a l c u l a t e d expected p a y ment a n d the a c t u a l p a y m e n t i n current a u c t i o n s are p r o v i d e d i n T a b l e 1.5. T h e difference between two payments i n d i c a t e s the a m o u n t a government w o u l d have saved if he h a d i m p l e m e n t e d the o p t i m a l a u c t i o n s specified as Scenario I I . O u r results i n d i c a t e t h a t , for a l l the a u c t i o n s i n t h e s a m p l e , the government w o u l d have p a i d less i f the o p t i m a l a u c t i o n s were i m p l e m e n t e d . F o r some of the a u c t i o n s , the i m p r o v e m e n t c a n be u p to 1 0 - 1 5 % .  1.4  More Examples on Applications  T h i s s e c t i o n lists some other s i t u a t i o n s w h e r e the p r o p o s e d e s t i m a t i o n m e t h o d o l o g y c a n be a p p l i e d . F o r the first several scenarios, we e x t e n d the a u c t i o n f r a m e w o r k e l a b o r a t e d above to other e c o n o m i c fields, because a n a u c t i o n is j u s t a m e c h a n i s m for t r a n s a c t i o n s .  W e t h e n p r o v i d e another  e x a m p l e where no strategic s t r u c t u r e is i n v o l v e d . T h r o u g h t h i s , we a t t e m p t to a p p l y o u r p r o p o s e d e s t i m a t i o n m e t h o d for a broader range of a p p l i c a t i o n s i n e m p i r i c a l studies.  Menu economy.  Costs  Suppose there are N firms c o n d u c t i n g p r o d u c t i o n i n the  I n one p e r i o d , each f i r m is subject to a r a n d o m shock e w h i c h  follows a d i s t r i b u t i o n . If the r e a l i z a t i o n of e is below a t h r e s h o l d Cp, the f i r m does not need to proceed w i t h any significant change o n its " m e n u " .  O t h e r w i s e , the f i r m has to p a y K for " r e p r i n t i n g m e n u s " . Therefore, at the end,  we observe n firms a c t u a l l y make the changes of t h e i r menus.  How  m u c h change t h e f i r m w i l l take is d e t e r m i n e d b y the r e a l i z a t i o n of e. It is i n f o r m a t i v e for a government to k n o w this K for p o l i c y - m a k i n g purposes. Efficiency  C o n s i d e r a monopsonist employer t h a t wants to p r o -  Wages  v i d e incentives for workers to make efforts to acquire a specific s k i l l i n their l a b o r m a r k e t . S u p p o s e a l l p o t e n t i a l workers are w i t h s i m i l a r b a c k g r o u n d s , say w i t h at least a h i g h school e d u c a t i o n . W o r k i n g experience i n a related area a n d a h i g h e r degree c a n p r o m o t e the worker's p r o d u c t i v i t y . T h i s labor m a r k e t p a r t i c i p a t i o n p r o b l e m is i d e n t i c a l to o u r first-price a u c t i o n m o d e l w i t h a n e n t r y cost. W i t h our e s t i m a t i o n f r a m e w o r k , we c a n estimate how m u c h "effective i n v e s t m e n t " a p o t e n t i a l worker w i l l have to i n c u r to j o i n the m a r k e t .  M o r e o v e r , we c a n help the employer to i d e n t i f y the o p t i m a l  incentive t h e y w i l l p r o v i d e to m a x i m i z e its benefits. T h i s c a n p r o v i d e e m 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 capital.  Jump at a discontinuity  point  C o n s i d e r a s o c i a l p r o g r a m t h a t a govern-  m e n t c o n d u c t s . O n l y households w i t h a n a n n u a l income lower t h a n $30,000 are q u a l i f i e d . N o w the government is interested i n k n o w i n g w h e t h e r the t h r e s h o l d is c o r r e c t l y i m p o s e d i n the f o l l o w i n g senses:  (1) F o r the house-  holds w i t h a n i n c o m e close to $30,000, the p r o g r a m is not h e l p i n g t h e m to a higher u t i l i t y t h a n the one reached by the houesehold whose i n c o m e is above $30,000 b u t close.  Because otherwise, the p r o g r a m w i l l encourage c e r t a i n  people p u t t i n g less effort i n t o the l a b o r m a r k e t . (2) T h e p r o g r a m is p r o v i d i n g a i d to the households w i t h a low income so t h a t t h e i r l i v i n g s t a n d a r d s are not far off f r o m the people outside the p r o g r a m . T h e s e purposes e n t a i l to evaluate the b e h a v i o r of households f r o m b o t h sides of the t h r e s h o l d . T h e difference t h e n estimates the " j u m p " at the d i s c o n t i n u i t y p o i n t . I n such a s i t u a t i o n , o u r p r o p o s e d e s t i m a t i o n m e t h o d applies.  W e also note t h a t , i n t h i s case, o u r e s t i m a t i o n s t r a t e g y i m p r o v e s the a c c u r a c y of estimates i n the e x i s t i n g methods.  C u r r e n t l y , e m p i r i c a l economists  use a p a r a m e t r i c a p p r o a c h to estimate b o t h groups ( w i t h i n a n d outside the p r o g r a m ) . T h e y t h e n evaluate the two e s t i m a t e d functions at $30,000. O u r a p p r o a c h does not lose a c c u r a c y i n estimates i n order to fit t h e entire functions.  O u r interest is to k n o w the difference between t w o p o i n t s at the  b o u n d a r i e s anyway.  1.5  Conclusion and Discussion  I n t h i s c h a p t e r , we propose using the c o m b i n a t i o n of o n e - s i d e d  nearest  neighbor e s t i m a t o r w i t h extreme order s t a t i s t i c i n t h e s a m p l e t o estimate t r u n c a t e d u n i v a r i a t e p r o b a b i l i t y density functions (pdf)  at t h e i r u n k n o w n  t r u n c a t i o n p o i n t s . I n p r a c t i c e , the p a r a m e t e r of interest is often a s m o o t h f u n c t i o n of the u n k n o w n p d f a n d some other a t t r i b u t e s of t h e d i s t r i b u t i o n at the t r u n c a t i o n p o i n t . W e also suggest a p l u g - i n e s t i m a t o r i n w h i c h such p a r a m e t e r is e s t i m a t e d b y first e s t i m a t i n g the p d f a n d some other relevant a t t r i b u t e s of the d i s t r i b u t i o n at the t r u n c a t i o n p o i n t , a n d t h e n p l u g g i n g the e s t i m a t e d values i n t o the s m o o t h f u n c t i o n . W e s t u d y the large sample p r o p e r t i e s of t h e p l u g - i n e s t i m a t o r to establish sets of c o n d i t i o n s sufficient for i t s consistency a n d a s y m p t o t i c n o r m a l i t y .  F u r t h e r , we a p p l y the proposed m e t h o d to estimate the p a r t i c i p a t i o n costs i n c u r r e d t o t h e b i d d e r s i n order to j o i n the M i c h i g a n H i g h w a y P r o c u r e m e n t Auctions.  O u r e m p i r i c a l result suggests t h a t the p a r t i c i p a t i o n costs are  s t a t i s t i c a l l y significant f r o m zero. B a s e d o n this e s t i m a t e , we infer how the o p t i m a l a u c t i o n o u t c o m e s c a n be i m p l e m e n t e d b y u s i n g the r e g u l a r p o l i c y tools. W e e x p l i c i t l y show the i m p r o v e m e n t the M i c h i g a n government c o u l d have m a d e o n p a y m e n t s , i f the entry subsidies were p r o v i d e d w h i l e enforcing the c u r r e n t reserve p r i c e policy. O u r e s t i m a t i o n m e t h o d is easy to i m p l e m e n t a n d not l i m i t e d to e m p i r i c a l a u c t i o n works. W e have l i s t e d some examples where the m e t h o d is a p p l i c a b l e . W i t h more successful a p p l i c a t i o n s of game t h e o r e t i c a l tools to other economic topics, we are confident to foresee more  e m p l o y m e n t of t h i s e s t i m a t i o n m e t h o d t o address questions t h a t arise i n other e c o n o m i c fields. I n o u r e m p i r i c a l a p p l i c a t i o n s , we o n l y considered homogeneous auctions in this chapter.  Heterogeneous  auctions have been m a n i p u l a t e d t h r o u g h  c o n t r o l l i n g some covariates i n the l i t e r a t u r e . (See, for e x a m p l e , G P V . ) I n formally, we c a n reconcile a heterogeneous case to o u r framework;  if the  covariates o n l y affect t h e locations of F r a t h e r t h a n its shapes, one c a n a l ways n o r m a l i z e the bids to c o n t r o l the shifts of F b y covariates so t h a t the heterogeneous case is t r a n s f o r m e d i n t o a homogeneous case. T h i s s o l u t i o n , however, is h e u r i s t i c . W e expect a more rigorous t r e a t m e n t to this p r o b l e m i n the f u t u r e research.  I n t e r m s of b i d d i n g strategies, we focus o n the u n i q u e s y m m e t r i c e q u i h b r i u m i n t h i s c h a p t e r . However, a l o n g the line of o p t i m a l a u c t i o n s , C e l i k a n d Y i l a n k a y a (2006) [8] p o i n t out t h a t there are s i t u a t i o n s where the e q u i l i b r i a w i t h a s y m m e t r i c cutoffs m a y further i m p r o v e the expected revenue (or p a y ment).  T h e a u t h o r s p r o v i d e sufficient c o n d i t i o n s for the existence of such  e q u i l i b r i a : the b e h a v i o r of T{v)  := YZ^^-  W e r e p o r t the p l o t s of T(v)  for  G r o u p s N = 9 a n d N = 1 2 as F i g u r e s 1.3 a n d 1.4. T h e plots i n d i c a t e t h a t for a u c t i o n s w i t h n i n e p o t e n t i a l b i d d e r s , T{v)  increases g l o b a l l y , w h i c h i n t u r n  m a y suggest t h e existence of e q u i l i b r i a for o p t i m a l a u c t i o n s w i t h a s y m m e t ric cutoffs. (See P r o p o s i t i o n 3 i n C e U k a n d Y i l a n k a y a (2006)[8] for details.). O n the c o n t r a r y , t h o u g h F i g u r e 1.4 does not show a n y g l o b a l m o n o t o n i c i t y 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 s y m m e t r i c e q u i l i b r i u m locates for t h i s g r o u p .  T h i s m a y also  i m p l y t h e e q u i l i b r i a for o p t i m a l a s y m m e t r i c auctions t h r o u g h a sufficient c o n d i t i o n o n the l o c a l behavior of T{v)  at v*.  (See P r o p o s i t i o n 2 i n C e l i k  a n d Y i l a n k a y a (2006) [8] for details.) T h i s is j u s t a p r e l i m i n a r y t r e a t m e n t o n c h e c k i n g t h e existence of e q u i l i b r i a w i t h a s y m m e t r i c cutoffs. T h e interest of our w o r k is of e s t i m a t i n g entry cost a n d s h o w i n g its a p p l i c a t i o n s . T h e focus of t h i s c h a p t e r is, therefore, l i m i t e d to the w o r l d of s y m m e t r i c e q u i l i b r i u m . M o r e r i g o r o u s e m p i r i c a l e x a m i n a t i o n o n the existence a n d i m p l e m e n t a b i l -  i t y of e q u i l i b r i a w i t h a s y m m e t r i c cutoffs m a y be w o r t h e x p l o r i n g i n future researches.  1.6  Tables and Figures  T a b l e 1.1: S i m u l a t i o n R e s u l t s o n Coverage  200  5% 500  1000  200  10% 500  1000  200  25% 500  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 95% Interval  0.9872 0.9999  0.9514 0.9998  0.9142 0.9821  0.9852 0.999  0.9499 0.9983  0.9195 0.9847  0.9866 0.9998  0.9276 0.999  0.8675 0.9735  1000  T a b l e 1.2: S i m u l a t i o n R e s u l t s , L o g n o r m a l ( 4 , l ) , k ^ T ^ / ^  100  5% 200  100  10% 200  100  25% 200  500  0.003447 0.002427 0.002449  500 0.002744 0.002143 0.001714  0.004360 0.002847 0.003302  0.005496 0.003690 0.004074  0.004641 0.00354 0.003003  0.004038 0.003429 0.002133  NN  RMSE Bias STD  0.004693 0.003705 0.002880  0.003877 0.003241 0.002128  500 0.003034 0.002660 0.001459  LLF  RMSE Bias STD  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  RMSE Bias STD  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  RMSE Bias STD  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 A n a l y s i s Variable Log(Estimate) Log(Back!og) Number of Days Distance Out-of-State F i r m (dummy) Letting Date (dummy) Constant  Dependent Variable: Number of observations: Number of Potential Bidders: Projects: Bidders:  OLS 0.9843* (.0119) -0.0066 (.0120) -0.0001 (.0000) 0.0191 (.0109) -0.0336 (.0434) -0,0001 (.0001) 2.2313 (2.6427)  Between Effects 0.9939* (.0164) -0.0145 (.0202) -0.0001 (.0001) 0.0168 (.0185) -0.0449 (.0768) -0.0000 (.0001) 1.3474 (3.4205)  Random Effects 0.9933* (.0157) -0.0012 (.0091) -.00001 (.0001) .0239 (.0093) -0.0110 (.0324) -0.0000 (.0001) 1.1930 (3.3130)  0.9780  0.9779  0.9779  Log(Bid) 171 5 Paving and Grading non-fringe firms  * Statistically significant at 5% level.  T a b l e 1.4: E s t i m a t i o n R e s u l t s o n E n t r y C o s t s pot* 3 4 5 6 7 8 9 10 11 12 13 14 15  nob" 257 354 473 503 468 410 356 119 228 249 215 145 191  .79 .77 .73 .72 .68 .64 .62 .51 .53 .52 .52 .47 .49  .02391 .01318 .00341 .00126 .00156 .00055 .00013 .00059 .00017 .00009 .00003 .00004 .00001  * Number of potential bidders. Sample size. § Rate of participation. t Estimated levels of participation costs(in engineer's * Estimated levels of participation costs (in Dollars).  estimate).  K* 13416 11557 3721 2916 3446 2007 1326 1871 1621 1226 1043 1201 1042  T a b l e 1.5: P o l i c y T o o l s T o w a r d s O p t i m a l A u c t i o n s  (i) (ii)  pot  v'  3 4 5 6 7 8 9 10 11 12 13 14 15  0.9448 1.0458 1.0625 1.1104 1.0867 1.0223 1.0706 1.0744 1.1016 1.0959 1.0787 1.0396 1.1108  fee*(i) -0.01496 -0.01463 -0.00369 -0.00152 -0.00202 -0.00064 -0.00013 -0.00057 -0.00019 -0.00010 -0.00001 0.00002 -0.00001  Max{b} 1.8447 2.9872 2.3447 2.7703 3.9949 2.6707 1.7973 1.9711 1.9684 1.8021 1.5400 1.4922 1.5305  r*  AP  EP  1.3707 2.2567 1.6640 1.8178 3.2271 2.0821 1.2658 1.2857 1.3002 1.3074 1.1252 1.0767 1.1527  0.9413 0.9503 0.9166 0.9325 0.9287 0.8799 0.9132 0.9473 0.9422 0.9319 0.9129 0.8901 0.9259  0.8395 0.9007 0.8676 0.8699 0.8933 0.8102 0.8918 0.8163 0.8280 0.9105 0.7585 0.8669 0.8205  pot: Number of potential bidders. v':  O p t i m a l cutoffs, calculating according to equation (1.11).  (iii) Max{v}: (iv)  Max{ÏÏ} 0.9514 1.0468 1.0692 1.1306 1.0877 1.0233 1.0875 1.0940 1.1435 1.1019 1.1214 1.1018 1.1492  M a x i m u m of recovered valuations, i;(after trimming near boundaries).  fee*((5): E n t r y fee/subsidy in the counterfactual experiment, where the reserve price is set as 1.1.  (v) (vi) (vii) (viii)  Max{b}: M a x i m u m of observed bids. r*: O p t i m a l reserve price, when no entry fee or subsidy is in effect. A P (Actual Payment): Average actual payments (buying prices from data). E P (Expected Payment): Expected payment in the counterfactual experiment, where the reserve price is 1.1 and entry fee/subsidy is in effect.  F i g u r e 1.1: R a t e of P a r t i c i p a t i o n Rate of Participation *  7  rate of participation (p) 90% confidence band  9 11 13 15 17 Number of Potential Bidders  19  21  F i g u r e 1.2: P a r t i c i p a t i o n C o s t s 0.031  Participation Costs 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 ) f o r 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 r e a l - w o r l d auctions is t h a t not a l l the eligible b i d d e r s act u a l l y s u b m i t a b i d , even t h o u g h they have i n i t i a l l y s h o w n t h e i r interest o n the a u c t i o n e d o b j e c t .  T h i s observation suggests t h a t i t m a y be costly  p a r t i c i p a t i n g the a u c t i o n . M o s t recently, several e m p i r i c a l works have a t t e m p t e d to s t u d y the auct i o n m o d e l s w i t h endogenous entry. A t h e y , L e v i n a n d S e i r a (2004) [4] s t u d y the p a r t i c i p a t i o n p a t t e r n i n the t i m b e r auctions w i t h costly entry.  Kras-  n o k u t s k a y a a n d S e i m (2006) [27] consider two types of b i d d e r s d e c i d i n g to enter the a u c t i o n s b y i n t r o d u c i n g b i d preference p r o g r a m s .  L i and Zheng  (2005) [34] p r o p o s e a s e m i - p a r a m e t r i c B a y e s i a n m e t h o d to j o i n t l y estimate e n t r y a n d b i d d i n g . B a j a r i , H o n g a n d R y a n (2004) [6] use a l i k e h h o o d - b a s e d e s t i m a t i o n a p p r o a c h i n the presence of m u l t i p l e e q u i l i b r i a . A l l of these works e s t i m a t e v a r i a n t s of a n e n t r y m o d e l based o n L e v i n a n d S m i t h (1994) [31] [hereafter, L S ] , i n w h i c h a l l the p o t e n t i a l bidders have no i n f o r m a t i o n on t h e i r v a l u a t i o n of the a u c t i o n e d object w h e n d e c i d i n g w h e t h e r or not to enter.  B y i n c u r r i n g a cost, they t h e n c a n become i n f o r m e d o n the v a l u a -  t i o n s a n d s u b m i t a b i d . T h e e q u i l i b r i u m of the m o d e l p r e d i c t s t h a t a l l the p o t e n t i a l b i d d e r s r a n d o m i z e t h e i r decisions o n e n t r y so t h a t t h e i r expected  payoffs f r o m e n t r y become zero. I n m a n y a p p l i c a t i o n s , fiowever, tfie e q u i l i b r i u m c o n d i t i o n s for L S m o d e l are h a r d t o j u s t i f y . F o r example, consider the h i g h w a y p r o c u r e m e n t auctions i n U S , w h i c h are the a p p l i c a t i o n s s t u d i e d i n those recent e m p i r i c a l w o r k s . T h e D e p a r t m e n t of T r a n s p o r t a t i o n ( D O T ) i n the state u s u a l l y announces the p r o j e c t u n d e r a u c t i o n a few weeks before t h e l e t t i n g date. T h e D O T also describes t h e j o b s a n d t i m e frame involved i n the project.  Though actual  p a r t i c i p a n t s i n the a u c t i o n o b t a i n more i n f o r m a t i o n later, the d e s c r i p t i o n p r o v i d e d at t h i s stage s t i l l contains valuable i n f o r m a t i o n for t y p i c a l p o t e n t i a l bidders, w h o are experienced c o n s t r u c t i o n companies. Therefore, i t does not seem reasonable to assume t h a t p o t e n t i a l bidders have n o i n f o r m a t i o n on the p r o j e c t s u p o n t h e i r decision o n p a r t i c i p a t i o n .  T h i s c h a p t e r considers a n a l t e r n a t i v e e n t r y m o d e l i n a u c t i o n s . O u r m o d e l is based o n a t h e o r e t i c a l work b y Samuelson (1985) [53], i n w h i c h the b i d ders a l r e a d y k n o w t h e i r v a l u a t i o n s w h e n m a k i n g the e n t r y decisions. endogenous e n t r y is due to a cost i n c u r r e d i n b i d s u b m i s s i o n .  The  In equilib-  r i u m , the b i d d e r does not s u b m i t a b i d i f a n d o n l y i f his e x p e c t e d profit f r o m 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 e x p e c t e d profit from p a r t i c i p a t i o n i n the L S m o d e l is d i s c a r d e d i n our framework.  C h a p t e r 1 adopts the same e n t r y m o d e l to s t u d y M i c h i g a n H i g h w a y p r o curement a u c t i o n s .  T h e r e we propose a n o n - p a r a m e t r i c e s t i m a t e for the  e n t r y cost, w i t h w h i c h we further suggest how to i m p l e m e n t o p t i m a l auctions. W i t h i n the n o n p a r a m e t r i c framework, c e r t a i n types of c o u n t e r f a c t u a l e x p e r i m e n t s are i m p o s s i b l e , because we have no way t o n o n p a r a m e t r i c a l l y recover the d i s t r i b u t i o n f u n c t i o n below the cut-off p o i n t .  F o r t h i s reason,  we sometimes e m p l o y a p a r a m e t r i c m o d e l to investigate the consequences of p o l i c y changes i n an a u c t i o n . I n t h i s chapter, we develop a m e t h o d to e s t i m a t e the a u c t i o n m o d e l w i t h endogenous e n t r y based o n a p a r a m e t r i c a l l y specified value d i s t r i b u t i o n . W i t h the e s t i m a t e d p a r a m e t r i c a u c t i o n  m o d e l , we c a n c o n d u c t various counterfactual e x p e r i m e n t s i n v o l v i n g the e n t i r e 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 a s y m m e t r i c o p t i m a l m e c h a n i s m ( C e l i k a n d Y i l a n k a y a (2006) [8]). A challenge i n e s t i m a t i n g a n a u c t i o n m o d e l is c o m p u t a t i o n .  W e need  t o c a l c u l a t e m u l t i - f o l d i n t e g r a l each t i m e we evaluate the o p t i m a l b i d d i n g f u n c t i o n , for e x a m p l e .  A remedy for this t y p e of p r o b l e m s is the use of a  s i m u l a t i o n - b a s e d m e t h o d . I n a s e m i n a l work b y L a f f o n t , O s s a r d a n d V u o n g (1995) [28] [hereafter, L O V ] , a s i m u l a t i o n - b a s e d e s t i m a t o r is p r o p o s e d to estim a t e one of the simplest t h e o r e t i c a l a u c t i o n models, the f i r s t - p r i c e sealed b i d i n d e p e n d e n t p r i v a t e - v a l u e a u c t i o n s ( w i t h o u t e n t r y ) . I n t e r m s of e s t i m a t i o n m e t h o d , t h i s c h a p t e r can be viewed as an extension of L O V .  H o w e v e r , there is an i m p o r t a n t difference between L O V a n d our w o r k . I n L O V ' s m o d e l , t h e d i s t r i b u t i o n of the observed bids is t r u n c a t e d b y a constant ( a n d observed) reserve price, w h i c h does not d e p e n d o n the p a r a m e t e r s to e s t i m a t e . O n t h e c o n t r a r y , t h e t r u n c a t i o n p o i n t of the b i d d i s t r i b u t i o n i n our m o d e l depends o n the parameters i n e s t i m a t i o n , because the e n t r y decision depends o n the p r i v a t e - v a l u e d i s t r i b u t i o n . It t u r n s out t h a t t h i s feature of o u r p r o b l e m c o m p l i c a t e s the large sample analysis of the s i m u l a t e d least squares m e t h o d , m a k i n g the e s t i m a t i o n objective f u n c t i o n n o n - s m o o t h i n the p a r a m e t e r s . F o l l o w i n g P o l l a r d (1985) [51], we derive a set of sufficient c o n d i t i o n s for t h e a s y m p t o t i c properties of our e s t i m a t o r .  Therefore, our  w o r k c o m p l i m e n t s to the l i t e r a t u r e b y showing the a p p l i c a b i l i t y of L O V ' s f r a m e w o r k to t h e context of endogenous entry.  T h e rest o f t h i s chapter is o r g a n i z e d as follows.  W e first describe our  a u c t i o n m o d e l w i t h endogenous e n t r y a n d develop a s t r u c t u r a l econometric m o d e l based o n i t . W e t h e n propose a m e t h o d to e s t i m a t e the econometric m o d e l a n d s t u d y the a s y m p t o t i c properties of the e s t i m a t o r .  I n the last  section we c o n c l u d e the chapter w i t h some discussion o n possible extensions of t h i s w o r k . T h e m a t h e m a t i c a l proofs are collected i n t h e a p p e n d i x .  2.2  Methodology  2.2.1  The  first-price  auction model with entry  W e consider a f i r s t - p r i c e sealed-bid a u c t i o n of a single i n d i v i s i b l e good. H o w e v e r , d e v i a t i n g f r o m the s t a n d a r d I P V f r a m e w o r k , we e x p l i c i t l y m o d e l a n e n t r y stage p r i o r to the b i d d i n g . T h e b i d d i n g process evolves as follows. F i r s t , w i t h i n the independent p r i v a t e - v a l u e ( I P V ) f r a m e w o r k , each p o t e n t i a l risk n e u t r a l p a r t i c i p a n t i € { 1 , 2 , . . . , A''} k n o w s her o w n value Vi for the o b j e c t , b u t o n l y k n o w s the d i s t r i b u t i o n of the values t o the other p o t e n t i a l b i d d e r s . It is assumed t h a t the values to i n d i v i d u a l s are i n d e p e n d e n t l y d r a w n f r o m the a b s o l u t e l y continuous d i s t r i b u t i o n F{v)  ( w i t h its p d f  fy)  w i t h s u p p o r t [v,v] C K + . I n the second stage, there presents a c o m m o n 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 i d d e r has to pay to p a r t i c i p a t e i n the a u c t i o n . G i v e n her p r i vate value, t h e b i d d e r decides w h e t h e r or not to s u b m i t a b i d ( p a y i n g K) a n d becomes a n a c t u a l b i d d e r . A l l p o t e n t i a l bidders m a k e t h i s d e c i s i o n s i m u l t a neously. T h e y therefore make t h e i r p a r t i c i p a t i o n decisions w i t h o u t k n o w i n g how m a n y c o m p e t i t o r s t h e y are a c t u a l l y going to face.  F i n a l l y , b i d d e r s s u b m i t t h e i r bids s i m u l t a n e o u s l y a n d the o b j e c t goes to the highest b i d d e r . T h e w i n n e r pays her b i d to the seller, p r o v i d e d t h a t the b i d is no less t h a n the reserve price r , w h i c h is a s s u m e d to be zero i n this p a p e r w i t h o u t loss of generality. W e focus o n the u n i q u e s y m m e t r i c B a y e s i a n N a s h e q u i l i b r i u m of the auct i o n game, (see Milgrom(2004)[37]).  I n e q u i l i b r i u m , each p o t e n t i a l p a r t i c -  i p a n t j o i n s the a u c t i o n if her value is no less t h a n , Vp, t h e cut-off p o i n t ( c o m m o n to a l l b i d d e r s ) , otherwise chooses not to p a r t i c i p a t e . T h e cut-off p o i n t Vp is such t h a t i t is indifferent to the p a r t i c i p a n t w i t h value Vp w h e t h e r  or not to enter the a u c t i o n . T h u s , Vp s h o u l d solve the e q u a t i o n - K - 0.  VpF{vpf-'  (2.1)  T h r o u g h o u t the chapter, we assume t h a t the e n t r y cost is m o d e r a t e so t h a t Vp is always a n i n t e r i o r p o i n t i n [v,v]. T o derive the e q u i l i b r i u m b i d d i n g strategy for b i d d e r i, define i's exp e c t e d profits ( I i i ) f r o m p a r t i c i p a t i o n as follows, Uiivi^y,  {bj)j^i)  = {vi - y)[F{max{vp,b-\y)))f-^  - AC,  where y is the b i d d e r i ' s b i d given Vi a n d a l l other parameters.  b~^ is the  inverse b i d d i n g s t r a t e g y for bidders. M a x i m i z a t i o n of 11^ 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  A p o t e n t i a l b i d d e r j p a r t i c i p a t e s if a n d o n l y i f Vj >  if  (2.2)  v > Vp  Vp.  A n important  p r o p e r t y is t h a t 6 as a f u n c t i o n of v, is s t r i c t l y i n c r e a s i n g o n [vp,v].  Hence,  the e q u i l i b r i u m is r e v e a l i n g , p r o v i d e d t h a t Vi > Vp. D u e to t h e s t r i c t m o n o t o n i c i t y of b i d d i n g strategy (2.2), the w i n n e r of the a u c t i o n m u s t be t h e b i d d e r w i t h the highest p r i v a t e value, p r o v i d e d t h a t 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^ c a n be w r i t t e n as: 6- = e(«(i),iV,F), where ^(i) denotes the largest order s t a t i s t i c a m o n g vi, ...v^. If  (2.3) < Vp, i t  represents a case t h a t no p a r t i c i p a n t s s u b m i t s a b i d a n d , therefore, w i n n i n g bids are undefined. T h e well-known Revenue Equivalence Theorem implies that b"" = e{v,  N, F)  =  E[max(^;(2), F - i ( p ( A r ) ) ) | i ; ( i ) = v, N, F]  (2.4)  (see, for e x a m p l e , M i l g r o m a n d W e b e r (1982)[38], M i l g r o m (2004)[37]). v^2) denotes t h e s e c o n d largest order s t a t i s t i c a m o n g vi, ...VN, f u n c t i o n of F,  a n d p{N)  c o n d i t i o n a l o n N.  =  F{vp\N)  is the inverse  represents the r a t e of p a r t i c i p a t i o n  A n i n t u i t i v e e x p l a n a t i o n t o e q u a t i o n (2.4) is t h a t , c o n d i -  t i o n a l o n w i n n i n g , the b i d d e r w o u l d j u s t b i d the e x p e c t e d level of the second highest p r i v a t e values. T h e difference between her b i d a n d t h e p r i v a t e value is n o r m a l l y c a l l e d the i n f o r m a t i o n rent for the b i d d e r .  T a k i n g t h e e x p e c t a t i o n w i t h respect to v, (2.4) gives B[b^\N,F]  =  E[max{vç2),F'\piN))}].  T h i s c o n d i t i o n a l e x p e c t a t i o n c a n be v i e w e d as a n i n t e g r a l w i t h respect to the d e n s i t y  E{b'^\N,F)  i n the f o l l o w i n g way:  = j^max{u(2),F-\p{N))}U{ui)...U{uN)dui...duN  (2.5)  where U(2)is the second largest order s t a t i s t i c a m o n g u i , ...ujy- (2.5) provides us a w a y to s i m u l a t e the m o m e n t of w i n n i n g b i d s i n our e s t i m a t i o n m e t h o d .  2.2.2  T h e structural econometric  model  W e now specify a n econometric m o d e l based o n the t h e o r e t i c a l a u c t i o n m o d e l w i t h endogenous e n t r y discussed above. W e here i m p o s e a s i m p l i f y i n g a s s u m p t i o n t h a t the n u m b e r of p o t e n t i a l b i d d e r s n is the same across a l l a u c t i o n s u n d e r s t u d y . W h i l e t h i s a s s u m p t i o n is not essential i n order for the e s t i m a t i o n m e t h o d described below to w o r k , i t allows us to a v o i d excessive c o m p l i c a t i o n i n o u r large sample analysis. A l s o , the c o n s t a n c y of t h e n u m b e r of p o t e n t i a l b i d d e r s is often reasonable i n p r a c t i c e .  F o r e x a m p l e , p r i o r to  the h i g h w a y p r o c u r e m e n t auctions, D O T requires the q u a l i f i c a t i o n check for the c o n s t r u c t i o n firms to be eligible for a n y p a r t i c u l a r types of j o b s at the b e g i n n i n g of each year. T h e n , w h e n l e t t i n g process s t a r t s , o n l y those eligible b i d d e r s c a n get i n v o l v e d . T h i s q u a l i f i c a t i o n stage effectively r e s t r i c t s the set of p o t e n t i a l b i d d e r s for the year.  L e t Vu d e n o t e the value of the a u c t i o n e d object for the i t h p o t e n t i a l b i d d e r i n a u c t i o n t. O u r a u c t i o n m o d e l requires t h a t va be i n d e p e n d e n t l y a n d i d e n t i c a l l y d i s t r i b u t e d across p o t e n t i a l bidders. W e here a d d the independence of Vit across a u c t i o n s . N a m e l y : {vu  Assumption  2.1  dom variables  absolutely  strictly  positive  non-empty  : i =  2,...  continuously  and continuous  subset  ,n, t £ N} distributed  is an i.i.d.  array  with a pdf fviv;ôo)  on [v,v], where OQ lives in O , a  of  ran-  that is compact,  ofW.  T h e w i n n i n g b i d is g i v e n b y (2.4), as l o n g as at least one p o t e n t i a l b i d d e r a c t u a l l y p a r t i c i p a t e s i n the a u c t i o n . However, the w i n n i n g b i d is not defined w h e n the a u c t i o n e d o b j e c t is u n s o l d , t h a t is, w h e n t e c h n i c a l convenience for the d i s c o n t i n u i t y p o i n t at v^iyt  < Vp^t- T o p r o v i d e — '^p,t,  we e x t e n d  (2.4) to M i n the f o l l o w i n g way.  br  =  E [ m a x { t ; ( 2 ) , F - i ( / 5 ) } | T ; ( i ) = v]l{v^,^ > +F-\p)l{v<2)  where l { . }  F-\p)}  < F-\p)},  (2.6)  is n o t a t i o n for the i n d i c a t o r f u n c t i o n w h i c h takes the value of  one i f the l o g i c a l c o n d i t i o n inside i t is satisfied, a n d zero elsewhere.  L e t H{-\n) denote the c u m u l a t i v e d i s t r i b u t i o n of 6^ c o n d i t i o n a l o n n for n > 1. It satisfies t h a t  ^ ^ '  1 -  F(up)"  I n t h i s e q u a l i t y , H{-\n) c a n be e s t i m a t e d b y u s i n g the observations of w i n n i n g b i d s for each n > 1, a n d F{vp),  w h i c h is the p r o b a b i l i t y of p a r t i c i p a t i o n i n  the a u c t i o n , c a n be e s t i m a t e d i f the n u m b e r of a c t u a l b i d d e r s is observed for each a u c t i o n a l o n g w i t h the n u m b e r of p o t e n t i a l p a r t i c i p a n t s .  W h e n e s t i m a t i n g a p a r a m e t r i c p r o b a b i l i t y m o d e l like ours, the m a x i m u m l i k e l i h o o d ( M L ) e s t i m a t i o n m e t h o d is u s u a l l y t a k e n as the first choice, as the  M L m e t h o d provides a more efficient estimates t h a n a n y other alternatives. U n f o r t u n a t e l y , M L e s t i m a t i o n involves t w o m a j o r difficulties i n a u c t i o n a p p l i c a t i o n s , w h i c h makes i t h a r d to i m p l e m e n t . F i r s t , i t has been shown t h a t the s u p p o r t of t h e b i d s ' d i s t r i b u t i o n depends o n t h e p a r a m e t e r 9, w h i c h violates t h e r e g u l a r i t y assumptions u n d e r l y i n g t h e f a m i l i a r nice properties of M L e s t i m a t i o n . D o n a l d a n d P a a r s c h (1993) [14] address t h e issue i n detail.  I n p a r t i c u l a r , they show t h a t t h e M L e s t i m a t o r h a s a n o n s t a n d a r d  asymptotic distribution. T h e s e c o n d difficulty, w h i c h is not l i m i t e d t o t h e M L m e t h o d , lies i n c o m p u t a t i o n . I n general, the e q u i l i b r i u m b i d d i n g f u n c t i o n (2.2) is h i g h l y n o n l i n ear i n p r i v a t e values. E v a l u a t i o n of the b i d d i n g f u n c t i o n (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-  t u r e makes i t n e a r l y i m p r a c t i c a l to estimate t h e m o d e l b y t h e M L m e t h o d or a n y other m e t h o d s t h a t require exact e v a l u a t i o n o f t h e inverse b i d d i n g function.  T o overcome t h e above-mentioned  problems,  we propose u s i n g L O V ' s  s i m u l a t i o n - b a s e d e s t i m a t i o n m e t h o d , w h i c h does n o t require t h e exact c o m p u t a t i o n o f either t h e e q u i l i b r i u m b i d d i n g strategy (2.2) o r t h e m o m e n t s 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 s q u a r e s e s t i m a t o r  W e n o w describe o u r s i m u l a t e d nonlinear least squares e s t i m a t i o n m e t h o d , w h i c h is based o n t h e first m o m e n t of w i n n i n g bids. expression o b t a i n e d b y r e p l a c i n g fy w i t h fy{-,0)  L e t l{9) denote the  i n t h e r i g h t - h a n d side of  (2.5). T h e u s u a l N L L S estimator s h o u l d m i n i m i z e s a n o b j e c t i v e f u n c t i o n QNLLS  ^  ( l / T ) ^ ^ ^ i [ 6 J " - l{9)]^ w i t h respect t o 9.  G i v e n t h a t l{9) is  difficult t o evaluate, we replace l{9) w i t h a n unbiased s i m u l a t o r X{9), i.e., a s i m u l a t o r such t h a t E [ X ( e ) ] = l{9). T h e r e are several ways to simulate the c o n d i t i o n a l m e a n o f w i n n i n g bids. F o l l o w i n g L O V , we use i m p o r t a n c e s a m p l i n g t o s i m u l a t e t h e integrals i n  (2.5).  A s n o t e d i n L O V , t h i s s i m u l a t i o n technique c a n reduce t h e s a m -  p l i n g v a r i a b i h t y o f t h e r a n d o m draws a n d hence i m p r o v e the p r e c i s i o n of simulation-based estimators. L e t 3 be a k n o w n continuous d e n s i t y w i t h a s u p p o r t V g at least c o n t a i n i n g the s u p p o r t of fviv). 1(6)  =  T h e n , (2.5) c a n be r e w r i t t e n as f max{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 d r a w S independent samples, each o f size n . T h e n , for every t, lt{d) c a n be e s t i m a t e d b y t h e sample m e a n :  X(«",...,«^*,^) =  i5^X(««*,^),  where u * ' (s = 1 , 2 , . . . , S ) is a n i.i.d. n x 1 r a n d o m vectors, whose c o m p o nents are i n d e p e n d e n t l y a n d i d e n t i c a l l y d i s t r i b u t e d w i t h d e n s i t y g, a n d Xiu-,e)  =  maxK^),F-^(pO,,)]/-K^g--/^^^  (2.7)  T w o features r e g a r d i n g t h e s i m u l a t i o n are w o r t h n o t i n g . F i r s t , E [ X ( u ^ ' , u^^,6)] = l{9) b y t h e c o n s t r u c t i o n .  Second, a l l t h e r a n d o m d r a w s u are  i n d e p e n d e n t of 6 a n d are d r a w n before e s t i m a t i o n . T h u s , for a n y given 6, the v a r i a b l e s X ( u * ' , ^ ) , (therefore, X ( u ^ * , ...u^*,6)), are i n d e p e n d e n t of bf.  N o w , w e are r e a d y t o define o u r s i m u l a t e d n o n l i n e a r least square e s t i m a t o r , 6, w h i c h s h o u l d m i n i m i z e t h e objective f u n c t i o n , for a n y fixed n u m b e r  of s i m u l a t i o n s S:  1  ^  - s { s ^ )  _  ^  ~  (  ^  -  ^  ^  T h e second t e r m o n the r i g h t h a n d side of (2.8) is t h e bias c o r r e c t i o n t e r m . W e show i n t h e next section t h a t i n c o r p o r a t i n g such a s i m p l e a d j u s t m e n t t e r m enables QT u n i f o r m l y converges to Q{0) 2.2.4  = E[6J" - 1(0)]^ a.s.  A s y m p t o t i c P r o p e r t y of 6  T h e r e exists a significant difference between o u r objective f u n c t i o n  (2.8)  a n d the one used i n L O V . D u e to the p a r t i c u l a r t r u n c a t i o n feature i n d u c e d b y the e n t r y cost i n o u r e n t r y m o d e l , 6 appears i n the " m a x " o p e r a t o r i n (2.7). H e n c e , our objective f u n c t i o n defined i n (2.8) is not twice c o n t i n u o u s l y differentiable i n 6.  Therefore, a separate i n v e s t i g a t i o n o n the a s y m p t o t i c  p r o p e r t i e s of 6 is needed.  I n this subsection, we derive a set of sufficient  c o n d i t i o n s t h a t ensures 6 is b o t h consistent a n d a s y m p t o t i c a l l y n o r m a l l y distributed.  T o ease the e x p o s i t i o n , we use the following n o t a t i o n s . /(, is the density f u n c t i o n of the d i s t r i b u t i o n for observed b i d s whose s u p p o r t is d e n o t e d as V b . |{.|| denotes the E u c l i d e a n n o r m . M o r e o v e r ,  1  ^  —  W e o m i t the s u b s c r i p t a n d superscript i n the proofs whenever there is no confusion.  Consistency W e assume the following for the the desired consistency property. (Consistency:)  A s s u m p t i o n 2.2 (i) ^0 uniquely  minimizes  Q over 9 .  (ii) 7^ = sup0çe,^Glt!,ïï] (iii)  < oo.  g = sup„gVg 5(w) < oo.  (iv) g = i n f „ e v g 9{u) > 0. (v) S is a fixed natural As  number no less than 2 and independent  oft.  Q is c o n t i n u o u s o n the c o m p a c t p a r a m e t e r set 6 , Q a t t a i n s i t s m i n -  i m u m o n O . A s s u m p t i o n 2.2(i) requires there be o n l y one m i n i m i z e r . A s s u m p t i o n 2.2(ii) states t h a t  is b o u n d e d .  T h i s a s s u m p t i o n is generally  a s s u m e d i n t h e a u c t i o n setup, as otherwise the b i d d i n g s t r a t e g y (2.2) w i l l not b e w e l l - d e f i n e d . A s s u m p t i o n 2.2(hi)-(iv) is not r e s t r i c t i v e i n t h e sense t h a t g is chosen b y the researcher. T h e r e s t r i c t i o n c a n h o l d w i t h o u t m u c h difficulty i f g is specified i n most of t h e c o m m o n d i s t r i b u t i o n f a m i l y . T h e last i t e m i n a s s u m p t i o n 2.2 reinforces t h a t o u r a s y m p t o t i c a n a l y s i s is for a fixed n u m b e r o f s i m u l a t i o n s as T —» oo.  T h e f o l l o w i n g l e m m a shows t h a t o u r sample objective  f u n c t i o n QT is  u n i f o r m l y consistent t o 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 t o prove t h e consistency result. L e m m a 2.1 (Uniform  Consistency:)  Given  Assumptions  2.1,  2.2(ii)-(v),  supegelQT(^)-Q(^)HO. T h e f o l l o w i n g p r o p o s i t i o n states the consistency p r o p e r t y of 6. P r o p o s i t i o n 2.2 estimator  Given Assumptions  of OQ as T ^ OO.  2.1 and 2.2, 9 is a strongly  consistent  Asymptotic  Normality  T o derive t h e a s y m p t o t i c n o r m a l i t y result, we first i n t r o d u c e t h e f o l l o w i n g notation.  L e t /io = V i ( ^ o ) , (TQ = Var[6]. L e t Xso = X{u^,6o)  a n d Yso =  VgXiu^Oo). W e f u r t h e r i m p o s e t h e following a s s u m p t i o n for t h e a s y m p t o t i c d i s t r i b u tion. A s s u m p t i o n 2.3 (Asymptotic (i) ^0 is an interior  Normality:)  point of Q.  (ii) { w i , . . . , u „ } are a random that is (iii)  with density g  continuous.  m{0^e,ve[v,v]  (iv) fv{v,.)  sample from the distribution  f{v,0)  =  > 0.  : Q i-^R is Lipschitz  (v) For all v e [v,v], fy{v,.) and the Henssian  matrix  uniformly  :O  in v e  [v,v].  R is twice continuously  V'^fviv,6)  satisfies  that  differentiable  supy^^^g^Q  \\v^fy{v,e)\\<oo. (vi) For all v e Henssian  [v,v],  matrix  F{v;.)  V'^F{v,6)  : © i—» R i s twice differentiable satisfies  and the  that sup„g[j,_^] ^ I ^ Q | | V ^ F ( i i , ^)|| <  oo. (vii) For all v £ [v,v] and O&Q, VeF{v,e) (viii) C = /XO/ZQ is  ^ 0.  nonsingular.  A s s u m p t i o n 2.3(iii) asks fy t o be b o u n d e d away f r o m zero. I n t h e auct i o n s e t t i n g , i t is u s u a l l y assumed so. O t h e r w i s e t h e b i d d i n g s t r a t e g y (2.2) is n o t s t r i c t l y i n c r e a s i n g , w h i c h further hurts b o t h t h e uniqueness of b i d d i n g e q u i l i b r i u m a n d t h e i d e n t i f i c a t i o n of s t r u c t u r a l a n a l y s i s . A s s u m p t i o n s 2.3(iv)-(vii) i m p o s e c o n d i t i o n s o n fy a n d F over t h e p a r a m e t e r set 0 . These  r e s t r i c t i o n s c a n be satisfied i f fy is d i o s e n f r o m ttie c o m m o n l y used d i s t r i b u t i o n s . E s p e c i a l l y , i n most of e m p i r i c a l a u c t i o n w o r k s t h a t use p a r a m e t r i c approaches, t h e p r i v a t e values are u s u a l l y specified to follow n o r m a l or logn o r m a l d i s t r i b u t i o n s . T h e n these a s s u m p t i o n s h o l d . Before i n t r o d u c i n g the a s y m p t o t i c d i s t r i b u t i o n r e s u l t , we present the following lemma. L e m m a 2.3 second  Given  Assumptions  2.1, 2.2, and 2.3, Xgo and Yso have  finite  moments.  Together w i t h A s s u m p t i o n 2 . 3 ( v i i i ) , the above l e m m a ensures t h a t the covariance m a t r i x i n the p r o p o s i t i o n 2.4 is well-defined. W e now present the following p r o p o s i t i o n , w h i c h states the a s y m p t o t i c n o r m a l i t y for 6. P r o p o s i t i o n 2.4 asymptotically  Given  distributed  Assumptions with N{0,ïp)  2.1, 2.2 and 2.3, T,~'^/'^VT(6-Oo) as T  oo, where E =  is  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 p r o o f for P r o p o s i t i o n 2.4 makes a c o n n e c t i o n i n the l i t e r a t u r e . P o l l a r d (1985) [51] relaxes the d i f f e r e n t i a b i l i t y a s s u m p t i o n s o n the s t a t i s t i c a l o b j e c t i v e f u n c t i o n . T h i s effectively breaks d o w n the a s y m p t o t i c n o r m a l i t y for the e x t r e m e e s t i m a t o r s under w e l l - k n o w n proofs. H e describes some new techniques t o prove the c e n t r a l l i m i t theorems.  W e benefit f r o m his w o r k  b y u s i n g t h e t h e o r e m 2 i n his p a p e r for the d e s i r e d a s y m p t o t i c r e s u l t . H o w ever, w h e n c h e c k i n g the stochastic d i f f e r e n t i a b i l i t y c o n d i t i o n r e q u i r e d b y his t h e o r e m , t w o m e t h o d s suggested i n his w o r k , b r a c k e t i n g a n d c o m b i n a t o r i a l m e t h o d s are h a r d to a p p l y i n our context, if at a l l . T o a v o i d t h i s diffic u l t y , we i n s t e a d e m p l o y a n o t h e r a p p r o a c h m e n t i o n e d i n P a k e s a n d P o l l a r d (1989) [46], b y v e r i f y i n g the E u c l i d e a n p r o p e r t y of a class of f u n c t i o n s .  In t e r m s o f efficiency, i t is possible t o c o m p a r e o u r S N L L S e s t i m a t o r t o the u s u a l N L L S e s t i m a t o r , w h i c h has t o c o m p u t e l{8).  T h e asymptotic  variance-covariance m a t r i x of N L L S estimator w o u l d b e C ~ M C " ^ A = (To/ioA'o-  where  T h e r e f o r e , o u r S N L L S e s t i m a t o r is less efficient t h a n N L L S  e s t i m a t o r b y a t e r m of order (l/S).  T h i s efficiency loss is due t o the s i m u l a -  t i o n i n v o l v e d i n o u r e s t i m a t i o n . If, however, the n u m b e r of s i m u l a t i o n draws S can b e increased t o infinity, t h e n our S N L L S e s t i m a t o r 6 becomes a s y m p t o t i c a l l y efficient as the u s u a l N L L S estimator. M o r e o v e r , as w e ignore t h e a u c t i o n heterogeneity i n analysis, t h e N L L S e s t i m a t o r is a s y m p t o t i c a l l y efficient i n t h e class of e s t i m a t o r s t h a t uses m o m e n t r e s t r i c t i o n Efè*"] = IÇOQ).  For the p u r p o s e of c o n d u c t i n g inferences o n ^o, i t is a n essential question o n h o w t o c o n s i s t e n t l y estimate t h e 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 t h e variances a n d covariances of Xgo a n d YSQ. A l l the v a r i a n c e a n d covariance terms can be easily e s t i m a t e d , b u t /XQA s /xo = V / ( ^ o ) , i t cannot be e x p l i c i t l y d e t e r m i n e d for the same reason as l{6o) c a n n o t . W e follow L O V ' s p r o p o s a l t o s i m u l a t e /UQ t h r o u g h Ygo, w h i c h is also consistent for a fixed n u m b e r of s i m u l a t i o n s as T goes t o infinity. W e define _  _  e = Y{e)Y{êy  1  ^  _  - j ^ ^ ^ ^ ( n w - Y{e)){Ys{e) -  _  Y{ê)y  N o t e t h a t t h e w a y we construct C is e x a c t l y same as w e define QT- It t h e n follows t h a t C7 —» C for any fixed S. Denote A = A A ' , where _  _   = {(6 - xiê))Y{ê)  1  + gj^rrY)  ^  _  -  ^(oWsiê)]}-  T h e n , for a n y fixed S, our e s t i m a t o r S : = C ~ ^ A C ~ ^ is consistent for S , as shown i n L O V .  2.3  Conclusion and extensions  A s e m i n a l w o r k b y L O V provides a new p a r a m e t r i c e s t i m a t i o n strategy for a n a l y z i n g a u c t i o n d a t a . T h e y propose a s i m u l a t e d N L L S e s t i m a t o r to a p p r o x i m a t e the bidders p r i v a t e - v a l u e d i s t r i b u t i o n s .  T h e i r work greatly  b r o a d e n the class of d i s t r i b u t i o n s t h a t e m p i r i c a l researchers c a n h a n d l e w h e n a n a l y z i n g a u c t i o n d a t a sets. T h i s c h a p t e r c a n be v i e w e d as a n extension to L O V ' s w o r k . W e a p p l y t h e i r m e t h o d to a 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 endogenous entry. T h e p a r t i c u l a r e n t r y p a t t e r n i n such models makes L O V ' s objective f u n c t i o n no longer twice c o n t i n u o u s l y differentiable. T h e loss of smoothness on the s t a t i s t i c a l objective d e m a n d s another i n v e s t i g a t i o n o n the a s y m p t o t i c properties for L O V ' s e s t i m a t o r s . W e derive a set of sufficient c o n d i t i o n s for the consistency a n d a s y m p t o t i c n o r m a l i t y to r e m a i n v a l i d . T h e r e are several lines a l o n g w h i c h this w o r k c a n be e x t e n d e d . F i r s t , i n some a p p l i c a t i o n s , the econometricians c a n 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 i d d i n g , they surely are i n f o r m a t i v e o n the p r i v a t e - v a l u e d i s t r i b u t i o n as w e l l .  However,  w i t h i n t h e f r a m e w o r k of endogenous entry, i n c l u d i n g these bids i n the estim a t i o n w i l l i n t r o d u c e other c o m p l i c a t i o n i n a s y m p t o t i c a n a l y s i s . Because the b i d s are no longer c o m p l e t e l y i n d e p e n d e n t l y d i s t r i b u t e d , the development of the p r o p o s e d e s t i m a t i o n s t r a t e g y for other types of a u c t i o n s seems appealing. T h i s c h a p t e r assumes the a u c t i o n s are homogeneous. I n e m p i r i c a l w o r k , t h i s a s s u m p t i o n c a n t y p i c a l l y be j u s t i f i e d b y c o n t r o l l i n g the a u c t i o n heterogeneity t h r o u g h one r a n d o m v a r i a b l e . T h e n a l l the observed bids c a n be n o r m a l i z e d so t h a t a l l a u c t i o n s are t a k e n as homogeneous. H o w e v e r , i n most of t h e a p p l i c a t i o n s , f i n d i n g such a o n e - d i m e n s i o n a l v a r i a b l e to h o m o g enize t h e a u c t i o n s m a y be a h a r d m i s s i o n , if possible at a l l . T h u s t a k i n g care of b o t h t h e observed a n d unobserved a u c t i o n heterogeneity i n e m p i r i c a l  analysis c a n be a n i n t e r e s t i n g d i r e c t i o n to pursue i n the f u t u r e . If t h e a u c t i o n heterogeneity is e x p h c i t l y considered, t h e e s t i m a t i o n m e t h o d of N L L S m a y not be a s y m p t o t i c a l l y efficient i n general. O n e possible w a y to i m p r o v e efficiency, as p r o p o s e d i n C h a m b e r l a i n (1987) [9], is to c o n s t r u c t a n e s t i m a t o r e x p l o i t i n g the o r t h o g o n a l i t y c o n d i t i o n s b y E [ W [ 6 - i(^o))] = 0. W c a n be a vector of auction-specific variables. S i m i l a r l y , P a k e s a n d P o l l a r d (1989)[46] consider the m e t h o d of s i m u l a t e d m o m e n t s i n t h e i r a p p l i c a t i o n s . H o w e v e r , how one c a n carefully choose W to g a i n efficiency is a n u n s e t t l e d question.  T h e p r o p o s e d e s t i m a t i o n m e t h o d involves s i m u l a t i n g t h e m o m e n t s of the observed b i d s . It is a v i t a l role t h a t the n u m b e r of r a n d o m d r a w s S plays i n the s i m u l a t i o n . Specifically, S appears i n the variance-covariance m a t r i x . T h e r e f o r e , i t is not a t r i v i a l question how the choice of S is g o i n g to affect the e s t i m a t i o n a c c u r a c y i n the finite sample. W e leave t h i s for the future research.  Chapter 3  W h a t Model for Entry in First-Price Auctions?  A  Nonparametric Approach 3.1  Introduction  Sealed tenders are a w i d e s p r e a d m e c h a n i s m for p r o c u r i n g goods a n d services i n the U n i t e d States.  T h i s is a large a n d i m p o r t a n t m a r k e t , a n d  u n d e r s t a n d i n g its w o r k i n g s is a t o p i c of general interest. well-documented  A robust and  feature of m a n y r e a l - w o r l d auctions is t h a t not a l l b i d -  ders w h o are eligible to s u b m i t a b i d choose to do so, suggesting t h a t e n t r y i n t o t h e a u c t i o n m a y be costly. I n t h i s c h a p t e r , we develop n o n p a r a m e t r i c approaches t h a t w i l l a l l o w the e m p i r i c a l researcher t o d i s c r i m i n a t e a m o n g different m o d e l s of entry. M o s t of the e m p i r i c a l auctions l i t e r a t u r e to date is based o n the t h e o r e t i c a l w o r k of L e v i n a n d S m i t h (1994) [31] ( L S hereafter). I n t h e i r m o d e l , p o t e n t i a l b i d d e r s are i n i t i a l l y u n i n f o r m e d a b o u t t h e i r v a l u a t i o n s of the g o o d , b u t m a y become i n f o r m e d a n d s u b m i t a b i d at a cost. I n e q u i l i b r i u m , t h e p o t e n t i a l e n t r a n t s r a n d o m i z e t h e i r e n t r y decisions a n d e a r n zero e x p e c t e d profit. Several e m p i r i c a l papers, most of t h e m recent, e s t i m a t e d v a r i a n t s of t h i s model.  B a j a r i a n d H o r t a c s u (2003) [5] have s t u d i e d e n t r y a n d b i d d i n g i n  e B a y a u c t i o n s , w i t h i n a c o m m o n value framework.  A Bayesian estimation  m e t h o d is i m p l e m e n t e d u s i n g a dataset of m i n t a n d p r o o f sets of U S coins. T h e m a g n i t u d e of the e n t r y cost is e s t i m a t e d , a n d e x p e c t e d seller revenues  axe s i m u l a t e d u n d e r different reserve prices. A t h e y , L e v i n a n d S e i r a (2004) [4] estimate a m o d e l of b i d d i n g i n t i m b e r auctions w i t h costly entry. T h e e n t r y cost is a s s u m e d t o be p r i v a t e i n f o r m a t i o n of the p o t e n t i a l b i d d e r s , w h o sort i n t o the p o o l of e n t r a n t s based o n t h e i r draws of the e n t r y cost. L i a n d Z h e n g (2005) [34] s t u d y e n t r y a n d b i d d i n g for l a w n m o w i n g c o n t r a c t s u s i n g the L S m o d e l .  T o our knowledge, it is t h e first p a p e r i n the  l i t e r a t u r e t h a t u t i l i z e s the n u m b e r of p l a n h o l d e r s as a measure of p o t e n t i a l c o m p e t i t i o n i n h i g h w a y p r o c u r e m e n t . L i a n d Z h e n g (2005) [34] propose a n d i m p l e m e n t a B a y e s i a n e s t i m a t i o n m e t h o d a n d use t h e i r s t r u c t u r a l estimates t o investigate t h e effect of r e s t r i c t i n g p o t e n t i a l c o m p e t i t i o n on the e x p e c t e d revenue. I n a d d i t i o n , L i (2005) [32] develops a general p a r a m e t r i c a p p r o a c h for a u c t i o n s w i t h entry. K r a s n o k u t s k a y a a n d S e i m (2006) [27] s t u d y b i d preference p r o g r a m s a n d b i d d e r p a r t i c i p a t i o n u s i n g C a l i f o r n i a d a t a . T h e i r p a p e r also uses the L S m o d e l , a n d as i n A t h e y , L e v i n a n d S e i r a (2004) [4], the focus is o n a s y m m e t r i c e q u i l i b r i a . B a j a r i , H o n g a n d R y a n (2004) [6] propose a p a r a m e t r i c l i k e l i h o o d - b a s e d e s t i m a t i o n strategy i n the presence of m u l t i ple e q u i l i b r i a , a n d a p p l y it to highway p r o c u r e m e n t a u c t i o n s , u s i n g the L S model.  A n a l t e r n a t i v e m o d e l of e n t r y was developed i n S a m u e l s o n (1985) [53] (S hereafter). I n t h i s m o d e l , bidders make t h e i r e n t r y decisions after t h e y have learned their valuations.  T h e entry cost is i n t e r p r e t e d solely as the cost  of p r e p a r i n g a b i d , a n d bidders choose to enter if t h e i r v a l u a t i o n s exceed a c e r t a i n cutoff. T h e set of entrants is therefore a selected s a m p l e , biased t o w a r d s b i d d e r s w i t h higher v a l u a t i o n s .  T o the best knowledge of ours,  chapter 1 of t h i s thesis is the o n l y w o r k a p p l y i n g S a m u e l s o n m o d e l to d a t a so far. B o t h L S a n d S models are s t y l i z e d to c a p t u r e the a m o u n t of i n f o r m a t i o n available t o b i d d e r s at the e n t r y stage: no i n f o r m a t i o n is available i n L S , w h i l e the i n f o r m a t i o n is perfect i n S. These p o l a r a s s u m p t i o n s lead to d r a s t i c a l l y differing p o l i c y i m p l i c a t i o n s .  O n e of the m o s t i m p o r t a n t a n d  w e l l - s t u d i e d p o l i c y i n s t r u m e n t s i n auctions is the reserve price. I n a s e m i n a l p a p e r , R i l e y a n d S a m u e l s o n (1981) [52] show t h a t , w h e n the e n t r y costs are n u l l , t h e o p t i m a l p o l i c y for the seller is to set the reserve price above the level t h a t he w o u l d be w i l l i n g to accept. M o r e o v e r , the o p t i m a l reserve price does not d e p e n d o n the n u m b e r of p o t e n t i a l bidders N . I n the S m o d e l , w h i l e the o p t i m a l reserve price is also above the seller's willingness to accept, it increases  w i t h A''. L S , o n the other h a n d , reach a s t r i k i n g conclusion t h a t  it is o p t i m a l to set the reserve price at the m a x i m a l willingness to accept level.  G i v e n t h a t p o l i c y i m p l i c a t i o n s are so different, i t is i m p o r t a n t t o be able to d i s c r i m i n a t e between these models e m p i r i c a l l y . W e b u i l d o n the insight i n H a i l e , H o n g a n d S h u m (2003) [19] ( H S S hereafter) exogenous v a r i a t i o n i n A/' as a basis for such a test.  a n d propose to use L e t F{v)  denote the  c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n ( C D F ) of v a l u a t i o n s , a n d let F*{v\N)  de-  note t h e C D F for those p o t e n t i a l bidders t h a t have s u b m i t t e d a b i d .  The  C D F F*{v\N)  is a c r u c i a l p a r a m e t e r whose b e h a v i o r across N allows us to  d i s c r i m i n a t e a m o n g the a l t e r n a t i v e m o d e l s of entry. F o l l o w i n g the a p p r o a c h of G u e r r e , P e r r i g n e a n d V u o n g (2000)[17] ( G P V hereafter), we show t h a t t h i s d i s t r i b u t i o n c a n be n o n p a r a m e t r i c a l l y identified i n b o t h m o d e l s i f the n u m b e r of p o t e n t i a l bidders a n d a l l bids i n each a u c t i o n are observed. W e show t h a t , w h i l e F*(i;|7V) does not d e p e n d o n TV i n the L S m o d e l , it does i n the S m o d e l . T h e i n t u i t i o n here is s i m p l y t h a t , i n the S m o d e l , the v a l u a t i o n s of active b i d d e r s are t r u n c a t e d by the e n t r y cutoffs v*{N)  that depend  o n TV, b u t a l l share the same parent d i s t r i b u t i o n across TV. T h i s imposes a r e s t r i c t i o n o n F*{v\N)  across TV. It is not too h a r d to show t h a t this  r e s t r i c t i o n i m p l i e s a stochastic d o m i n a n c e o r d e r i n g for  F*iv\N)  > F*{v\N')  for  N' > TV.  F*{v\N):  (3.1)  I n other w o r d s , as TV becomes larger, the d i s t r i b u t i o n b e c o m e more t i l t e d t o w a r d s b i d d e r s w i t h higher valuations. T h i s is of course a n i n t u i t i v e i m p l i c a t i o n of selective entry. It is also t r i v i a l l y satisfied by the L S m o d e l , w i t h  e q u a l i t y signs for a l l N. I n t h i s c h a p t e r , we also propose a generalized m o d e l t h a t allows for select i v e e n t r y b u t dispenses w i t h the s t a r k a s s u m p t i o n t h a t p o t e n t i a l bidders perfectly k n o w t h e i r v a l u a t i o n s at the entry stage as i n S, t h u s s h a r i n g w i t h the L S m o d e l a c o s t l y v a l u a t i o n discovery stage.  It f o r m a l l y nests the L S  m o d e l . T h i s m o d e l , w h i c h we refer to as a n affiliated m o d e l of e n t r y ( A M E hereafter), is as follows. A t the e n t r y stage, the p o t e n t i a l b i d d e r s each observe a p r i v a t e s i g n a l correlated w i t h t h e i r yet u n k n o w n v a l u a t i o n of the g o o d . B a s e d o n t h i s p r i v a t e s i g n a l , a bidder m a y l e a r n t h e v a l u a t i o n u p o n i n c u r r i n g a n e n t r y cost k. T h e b i d d e r w h o entered w i l l o n l y b i d if the v a l u a t i o n exceeds the reserve price.  T h e signals m a y be i n f o r m a t i v e a b o u t  the v a l u a t i o n s , however unlike i n the S m o d e l , t h e y are not p e r f e c t l y informative.  B o t h L S a n d S models c a n be viewed as its l i m i t cases:  the L S  m o d e l corresponds to u n i n f o r m a t i v e signals, w h i l e the S m o d e l corresponds to p e r f e c t l y i n f o r m a t i v e signals.  M o d e l s s i m i l a r t o A M E have been looked at i n the l i t e r a t u r e . H e n d r i c k s , P i n k s e a n d P o r t e r (2003) [21] estimate a m o d e l of b i d d i n g for off-shore o i l . T h e y sketch a m o d e l of e n t r y t h a t is i n some respects s i m i l a r to ours, b u t w i t h a c o m m o n - v a l u e component.  T h e focus of t h e i r p a p e r is however not  o n e n t r y b u t o n t e s t i n g a n e q u i l i b r i u m m o d e l of b i d d i n g . T h e m o d e l is also o u t l i n e d i n the c o n c l u d i n g section of Y e (2007) [62]. T o i m p l e m e n t the tests, we follow the a p p r o a c h of G P V a n d show t h a t t h e d i s t r i b u t i o n of e n t r a n t s ' v a l u a t i o n s can be n o n p a r a m e t r i c a l l y identified f r o m the d a t a i f N a n d a l l bids i n each a u c t i o n are observed. T h i s enables us to develop a n o n p a r a m e t r i c quantile-based test of selective e n t r y i n the s p i r 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 o u r a p p r o a c h shares w i t h H a i l e , H o n g a n d S h u m (2003) [19] the basic i d e a t h a t exogenous v a r i a t i o n i n the n u m b e r of b i d d e r s c a n be used for t e s t i n g the i n f o r m a t i o n e n v i r o n m e n t of the game, there is a n u m b e r of  i m p o r t a n t differences. H a i l e , H o n g a n d S h u m (2003) [19] consider a different m o d e l i n w h i c h b i d d e r s ' valuations m a y have a c o m m o n c o m p o n e n t .  They  propose a test for c o m m o n values based o n the v a r i a t i o n i n the n u m b e r of actual  b i d d e r s , w h i l e we test for selective e n t r y u s i n g the v a r i a t i o n i n the  n u m b e r of potential  bidders.  O u r a p p r o a c h is also different i n the i m p l e m e n t a t i o n i n t h a t we use a direct quantile estimation method.  T h e m e t h o d is easy to i m p l e m e n t , does not  require the c o m p u t a t i o n of pseudo values of G P V , a n d also allows a r b i t r a r y f o r m of dependence o n covaxiates. T h i s last feature is p a r t i c u l a r l y i m p o r t a n t since the m e t h o d of covariate c o n t r o l i n H a i l e , H o n g a n d S h u m (2003) [19] is not a p p l i c a b l e i n the s e t t i n g w i t h e n t r y considered i n t h i s chapter.  W e m a k e a n u m b e r of observations about the i d e n t i f i c a t i o n of m o d e l p r i m itives i n the L S a n d S models.  A s t a n d a r d reference for i d e n t i f i c a t i o n i n  auctions is A t h e y a n d H a i l e (2002)[2].  However, t h e y do not address i d e n -  t i f i c a t i o n i n m o d e l s w i t h endogenous entry. These observations are discussed i n S e c t i o n 3.3. I n p a r t i c u l a r , i f the reserve price is b i n d i n g a n d there is no v a r i a t i o n i n the n u m b e r of p o t e n t i a l bidders, the e n t r y cost i n the L S m o d e l is not i d e n t i f i e d . T h e reason is t h a t d a t a allow a n equivalent i n t e r p r e t a t i o n as b e i n g generated i n a m o d e l w i t h zero e n t r y cost, a n d n o n p a r t i c i p a t i o n is s i m p l y e x p l a i n e d b y the fact t h a t some b i d d e r s d r a w v a l u a t i o n s below the reserve p r i c e . N o t e t h a t t h i s e x p l a n a t i o n was o r i g i n a l l y p u t f o r w a r d i n P a a r s c h (1997) [44]. If there is v a r i a t i o n i n the n u m b e r of p o t e n t i a l bidders, t h e n t h e e n t r y cost m a y be identified. W e observe t h a t a sufficient c o n d i t i o n for i d e n t i f i c a t i o n is t h a t the p a t t e r n of the p r o b a b i l i t y 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 c e r t a i n t h a t bidders enter w i t h p r o b a b i l i t y 1 a n d n o n p a r t i c i p a t i o n is due to the t r u n c a t i n g effect of the reserve price ^See also A t h e y a n d H a i l e (2005)[3].  only, a n d therefore are able to identify F{r)  = 1 -p{N).  O n the decreasing  segment, we are c e r t a i n t h a t bidders are indifferent between e n t e r i n g or not, a n d are able to i d e n t i f y the e n t r y cost f r o m the indifference c o n d i t i o n given the knowledge of  F{r).  H o w e v e r , t h e e s t i m a t e of the e n t r y cost m a y be sensitive t o m o d e l m i s s p e c i f i c a t i o n . W e show t h a t , i f t h e d a t a are generated a c c o r d i n g to a m o d e l w i t h selective e n t r y (either S or our m o d e l ) , b u t the researcher uses the L S m o d e l , t h e e s t i m a t e d e n t r y cost w i l l be u p w a r d biased. M o r e o v e r , we show b y the w a y of a n e x a m p l e t h a t the bias m a y be severe. T h e i n t u i t i o n for t h i s result is t h e f o l l o w i n g . W h e n the e n t r y cost is e s t i m a t e d i n the L S m o d e l , it is a s s u m e d t h a t each p o t e n t i a l entrant is indifferent between entering or not, so t h a t t h e e n t r y cost is equal to the e x p e c t e d profit of a b i d d e r w h o w i l l d r a w , so to speak, a n average v a l u a t i o n u p o n entry. I n the models w i t h selectivity, the S a n d A M E , the entry cost is equal to the e x p e c t e d profit of a m a r g i n a l b i d d e r . W h e n the signals are p o s i t i v e l y c o r r e l a t e d w i t h v a l u a t i o n s , the v a l u a t i o n t h a t a m a r g i n a l b i d d e r w i l l d r a w m a y p l a u s i b l y be less t h a n the average v a l u a t i o n .  I n o u r e m p i r i c a l a p p l i c a t i o n , we use a dataset of a u c t i o n s c o n d u c t e d  by  the O k l a h o m a D e p a r t m e n t of T r a n s p o r t a t i o n ( O D O T ) . I n a d d i t i o n to a l l w i n n i n g a n d l o s i n g bids a n d c e r t a i n p r o j e c t characteristics, we also observe the n u m b e r of firms t h a t o b t a i n e d c o n s t r u c t i o n p l a n s , a v a r i a b l e t h a t c a n serve as a reasonable p r o x y for the n u m b e r of p o t e n t i a l b i d d e r s . W e argue t h a t , because t h e q u a l i f i c a t i o n process essentially selects b i d d e r s based on w o r k i n g c a p i t a l requirements, the n u m b e r of p l a n h o l d e r s m a y be assumed to be exogenous. T h e e m p i r i c a l results are somewhat m i x e d , b u t we do have a n u m b e r of findings. F i r s t , the S m o d e l is r o b u s t l y rejected. Second, there is some s u p p o r t for the L S m o d e l , b u t somewhat more s u p p o r t 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 m o d e l s share a c o m m o n s t r u c t u r e . T h e r e is a n e n t r y stage i n w h i c h N p o t e n t i a l bidders contemplate entry into the a u c t i o n . A t the a u c t i o n stage, a b i d d i n g g a m e transpires a m o n g those bidders t h a t have entered. T h e a u c t i o n is f i r s t - p r i c e sealed b i d , p o s s i b l y w i t h a reserve p r i c e r.  O n l y the  bidders w i t h v a l u a t i o n s above t h e reserve price a c t u a l l y s u b m i t bids. W e c a l l t h e m actual b i d d e r s . W e assume t h e Independent P r i v a t e V a l u e s ( I P V ) e n v i r o n m e n t . T h e b i d d e r s ' v a l u a t i o n s are d i s t r i b u t e d a c c o r d i n g to t h e C D F F{-)  t h a t h a s s u p p o r t [v,v], a c o r r e s p o n d i n g d e n s i t y /(•) p o s i t i v e o n the  s u p p o r t . E n t r y is costly; o n l y t h e bidders t h a t have i n c u r r e d t h e e n t r y cost k can b i d i n the auction.  T h e t w o m o d e l s differ i n the i n f o r m a t i o n available at t h e e n t r y stage. T h e L S m o d e l assumes t h a t no i n f o r m a t i o n is available. U p o n i n c u r r i n g the e n t r y cost, t h e b i d d e r s l e a r n t h e i r v a l u a t i o n s a n d proceed t o t h e b i d d i n g stage. O n l y t h e e n t r a n t s w i t h v > r s u b m i t a b i d . L e v i n a n d S m i t h characterize a s y m m e t r i c p e r f e c t - B a y e s i a n e q u i l i b r i u m o f this game i n w h i c h bidders s u b m i t a b i d w i t h p r o b a b i l i t y p G [0,1]. f u n c t i o n o f N i s d e n o t e d as p{N).  T h e e q u i l i b r i u m value of p as a  T h e e q u i l i b r i u m is c h a r a c t e r i z e d i n t h e  f o l l o w i n g p r o p o s i t i o n . W e assume t h a t t h e reserve price is b i n d i n g , b u t t h e result carries over w i t h m i n o r changes t o t h e case w h e n i t is n o t b i n d i n g .  Proposition 3.1 A symmetric  (Levin  equilibrium  bidp and bidding  and Smith, is characterized  strategy B{v).  1994[31j;  Milgrom,  by the probability  The ex-ante equilibrium  2004[37])  of submitting  profit from  a  bidding  is equal to  n ( p , N) - j\l  - F{v)){l  - p + pF*{v)f~^dv.  (3.2)  The equilibrium  distribution F*{v)  of active bidders valuations = {{Fiv)  - F(r))/(1 -  and does not depend on N.  Denote the equilibrium  of N asp{N).  > 0, then p{N)  p{N)  = 0.  profit  IfU{l,N) Otherwise  € (0,1)  p{N)  is given  by  F(r))) probability  p as a  = 1, and ifn{0,N)  is determined  from  function  < k,  then  the zero expected  equation U{p{N),N)  (3.3)  = 0.  T h e r e is a q u a h f i c a t i o n to be added to the above p r o p o s i t i o n , as w e l l as to s i m i l a r results for other models.  T h r o u g h o u t the c h a p t e r , we  assume  away the u n i n t e r e s t i n g case of the entry cost so large t h a t there is no entry, p{N)  = 0.  T h e e q u i l i b r i u m b i d d i n g strategy B{v)  is e x p l i c i t l y derived i n  L S . T h e L S m o d e l has the following i m p l i c a t i o n s (we w i l l show later i n this chapter t h a t these i m p l i c a t i o n s are testable). F i r s t , since t h e profit f u n c t i o n i n (3.2)  is decreasing i n the r i v a l b i d d i n g p r o b a b i l i t y p as w e l l as i n the  n u m b e r of p o t e n t i a l rivals N, we c a n see t h a t the e q u i l i b r i u m p r o b a b 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')  (3.4)  y N <N',  w i t h s t r i c t i n e q u a l i t y if N' is sufficiently large. Second, the d i s t r i b u t i o n of e n t r a n t s v a l u a t i o n s coincides w i t h the d i s t r i b u t i o n of p o t e n t i a l b i d d e r s v a l u a t i o n s . T h e C D F of v a l u a t i o n s c o n d i t i o n a l o n e n t r y F*{v) [r, v] a n d is i n d e p e n d e n t of  has t h e s u p p o r t  N,  F*{v\N)  = F*{v\N')  V N,N'.  (3.5)  I n the S m o d e l , the p o t e n t i a l bidders k n o w their v a l u a t i o n s a l r e a d y at the e n t r y stage.  I n any s y m m e t r i c e q u i l i b r i u m , a b i d d e r w h o s e v a l u a t i o n  is at the lower e n d of the s u p p o r t , i» = u , is unable to w i n w i t h a positive p r o b a b i l i t y , a n d w i l l not enter. Samuelson shows t h a t there is a cutoff such t h a t a b i d d e r s t r i c t l y prefers to enter i f a n d o n l y if v > v*{N),  v*{N) so t h a t  the e q u i U b r i u m p r o b a b i h t y of entry isp(7V) = l-F{v*{N)). v*{N)  N o t e t h a t , since  > r , i n the S m o d e l this is the same as the p r o b a b i l i t y 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 f o r m a l l y characterized i n the following p r o p o s i t i o n . P r o p o s i t i o n 3.2 symmetric  (Samuelson,  equilibrium,  and continuous  p{N)  =  1 -  determined expected  F{v*{N))  bidding stage has a  in which the bidding strategy B(v)  function.  entrant with valuation  1985[53])The  The profit at the bidding  is an  increasing  stage of the  is given by {v*{N) - r ) ( l - p{N))^~^,  v*{N)  is the probability  by the requirement  of bidding.  unique  marginal where  The cutoff v*{N)  that bidder with valuation  v* (N)  is  makes zero  profit: k = {v*{N) - r ) ( l - p{N))^-\  There is always  entry with probability  less than 1, i.e.  (3.6) e {r,v  v*{N)  T h e S m o d e l shares w i t h the L S m o d e l r e s t r i c t i o n (3.4)  — k).  that bidding  p r o b a b i l i t i e s are n o n - i n c r e a s i n g (they m u s t a c t u a l l y be s t r i c t l y decreasing i n the S m o d e l ) . B u t it i m p l i e s a different r e s t r i c t i o n for the d i s t r i b u t i o n of active b i d d e r s v a l u a t i o n s F*{v\N).  For v >  1 -  v*{N),  F{v*{N))  Fiv)-{l-p{N))  (3.7)  piN) where we u s e d the fact t h a t the entry p r o b a b i l i t y p{N) F{v*{N)).  is e q u a l to 1  -  S i n c e the d i s t r i b u t i o n F does not d e p e n d o n A'', a m a n i p u l a t i o n  of (3.7) leads to the f o l l o w i n g r e s t r i c t i o n of the S m o d e l : p{N)F*iv\N) 3.2.2  + 1 - p{N)  = piN')F*{v\N')  T h e affiliated m o d e l of e n t r y  + 1 - p{N')  V N, N'.  (3.8)  (AME)  A m o d e l of selective e n t r y proposed i n t h i s chapter o c c u p i e s a m i d d l e g r o u n d between  S a n d L S . Specifically, i t shares w i t h S t h e a s s u m p t i o n  t h a t i n f o r m a t i o n a b o u t the v a l u a t i o n is available at the b i d d i n g stage, b u t  dispenses w i t h t h e s t a r k a s s u m p t i o n t h a t this i n f o r m a t i o n is perfect. T h e game begins w i t h t h e e n t r y stage i n w h i c h TV p o t e n t i a l r i s k - n e u t r a l bidders o b t a i n p r e l i m i n a r y estimates (signals) Si of their t r u e values Vi; i t is assumed t h a t t h i s i n f o r m a t i o n is available to t h e m for free.  U p o n o b s e r v i n g Si, a  b i d d e r m a y e x p e n d a n e n t r y cost k, w h i c h results i n o b s e r v i n g t h e true value Vi a n d entering t h e a u c t i o n . O n l y the bidders t h a t have learned Vi are eligible t o s u b m i t a b i d i n t h e a u c t i o n . M o r e o v e r , o n l y those w i t h valuations at o r above t h e reserve price r s u b m i t a b i d .  W e assume t h a t t h e pairs {Vi,Si)  are i d e n t i c a l l y a n d i n d e p e n d e n t l y d i s -  t r i b u t e d across p o t e n t i a l bidders i = 1, ...,7V a n d are d r a w n f r o m d i s t r i b u t i o n F{v, s) w i t h s u p p o r t [v,v] x [0,1] a n d density f{v, s). F o r convenience, we assume t h a t t h e m a r g i n a l d i s t r i b u t i o n of the signals is u n i f o r m o n [0,1]. Since t h e i n f o r m a t i o n a l content of signals is preserved under a m o n o t o n e t r a n s f o r m a t i o n , t h i s a s s u m p t i o n is w i t h o u t loss of generality.  T h e e n t r y stage is followed b y the b i d d i n g stage. A c t i v e bidders d r a w their values Vi, a n d t h e n s i m u l t a n e o u s l y a n d i n d e p e n d e n t l y s u b m i t sealed bids. A c t i v e b i d d e r s d o n o t k n o w t h e n u m b e r of active bidders, o n l y t h e n u m b e r of p o t e n t i a l b i d d e r s TV. T h e g o o d is awarded to the highest b i d d e r w h o pays its b i d . W e assume t h a t t h e signals are i n f o r m a t i v e a n d t h a t higher signals are " g o o d n e w s " .  F o r m a l l y , we assume affiliation, i n t h e sense of M i l g r o m  a n d W e b e r (1982) [38]. Assumption  3.1 For each bidder i, the variables  any z — {v,s)  and z' = {v',s'),  {Vi,Si)  are affiliated:  / ( m a x { 2 , ^ ; ' } ) / ( m i n { z , z'}) =  for  f{z)f{z').  N o t e t h a t b o t h t h e L S a n d S models c a n be viewed as l i m i t cases of the A M E . T h e L S m o d e l is f o r m a l l y nested since i t corresponds to signals b e i n g i n d e p e n d e n t of the valuations; t h i s w o u l d effectively p u r i f y the m i x e d s t r a t e g y e q u i l i b r i u m . T h e S m o d e l corresponds to the other e x t r e m e , n a m e l y the signals a n d v a l u a t i o n s b e i n g perfectly correlated. B u t because we assume existence o f a j o i n t density of valuations a n d signals f{v, s), t h e S m o d e l is  not nested. ^ A s y m m e t r i c e q u i l i b r i u m of the A M E m o d e l c a n be c h a r a c t e r i z e d i n a m a n n e r s i m i l a r to the L S m o d e l .  T h i s is done i n the p r o p o s i t i o n below,  whose p r o o f is i n the A p p e n d i x (the p r o o f also contains a f o r m u l a for the b i d d i n g s t r a t e g y ) . O n c e a g a i n , we assume t h a t the reserve p r i c e is b i n d i n g , b u t t h e result carries over w i t h m i n o r changes to the case w h e n i t is not binding. P r o p o s i t i o n 3 . 3 A symmetric  equilibrium  off s such that only those potential equilibrium  probability  is characterized  bidders with Si>s  of submitting  by a signal cut-  choose to enter.  a bid is (3.9)  P{s)^Pr{Si>s,V^>r}, and the distribution  of active bidders valuations F*{v\s) = Pr{Vi  is  < v\Si >s,Vi>  r}.  For any bidder i with signal Si = s, the equilibrium  n{s,s,N)  = j\l  - F{v\s)){l  - P{s)  profit from is equal to  + P{s)F*{v\s)f-^dv  / / n ( 0 , 0 , TV) > 0, then s = 0, and all potential  bidders  n ( 0 , 0 , TV) < 0, then s = 1 and there is no entry.  Otherwise,  signal s is indifferent  between entering n{s,s,N)  Define  the cutoffs  a function  >  - k.  always  (3.10)  enter.  If  the bidder with  or not so that s is determined  from (3.11)  = 0.  of N as s{N).  (0,1) also for N' > TV, and s(N')  The  Ifs(N)  G ( 0 , 1 ) , then s(TV') G  s{N).  * A n i n t e r e s t i n g question t h a t we d o not address is whether the e q u i l i b r i u m of the S m o d e l c a n be s u p p o r t e d as a l i m i t point of o u r class of models.  If the reserve price is n o n - b i n d i n g , i.e.  r < v, some m o d i f i c a t i o n s  are  necessary. I n the A p p e n d i x , we show t h a t the e n t r y e q u a t i o n (3.11) becomes  = (1 - Pis))^-\v  Hi Jv  -r)+  - Fiv\s)){l - P{s)  +  Pis)F*{v\s)f-^dv, (3.12)  where the presence of the first t e r m reflects the fact t h a t the b i d d e r w i t h t y p e V makes profit by b i d d i n g the reserve price r a n d w i n n i n g the a u c t i o n o n l y w h e n no one else enters. T h e b i d d i n g strategy also needs t o be modified 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 i d d i n g p(TV) =  P{s{N))  is also a non-decreasing f u n c t i o n of TV. T h i s is the r e s t r i c t i o n (3.4) we have seen before. i n t h i s chapter.  It is shared w i t h other models of e n t r y  considered  B u t the r e s t r i c t i o n o n active b i d d e r s ' C D F F*(z;|TV) is  different f r o m either L S or S. T o derive t h i s c o n d i t i o n , n o t e t h a t is e q u a l to Pr{Vi Pr{Vi  that  < v\Si > s(TV), Vi > r}.  < v\Si > s,Vi  F*{v\N)  T h e a s s u m p t i o n 3.1 i m p l i e s t h a t  > r} is non-decreasing i n s ( T h e o r e m 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 v a l u a t i o n s : F*{v\N)  > F*iv\N')  V TV < AT'  (3.13)  N o t e t h a t t h i s r e s t r i c t i o n is i m p l i e d by restrictions (3.5) of the L S m o d e l as w e l l as r e s t r i c t i o n (3.8) of the S m o d e l , b u t is clearly weaker.  3.3  Nonparametric identification  I n order t o be able to test the restrictions derived i n the p r e v i o u s section w i t h o u t m a k i n g p a r a m e t r i c assumptions, it is necessary t o show t h a t the r e q u i r e d q u a n t i t i e s are n o n p a r a m e t r i c a l l y identified. T h i s section is devoted  to i d e n t i f i c a t i o n of the m o d e l s of e n t r y considered i n t h i s chapter. It is a s s u m e d t h a t the econometrician can observe a l l the bids a n d therefore also t h e n u m b e r of active bidders n. A n i m p o r t a n t a d d i t i o n a l i n f o r m a t i o n t h a t is a s s u m e d to be also available is the n u m b e r of p o t e n t i a l bidders I n other w o r d s , we assume t h a t the d a t a generating process identifies  N. p{N)  a n d G*{-\N) where p(N)  = E[n\N]/N  is the p r o b a b i l i t y of s u b m i t t i n g  a b i d a n d G*{b\N) is the d i s t r i b u t i o n of e n t r a n t s ' b i d s c o n d i t i o n a l o n  N.  W e now show t h a t i n a l l m o d e l s considered i n t h i s chapter, the d i s t r i b u t i o n F*{v\N)  c a n be recovered from the first-order c o n d i t i o n s .  O u r identifica-  t i o n s t r a t e g y follows G P V . C o n s i d e r first-order e q u i l i b r i u m c o n d i t i o n s of the b i d d i n g game. A b i d d e r w i t h value v w h o s u b m i t s a b i d b has a p r o b a b i l i t y of w i n n i n g over a given r i v a l equal to 1 - p{N)  + p{N)G*{b\N).  Since there  are i V — 1 i d e n t i c a l rivals, it follows by independence t h a t the p r o b a b i l i t y of w i n n i n g is (1 - p{N) n(b,v)  a n d the e x p e c t e d profit is  +p{N)G*{b\N))^-'^, =  ib-v){l~piN)+p{N)G*{b\N)f-\  W r i t i n g out the first-order c o n d i t i o n , i.e.  t a k i n g the d e r i v a t i v e of Û{b, v)  w i t h respect to b a n d s e t t i n g i t equal to 0, gives the inverse b i d d i n g strategy  mN)  . . = b+  l-piN)+p{N)G*{b\N)  .  T h e inverse b i d d i n g strategy ^(•|A'^) is identified f r o m t h e a n d i t s inverse, the b i d d i n g strategy B{v\N),  is also i d e n t i f i e d .  d i s t r i b u t i o n of active b i d d e r s ' v a l u a t i o n s F*{v\N) F*{v\N)  =  (3.14)  observables, T h e n the  is identified a c c o r d i n g to  G*{B{v)\N).  It is i n t e r e s t i n g to note t h a t , i f there is no v a r i a t i o n i n A^, the L S a n d A M E m o d e l s are o b s e r v a t i o n a l l y equivalent.  T h i s is because, even i f the t r u e  d a t a g e n e r a t i n g process corresponds to the A M E m o d e l , the d i s t r i b u t i o n F*{v\N)  c a n be i n t e r p r e t e d as the d i s t r i b u t i o n of v a l u a t i o n s Vi c o n d i t i o n a l  o n V i > r t h a t w o u l d arise i f the true m o d e l w a s L S . If t h e reserve price is n o n - b i n d i n g , r < v, t h e n t h e S m o d e l is also o b s e r v a t i o n a l l y equivalent. If, on t h e o t h e r h a n d , t h e reserve price is b i n d i n g , r e {v,v), t h e n t h e S m o d e l is n o t o b s e r v a t i o n a l l y equivalent since t h e lower b o u n d of t h e s u p p o r t of i d e n t i f i e d as ^{b\N), must be greater t h a n r i n t h e S m o d e l , b u t is  F*{-\N),  equal t o r i n b o t h L S a n d A M E models.  C o n t i n u i n g t o assume t h a t N is fixed, a further i n t e r e s t i n g q u e s t i o n i f the m o d e l p r i m i t i v e s are identified. I n t h e A M E m o d e l , t h e p r i m i t i v e s are t h e entry cost k a n d t h e d i s t r i b u t i o n F{v\s).  N e i t h e r is n o n p a r a m e t r i c a l l y i d e n -  tified. T h e reason is t h a t t h e d a t a generating process o n l y reveals t h e d i s t r i b u t i o n o f b i d d e r s ' v a l u a t i o n s , i.e. F*{v\N) b u t n o t F{v\s).  = Pr{Vi  < v\Si > s{N), Vi > r},  T h e knowledge of F{v\s) w o u l d also be needed t o identify  the e n t r y cost a c c o r d i n g t o (3.11).  T h e L S m o d e l is fully identified i f r < u a n d p{N)  6 (0,1). T h e distri-  b u t i o n F*{v) is e q u a l t o t h e d i s t r i b u t i o n of v a l u a t i o n s o f p o t e n t i a l bidders, a n d since Pr{Si indifference  > s} = p{N) e ( 0 , 1 ) , t h e entry cost is i d e n t i f i e d from t h e  condition  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, t h e n we c a n o n l y c o n c l u d e t h a t t h e reserve price is b o u n d e d f r o m above b y t h e expected profit t h a t appears i n t h e r i g h t - h a n d side of (3.15). S i m i l a r l y , i f the reserve price is b i n d i n g , r G {v,v),  the entry  cost is n o t i d e n t i f i e d . T h i s is because n o w k p{N)  =  {l-Fir))J\l-F\v)){l-piN)+p{N)F*iv)f-'di^.l6)  =  {l-F{r))Pr{Si  = s{N)},  a n d t h e m o d e l is o b s e r v a t i o n a l l y equivalent to t h e one w i t h Pf{Si  > s{N)}  of d r a w i n g  (3.17) = 0 so t h a t  — 1 a n d the p r o b a b i l i t y of b i d d i n g is e q u a l t o the p r o b a b i l i t y  Vi>r.  I n t h e S m o d e l , the d i s t r i b u t i o n of p o t e n t i a l b i d d e r s ' v a l u a t i o n s is t r u n cated (at v*{N)  > v) even if r < v. Since the p r o b a b i l i t y of b i d d i n g is now  equal t o 1 - F{v*{N)), c a n i d e n t i f y F{v),  a n d v*{N)  is identified as ^{b\N), i t follows t h a t we  b u t o n l y for v > v*{N).  O n the other h a n d , the entry  cost is i d e n t i f i e d , k = {v*{N) - r ) ( l - p{N))^-\  (3.18)  N o t e t h a t for the identification of the S m o d e l , whether or not the reserve price is b i n d i n g plays no role. N o w assume t h a t the n u m b e r of p o t e n t i a l bidders N G N,N_ + I, where T£ < N.  I n other words, there is v a r i a t i o n i n N.  ...jN  It is easy to show  t h a t t h e v a r i a t i o n i n N does not lead to the i d e n t i f i c a t i o n of the p r i m i t i v e s of the A M E m o d e l .  I n the L S m o d e l , t h i s v a r i a t i o n c a n sometimes (but  not always) lead t o i d e n t i f i c a t i o n of the entry cost even i f the reserve price is b i n d i n g . bidding  C o n s i d e r the following general p a t t e r n for the p r o b a b i l i t i e s of  p{N): p{N)  where  < M.  =  ...^piN.)<...<p(N)  W e allow for b o t h Nt = N a n d A^, = N.  I n the A p p e n d i x ,  we prove t h e following p r o p o s i t i o n . P r o p o s i t i o n 3.4 k is identified  In the LS model with a binding  if and only if  G {N_ + 1 , N  T h e i n t u i t i o n for t h i s result is as follows.  reserve price,  r G  {v,v),  — 1}. W h e n TV belongs t o the fiat  segment. A'" < 7V», we are c e r t a i n t h a t bidders enter w i t h p r o b a b i l i t y 1 a n d n o n p a r t i c i p a t i o n is due to the t r u n c a t i n g 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 c e r t a i n t h a t b i d d e r s are indifferent between e n t e r i n g or not, a n d are able to identify the e n t r y cost f r o m the indifference c o n d i t i o n given the knowledge of F{r),  a c c o r d i n g t o (3.16).  A l s o note t h a t i n the S m o d e l , the o n l y i m p l i c a t i o n of the v a r i a t i o n i n N is t h a t we c a n identify the d i s t r i b u t i o n F ( - ) for a l l v =  v*{N).  O u r final r e m a r k i n t h i s section concerns the bias i n t h e e s t i m a t i o n of the e n t r y cost.  S u p p o s e t h a t the reserve price is n o n - b i n d i n g a n d p{N)  <  1,  so t h a t the e n t r y cost is identified i n the L S m o d e l . H o w e v e r , the d a t a are generated a c c o r d i n g to A M E , w i t h s t r i c t affiliation i n the sense t h a t F{v\s) < F{v\s'),  V U G [V,V],S'  (3.19)  > s.  F o r s i m p l i c i t y , assume t h a t the n u m b e r of p o t e n t i a l b i d d e r s N is fixed (this is not c r u c i a l ) , a n d t h a t the researcher estimates the e n t r y cost w r o n g l y a s s u m i n g t h a t the d a t a are generated according to the L S m o d e l , i.e. if the cost was d e t e r m i n e d by equation (3.15).  T h e difference  c o r r e c t l y specified m o d e l is t h a t the researcher uses the w r o n g 1 — F*{v)  i n s t e a d of the correct one 1 — F{v\s{N)),  f r o m the expression  because t h e t r u e cost is  g i v e n b y (3.12). B u t w i t h the s t r i c t affiliation a s s u m p t i o n (3.19), F*{v\N) F{v\s{N)),  as  >  so t h a t i n the case of misspecification, t r u e k is s m a l l e r t h a n the  one p r o d u c e d b y (3.15).  T h e i n t u i t i o n for the existence of t h i s bias is as follows.  T h e e n t r y cost  i n the L S m o d e l is equal to the e q u i l i b r i u m e x p e c t e d profit of the average p o t e n t i a l b i d d e r , w h i l e i t is equal to the e x p e c t e d profit of the m a r g i n a l p o t e n t i a l b i d d e r ( w i t h s i g n a l s{N))  i n our m o d e l .  B e c a u s e the signals are  p o s i t i v e l y r e l a t e d t o the v a l u a t i o n s , the m a r g i n a l b i d d e r m a y p l a u s i b l y have a n e n t r y cost o n average smaller t h a n the average b i d d e r , so the e n t r y cost w o u l d be  overestimated.  T h e same bias is present also w h e n the d a t a are generated a c c o r d i n g to the S m o d e l , so t h a t the t r u e e n t r y cost is given b y e q u a t i o n (3.18). F r o m (3.15), the e s t i m a t e d e n t r y cost i n the misspecified L S m o d e l is greater t h a n the t r u e k. T h e following e x a m p l e helps i l l u s t r a t e t h a t the bias m a y be very severe. E x a m p l e . Suppose t h a t the valuations are u n i f o r m l y d i s t r i b u t e d o n [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  m o d e l is S. T h e cutoff v*{N)  is d e t e r m i n e d b y (3.18), a n d v*(N)  T h u s t h e d i s t r i b u t i o n o f active b i d d e r s ' v a l u a t i o n s F*{v\N) u n i f o r m [0,1], w i t h a t r u n c a t i o n p o i n t given b y v*{N). of s u b m i t t i n g a b i d is p{N)  = 1 — v*{N).  = k^l^.  is a t r u n c a t e d T h e probability  T h e researcher misspecifies the  m o d e l as L S a n d estimates the entry cost a c c o r d i n g t o (3.15), s u b s t i t u t i n g F4v\N)  = {v - v*{N))/{l - v*{N))  a n d v = v''{N).  A f t e r e v a l u a t i n g the  i n t e g r a l i n (3.15), one obtains o n the right h a n d side  ~^l-v*iN)\  N  N+1  J'  i n s t e a d o f k, w h e r e t h e second s u m m a n d is t h e bias t e r m . Observe t h a t the bias c a n be s u b s t a n t i a l w h e n TV is s m a l l , even i f t r u e cost is negligible. Since limk-^QV*{N)  3.4  = 0, t h e bias becomes 1/(TV(TV + 1)) w h e n A; - » 0.  Econometric  implementation  I n w h a t follows, we allow for auctions heterogeneity b y i n t r o d u c i n g the vector o f a u c t i o n specific covariates x. W e assume n o w t h a t the d i s t r i b u t i o n of v a l u a t i o n s c a n change from a u c t i o n t o a u c t i o n d e p e n d i n g o n the value of X a n d is d e n o t e d b y F{v\x).  S i m i l a r l y , t h e d i s t r i b u t i o n of v a l u a t i o n s  c o n d i t i o n a l o n e n t r y is n o w denoted as F*{v\N,x), s u b m i t t i n g a b i d as p{N,x).  a n d t h e p r o b a b i l i t y of  T h e m o d e l selection is also c o n d i t i o n a l o n x,  i.e. different m o d e l s m a y be true for different values of x.  3.4.1  Hypotheses  T h e p r e v i o u s section shows t h a t the d i s t r i b u t i o n s of v a l u a t i o n s c o n d i t i o n a l o n b i d d i n g are identified for t h e three a l t e r n a t i v e models, a n d therefore i n p r i n c i p l e , m o d e l selection tests c a n be f o r m u l a t e d i n t e r m s oî F*{v\N,x)  or  equivalently, i n t e r m s of quantiles of t h i s d i s t r i b u t i o n , as suggested i n H a i l e , H o n g a n d S h u m (2003) [19]. Define Q*(r|TV,a;) = F * - i ( r | T V , x )  t o b e the r - t h q u a n t i l e o f the d i s t r i b u t i o n o f e n t r a n t s ' v a l u a t i o n s .  Assume  t h a t N varies between the lower b o u n d N a n d the u p p e r b o u n d TV. I n terms of the q u a n t i l e s , the testable r e s t r i c t i o n o f the L S m o d e l is HLS--Q*iT\N,x)  = ... = Q*{T\N,x),  V r e [0,1],  w h i l e t h e A M E m o d e l i m p l i e s the r e s t r i c t i o n HAME  : Q*{T\N,X)  =<  ... < Q*{T\N,X),  V T G [0,1].  (3.20)  T h e testable r e s t r i c t i o n (3.8) of the Samuelson m o d e l c a n also be expressed using the quantiles function  Q*{-\N,x)  as follows.  F i r s t , b y the d e f i n i t i o n  a n d since F has a c o m p a c t s u p p o r t , for any r € [0,1], F{Q{T\X)\X) N e x t , for those quantiles o f F ( - ) t h a t c o r r e s p o n d cutoffs  v*{N),  i.e. for r > 1  -p{N,x),  = r.  t o v a l u a t i o n s above the  e q u a t i o n (3.7)  implies  w h i c h in t u r n implies that  Define a f u n c t i o n air N x ) (^ir,N,x)-  - - i ' - ^^^^^^ P i ^ ^ ^ ) )  T h e q u a n t i l e s i n the l e f t - h a n d side o f (3.21) d o not d e p e n d o n TV because they correspond  t o t h e d i s t r i b u t i o n of potential  bidders' valuations, and  we t h e n have t h a t Q*(û:(r, TV,x)|TV, 2;) m u s t b e constant across TV's for a l l T > 1 -p(TV,x): Hs  : Q * ( a ( T , i V , a ; ) | T V , x ) = ... = Q*(a(r,TV,a;)|TV,a;),  V r > 1 -p(TV,i).  T h e r e s t r i c t i o n i n Hs is l i m i t e d t o a p a r t i c u l a r range o f r ' s .  A similar  r e s t r i c t i o n , however w i t h r G [0,1], c a n b e o b t a i n e d f r o m (3.8) directly.  Define a f u n c t i o n  N o t e t h a t , since p(N,x)  < p{N,x),0  < 1 for a l l r G [0,1], a n d  < /3{T,N,X)  therefore c a n be i n t e r p r e t e d as a l e g i t i m a t e t r a n s f o r m a t i o n of the q u a n t i l e order r . ^ T h e c o n d i t i o n i n (3.8) i m p h e s t h a t for a l l TV, F*{v\N, x) = PiF*iv\N,  x), TV, x),  (3.22)  a n d , b y the same a r g u m e n t as before, we o b t a i n the f o l l o w i n g r e s t r i c t i o n i n terms of t h e t r a n s f o r m e d 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 p r a c t i c a l p o i n t of v i e w , t e s t i n g Hs is s i m i l a r to t e s t i n g H'g; h o w ever, the last one does not require t r u n c a t i o n of r ' s . T h e r e f o r e , we focus o n l y o n H'g. N o t e also t h a t because /3(r, TV, x) is decreasing i n TV, t h e r e s t r i c t i o n s under Hs  a n d H'g are consistent w i t h the r e s t r i c t i o n of A M E (3.20) o n the  quantiles Q*{T\N,X)  w i t h o u t the t r a n s f o r m a t i o n (3, b u t are stronger.  I n t h i s c h a p t e r , we consider independent t e s t i n g of His,  H A M E , a n d H'g  against t h e i r c o r r e s p o n d i n g u n r e s t r i c t e d a l t e r n a t i v e s . I n a d d i t i o n , we also consider t e s t i n g w h e t h e r the e n t r y p r o b a b i l i t i e s p(TV, x) are n o n - i n c r e a s i n g i n TV. T h e n u l l h y p o t h e s i s , for a g i v e n value of x, is Hp:l>  p{N,  x)>...>  p{N,  x) > 0,  a n d i t is also tested against its c o r r e s p o n d i n g u n r e s t r i c t e d a l t e r n a t i v e .  T h e last test is of independent interest.  T h e fact t h a t the e q u i l i b r i u m  p r o b a b i l i t i e s of s u b m i t t i n g a b i d decline i n the n u m b e r of p o t e n t i a l b i d d e r s ^ W h i l e any other fixed value of N c a n be used i n the place of  i n t h e definition of 0,  the choice N = N ensures t h a t (3 takes o n values i n the zero-one interval.  is p r o b a b l y a c o m m o n feature o f m a n y other models of entry. W h e n e v e r a m o d e l w i t h c o s t l y e n t r y is brought to e x p l a i n w h y some p o t e n t i a l bidders do n o t b i d , one m u s t confront a n a l t e r n a t i v e e x p l a n a t i o n . N a m e l y , following P a a r s c h (1997) [44], even i f there is no e n t r y cost, n o n - p a r t i c i p a t i o n m a y s t i l l b e e x p l a i n e d b y t h e fact t h a t some bidders d r a w t h e i r v a l u a t i o n s below the reserve price. B u t i n t h a t case, t h e p r o b a b i l i t y o f b i d d i n g is e q u a l t o PriVi  < r}  a n d therefore does n o t depend o n t h e n u m b e r of p o t e n t i a l  b i d d e r s . V i e w e d t h i s way, t h e above hypothesis Hp is a testable r e s t r i c t i o n of c o s t l y versus costless entry.  3.4.2  T h e data generating  process  W e assume t h a t a sample of L auctions is available, a n d i n d e x a u c t i o n s b y / = 1,.., L. E a c h a u c t i o n is characterized b y t h e vector of covariates x ; € X. W e assume t h a t t h e covaxiates xi are d r a w n i n d e p e n d e n t l y for each a u c t i o n f r o m 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 o n x / , t h e n u m b e r of p o t e n t i a l b i d d e r s , Ni, is d r a w n i n d e p e n d e n t l y f r o m the d i s t r i b u t i o n 7r(7V|x/); it is a s s u m e d t h a t 7r(-|x;) has s u p p o r t A/" = {N,N}}°  T h e e n t r y cost k{x)  is a s s u m e d t o b e a d e t e r m i n i s t i c f u n c t i o n of x . T h e r e is a b i n d i n g reserve price r / a n d i t is observable.  C o n d i t i o n a l o n x ; = x a n d TV; = TV, the v a l u a t i o n s Vu of p o t e n t i a l bidders i = 1 , T V ; are d r a w n i n d e p e n d e n t l y f r o m a d i s t r i b u t i o n w i t h d e n s i t y / ( - [ x ) t h a t does n o t d e p e n d o n TV. T h e s u p p o r t of Vu is [v{x),v{x)], 1,...,TV;. T h e s e v a l u a t i o n s are unobservable.  where i =  T h e central to our approach  is t h e a s s u m p t i o n t h a t t h e n u m b e r of p o t e n t i a l bidders TV is exogenous c o n d i t i o n a l o n x ; = x . T h i s a s s u m p t i o n allows us to use t h e v a r i a t i o n i n TV for t h e p u r p o s e of t e s t i n g . I n Section 3.6, we e x p l a i n w h y t h i s a s s u m p t i o n is p l a u s i b l e i n t h e context o f our e m p i r i c a l a p p l i c a t i o n . A s s u m p t i o n 3.2 Vu and Ni are independent  conditional  onxi.  ^°FoT s i m p l i c i t y , we assume that the s u p p o r t does not d e p e n d o n x, b u t t h e results continue t o h o l d even w i t h o u t this a s s u m p t i o n .  T h e b i d bu c o r r e s p o n d i n g to the v a l u a t i o n Vu is generated a c c o r d i n g to the b i d d i n g s t r a t e g y bu =  B{Vu\Ni,xi).  T h e decisions t o s u b m i t a b i d , yu G { 0 , 1 } , are generated a c c o r d i n g to the cutoff strategy yu = 1  if  Su > s{Ni,xi)  and  Vu > ri,  where the signals are u n i f o r m l y d i s t r i b u t e d . Su ~  t^[0,1].  The bidding  strategy B a n d t h e cutoff f u n c t i o n s depend o n the m o d e l ' s p r i m i t i v e s / a n d k t h r o u g h the e q u i h b r i u m conditions of each m o d e l ; s is available i n closed f o r m . ni = J Z i ^ i yu-  neither B  nor  T h e n u m b e r of a c t u a l b i d d e r s is given b y  N o t e t h a t ni is p o s i t i v e l y correlated w i t h v a l u a t i o n s if the  m o d e l is not L S . O u r f o r m a l a s s u m p t i o n s guarantee t h a t the d i s t r i b u t i o n of b i d s has a density g*{b\N, x).  F u r t h e r m o r e , the following l e m m a s i m i l a r to P r o p o s i t i o n  1 of G P V h o l d s . L e m m a 3.5 tribution  Under Assumption  from  for all N e M and x e X,  of bids has the compact support [b{N,x),b(N,x)],  if there is a binding partial  C.l(f),  derivatives  reserve price, andg*{-\N, on its interior.  Furthermore,  with b{N,x)  •) has at least R-\-l g*{b\N,x)  the dis= r  continuous  is bounded away  zero.  A f u l l list of t e c h n i c a l econometric  a s s u m p t i o n o n 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 of  quantiles  In t h i s s e c t i o n , we present o u r n o n p a r a m e t r i c e s t i m a t i o n m e t h o d Q*{T\N,X).  for  O u r e s t i m a t i o n m e t h o d is based o n the fact t h a t , since the  b i d d i n g strategies are increasing, the quantiles of v a l u a t i o n s Q*{T\N,  X) a n d  bids q*iT\N,x)  G*-\T\N,X)  = =  mi{b:G*{b\N,x)>T}  are l i n k e d t h r o u g h the (inverse) b i d d i n g strategy, Q*{T\N,x)  =  Since b o t h ^(.[TV, x ) a n d q*{T\N,x)  aq*ir\N,x)\N,x). can be e s t i m a t e d n o n p a r a m e t r i c a l l y , we  consider a n a t u r a l p l u g - i n e s t i m a t o r Q*iT\N,  x) = ê(q*(T|7V, x)\N, x).  (3.23)  T h e n o n p a r a m e t r i c e s t i m a t o r s for Ç a n d q* axe c o n s t r u c t e d as follows. R e c a l l i n g t h a t the inverse b i d d i n g strategy Ç(-|Af, x ) is g i v e n b y  i;WJ^,x)-0+  ^N-l)p{N,x)9*{b\N,x)  '  our e s t i m a t o r ^ ( - l i V , x ) is o b t a i n e d by r e p l a c i n g p{N,x), g*ib\N,  x) w i t h n o n p a r a m e t r i c estimators, p{N, x ) , G*{b\N,  T h e c o n d i t i o n a l q u a n t i l e q*{T\N,x)  G*{b\N,x),  and  x ) , and g*{b\N,  x).  is e s t i m a t e d b y i n v e r t i n g the n o n p a r a -  m e t r i c e s t i m a t o r for the C D F G(6|A'',x): q*{T\N,x)  = i n f { 6 : G(fc|7V,x) >  r}.  T h e t r a n s f o r m a t i o n /3(T, N, X) can be s i m i l a r l y e s t i m a t e d b y u s i n g the e s t i m a t o r s p{N,  x),  a n d t h e t r a n s f o r m e d quantiles e s t i m a t e d as  Q*{/3{T,N,x)\N,x).  O u r n o n p a r a m e t r i c estimators for tlie r e q u i r e d i n p u t functions g*{b\N, x), G*{b\N, x), a n d p{N, x) are based o n the kernel m e t h o d . Specifically, we use the f o l l o w i n g e s t i m a t o r s :  TriN\x) E f = i n L i ^ ( ^ ) T.Lnil{Ni  p{N,x)  =  N}nUK{^-M^)  where h is t h e b a n d w i d t h p a r a m e t e r , iiT is a kernel f u n c t i o n satisfying A s s u m p t i o n C . 2 i n t h e A p p e n d i x , a n d n / = l ^ S i Vu  ^he n u m b e r of a c t u a l  b i d d e r s i n a u c t i o n I. Since the p r o b a b i l i t y of o b s e r v i n g N c o n d i t i o n a l o n x a n d t h e p r o b a b i l i t y of s u b m i t t i n g a b i d c o n d i t i o n a l o n N a n d x c a n b e 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 t a n d a r d n o n p a r a m e t r i c regression e s t i m a t o r s . I n 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 t h a t the e s t i m a t o r of b i d subm i s s i o n p r o b a b i l i t y p{N, x) is a s y m p t o t i c a l l y n o r m a l a n d derive i t s a s y m p t o t i c variance  M o r e o v e r , we show t h a t the estimators p{N, x) are a s y m p t o t i c a l l y i n d e p e n dent for a n y d i s t i n c t N,N'  e{N,...,N}  a n d x,x'  e A't.  T h e c o n d i t i o n a l b i d densities a n d d i s t r i b u t i o n s are e s t i m a t e d b y a kernel m e t h o d , w i t h a n adjustment needed t o account for a r a n d o m n u m b e r of observations w i t h i n each a u c t i o n . W e estimate first t h e e x p e c t e d n u m b e r of b i d observations t h a t correspond t o TV-bidder auctions i n t h e s a m p l e w i t h covariates x as ê{N, x) = p{N,  x)9{N\x)NL.  T h e p r o p o s e d e s t i m a t o r s of g* a n d G* are  ^^^'^'''^ where (p{x)  -  h<iê{N,x)ip{x)  '  is the s t a n d a r d m u l t i v a r i a t e kernel d e n s i t y e s t i m a t o r . T h e e s t i -  m a t o r s g* a n d G* are essentially s t a n d a r d n o n p a r a m e t r i c c o n d i t i o n a l density a n d C D F e s t i m a t o r s w i t h the n u m b e r of bids observations Ylf=i E i ^ i Vu l{Ni  = N}  r e p l a c e d b y its e s t i m a t e d expected value  ê{N,x).  I n t h e A p p e n d i x , we prove t h a t the e s t i m a t o r s Q*{r\N, x) a n d Q*{P{T,  N, X)  \N, x) are a s y m p t o t i c a l l y n o r m a l . Specifically, we prove t h a t , u n d e r c e r t a i n t e c h n i c a l b u t s t a n d a r d econometric a s s u m p t i o n s , -  VLh'i+^Q*{T\N,x)  Q*{T\N,X))  is a s y m p t o t i c a l l y n o r m a l w i t h m e a n zero a n d variance  VQ{N,r,x)=  /  /•  (^J  -  \'^+'  K{ufdi^  (l-p(7V,x)(l-r))2  (N -  A consistent e s t i m a t o r VQ{N,T,X) q*{T\N,x)  l)2Np3{N,x)g*^{q*{T\N,x)\N,x)Tr{N\x)ip{x) c a n be o b t a i n e d b y r e p l a c i n g  p{N,x),  a n d other u n k n o w n functions b y t h e i r e s t i m a t o r s . 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 r a n d o m v a r i a b l e w i t h m e a n zero a n d variance VQ{N, /3(r, N, x ) , x ) ) . M o r e o v e r , for any d i s t i n c t A'', N' £ {T£, . . . , Î V } , T,T'  e  T , and x , x '  £ X\  the estimators Q*{T\N,X)  i n d e p e n d e n t , as are the e s t i m a t o r s Q*{^{T,N,x)\N,x). c o n t a i n e d 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 .  are a s y m p t o t i c a l l y T h e s e results are  3.4.4  Comparison with the estimation method of Haile, Hong and Shum (2003)  It is i n t e r e s t i n g t o c o m p a r e o u r e s t i m a t o r Q*{T\N,X)  w i t h t h a t of H H S ,  w h o present t h e i r m e t h o d i n a m o d e l different f r o m ours i n t h a t they allow for c o m m o n value effects. T h e i r m e t h o d is s e m i p a r a m e t r i c a n d i n the present context, reduces to the following. O n e begins w i t h r e m o v i n g the effect of covariates on v a l u a t i o n s b y p e r f o r m i n g a p r e l i m i n a r y regression. T h e m a i n a s s u m p t i o n i n H H S is t h a t the v a l u a t i o n s d e p e n d o n covariates a d d i t i v e l y , w i t h the m e a n v a l u a t i o n specified b y some f u n c t i o n r{xi;d)  t h a t depends o n x a n d a f i n i t e - d i m e n s i o n a l  p a r a m e t e r 9: (3.24)  Vii=rixi;e)+eii.  T h e error t e r m en is mean-zero a n d d i s t r i b u t e d i n d e p e n d e n t l y of xi w i t h C D F Fe(-). It is also assumed t h a t the reserve p r i c e has the same a d d i t i v e f o r m , ri =  O n e c a n always w r i t e the b i d d i n g strategy as  ro + r{xi;9).  the v a l u a t i o n m i n u s the m a r k u p , B{vii\Ni,xi)  = vu - m{vu,xi,Ni).  HSS  show t h a t i n t h e i r s e t t i n g , the m a r k u p Tn{vu,xi, Ni) does not d e p e n d o n xi: m{vu,xi,Ni)  = mo{£u,Ni).  It is t h e n easy to show t h a t the b i d s regression  takes t h e a d d i t i v e f o r m bu = a{Ni) where a(Ni)  = E[moieu,Ni)\Ni]  + r{xi;9)  (3.25)  + e'u,  a n d e^, = eu - moiu,Ni)  + a(A^()-  The  p a r a m e t e r c a n t h e n be e s t i m a t e d b y any n o n l i n e a r regression m e t h o d , a n d the bids "homogenized"  a c c o r d i n g t o bu = bu —  T{xi;9).  N e x t , the inverse b i d d i n g strategy ^{b\N) is e s t i m a t e d n o n p a r a m e t r i c a l l y , i n essentially the same w a y as i n our m e t h o d , a n d a pseudo s a m p l e of bids (cf. G P V ) is formed a c c o r d i n g to vu = C{bu\Ni), t r i m m e d a p p r o p r i a t e l y to a v o i d b i a s i n g b o u n d a r y effects.  L a s t l y , the quantiles Q*{T\N)  are e s t i m a t e d as  s a m p l e quantiles of the t r i m m e d s a m p l e . If desired, the c o n d i t i o n a l quantiles  c a n be e s t i m a t e d as Q*{T\N,X)  = T{x;9)  +  Q*{T\N).  T h i s " h o m o g e n i z a t i o n " is n o t feasible i n o u r s e t t i n g , since even i f one a s sumes t h e a d d i t i v e f o r m (3.24), t h e m a r k u p m ( u j ; , xi,Ni)  i n general depends  o n xi- T h i s is because, u n l i k e i n t h e models considered i n H S S , i n o u r case the e q u a t i o n for t h e b i d d i n g strategy ( C . 5 ) also c o n t a i n s t h e e n t r y p r o b a b i l i t y p{N,x),  a n u n k n o w n f u n c t i o n of x. Therefore t h e regression (3.25)  p r o d u c e s i n c o n s i s t e n t estimates of 9. O u r m e t h o d effectively changes the o r der o f steps o f the H S S e s t i m a t o r . U n l i k e H H S , o u r t r e a t m e n t of covariates is f u l l y n o n p a r a m e t r i c . W e first n o n p a r a m e t r i c a l l y e s t i m a t e t h e quantiles o f b i d s q(T\N,x)  a n d t h e n insert t h e m i n t o t h e e s t i m a t o r ^{-{NjX)  our c o n d i t i o n a l q u a n t i l e e s t i m a t o r Q*{T\N,X)  3.4.5  = ^{q{T\N,x)\N,  to obtain x).  Tests  I n v i e w o f t h e results of subsection 3.4.3, q u a n t i l e r e s t r i c t i o n s d e r i v e d f r o m the L S , S, a n d A M E models c a n b e tested i n a s t a n d a r d m a n n e r as e q u a l i t y or i n e q u a l i t y c o n s t r a i n t s . W e i m p l e m e n t t h e tests u s i n g a finite set of r ' s from (0,1) interval, T = { n , r 2 , T j t } .  T h e tests of H L S a n d H'g are based  o n t h e c o r r e s p o n d i n g s t a t i s t i c s T'^^lx) a n d T^{x) o f t h e e s t i m a t e d q u a n t i l e s Q*{T\N,X) means:  t h a t measure d e v i a t i o n s  a n d Q*0{T,N,X)\N,X)  f r o m their  11  B y t h e results of P r o p o s i t i o n C . 5 , Q*{T\N,X)  a n d Q*(P{T,N,X)\N,X)  are  a p p r o x i m a t e l y n o r m a l i n the large samples a n d i n d e p e n d e n t across N; therefore, t h e L S m o d e l is rejected a t level a for t h e a u c t i o n s w i t h covariates' values X whenever  T^^{x)  ' ^ R e c a l l t h a t Q'{T\N,X) t h e same for all  > xf#j^-,i)k,i-a'  ^^^^^ ^\#M-\)k,i-<x  is the same for all N u n d e r HLS, a n d Q'{P{T,  u n d e r H's.  denotes  N,x)\N,x)  is  the 1 — a q u a n t i l e of the chi-squaxe d i s t r i b u t i o n w i t h degrees of {#J\f ~l)k,  w h e r e #A/' denotes the n u m b e r of elements i n M.  rejects H'g i f T^{x)  > X(^jv_i)A: i-a  freedom  S i m i l a r l y , one  N o t e t h a t due to a s y m p t o t i c n o r m a l i t y  a n d independence of the q u a n t i l e estimators, T^^{x)  a n d T^{x)  can be also  v i e w e d as l i k e l i h o o d r a t i o ( L R ) s t a t i s t i c s . W e n o w t u r n to t e s t i n g of the A M E m o d e l . I n t h i s case, the distance or L R s t a t i s t i c is min  T^^^ix)^  and  V  Lh'^^'  VK.<-<m.r^rer  (Q^WAT, x) -  ,;.,.)^  VQ(7V,T,X)  ^ ^ ^ ^ ^  one rejects t h e n u l l of A M E i n favor of the general a l t e r n a t i v e w h e n  T^^^{x)  takes o n large values.  T h e A M E m o d e l does not determine u n i q u e l y the n u l l d i s t r i b u t i o n of the T^^^{x)  s t a t i s t i c ; however, we show i n P r o p o s i t i o n C . 6 i n the A p p e n d i x  t h a t the p r o b a b i l i t y of t y p e I error is m a x i m i z e d w h e n a l l inequalities are r e p l a c e d w i t h the equalities, i.e. the same r e s t r i c t i o n s as i n the L S m o d e l . T h i s p r o p o s i t i o n also shows t h a t , under the r e s t r i c t i o n of the L S m o d e l , the s t a t i s t i c T^^^{x)  is a s y m p t o t i c a l l y equivalent t o the r a n d o m variable  defined b y min  T^M'^{x)^y  where Zr ~ A''(0, I#Af-i) VQ{T,  X) =  diag(VQ{N,  (3.27)  a n d independent across r ' s , Q{T, X) = RVQ{T, r , x),VQ{N,  r , x)),  and  R is the  ( # A / ' - 1) x  X)R', (#A^)  differencing m a t r i x  R=  1  -1  '  '  ^ 0 0  0  0  \  « ...  C o n s e q u e n t l y , a test t h a t rejects HAME where c{x)AME,i-a  ...  1  -1  y  w h e n T'^^^{x)  >  is the 1 - a q u a n t i l e of the d i s t r i b u t i o n of  c{x)AME,i-a, T^^^(a;),  has a s y m p t o t i c size a.  T h e d i s t r i b u t i o n of T ' ^ ^ ^ ( a ; ) depends o n a s y m p t o t i c variances of the q u a n t i l e e s t i m a t o r s ; however, the c r i t i c a l values c a n be s i m u l a t e d as follows. F i r s t , f r o m the e s t i m a t e d a s y m p t o t i c variances VQ{N_,T,X), c o n s t r u c t the m a t r i c e s VQ{T,X) 1 , M ,  a n d CI{T,X)  for T i , . . . , T f c .  generate i n d e p e n d e n t i V ( 0 , I # A / ' - I ) vectors Zri,m,  p u t e f^^^{x)  as defined i n ( ? ? ) ,  ...,VQ{N,T,X) S e c o n d , for m  =  Zri^m, a n d c o m -  b u t w i t h Zr replaced b y Zr,m,  a n d ft  r e p l a c e d w i t h Q . T h e s i m u l a t e d c r i t i c a l value for a test w i t h a s y m p t o t i c size a, d e n o t e d b y c{x)AME,i-a, of {f^^^ix) T^^^ix)  : m = >  is t h e n c o m p u t e d as the 1 — a s a m p l e q u a n t i l e O n e s h o u l d reject the n u l l of A M E w h e n  1,...,M}.  c{x)AME,l-a-  T e s t i n g w h e t h e r the e n t r y p r o b a b i l i t i e s are n o n - i n c r e a s i n g i n A'', the Hp h y p o t h e s i s , c a n be p e r f o r m e d s i m i l a r l y to t e s t i n g H A M E -  min  T^ix)=  m>->v^ O n e s h o u l d reject Hp whenever TP{x) 1 - a q u a n t i l e of TP(x)  Define  ^  VpiN,x) > c{x)p^i-a,  = rnm^y2^^^^^^ \\Z - fif,  where c(x)pj-a fip(i)  is the  = RVp{x)R\  is a d i a g o n a l m a t r i x w i t h the m a i n d i a g o n a l elements Vp{N,, x ) , V p { N , and Z ~ A / ' ( 0 , S u c h A s i n t h e case of T^'^^[x),  Vp{x) i),  a test has a s y m p t o t i c size a a n d is consistent. t h e c r i t i c a l values for the T P ( X ) test c a n be  s i m u l a t e d f o l l o w i n g the steps described above.  3.5  Monte-Carlo experiment  I n t h i s s e c t i o n we present a M o n t e - C a r l o s t u d y of the s m a l l s a m p l e p r o p e r ties of the tests. I n p a r t i c u l a r , we are interested how the choice of quantiles T affects size a n d power of the tests. I n our s i m u l a t i o n s , we focus o n t e s t i n g t h e A M E m o d e l w i t h o u t covariates x.  W e s i m u l a t e the r a n d o m signals S a n d v a l u a t i o n s V u s i n g the G a u s s i a n c o p u l a . L e t {Zi,Z-i)  be bivariate n o r m a l w i t h zero m e a n s , variances equal  to one, a n d t h e c o r r e l a t i o n coefficient p. L e t $ denote t h e s t a n d a r d n o r m a l C D F . A pair ( 5 , y )  is generated as 5 = « > ( Z i ) , a n d V = $(^2).  Nonzero  values of p c o r r e s p o n d to the case of i n f o r m a t i v e signals a n d selective entry; w h i l e /9 = 0 corresponds to the case of the L S m o d e l . T h e details of the c o m p u t a t i o n of the d i s t r i b u t i o n s F{v\S') a n d  F*{v\N)  t h a t are needed i n order to solve for the e q u i l i b r i u m of the a u c t i o n are as follows.  First, recall that  ^ N{pzi,l  - p^),  a n d , consequently,  the  c o n d i t i o n a l d i s t r i b u t i o n of V given 5 is given b y F{v\S)  =  P{V<v\S)  =  P{Z2<^-\v)\^-\S))  \ N e x t , n o t e t h a t t h e m a r g i n a l d i s t r i b u t i o n of 5" is u n i f o r m o n the [0,1] i n t e r val, and F*{v\N)  =  F{v\S>-s{N))  1 - s  h{N)  /  where the cutoff s i g n a l 's{N^ c a n be found, given the value of TV, as a s o l u t i o n to e q u a t i o n (3.11). L a s t l y , for S > s{N),  the bids are c o m p u t e d  according  t o the b i d d i n g s t r a t e g y ( C . 5 ) . I n o u r s i m u l a t i o n s , 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 n u m b e r of M o n t e C a r l o r e p l i c a t i o n s is 1,000;  i n each r e p l i c a t i o n , the c r i t i c a l values for the T^'^^  test are o b t a i n e d u s i n g  999 r e p l i c a t i o n s . W e use the triweight kernel f u n c t i o n K{u)  = (35/32)(l  -  u'^)^l{|u| < 1} for n o n p a r a m e t r i c kernel e s t i m a t i o n . T o reflect the fact t h a t the n u m b e r of active bidders varies from a u c t i o n to a u c t i o n d e p e n d i n g o n the  n u m b e r of p o t e n t i a l b i d d e r s i n the a u c t i o n N a n d p{N), a b a n d w i d t h t h a t depends o n N.  we d e c i d e d to use  Specifically, we used h =  {LNp{N))~'^/^.  T a b l e 3.2 r e p o r t s the results of size s i m u l a t i o n s for p = 0 , 0 . 5 , a n d the f o l l o w i n g sets of quantiles: {0.5}, {0.3,0.5,0.7}, a n d { 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 7 } . W h i l e t h e a s y m p t o t i c a p p r o x i m a t i o n works reasonable w e l l for a s m a l l n u m ber q u a n t i l e s , the finite s a m p l e size p r o p e r t i e s of the test d e t e r i o r a t e w h e n the n u m b e r of quantiles used to c o n s t r u c t the test increases. F o r e x a m p l e , the T^M^  test over rejects the n u l l w h e n 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%, a n d 1% the s i m u l a t e d rejection rates are a p p r o x i m a t e l y 16%, 1 0 % a n d 4% respectively. N o t e also t h a t the rejection rates for p = 0.5 are s m a l l e r t h a n for p = 0.5 a n d below the n o m i n a l rej e c t i o n rates. T h i s reflects the fact t h a t t h e p r o b a b i l i t y of t y p e I error is m a x i m i z e d at p = 0.  T a b l e 3.3 r e p o r t s the size corrected power results (the c r i t i c a l values are c o m p u t e d f r o m the s i m u l a t e d d i s t r i b u t i o n of the test s t a t i s t i c u n d e r the n u l l ) . T o address the power issue, i t is necessary to come u p w i t h a n a l t e r n a t i v e . Ideally, t h i s w o u l d be achieved b y c o n s i d e r i n g a s t r u c t u r a l m o d e l . A b s e n t a s t r u c t u r a l m o d e l , however, we are allowed to consider a n y configur a t i o n of b i d d i n g quantiles, i n p a r t i c u l a r we m a y reverse t h e i r order, m a k i n g decreasing  as o p p o s e d to increasing i n A''. T o do t h i s i n the s i m p l e s t fashion  possible, we m u l t i p l y each q u a n t i l e b y m i n u s one a n d t h e n a d d a constant to a l l q u a n t i l e s to assure t h a t they are positive.  T a b l e 3.3 shows t h a t the power increases w i t h the distance f r o m the n u l l . T h e power also increases w h e n we use quantiles 0.3 a n d 0.7 i n a d d i t i o n to the m e d i a n . However, we also observe t h a t i n some cases the size corrected power decreases w i t h the n u m b e r of quantiles, w h e n the n u m b e r of quantiles used t o c o n s t r u c t the test is large. F o r e x a m p l e , i n the case of {0.3,0.5,0.7} q u a n t i l e s , p = 0.9, a n d t h e n o m i n a l size of 5%, the s i m u l a t e d rejection rate is a b o u t 2 0 % ; however, w h e n we use i n a d d i t i o n the quantiles 0.4 a n d 0.6, the r e j e c t i o n r a t e is o n l y a b o u t 18%. In practice, given samples of moderate  size, a r u l e of t h u m b w o u l d be to use 3 fixed quantiles i n order to m a i n t a i n g o o d size a n d reasonable power.  3.6  Empirical application  O u r dataset consists of 547 auctions for surface p a v i n g a n d g r a d i n g c o n t r a c t s let b y O k l a h o m a D e p a r t m e n t of T r a n s p o r t a t i o n ( O D O T ) d u r i n g the p e r i o d of J a n u a r y , 2002 to December, 2005. ^^^^ T h e available d a t a items i n c l u d e a l l b i d s , t h e engineer's estimate, the t i m e l e n g t h of the c o n t r a c t (in d a y s ) , the n u m b e r of items i n the p r o p o s a l a n d the l e n g t h of the r o a d . T h e O D O T i m p l e m e n t s a p o l i c y under w h i c h a l l bids over 7% of the engineer's e s t i m a t e are t y p i c a l l y rejected, so there is a b i n d i n g reserve price. I n reality, we d o observe b i d s above the reserve price ( a l t h o u g h e x t r e m e l y few w i n n i n g b i d s were above the reserve price). W e treat these bids as non-serious. T h e y are o n l y used i n the e s t i m a t i o n of b i d d i n g p r o b a b i l i t i e s , b u t are otherwise e l i m i n a t e d f r o m t h e sample of b i d s .  I m p o r t a n t l y , we observe the list of eligible bidders (planholders) for each auction.  I n the v o c a b u l a r y of this chapter, these eligible bidders are the  p o t e n t i a l b i d d e r s . T h e list of p l a n h o l d e r s is p u b l i s h e d o n the O D O T website p r i o r to b i d d i n g . A f i r m becomes a p l a n h o l d e r t h r o u g h t h e f o l l o w i n g process. A l l projects to be a u c t i o n e d are advertised b y the O D O T 4 to 10 weeks p r i o r t o the l e t t i n g d a t e . T h e s e advertisements include the engineer's e s t i m a t e , a brief s u m m a r y of the project, l o c a t i o n of the w o r k a n d the t y p e of the w o r k i n v o l v e d . B u t the advertisements lack detailed schedules of w o r k items w h i c h are o n l y revealed i n c o n s t r u c t i o n p l a n s . ' ^ T h e d a t a were o b t a i n e d from the O D O T website,  http://www.okladot.state.ok.us/.  ^ ' O u r choice o f surface p a v i n g a n d g r a d i n g contracts H o n g a n d S h u m (2002) [22],  is m o t i v a t e d  by the fact  that  i n their s t u d y of highway p r o c u r e m e n t auctions i n N e w Jer-  sey, find little s u p p o r t for c o m m o n values for this type of contracts.  See also D e Silva,  K a n k a n a m g e , a n d K o s m o p o u l o u (2007)[13l. T h i s is i m p o r t a n t because i n t h i s chapter, we assume independent private values (costs).  Interested firms c a n t h e n s u b m i t a request for p l a n s a n d b i d d i n g proposals, the d o c u m e n t s t h a t c o n t a i n the specifics o f the project ( i n p a r t i c u l a r , t h e items schedule).  A n i m p o r t a n t feature o f the q u a l i f i c a t i o n process is t h a t  o n l y eligible firms are allowed access t o these d o c u m e n t s . A firm is deemed eligible i f i t satisfies c e r t a i n q u a l i f i c a t i o n r e q u i r e m e n t s .  T h e g o a l o f the  q u a l i f i c a t i o n process is t o ensure t h a t the w i n n i n g firm w i l l have sufficient expertise a n d c a p a c i t y t o u n d e r t a k e the project.  W h i l e the expertise p a r t  is t y p i c a l l y d e t e r m i n e d a t the p r e - q u a l i f i c a t i o n stage ( i n most cases, once per y e a r ) , t h e c a p a c i t y p a r t is project-specific.  A n i m p o r t a n t requirement  is t h a t the p r o s p e c t i v e bidder is n o t qualified for the aggregate a m o u n t o f w o r k t h a t exceeds 2.5 times its current w o r k i n g c a p i t a l .  G i v e n t h a t the  bidders k n o w the sizes o f a l l projects t o be let b u t not the p r o j e c t specifics, it is p l a u s i b l e t h a t the decision o f a firm t o request the p l a n for a p a r t i c u l a r project is p r i m a r i l y d e t e r m i n e d b y the project size as w e l l as the sizes of other projects for w h i c h i t is p r e - q u a l i f i e d , i n r e l a t i o n t o the available c a p a c i t y o f the firm. T h e c a p a c i t y m a y b e d e t e r m i n e d b y a n u m b e r o f factors, such as for e x a m p l e the a m o u n t of resources c o m m i t t e d t o other p r o j e c t s , i n c l u d i n g b u t not l i m i t e d t o those p r e v i o u s l y contracted w i t h O D O T .  Before t u r n i n g t o our n o n p a r a m e t r i c tests, we investigate the i m p o r t a n c e of v a r i o u s observable covariates o n b i d levels a n d the decisions t o s u b m i t a b i d w i t h the help o f u s u a l O L S a n d logit regressions. T h e variables used i n the regressions are described i n T a b l e 3.4.  T h e results of t h e e n t r y logit  regression a n d O L S b i d d i n g regression are presented i n T a b l e 3.5. O L S regression, the dependent variable is log{bid),  I n the  w h e r e b i d is t h e a m o u n t  of b i d i n m i l l i o n s o f dollars. T h e size o f the p r o j e c t has a s t r o n g l y p o s i t i v e effect o n t h e b i d s .  C l e a r l y , i t is t h e most i m p o r t a n t v a r i a b l e i n t h e O L S  regression. U s i n g i t alone p r o d u c e d  o f about 0.79, so the i m p a c t of the  other v a r i a b l e s is m u c h smaller. I n the order of i m p o r t a n c e , t h e n e x t v a r i a b l e is the n u m b e r o f p o t e n t i a l bidders N; i f i t is i n c l u d e d i n t h e regression, R"^ " T h i s r e q u i r e m e n t is explicitly stated i n the O D O T rule O A C 730:25. ' ^ T h e covariates a r e basically t h e same as i n other papers o n p r o c u r e m e n t  (e.g.  auctions  B a j a r i a n d Y e (2003) [7]; Pesendorfer a n d Jofre-Bonet (2003) [48]; K r a s n o k u t s k a y a  (2003)[26]; K r a s n o k u t s k a y a and Seim (2006)[27l; L i a n d Z h e n g (2005)[34l).  increases t o a b o u t 0.94. W e also m e n t i o n t h a t the p r o j e c t size has a negative (but not s t a t i s t i c a l l y significant) effect o n the p r o b a b i l i t y of s u b m i t t i n g a bid. T h e effect of the n u m b e r of p o t e n t i a l bidders is s t a t i s t i c a l l y significant i n a l l regressions. H a v i n g more p o t e n t i a l bidders reduces the b i d s u b m i s s i o n rate: i n c r e a s i n g A'' b y 1 reduces the odds of s u b m i t t i n g a b i d b y a b o u t 4%. H a v i n g m o r e b i d d e r s also results i n lower bids. T h i s is of course consistent w i t h the m o d e l s considered i n t h i s chapter. T h e logit regression also shows t h a t p r o j e c t size has a negative effect o n the p r o b a b i l i t y of b i d d i n g . T h i s effect, however, is not s t a t i s t i c a l l y significant.  T h e c o m p l e x i t y of the project is c a p t u r e d b y the n u m b e r of i t e m s i n the c o n s t r u c t i o n p l a n . T h i s is the variable Nitems. is a s u b s t a n t i a l v a r i a t i o n i n Nitems;  T a b l e 3.4 shows t h a t there  the m e a n is 72 a n d the s t a n d a r d d e v i -  a t i o n is 71 i t e m s . O n e m i g h t expect t h a t the cost of p r e p a r i n g a b i d is a n i n c r e a s i n g f u n c t i o n of Nitems, T h e c o n j e c t u r e d effect of Nitems  even c o n t r o l l i n g for the size of the project. is therefore to reduce the p r o b a b i l i t y 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 t h e logit regres-  s i o n confirms t h i s conjecture. However, the effect is q u i t e s m a l l : increasing Nitems  b y one s t a n d a r d d e v i a t i o n , i.e. a d d i n g a b o u t 70 p a y i t e m s , reduces  the o d d s b y o n l y a b o u t 1%.  I n c l u d e d i n b o t h regressions are d u m m y variables for 20 firms t h a t appear o n the p l a n h o l d e r s list most frequently. T h e other firms are t r e a t e d as fringe firms.  O b s e r v e t h a t even t h o u g h not a l l firms enter at the same rate a n d  b i d s i m i l a r l y , t h e e m p i r i c a l evidence of a s y m m e t r i e s is s t r o n g o n l y for o u t of-state f i r m s (the f i r m s w i t h headquarters outside the state of O k l a h o m a ) t h a t enter less f r e q u e n t l y a n d also b i d less, a n d for the f o l l o w i n g three firms: B r o c e C o n s t r u c t i o n , G l o v e r C o n s t r u c t i o n a n d B e c c o C o n t r a c t o r s . Since our m o d e l assumes b i d d e r s y m m e t r y , we decided t o exclude the auctions i n w h i c h e i t h e r out-of-state firms or these three firms were o n the p l a n h o l d e r s list.  I n the p r a c t i c a l i m p l e m e n t a t i o n of our estimators, we are confronted w i t h the u s u a l bias a n d variance trade-off.  I n c l u d i n g m o r e variables w i l l reduce  the bias, b u t at the same t i m e w i l l increase the sample v a r i a b i l i t y of our e s t i m a t o r s . G i v e n the p r e l i m i n a r y regression results, we o n l y c o n d i t i o n o n the p r o j e c t size. F i g u r e 3.1 shows the e m p i r i c a l frequencies (i.e. the h i s t o g r a m ) of project sizes. T h e p a t t e r n is h i g h l y skewed towards smaller projects:  the average  p r o j e c t size is a b o u t $3.6 m i l l i o n (from T a b l e 3.4), b u t the projects for w h i c h there are at least 10 auctions i n the dataset have sizes not exceeding $2.5 million.  Since we need a m o d e r a t e n u m b e r of observations t o i m p l e m e n t  o u r n o n p a r a m e t r i c tests, we have chosen the set of p r o j e c t sizes X to be the e q u a l p a r t i t i o n of the i n t e r v a l [0,2.5], i.e. i n m i l l i o n s of d o l l a r s , X  =  {0.5,1,1.5,2,2.5}.  T h e projects of larger size m a y be, other t h i n g s e q u a l , m o r e a t t r a c t i v e to the b i d d e r . scale.  F o r e x a m p l e , t h i s w o u l d be so i f there are economies of  T h e r e is some evidence of t h i s i n the d a t a .  T a b l e 3.6 shows e s t i -  m a t e d c o n d i t i o n a l p r o b a b i l i t i e s 7r(7V|a;), where as before N is the n u m b e r of p o t e n t i a l b i d d e r s a n d x is the size of the project as m e a s u r e d b y the engineer's e s t i m a t e . w i t h X from E{N\x E[N\x]  O n e c a n see t h a t the e s t i m a t e d m e a n E[N\x] = 2]=  4.69 to E[N\x  increases  = 2.5] = 7.97, a n d the differences  - £;[iV|a;'], x > x', are s t a t i s t i c a l l y significant for a l l a; G X .  T a b l e 3.6 also shows t h a t there s t i l l r e m a i n s a s u b s t a n t i a l v a r i a t i o n i n the n u m b e r of p o t e n t i a l bidders even after c o n t r o l l i n g for project size. T h i s v a r i a t i o n w i l l i m p o r t a n t for o u r tests. E q u a l l y i m p o r t a n t is t h a t the v a r i a t i o n i n the n u m b e r of p l a n h o l d e r s is likely to be exogenous, u n c o r r e l a t e d w i t h unobservable project characteristics since the l a t t e r o n l y become available i n the p l a n s . A n o t h e r p r a c t i c a l issue is t h a t , as is w e l l k n o w n , n o n p a r a m e t r i c e s t i m a t o r s suffer from s u b s t a n t i a l loss of p r e c i s i o n w h e n sample size is very s m a l l . W h e n  we t r i e d t o i n c l u d e a l l a u c t i o n s , the estimates of the q u a n t i l e s Q{T\N,  X) were  h i g h l y e r r a t i c . T h e p r o b l e m is t h a t , because the d a t a is sparse, for some {N,x)  p a i r s t h e e s t i m a t e d p r o b a b i l i t y 7r(A''|a;) is v e r y close to 0. T o m a k e  o u r e s t i m a t o r s s t a b l e , we decided to exclude those p a i r s {N,x) n u m b e r of o b s e r v a t i o n s t h a t TV; = TV a n d xi G [x - h,x  where the  + h] is less t h a n 15.  T h e w o r k i n g s a m p l e u l t i m a t e l y consisted of 258 a u c t i o n s , a n d a l l the results discussed b e l o w were o b t a i n e d u s i n g t h i s s m a l l e r s a m p l e .  T h e tests are p e r f o r m e d c o n d i t i o n a l on project size x G X.  W e first  test the p r e d i c t i o n s h a r e d by a l l models considered i n t h i s c h a p t e r , n a m e l y t h a t b i d s u b m i s s i o n p r o b a b i l i t i e s p{N,x) p o t e n t i a l b i d d e r s TV for each x.  are d e c l i n i n g i n the n u m b e r of  Refer to T a b l e 3.7, where the e s t i m a t e d  b i d d i n g p r o b a b i l i t i e s p(N, x) as w e l l as the results of t h e tests are r e p o r t e d . T h e average r a t e of b i d s u b m i s s i o n is a b o u t 6 2 % . E x c e p t for r e l a t i v e l y large p r o j e c t s , x = $2.5 m i l . , there is a s t r o n g l y d e c l i n i n g p a t t e r n over TV. F o r e x a m p l e , for m o d e r a t e l y sized projects, x = b i d d i n g p(N,  x)  $1.5 m i l . , the p r o b a b i l i t y of  falls f r o m a r e l a t i v e l y large value of 0.56 w h e n there are  3 p o t e n t i a l b i d d e r s , to a b o u t h a l f of t h a t , 0.28 w h e n there are 9 p o t e n t i a l bidders.  F o r o t h e r values of x G X \ { 2 . 5 } , the p a t t e r n is less p r o n o u n c e d ,  b u t the f o r m a l tests of the m o n o t o n i c i t y r e s t r i c t i o n s s t i l l do n o t reject the n u l l a t t h e c o n v e n t i o n a l 5% significance level.  W e n o w t u r n to the tests of the models:  L S , S, a n d A M E . F i r s t  t h a t , because the p r o c u r e m e n t auctions are l o w - b i d , the n u l l  note  hypothesis  t h a t c o r r e s p o n d s to the A M E m o d e l must be c h a n g e d a c c o r d i n g l y , i.e. the q u a n t i l e s m u s t decreasing r a t h e r t h a n i n c r e a s i n g . A l s o , the inverse strategy is now  c(6iTv,.) = 6 -  ^-p(^^-)o*m,x) {N-l)p{N,x)g*ib\N,x)'  a n d the a s y m p t o t i c variances of the quantiles are VQ{N,T,X)  =  (^j  K{ufdv^  (l-p(TV,x)r)2 {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 i n S e c t i o n 3.5.  b u t a l l other aspects of e s t i m a t i o n are unchanged. T h e results of t h e tests are presented i n T a b l e 3.9.  C o n s i d e r first the  results for j u s t one q u a n t i l e , the m e d i a n (Tables 3.7 a n d 3.8 r e p o r t the estim a t e d m e d i a n a n d t r a n s f o r m e d m e d i a n costs respectively a n d t h e i r s t a n d a r d errors). T h e A M E m o d e l is rejected for r e l a t i v e l y s m a l l projects, x = $0.5 a n d a; = $1 m i l . , b u t i t is not rejected for larger projects. N e i t h e r L S models are rejected for most project sizes. Since the findings are s o m e w h a t counter i n t u i t i v e ; b u t r e c a l l t h a t our M o n t e - C a r l o studies have s h o w n t h a t the power of the tests c a n be increased s u b s t a n t i a l l y i f we increase t h e n u m b e r of q u a n tiles f r o m one to three. W h e n 0.3, 0.5 a n d 0.7 q u a n t i l e s are s i m u l t a n e o u s l y considered, the s u p p o r t for the S m o d e l disappears c o m p l e t e l y at 5% significance for a l l p r o j e c t sizes. T h e largest p-value of the test is 0.02. T h e L S m o d e l fairs b e t t e r , b u t i t too is rejected for a l l projects b u t the largest ones, w i t h X = $2.5 m i l . T h e A M E m o d e l , on the other h a n d , is n o w rejected also for x = $1 m i l . , b u t s i m i l a r l y to the one q u a n t i l e case, is not rejected for p r o j e c t sizes x = $2.0 a n d x = $2.5 m i l .  Increasing the n u m b e r of quantiles f r o m three t o five (we have chosen the q u a n t i l e s 0.3, 0.4, 0.5, 0.6 a n d 0.7) leads to the same results: the S m o d e l is r o b u s t l y rejected, the L S m o d e l is rejected for a l l b u t the largest projects. T h e A M E m o d e l is not rejected for the two largest p r o j e c t sizes we c o n s i d ered. T h e reason t h a t a l l models are rejected for the s m a l l project c a n be as follows. It is possible t h a t for s m a l l projects, firms c o o r d i n a t e t h e i r decisions to request p l a n s a n d become p o t e n t i a l b i d d e r s , w h i l e these decisions are m a d e i n d e p e n d e n t l y for larger projects. C o n f i r m i n g t h i s hypothesis e m p i r i c a l l y is l i k e l y t o be difficult i n v i e w of p o t e n t i a l l y c o n f o u n d i n g effects of u n o b s e r v e d p r o j e c t heterogeneity. T h e fact t h a t the S m o d e l is always rejected is p r o b a b l y not v e r y s u r p r i s i n g . R e c a l l t h a t the e n t r y cost i n the S m o d e l is solely the cost of p r e p a r i n g a b i d r a t h e r t h a n the j o i n t cost t h a t also includes i n f o r m a t i o n a c q u i s i t i o n . E v e n for s m a l l projects, firms m a y face u n c e r t a i n t y a b o u t the exact level of  their c o n s t r u c t i o n costs t h a t can only be resolved t h r o u g h c o s t l y i n f o r m a t i o n a c q u i s i t i o n , so i t is o n l y n a t u r a l t h a t this is c o n f i r m e d e m p i r i c a l l y . T h e e m p i r i c a l evidence r e g a r d i n g the L S m o d e l c a n also be e x p l a i n e d i n t u i t i v e l y . It is p l a u s i b l e t h a t p r o j e c t c o m p l e x i t y increases w i t h p r o j e c t size. O u r i n a b i l i t y to reject t h e L S m o d e l for large projects m a y be due to t h e fact t h a t for these p r o j e c t s , the i n f o r m a t i o n received b y the bidders before t h e p l a n s are a v a i l a b l e is r e l a t i v e l y less precise.  3.7  Concluding remarks  I n t h i s c h a p t e r , we have proposed n o n p a r a m e t r i c tests to d i s c r i m i n a t e a m o n g a l t e r n a t i v e models of entry i n first-price auctions. T h e models c o n sidered are:  (a) t h e L e v i n a n d S m i t h (1994) [31] m o d e l w i t h r a n d o m i z e d  e n t r y strategies, (b) the Samuelson (1985) [53] m o d e l t h a t assumes t h a t b i d ders are p e r f e c t l y i n f o r m e d a b o u t t h e i r v a l u a t i o n s at the e n t r y stage, a n d select i n t o the p o o l of entrants based on t h i s i n f o r m a t i o n , a n d (c) a new m o d e l t h a t allows for selective e n t r y b u t i n a less s t a r k f o r m t h a n S a m u e l son (1985) [53]. Specifically, our m o d e l assumes t h a t bidders receive signals t h a t are i n f o r m a t i v e a b o u t t h e i r valuations a n d m a k e t h e i r e n t r y decisions based o n these signals.  I n t h e e m p i r i c a l a p p l i c a t i o n , we have tested the r e s t r i c t i o n s of each m o d e l against t h e u n r e s t r i c t e d alternatives using a dataset of h i g h w a y p r o c u r e m e n t a u c t i o n s f r o m the O k l a h o m a D e p a r t m e n t of T r a n s p o r t a t i o n ( O D O T ) . W e have f o u n d s t r o n g evidence for selective entry a c c o r d i n g to o u r m o d e l . W h i l e these f i n d i n g s are e n c o u r a g i n g a n d the t e s t i n g f r a m e w o r k c a n be used i n other a p p l i c a t i o n s , t h i s research c o u l d be extended i n a n u m b e r of directions. O n e i m p o r t a n t e x t e n s i o n w o u l d be developing more p o w e r f u l tests.  One  c o u l d i m p r o v e power b y c o n s i d e r i n g tests based o n a c o n t i n u u m of q u a n tiles r a t h e r t h a n a finite set. H S S have p u r s u e d t h i s a p p r o a c h , d e v e l o p i n g a K o l m o g o r o v - S m i r n o v t y p e test. A s we have a l r e a d y discussed, t h e i r est i m a t i o n m e t h o d c a n n o t be d i r e c t l y transferred to our s e t t i n g . T h i s w o u l d  be a n i m p o r t a n t extension of o u r approach left for future work. S i m i l a r h y potheses are considered i n the recent l i t e r a t u r e o n tests of stochastic d o m i nance a n d m o n o t o n i c i t y (see, for example, Lee, L i n t o n , W h a n g , S u n t o r y , for E c o n o m i c s , a n d D i s c i p l i n e s (2006)[29]); however, t h e i r a p p r o a c h cannot a p p l i e d d i r e c t l y i n our case, since private valuations are unobservable,  be and  o u r s t a t i s t i c s are based o n kernel density estimators. O u r t e s t i n g framework is q u i t e general a n d the e m p i r i c a l findings are by a n d large i n t u i t i v e . B u t there is also a n i m p o r t a n t l i m i t a t i o n t h a t the future research s h o u l d address.  I n a u c t i o n datasets, one t y p i c a l l y finds t h a t the  v a r i a t i o n i n b i d s is o n l y p a r t i a l l y e x p l a i n e d by their v a r i a t i o n w i t h i n auct i o n s . T h e b e t w e e n - a u c t i o n v a r i a t i o n is t y p i c a l l y present. It is also observed i n our dataset. T h i s p a t t e r n can be e x p l a i n e d w i t h i n the I P V framework o n l y b y unobserved p r o j e c t heterogeneity.  Recently, K r a s n o k u t s k a y a (2003) [26]  has developed a s t r u c t u r a l e s t i m a t i o n m e t h o d t h a t c a n be a p p l i e d even i n the presence of unobserved heterogeneity.  She assumes t h a t the heterogene-  i t y enters i n t o the specification of v a l u a t i o n s as a m u l t i p l y i n g factor.  Her  m e t h o d relies o n the fact t h a t the same m u l t i p l i c a t i v e s t r u c t u r e carries over t o the b i d s . U n f o r t u n a t e l y , t h i s is not true for the models w i t h e n t r y c o n s i d ered i n t h i s c h a p t e r , for reasons largely s i m i l a r to those d e s c r i b e d i n Section 3.4.4.  • A l t e r n a t i v e l y , one c a n e x p l a i n between-auction  variation using a model  w i t h afïiUated p r i v a t e values ( A P V ) , as i n L i , P e r r i g n e a n d V u o n g (2002) [33]. N o t e however t h a t the theoretical m o d e l s of e n t r y t h a t we b u i l d o n are a l l w i t h i n t h e I P V s e t t i n g . It is k n o w n t h a t the A P V m o d e l c a n lead to q u a l i t a t i v e l y different p r e d i c t i o n s ( P i n k s e a n d T a n (2005) [49]), so i t is a n o p e n q u e s t i o n if i t c a n l e a d t o testable i m p l i c a t i o n s s i m i l a r t o those considered here. W e leave t h i s for future research. A n o t h e r extension w o u l d be to allow b i d d e r a s y m m e t r i e s (e.g., a recent w o r k i n g p a p e r b y K r a s n o k u t s k a y a a n d S e i m (2006) [27]). T h e o b v i o u s diff i c u l t y here w o u l d the necessity to deal w i t h m u l t i p l e e q u i l i b r i a .  Bajari,  H o n g a n d R y a n (2004) [6] o b t a i n a n u m b e r of i d e n t i f i c a t i o n results i n t h i s d i r e c t i o n a n d e s t i m a t e a p a r a m e t r i c model w i t h m u l t i p l e e q u i l i b r i a for h i g h way procurement auctions.  F i n a l l y , i n c o r p o r a t i n g d y n a m i c features as i n  Pesendorfer a n d J o f r e - B o n e t (2003) [48] is also left for future research.  3.8  Tables and Figures T a b l e 3.1: Size of A M E Test  Quantiles  rho=0  rho=0.5  Nominal size  0.5  0.3, 0.5, 0.7  0.10  0.3, 0.4, 0.5, 0.6, 0.7  0.0760  0.1520  0.1580  0.05  0.0310  0.0730  0.1030  0.01  0.0070  0.0200  0.0360  0.10  0.0420  0.0660  0.0640  0.05  0.0240  0.0350  0.0310  0.01  0.0010  0.0130  0.0100  T a b l e 3.2: Size-corrected Power of the A M E Test  Quantiles  rho=0.5  rho=0.9  Nominal size  0.5  0.3, 0.5, 0.7  0.10  0.1622  0.2800  0.3, 0.4, 0.5, 0.6, 0.7 0.2683  0.05  0.0941  0.2030  0.1812  0.01  0.0210  0.0630  0.0651  0.10  0.2693  0.3624  0.4124  0.05  0.1842  0.2853  0.2983  0.01  0.0661  0.1231  0.1552  T a b l e 3.3: D e s c r i p t i o n o f V a r i a b l e s Min  Max  4.488  0.066  24.800  1.067  0.173  0.385  2.106  71.736  70.704  1.000  363.000  195.995  142.370  10.000  681.000  4.699  4.794  0.000  36.630  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  T h e total amount of unfinished work on a given day and normalized by the bidder-specific maximum, 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  dummy =1 if the firm has headquarters outside the state of Oklahoma  0.1358936  0.342929  0.000  1.000  Variable  Description  EngEst  T h e engineer's estimate for the project, i n m i l . dollars  Bid  B i d divided by the engineer's estimate  Nitems  Number of pay items in the project ad  Ndays  Number of business days to complete the project  Length  Length of the road (in miles)  Distance  Out-of-state  Mean 3.647  Std. Dev.  T a b l e 3.4: L o g i t a n d O L S Regressions Logit Variable  Coeff.  Intercept  3.565  OLS s.e.  Coeflf.  s.e.  1.148  0.497  0.123  0.034  0.970  0.010 0.002  Log(EngEst)  -0.01  Npotential  -0.043  0.019  -0.008  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  0.000  0.000  0.000  0.000  Distance 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  A P A C - O K L A H O M A , 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 0.033  BECCO CONTRACTORS, INC.  -1.818  0.876  -0.068  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  A L L E N CONTRACTING, INC.  0.245  0.493  0.014  0.035  DUIT CONSTRUCTION CO., INC.  1.275  0.707  0.027  0.037  -1.122  0.809  0.006  0.037  MUSKOGEE BRIDGE CO., INC. Observations Log-Iikelihood/fl2  4485 -1543.750  1860 0.983  Significemt coefficients (at 5% level) are marked in bold. T h e 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.  T a b l e 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 o n P r o j e c t Size x  P r o j e c t s i z e ($ m i l . ) x=0.5  x=l  x=1.5  x=2  x=2.5  s.e.  •K(N/X)  s.e.  7r(Ar/x)  s.e.  7r(iV/x)  s.e.  0.07  0.02  0.04  0.02  0.02  0.02  0.01  0.02  0.00  0.00  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 0.07  N  n{N/x)  2 3  s.e.  6  0.17  0.03  0.15  0.03  0.19  0.05  0.26  0.07  0.18  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  0.00  0.00  0.02  0.02  0.04  0.03  0.01  0.01  0.01  0.01  0.03  0.03  0.01  0.02  0.02  0.04  0.03  0.03  6.84  0.04  7.97  0.04  13 14  0.01  0.01  0.01  0.01  15 E{N/x)  4.69  0.02  5.23  0.02  5.95  T a b l e 3.6: E s t i m a t e d P r o b a b i l i t y o f B i d d i n g P{N, x)  P r o j e c t size ($ m i l . ) x=0.5 N  P{N,x)  x=l s.e.  P{N,x)  x=1.5 s.e.  P(iV,x)  x=2 s.e.  P{N,x)  x=2.5 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  0.37  0.04  0.36  0.05  0.34  0.05  0.30  0.06  0.27  0.08  0.30  0.06  0.31  0.06  0.27  0.07  0.28  0.07  0.35  0.09  0.45  0.11  0.21  0.08  7 8 9 10 11 Test Statistic  0.00  0.00  0.01  4.73  19.74  p-value  0.99  1.00  1.00  0.17  0.00  T a b l e 3.7: E s t i m a t e d M e d i a n s o f C o s t s  P r o j e c t size ($ m i l . ) x=0.5  x=l  x=1.5  x=2.5  x=2  Mediein  s.e.  Median  s.e.  Median  s.e.  Median  s.e.  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  0.90  0.06  0.90  0.06  0.88  0.07  0.87  0.90  0.07  0.92  0.07  0.93  0.09  0.76  0.22  0.82  N  Median  s.e.  0.08  0.84  0.13  0.08  0.80  0.03  0.81  0.11  2  7 /  8 9 10 11  T a b l e 3.8: E s t i m a t e d T r a n s f o r m e d M e d i a n s of C o s t s  P r o j e c t size ($ m i l . ) x=0.5 N  x=l  Trans. Median  s.e.  3  0.85  4  0.92  x=1.5  Trans. Median  s.e.  0.14  0.85  0.07  0.91  x=2  x=2.5  Trans. Median  s.e.  0.16  0.86  0.20  0.84  0.33  0.08  0.93  0.11  0.95  0.10  Trans. Median  s.e.  Treins. Median  s.e.  2  5  0.91  0.07  0.92  0.08  0.89  0.14  0.69  0.45  6  0.90  0.06  0.91  0.06  0.90  0.07  0.87  0.08  0.85  0.13  7 ( 8  0.90  0.07  0.92  0.07  0.93  0.09  0.76  0.22  0.82  0.08  0.84  0.07  0.81  0.11  9 10 11  T a b l e 3.9: Test R e s u l t s  P r o j e c t S i z e ($ m i l . )  A M E  statistic  A M E  p-value  L S Statistic  Quantiles  ; 0.5  LS  p-value  s  statistic  S p-value  0.5  9.56  0.02  9.56  0.05  8.35  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  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  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  Quantiles  Quantiles  0.08  : 0.3,0.5,0.7  : 0.3,0.4,0.5,0.6,0.7  Bibliography Bibliography [1] A n d r e w s , D .  G e n e r i c U n i f o r m Convergence.  Econometric  Theory,  8 ( 2 ) : 2 4 1 - 2 5 7 , 1992. [2] A t h e y , S. a n d P . , H a i l e . Identification of S t a n d a r d A u c t i o n M o d e l s . Econometrica,  70(6):2107-2140, 2002.  [3] A t h e y , S. a n d P . , H a i l e . N o n p a r a m e t r i c A p p r o a c h e s t o A u c t i o n s . book of Econometrics,  Hand-  6, 2005.  [4] A t h e y , S., J . L e v i n , a n d E . Seira.  C o m p a r i n g Sealed B i d a n d O p e n  A u c t i o n s : T h e o r y a n d E v i d e n c e f r o m T i m b e r A u c t i o n s . D e p a r t m e n t of E c o n o m i c s , S t a n f o r d U n i v e r s i t y , 2004. [5] B a j a r i , P . a n d A . H o r t a c s u . W i n n e r ' s C u r s e , Reserve P r i c e s a n d E n dogenous E n t r y : E m p i r i c a l Insights f r o m e B a y A u c t i o n s . RAND nal of Economics,  Jour-  34(2):329-355, 2003.  [6] B a j a r i , P . a n d H . H o n g a n d S. R y a n . I d e n t i f i c a t i o n a n d E s t i m a t i o n of D i s c r e t e G a m e s of C o m p l e t e I n f o r m a t i o n . 2004. N B E R W o r k i n g paper. [7] B a j a r i , P . a n d L . Y e . Review of Economics  Deciding Between Competition and Collusion. and Statistics,  85(4):971-989, 2003.  [8] C e l i k G . a n d O . Y i l a n k a y a . O p t i m a l A u c t i o n s w i t h P a r t i c i p a t i o n C o s t s . W o r k i n g P a p e r , U n i v e r s i t y of B r i t i s h C o l u m b i a , 2006. [9] C h a m b e r l a i n G . A s y m p t o t i c Efficiency i n E s t i m a t i o n w i t h C o n d i t i o n a l M o m e n t R e s t r i c t i o n s . Journal  of Econometrics,  34:305-334, 1987.  [10] C h e n g M . , J . F a n a n d J . M a r r o n . O n A u t o m a t i c B o u n d a r y C o r r e c t i o n s . The Annals  25:1691-1708, 1997.  of Statistics,  [11] C o w h n g A . a n d P . H a l l . O n P s e u d o d a t a M e t h o d s for R e m o v i n g B o u n d a r y Effects i n K e r n e l D e n s i t y E s t i m a t i o n . Journal tical Society.  of the Royal  Statis-  Series B., 58:551-563, 1996.  [12] D a v i d s o n , J . . Stochastic  Limit  Theory.  O x f o r d U n i v e r s i t y Press, N e w  Y o r k , 1994. [13] D e S i l v a , D . G . a n d T . D u n n e , a n d A . K a n k a n a m g e ,  and G . K o s -  m o p o u l o u . T h e I m p a c t of P u b l i c I n f o r m a t i o n o n B i d d i n g i n H i g h w a y P r o c u r e m e n t A u c t i o n s , forthcoming, [14] D o n a l d S. a n d H . P a a r s c h .  European  Economic  Review, 2007.  Piecewise P s e u d o - M a x i m u m L i k e l i h o o d  E s t i m a t i o n i n E m p i r i c a l M o d e l s of A u c t i o n s .  International  Economic  Review, 3 4 : 1 2 1 - 1 4 8 , 1993. [15] D o n a l d S. a n d H . P a a r s c h . I d e n t i f i c a t i o n , E s t i m a t i o n , a n d T e s t i n g i n P a r a m e t r i c E m p i r i c a l M o d e l s of A u c t i o n s w i t h i n the Independent vate V a l u e s P a r a d i g m . Econometric [16] G o u r i e r o u x , C . a n d A . M o n f o r t .  Pri-  Theory, 12:517-567, 1996. Statistics  and Econometric  Models.  C a m b r i d g e U n i v e r s i t y Press, C a m b r i d g e , 1995. [17] G u e r r e , E . a n d I. P e r r i g n e a n d Q . V u o n g .  Optimal Nonparametric  E s t i m a t i o n of F i r s t - P r i c e A u c t i o n s . Econometrica, [18] H a d l e W . . Applied  Nonparametric  Regression.  6 8 ( 3 ) : 5 2 5 - 7 4 , 2000. Cambridge University  P r e s s , N e w Y o r k , 1990. [19] H a i l e , P . a n d H . H o n g , a n d M . S h u m . N o n p a r a m e t r i c Tests for C o m m o n V a l u e s at F i r s t - P r i c e S e a l e d - B i d A u c t i o n s . N B E R W o r k i n g P a p e r 10105, 2003. [20] H a l l , P . a n d P a r k , B . and boundaries.  N e w m e t h o d s for bias c o r r e c t i o n at  The Annals  of Statistics,  endpoints  30:1460-1479, 2002.  [21] H e n d r i c k s , K . a n d J . P i n k s e a n d R . P o r t e r . E m p i r i c a l I m p l i c a t i o n s of E q u i l i b r i u m B i d d i n g i n First-Price, Symmetric, C o m m o n Value A u c tions. Review of Economic  Studies, 70(1):115-145, 2003.  [22] H o n g , H . a n d M . S h u m . Curse:  Increasing C o m p e t i t i o n a n d the W i n n e r ' s  Evidence from Procurement.  Review  of Economic  Studies,  6 9 ( 4 ) : 8 7 1 - 8 9 8 , 2002. [23] J e n n r i c h R . A s y m p t o t i c P r o p e r t i e s of N o n l i n e a r L e a s t Squares E s t i m a tors. Annals  of Mathematical  4 0 : 6 3 3 - 6 4 3 , 1969.  Statistics,  [24] Jones M . . S i m p l e B o u n d a r y C o r r e c t i o n for K e r n e l D e n s i t y E s t i m a t i o n . Statistics  and Computing,  3:135-146, 1993.  [25] K i e f e r J . . I n t e g r a t e d L o g a r i t h m A n a l o g u e s for S a m p l e Q u a n t i l e s W h e n Pn -> 0. Proc.  Sixth Berkeley  Symp. Math.  Statisti.  1:227-244,  Prob.,  1972. [26] K r a s n o k u t s k a y a , E .  Identification a n d E s t i m a t i o n i n H i g h w a y P r o -  curement A u c t i o n s under U n o b s e r v e d A u c t i o n Heterogeneity.  Working  P a p e r , U n i v e r s i t y of P e n n s y l v a n i a , 2003. [27] K r a s n o k u t s k a y a , E . a n d K . S e i m . B i d Preference P r o g r a m s a n d P a r ticipation i n Highway Procurement Auctions.  2006.  W o r k i n g paper.  U n i v e r s i t y of P e n n s y l v a n i a . [28] L a f f o n t J . , H . O s s a r d a n d O . V u o n g . E c o n o m e t r i c s of F i r s t - P r i c e A u c tions. Econometrica,  63(4):953-980, 1995.  [29] L e e , S. a n d O . L i n t o n , a n d Y . W h a n g .  T e s t i n g for S t o c h a s t i c  Mono-  t o n i c i t y . W o r k i n g P a p e r , L S E , 2006. [30] L e h m a n , E . L . a n d J . P . R o m a n o .  Testing  Statistical  Hypotheses.  S p r i n g e r , N e w Y o r k , 2005. [31] L e v i n , D . a n d J . S m i t h . American  Economic  Equilibrium in Auctions with Entry.  Review, 84(3):585-599, 1994.  The  [32] L i , T . E c o n o m e t r i c s of first-price auctions w i t f i e n t r y a n d b i n d i n g reserv a t i o n prices. Joural  of Econometrics,  1 2 6 ( l ) : 1 7 3 - 2 0 0 , 2005.  [33] L i , T . a n d L P e r r i g n e a n d Q . V u o n g . S t r u c t u r a l E s t i m a t i o n of the A f f i l i a t e d P r i v a t e V a l u e A u c t i o n M o d e l . The RAND  Journal  of  Economics,  3 3 ( 2 ) : 1 7 1 - 1 9 3 , 2002. [34] L i , T . a n d X . Z h e n g . P r o c u r e m e n t A u c t i o n s w i t h E n t r y a n d an U n c e r t a i n N u m b e r of A c t u a l B i d d e r s : T h e o r y , S t r u c t u r a l Inference, a n d a n A p p l i c a t i o n . 2005. W o r k i n g P a p e r , I n d i a n a U n i v e r s i t y . [35] L i T . , I. P e r r i g n e a n d Q . V u o n g . C o n d i t i o n a l l y I n d e p e n d e n t P r i v a t e I n f o r m a t i o n i n O C S W i l d c a t A u c t i o n s . Journal  98:129-  of Econometrics,  161, 2000. [36] L o f t s g a a r d e n D . a n d C . Quesenberry.  A N o n p a r a m e t r i c E s t i m a t e of a  M u l t i v a r i a t e D e n s i t y F u n c t i o n . The Annals  of Mathematical  Statistics,  3 6 : 1 0 4 9 - 1 0 5 1 , 1965. [37] M i l g r o m , P . Putting  Auction  Theory to Work.  Cambridge University  P r e s s , 2004. [38] M i l g r o m , P . a n d R . W e b e r . B i d d i n g . Econometrica,  A T h e o r y of A u c t i o n s a n d  Competitive  50(5):1089-1122, 1982.  [39] M o o r e D . a n d J . Y a c k e l . C o n s i s t e n c y P r o p e r t i e s of Nearest D e n s i t y F u n c t i o n E s t i m a t o r s . The Annals  of Statistics,  Neighbor  5:143-154, 1977.  [40] M o o r e D . a n d J . Y a c k e l . L a r g e S a m p l e P r o p e r t i e s of Nearest N e i g h b o r D e n s i t y F u n c t i o n E s t i m a t o r s . Statistical  Decision  Theory and  Related  Topics, pages 2 6 9 - 2 7 9 , 1976. [41] M u l l e r H . Biometrika,  Smooth  Optimum  Kernel Estimators Near  Endpoints.  78:521-530, 1991.  [42] M y e r s o n R . O p t i m a l A u c t i o n D e s i g n . Mathematics search, 6 : 5 8 - 7 3 , 1981.  of Operations  Re-  [43] Newey, W . K . .  K e r n e l E s t i m a t i o n of P a r t i a l M e a n s a n d a G e n e r a l  V a r i a n c e E s t i m a t o r . Econometric  Theory, 10:233-253,  1994.  [44] P a a r s c h , H . D e r i v i n g a n estimate of the o p t i m a l reserve price: A n app l i c a t i o n t o B r i t i s h C o l u m b i a n t i m b e r sales. Journal  of  Econometrics,  78(2):333-357, 1997. [45] P a g a n A . a n d A . U U a h . Nonparametric  Themes in M o d -  Econometrics.  ern E c o n o m e t r i c s . C a m b r i d g e U n i v e r s i t y P r e s s , N e w Y o r k , 1999. [46] P a k e s A . a n d D . P o l l a r d . S i m u l a t i o n a n d the A s y m p t o t i c s of O p t i m i z a t i o n E s t i m a t o r s . Econometrica,  57:1027-1057,  1989.  [47] P e r l m a n , M . O n e - S i d e d T e s t i n g P r o b l e m s i n M u l t i v a r i a t e A n a l y s i s . The Annals  of Mathematical  Statistics,  40(2):549-567, 1969.  [48] Pesendorfer, M . a n d M . J o f r e - B o n e t . E s t i m a t i o n of a D y n a m i c A u c t i o n G a m e . Econometrica,  71 (5): 1443-1489, 2003.  [49] P i n k s e , J . a n d G . T a n . T h e A f f i l i a t i o n Effect i n F i r s t - P r i c e A u c t i o n s . Econometrica,  73(l):263-277,  [50] P o l l a r d , D . Convergence  2005.  of Stochastic  Processes.  Springer-Verlag, New  Y o r k , 1984. [51] P o l l a r d D . N e w W a y s to P r o v e C e n t r a l L i m i t T h e o r e m s .  Econometric  Theory, 1:295-314, 1985. [52] R i l e y , J . a n d W . S a m u e l s o n .  O p t i m a l auctions.  The American  Eco-  nomic Review, 7 1 : 5 8 - 7 3 , 1981. [53] S a m u e l s o n , W . Letters,  Competitive Bidding with Entry Costs.  17(l):53-57,  [54] S i l v e r m a n B .  Economics  1985.  Density  Estimation  for  Statistics  and Data  Analysis.  C h a p m a n a n d H a l l , 1986. [55] S t e g e m a n M . . P a r t i c i p a t i o n C o s t s a n d Efficient A u c t i o n s . Journal Economic  Theory, 71:228-259,  1996.  of  [56] Stone C . C o n s i s t e n t N o n p a r a m e t r i c Regression. The Annals  of  Statis-  tics, 5: 5 9 5 - 6 2 0 , 1977. [57] T a n G . a n d O . Y i l a n k a y a . P a r t i c i p a t i o n C o s t s . Journal [58] v a n der V a a r t , A . W .  E q u i l i b r i a i n Second P r i c e Auctions with of Economic  Asymptotic  Theory.  Statistics.  Cambridge University  P r e s s , C a m b r i d g e , 1998. [59] W a g n e r T . S t r o n g C o n s i s t e n c y of a N o n - p a r a m e t r i c E s t i m a t e of a D e n s i t y F u n c t i o n . IEEE  Trans. Systems,  Man and Cybernetics,  3:289-290,  1973. [60] W h i t e H . Estimation,  Inference  and Specification  Analysis.  Economet-  rics Society M o n o g r a p h s . [61] W i l k s S. . Mathematical  Statistics.  W i l e y , N e w Y o r k , 1962.  [62] Y e , L . I n d i c a t i v e b i d d i n g a n d a t h e o r y of two-stage a u c t i o n s . and Economic  Behavior,  Games  pages 181-207, 2007.  [63] Z h a n g S., R . K a r u n a m u n i a n d M . Jones. A n I m p r o v e d E s t i m a t o r of the D e n s i t y F u n c t i o n at the B o u n d a r y . Journal Association,  94:1231-1241, 1999.  Of the American  Statistical  Appendix A  Proofs for chapter 1 A.l  Details of estimation method W e a k C o n s i s t e n c y : W e follow the hne  P r o o f o f P r o p o s i t i o n 1.1.  of p r o o f for t h e weak consistency of a regular N N estimate b y L o f t s g a a r d e n a n d Q u e s e n b e r r y (1965) [36]. T h e modifications l i e i n t w o folds: (i) we look at a u n i v a r i a t e case, a n d (ii) o u r proposed e s t i m a t o r is one-sided. N o t a t i o n : Sr : = {x\0 <x-x<r}.  T h e measure of Sr is d e n o t e d dr, w h i c h  s i m p l y equals r i n t h e u n i v a r i a t e case.  Therefore, t h e weak consistency i n  the p r o p o s i t i o n entails: (A.l)  4>{x) = limP{Sr)/dr, r—>0 i.e., for a n y a r b i t r a r y e > 0, there exists a R such t h a t iî r < R, \PiSr)/dr F u r t h e r denote r ^ ^ : =  - m\  (A.2)  < e.  I t is easy to show t h a t PiSr^^ ) — Ukj. has a  -x.  b e t a 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 ( T h e o r e m 8.7.1, p.236, W i l k s (1962)[61]). W e first show t h a t Ukj-ldr^^  —> 0 ( x ) i n p r o b a b i l i t y .  A n a p p l i c a t i o n of  T c h e b y c h e v i n e q u a l i t y y i e l d s Uk^. —» 0 i n p r o b a b i l i t y .  However, this can  o n l y h a p p e n w h e n dr^^ i î be as defined i n ( A . 2 ) . arbitrary  0 i n p r o b a b i l i t y , i.e., rfe^  0 i n probability. Let  T h e r e exists a t such t h a t for T > i a n d a n y  > 0, P{rk^  <R}>l-ri-  (A.3)  T h i s suffices to i m p l y t h a t UkT/dr,^^ —> (p{x) i n p r o b a b i l i t y . F o r the desired r e s u l t , we are left to show t h a t {T/kT}Ukj.  —> 1 i n p r o b a b i l -  ity. T h i s c a n be s h o w n by using the fact t h a t Ukj. has a b e t a d i s t r i b u t i o n along w i t h a n a p p l i c a t i o n of the T c h e b y c h e v inequality. S t r o n g C o n s i s t e n c y : W e follow the line of p r o o f for the a.s. consistency of a regular N N estimate by M o o r e a n d Y a c k e l (1976)[40]. W a g n e r (1973)[59] provides a n o t h e r p r o o f for the s t r o n g consistency  t h r o u g h a different  ap-  p r o a c h . H i s a p p r o a c h c a n also be modified for o u r one-sided N N estimates. B y n o t i c i n g t h a t Ukj. c a n be viewed as sample kr/T-tile  from T i i d uniform  r a n d o m v a r i a b l e o n the u n i t i n t e r v a l , i t c a n be s h o w n t h a t u n d e r A s s u m p t i o n 1.2(ii) '-^-.1 [cf.,  (A.4)  a..  K i e f e r (1973)[25], M o o r e a n d Y a c k e l (1976)[40]] T h i s f u r t h e r implies  t h a t Ukj, —» 0 a.s. W e first c l a i m t h a t -^0  = i n f { r : Ur > 0}  It is clear t h a t r^^, > 0 a.s. for each T.  a.s.  (A.5)  If for a n y a r b i t r a r y e > 0, rkj, > e  for a sequence of T at a sample point w, t h e n C/fe^ >  > 0 for these T  at Lj. Since Ukj, —> 0 a.s., t h i s c a n occur o n l y o n a set of ui h a v i n g zero probabilities. W e note t h a t Ur = P{x-x<r}=  I  (t>{x)dx.  (A.6)  A p p l y i n g the m e a n value t h e o r e m for integrals, there exists dx s a t i s f y i n g i n f (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 c o n t i n u i t y of results.  1 a.s. ( A . 5 ) together w i t h t h e  at x ensures t h a t dr —> (p{x) a.s. Therefore,  t h e desired  • F i r s t , we note t h a t t h e right differen-  P r o o f o f P r o p o s i t i o n 1.2. t i a b i l i t y of 0 at X allows us t o w r i t e  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 t h e A s s u m p t i o n 1.3 further impUes t h a t kll^[<Pixt)  - <P{^)] ^ 0.  ( A . 10)  B y t h e c o n t i n u i t y o f 0 a t x , as i n t h e p r o o f o f t h e previous  proposition,  we c a n w r i t e k It 4>i£.) = -Tj—<P{xt)  where xt~ x<  rfc„.  (A.ll)  UkT  M o r e o v e r , we have s h o w n t h a t rkj. —> 0 a n d  <p(x)  - ^(x)) = k ^ i ' - ^ - 1) + Ukr  1 i n probability.  k^ipl <P{x)  1 ) ^ . Ukr  (A.12)  Therefore, ( A . 1 0 ) i m p l i e s t h a t for the desired result, we o n l y need t o show 4 ' ' ( ^ - l ) - N ( 0 , l ) . Since Ukj, is t h e / j T - t h order statistic of T i i d u n i f o r m r a n d o m UuU2,...,UTon[Q,  1],  P„(a)  =  P[4/2(^_i)<a]  1 -I- akj. ' =  P[BT<kT],  (A.13) variables  w h e r e BT is t h e n u m b e r OÎUI,...,UT  f a l l i n g b e l o w TT^ =  —^''li/2  a n d has t h e  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 a s s u m p t i o n , Trr —» 0 a n d TTTT  oo,  so t h a t BT is a s y m p t o t i c a l l y n o r m a l . 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)]^''^, a n d n o t i c i n g t h a t kj. - TTTT O'T  (A.15)  a.  W e o b t a i n PT{O) —> ^ ( a ) , $ b e i n g t h e s t a n d a r d n o r m a l d i s t r i b u t i o n . T h i s completes t h e 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)  L o f t s g a a r d e n a n d Q u e n s e n b e r r y (1965) [36] show t h a t t h e second t e r m o n the r i g h t h a n d side of ( A . 1 6 ) converges t o zero i n p r o b a b i l i t y u n d e r A s s u m p t i o n s 1.1, 1.2(i) a n d 1.2(ii). If, i n a d d i t i o n . A s s u m p t i o n 1.2 ( i i i ) h o l d s , (1973) [59] establishes t h e a l m o s t sure consistency  Wagner  for t h e second t e r m o n  the r i g h t h a n d side of ( A . 1 6 ) . Therefore, for t h e desired result t o h o l d , i t suffices t o show t h e first t e r m o n t h e r i g h t h a n d side o f ( A . 1 6 ) t o converge.  T o t h i s e n d , we r e w r i t e the first t e r m on the r i g h t h a n d side of ( A . 16) kr/T  kr/T  kr/T  ^^^^^  kr/T  = where  x < x <  Xsr-  0,(fcj').  (A.17)  T h e second equaUty follows f r o m the m e a n value theo-  r e m a n d t h e second last e q u a l i t y follows f r o m the fact s h o w n i n t h e 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  4>a.T)  -  0(2)  — x =  Oa.s.{T~^)  •  1.5.  =  (?(^T) -  ^(2))  +  (0(£)  -  4>{^)  (A.18)  I n l i m i t , the b e h a v i o r of the t e r m i n the first bracket o n the r i g h t h a n d side of ( A . 1 8 ) is d o m i n a t e d b y the terms i n the second bracket as s h o w n i n the p r o o f of p r e v i o u s p r o p o s i t i o n .  {kT)2{<p{x)  — (j){x)) is s h o w n t o follow a n o r m a l  d i s t r i b u t i o n w i t h zero m e a n a n d variance of  (j){x)'^  b y P r o p o s i t i o n 1.2.  desired result therefore follows b y the a s y m p t o t i c equivalence l e m m a . P r o o f o f P r o p o s i t i o n 1.10.  The •  F o r the sake of consistency, we prove  the p r o p o s i t i o n i n the context of f i r s t ( h i g h ) - b i d a u c t i o n . C e l i k a n d Y i l a n k a y a (2006) [8] show t h a t for any ( s y m m e t r i c ) o p t i m a l auc-  t i o n , t h e o p t i m a l cutoff v* s h o u l d solve the f o l l o w i n g :  (A.19)  l^*~^-J^]F(^T''=^-  W h i l e , as s h o w n i n p r e v i o u s section, the e q u i l i b r i u m s h o u l d e n t a i l the cutoff, v^, as s o l v i n g the f o l l o w i n g : = «•  v'F{vT~^  (A.20)  F o r the desired r e s u l t , we need to show v* > v^. I n n e g a t i o n , we suppose v* < v^. T h e n it m u s t be the case t h a t F ( u * ) " ~ ^ < T h i s further implies to the s u p p o s i t i o n .  < [v* -  ^Tpp]  < v*, w h i c h c o n t r a d i c t s  • W e first show the a.s. convergence of X j .  P r o o f o f L e m m a 1.3.  t o X. N o t e t h a t Xj^^ > X^^^^ for every T > 1. T h i s means t h a t , for every s a m p l e p o i n t u i n the u n d e r l y i n g p r o b a b i l i t y space, X ^ j ( w ) > T h e r e f o r e , {Xj^^(a;),T' >  X^^^(a;).  l } i s a n o n - i n c r e a s i n g r e a l sequence, so i t has a  l i m i t , d e n o t e d b y X ( a ; ) . I n other words, the sequence of r a n d o m variables {X^^yT  > 1} converges a.s. to the r a n d o m v a r i a b l e X , a n d X  T  So i t w i l l suffice to prove t h a t X  >  I.  = £ a.s.  < X j ^ ^ for  T o t h i s e n d , for any  a r b i t r a r y e, ProhiXf^)  -x>e)  where F is X ' s C D F . S o , X ^ j —> i  =  Prob{Xf^^  =  [l-F{x  +  ->  0  T  as  >x  i n probability.  + e) e)f oo,  (A.21)  W e have j u s t s h o w n  t h a t X^^j —> X a.s., w h i c h also i m p l i e s X j ^ ^ —> X i n p r o b a b i l i t y . Since no sequence c a n have d i s t i n c t l i m i t s , X = ^ a.s. T h e r e f o r e , Xj^ ~^ £  N o w we are left to f i n d the convergence rate. D e n o t e QF as t h e q u a n t i l e f u n c t i o n of F a n d Yi = FiX'l'^^).  Then Xj^j = QF(I^I)-  B y m e a n value  t h e o r e m , we have Xf,)  -x  = QF{Y,)  -  QF{Q)  (A.22)  = Q'F{Y)YU  where t h e second e q u a U t y is b y the m e a n value t h e o r e m a n d Y is f r o m some n e i g h b o r h o o d o f 0. Since Yi is t h e smallest of T independent observations from the u n i f o r m d i s t r i b u t i o n o n t h e u n i t i n t e r v a l ,  "  ^  "  {T+l)...{T  + k) •  ^^-^^^  Therefore, i t follows t h a t t h e m e a n squared error of Yi is of order T ~ ^ , i.e., E F j ^ =  0(T-2).  T h i s further i m p l i e s t h a t Yi =  0{T-'^).  Since  Q'p{Y) — Q'p{0) —> 0 f r o m t h e fact t h a t (p (therefore, Q'p) is continuous i n a n e i g h b o r h o o d of 0. P u t t i n g a l l together, we have X j ^ - x = 0(^1) = o ( T ~ ^ ) . T h i s completes t h e proof.  A.2  •  Details of Identification Issues  G P V investigate some i d e n t i f i c a t i o n issues i n the first-price a u c t i o n s w i t h endogenous entry. T h e y consider a b i n d i n g reserve price i n first-price auction (Theorem 4 , p.548).  I n t h a t case, t h e y assume o n l y t h e n u m b e r of  a c t u a l b i d d e r s is observed, besides the bids. However, we e m p l o y a different f r a m e w o r k t o set u p t h e e n t r y process. Therefore, m o d i f i c a t i o n is needed t o a p p l y G P V ' s i d e n t i f i c a t i o n results to o u r e n t r y m o d e l .  L e t P denote t h e set of a l l absolutely c o n t i n u o u s d i s t r i b u t i o n s w i t h a n i n t e r v a l s u p p o r t i n R""". L e t G be the j o i n t d i s t r i b u t i o n o f  (61,6„),  where  n is t h e n u m b e r o f a c t u a l bidders. G a n d g denote t h e d i s t r i b u t i o n of 6j a n d its d e n s i t y f u n c t i o n , respectively. Define  Vi:=^{h,,G,N)  = hi + j ^ ^  if  bi>bp.  (A.24)  ( A . 2 4 ) c a n be v i e w s as the inverse b i d d i n g f u n c t i o n . T h e following p r o p o s i tions p r o v i d e the results for i d e n t i f i c a t i o n of F o n [vp, v] a n d the p a r t i c i p a t i o n cost K, respectively. P r o p o s i t i o n A . l (GPV,  2000) Assume  setting,  both the number of potential  bidders,  n, are observed.  auction  of bidders'  private  values F  private  of the observed equilibrium  with independent  participation  N, and the number  of  Let G * belong to the set P " with support  There exists a distribution is the distribution  in the endogenous  bidders,  bids in a  actual [0,6]".  S P such that G first-price  values and endogenous participation  sealed-hid if and  only if (1) G * ( 6 i , . . . , 6 „ ) = (2) The inverse inverse  nr=iG*(6i);  bidding function  is differentiable  (A.24)is strictly  on [vp,v] =  Moreover,  when F exists, it is unique  isfies F{v)  = [1 - F{vp)]G*{Ç-^{v,G*,N))  ^{•,G*,N)  is the quasi-inverse  ^{b, G*,N)  = b-\b,  increasing  on [0, b] and its  [^{bp,G*,N),^(b,G*,N)]. on the support for  of the equilibrium  of [vp,v] and it sat-  all v e [vp,v].  In  addition,  strategy in the sense that  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 .  D e n o t e u) as the event of p a r t i c i p a -  t i o n . T h e o r e m 1 i n G P V identifies c o n d i t i o n a l d i s t r i b u t i o n , F{-\u)) f r o m the observations.  Therefore, i t suffices to show t h a t there is a u n i q u e m a p p i n g  f r o m the c o n d i t i o n a l d i s t r i b u t i o n , F{-\uj), to the u n c o n d i t i o n a l d i s t r i b u t i o n , F.  Since we c a n observe n for each a u c t i o n , the n u m b e r of a c t u a l bidders, a n d TV, the n u m b e r of p o t e n t i a l bidders, i t is true t h a t F = F{-\w) • Pr{w) for any given a u c t i o n . T h e r a t i o , n/7V gives the p r o b a b i l i t y of p a r t i c i p a t i o n for the g i v e n a u c t i o n . Therefore, Pr{ui)  is now observed. Therefore, such a  u n i q u e m a p p i n g exists. F o r the second p a r t of the p r o p o s i t i o n , i t is needed to show t h a t (A.24) is i n fact a n inverse b i d d i n g f u n c t i o n .  T o this e n d , take the first-order  d e r i v a t i v e o f ( 1 . 3 ) w i t h respect t o v a n d rearrange t e r m s , t h e f o l l o w i n g is derived  l =  (..-M(iV-l)||g^  ^f  v.>v,  (A.25)  T h i s is i d e n t i c a l t o t h e e q u a t i o n ( 2 ) given i n G P V ( 2 0 0 0 ) [17]. T h e n t h e i r discussions a b o u t t h e inverse o f b i d d i n g f u n c t i o n follow. T h u s , t h e desired results are d e r i v e d b y e m p l o y i n g T h e o r e m 1 i n G P V . B P r o p o s i t i o n A . 2 Given cost, K, is identified borhood  the result  of Proposition  if and only if F is strictly  A.l,  increasing  the  participation  in a right  neigh-  ofvp.  B y t h e c o n s t r u c t i o n of t h e m o d e l ,  P r o o f for P r o p o s i t i o n A . 2 P r ( w ) = Pr[ivi  - bi){Fimax{vp,b'\bi)})f-^  -K>0].  (A.26)  Define _ Vi • [F(vp)]'^  Vi<Vp  1  D i f f e r e n t i a t e q{vi) w i t h respect t o vf.  F(up)^  *  Vi <Vp  T h u s , q{vi) is i n c r e a s i n g i n Vi a n d s t r i c t l y i n c r e a s i n g i n t h e r i g h t neighborh o o d o f Vp, so is t h e c d f o f qivi).  M o r e o v e r , Pr{uj)  a u c t i o n . T h e r e f o r e t h e desired result f o l l o w s . •  is observable for each  Appendix B  Proofs for chapter 2. P r o o f for L e m m a 2.1.  W e first show t h a t \QT - E ( 9 t ) l - » 0  u n i f o r m l y o n 0 as T —» oo.  T o this e n d , we a p p l y J e n n r i c h ' s u n i f o r m law  of large n u m b e r s ( J e n n r i c h (1969) [23], T h e o r e m 2).  S p e c i f i c a l l y , we need  to show there exists a f u n c t i o n D such t h a t \q{b,u,9)\ < D{b,u) 9 GQ  for a l l  a n d a l l b a n d u i n the s u p p o r t s of F a n d G. M o r e o v e r , E [ J D ( 6 , U ) ] =  / D{b,u)dF{b)dG{u)  < oo.  N o t i n g t h a t the second t e r m i n (2.9) is always  n o n - n e g a t i v e , we have \qib,u,9)\  < <  \{b-Xiu,9)f\ 2b'^ +  2X{u,9f /  <  2b'^ +  S  \ 2  S  g{u)  p^:=D{b,u).  F u r t h e r , we have  E [ Z ) ( 6 , u ) ] = E[262 + | ! ^ ]  = 2 j e{vf U{v;9)dv  + |ïï(/J"  where t h e last i n e q u a l i t y is guaranteed b y the finiteness of /  < 00,  and bidding  s t r a t e g y e.  N e x t , we w o r k o n the expression for E ( ç f ) . N o t e t h a t 'E,[qt{9)] is the same for every t € { 1 , 2 , ...T}  due to the p r o p e r t y of i.i.d.  i n our context.  We  therefore suppress the s u b s c r i p t t i n the analysis.  E(g)  =  _ E[b - Xie)f  =  E[b-i{9)f  -  +  1 ^ _ ^^ElY^iXM  E[6 -  =  nb-i{e)f  _ X{9))']  -E[i{e)-xie)]'^  +2E[6 =  -  - X{e)]  1(6)]' + ^Var[Xs{e)]  -  ^VarlXsid)] ~Var[Xs{e)]  = Qie),  where the second e q u a l i t y follows f r o m the u n b i a s e d e s t i m a t i o n of  Xs{6).  T h e t h i r d e q u a l i t y follows f r o m the fact t h a t b a n d s i m u l a t i o n d r a w s are i n d e p e n d e n t . T h e r e f o r e , 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.  F o r the desired r e s u l t , i t is equivalent  to s h o w i n g each c o m p o n e n t of A is finite.  T o t h i s e n d , it is enough  to  show CTQ, V a r [ X s o ] , Var[yso] a n d /io are finite. N o t e t h a t t h e finiteness of Coiv{XsQ,YsQ)  c a n be i m p l i e d if b o t h V a r f X s o ] < oo a n d V a r f Y s o ] < oo.  W e first show CTQ < oo. R e c a l l t h a t b is c o n t i n u o u s l y d i s t r i b u t e d w i t h o n t h e s u p p o r t V b - fb is finite a n d V b is a closed i n t e r v a l defined o n R + . T h e s e facts ensure t h a t CTQ < oo.  N e x t , we consider V a r [ X s o ] .  R e c a l l (2.7), the d e f i n i t i o n of X^o-  observe the f o l l o w i n g :  1^.^0  =  max[.^,),F-n/,^)]-^^^"^'^^-^''^""'^^ 5(uf)...^«)  5"  We  where u is the u p p e r b o u n d of V g . M o r e o v e r , g the d e n s i t y f u n c t i o n of u is finite.  T h e r e f o r e , t h e second m o m e n t of Xgo exists.  S i m i l a r l y , for V a r [ F s o ] , we notice iinoii  where V%  <  \\ui^)ym/g{u)\\  +  \\vmF-\e)-vF{e)\\  := sup„g[j,^^]_ege l|V/^|| a n d V F := sup„g[„_5]_e6e l|VF||.  The  existence of s u c h u p p e r b o u n d s are i m p l i e d b y the a s s u m p t i o n s . A g a i n , as u is c o n t i n u o u s l y d i s t r i b u t e d w i t h g, w h i c h is  finite.  T h e s e facts together  i n d i c a t e t h a t V a r [ y s o ] < ooL a s t l y , we show /io < oo. T o this end, we note t h a t KOo) =  /  ••• /  max{u?2),F  ^(p,eo)fviui;do)---fv{un;Oo)dui...dun,  w h i c h c a n be r e w r i t t e n as iieo) = Loieo) + Li{eo) + ... +  Lsieo),  where  LoiOo)  =  ... Ju  F-\p-M Ju  fv{ui;Ôo)...fv{un;9o)dui...dun  JF-^(P,0O)  JU  JU  fv{ui;9o):. fv{ui;9o)...fviun;9o)dui...dun F-^pfio)  L2{9o)  =  / JP'-^ipfio)  /  JF-^(pfio)  rF-^pfio)  ... /  Ju  Ju  fviui;9o)...fviun;9o)dui...dun  Lsi9o)  =  ... r  r Jp-Hpfio)  u^2) JF-Hpfio) fv{ui;9o)...fv{un;9o)dui...dun.  N o t e t h a t Lo(^o), •••,Ls{9o) are a l l differentiable w i t h respect to 9o.  Then  the d e r i v a t i v e of l{9o) is finite, p r o v i d e d t h a t L's a n d t h e i r the derivatives are finite. F u r t h e r note t h a t F~^,  f^, U(2), V / t , a n d V F are a l l finite. T h e s e  facts together are e n o u g h to i m p l y t h a t /XQ < oo.  P r o o f for P r o p o s i t i o n 2.2.  •  T h e p r o p o s i t i o n follows d i r e c t l y  f r o m the t h e o r e m due to W h i t e (1994) [60] [Theorem 3.4 o n 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 e x p o s i t i o n , we s i m -  p l i f y the n o t a t i o n s i n the f o l l o w i n g way. Xs{9) Xt{v},...,u^,n,9).  •  denotes X{u^,n,9).  X{9)  is  T h e s u b s c r i p t 0 denotes the r a n d o m variables are e v a l u -  a t e d at ^0- T h e s u b s c r i p t t is o m i t t e d . T h e a r g u m e n t s i n t h e f u n c t i o n s m a y be suppressed whenever there involves no confusion.  L e t Wsi0) = max[u%.,F-^p;e)].  q{e)  Then,  -, 2 u>s{e)fyie)  =  n2  1 «=1  and  = î^^^^^,  = 1 E l l Xs{e).  Denote f2)V/(^)/fl(«) _ J ""(2)  a n d Y = {1 /S) J2Li  Ys{0).  N o t e 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]-  -  F o l l o w i n g d i r e c t l y f r o m T h e o r e m 2 i n P o l l a r d (1985)[51], t h e p r o o f o f the p r o p o s i t i o n a m o u n t s t o checking t h e f o l l o w i n g r e g u l a r i t y c o n d i t i o n s are satisfied: (i) E[q(0)] h a s a n o n s i n g u l a r second d e r i v a t i v e J at 9Q. (ii) E A = 0 a n d E ( A A ' ) < oo, where A ( 6 , u ) = V g ( 6 , u , 6 l o ) . (iii) S t o c h a s t i c d i f f e r e n t i a b i l i t y c o n d i t i o n as defined i n t h e e q u a t i o n (4) o f P o l l a r d (1985) holds for r ( 6 , u , 6 l ) , where  . ( M , ^ ) 4 ^ ^ ^ ^ ' ^ - ^ ^ ' ° ^ - ^ ' - ' ° ^ ' ^ ^ ' ' \ 0  ^ = ^0  (B.l)  F i r s t , we show t h e stochastic d i f f e r e n t i a b i l i t y c o n d i t i o n holds for T o t h i s e n d , we first e s t a b l i s h a result t h a t 'E[r{.,d)'^]  0 as 9  r{.,9).  OQ. N o t e  that  where Hi  =  [b-X{9)f-[b-X{9o)f  H2  =  [Xsie)-Xi9)f-[Xsi9o)-Xi9o)]''  +  2{9-9oy{b-Xo)Yo +  2i9-9on{Xso-Xo)Yso]  D e p e n d e n t o n t h e r e a l i z e d values of (6, u), there exists a r e a l e > 0 such t h a t q is differentiable o n t h e 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 e n s u r e d b y t h e fact t h a t 9o is non-differentiable w i t h p r o b a b i l i t y zero. T h e n , t h e m e a n value t h e o r e m i m p l i e s t h a t there exists 9 i n t h e b a l l such t h a t  where t h e convergence happens because of t h e continuous d i f f e r e n t i a b i l i t y of / „ i n t h e a s s u m p t i o n a n d t h e fact t h a t (9 — 9o)/\\9 — 9o\\ is b o u n d e d b y a u n i t b a l l . S i m i l a r l y , i t c a n be s h o w n t h a t established t h a t r{.,9)  0. T h e r e f o r e , we have  —+ 0 as 9 —> ^o- F u r t h e r note i t is also true t h a t  r{.,9f^0as9^9o. N o w we s h o w t h a t there exists a f u n c t i o n W such t h a t \r{9)\ < W for a l l  6 e@  a n d E [ W j < oo. W e notice t h e f o l l o w i n g  IXsiOo) - Xs{e)\  1  \\e - eoW  \\e - 9o\\  Ws{9)Ue)/g{u)  ^  -  wsieo)fM/giu)  •i{uL.>F-\e)}ufMe)/9iu)  +i{u%, < ^'(2)  F-Ho)}F-\e)ue)/g{u)  ~l{ul2^>F-\6o)}ul2)Meo)/g{u)  <  2  -l{ii^2) <  F-'(9o)}F-\eo)Meo)/9{u)\  \F-H6)M9)/g{u)  -  ll^-^oll +2  where 1{.}  F-\eo)fM/g{u)\  l»(2)/.W/g(")-»(2)^(^o)/gM  ll^-^oll  is n o t a t i o n for t h e i n d i c a t o r f u n c t i o n w h i c h takes t h e value of  one i f t h e l o g i c a l c o n d i t i o n inside i t is satisfied, a n d zero elsewhere. B u t - F-\eQ)U{e(>)/g{u)\/\\e - 04 is n o t h i n g else t h a n t h e  \F-^{e)U{e)/9{u)  slope of t h e f u n c t i o n F"^{6)fy{6)/g{u) uniform Lipschitz property on  w i t h respect t o 6.  T h e assumed  ensures t h a t there m u s t exist W\ such  that \F-'{9)U{e)/g{u)  -  F-\eo)fvi9o)/9{u)  <  S i m i l a r l y , there m u s t also exist W2 such t h a t ul^^Ue)l9{u)-ul^^fMl9{u)\ < H'2-  T h e r e f o r e , we have \Xs{eQ) -  Xs{6)\  <2{Wi  + W2).  Wi.  N e x t , we consider \Xs{0) + Xs{Oo)\ for a n y 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 t h e u p p e r b o u n d o f Vg. / „ is t h e m a x ; i m u m of / „ , whose existence is a g a i n i m p h e d b y t h e a s s u m p t i o n of u n i f o r m L i p s c h i t z p r o p e r t y o n / „ . T h e last i n e q u a l i t y follows f r o m t h e fact t h a t Vg contains t h e set [u,v]W e w r i t e r{d) as t h e following  r{.,e) =  1  \\o-eo\\  1 SiS-1)  s  ^iC2-(^-eo)'A] s=l  where Ki  =  K2  =  [b-X{6)f-[b-Xo]^ [Xso-Xo?-[x,ie)-xie)]|2  Note that 1 -A[2b-Xo-x{e)][Xo-x{e)]\ we-eoV  1^11  \\9-eo\\ =  2\b\ ^ -  - x(e)) - ( X o - x ( ^ ) ) ( X o + x{e))\  pZ-ô^\\^^(^o  |X,o -  S ^i  Xs{6)\  P-^o\\  s  s  s  s=l  s=l  s=l  and 1 Mx,o \\6-eo\\'  1^21  \\e-eo\\  + Xs{e))-{Xo  + x{e))]  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 +  + ixo + x ( . ) | F ° -  S  s=l  52 s=l  s=l  F o l l o w i n g the facts t h a t fy is s t r i c t l y positive a n d u n i f o r m l y L i p s c h i t z a n d t h a t (6 - Oo)/\\0 - 6o\\ is b o u n d e d by a u n i t b a l l , there must exist  such  that \\0-ôo\\  -  '  N o t i n g the fact t h a t r is continuous i n K\,K2 s h o w n t h a t there exists a f u n c t i o n W \r{OY\ < r{.,9y  a n d A , we therefore  :=  have  W2, M^3i W^4) such t h a t  for a l l 6» G G a n d E[W'^] < 00. Together w i t h the result t h a t  —> 0 as 9 —» ^0, the d o m i n a t e d convergence t h e o r e m i m p l i e s t h a t  E[r(.,e)2]-.0  as  (B.2)  9 ^ 9o.  L e t i? be a class of functions R = {r(., 9) : 9 G © } . F o r the desired stochastic differentiability c o n d i t i o n , we are left to show t h a t r{.,9) has the stochastic e q u i c o n t i n u i t y p r o p e r t y , t h a t is for r G i? a n d each e > 0, there always exists a, Ô > 0 such t h a t l i m sup P r o 6 { sup |t>rr| > e} < e,  (B.3)  p{r)<S  where p{r)  = ( E r ^ ) ! / ^ a n d vrr = T~'^/^ Tj^iin-Er).  T o t h i s e n d , we next  show t h a t R has the E u c l i d e a n p r o p e r t y m e n t i o n e d i n P a k e s a n d P o l l a r d (1989) [46]. N o t e t h a t for a n y given {b,u)  G V b x V g ^ , the p o i n t s of 9 where q is  non-differentiable f o r m a closed set w i t h o u t i n t e r i o r s .  © is separated by  such a closed set into finite n u m b e r s of subsets. T h e r e s u l t i n g p a r t s c o n t a i n p o i n t s such t h a t q is always differentiable. U s e i G {1,2..., 2'^} to i n d e x these sets. W e first show t h a t for 9 G 6 ' , the H e s s i a n of q, V^q{b,u,9)  is b o u n d e d  b y a p X p m a t r i x (f){b, u). N o t e t h a t , for a n y g i v e n (6, u) a n d 9 G S  q{9)  n 2  ~  = s=l  S 5 ( 5 - 1) ^  r~ [  9{u)  where ûJs takes o n t h e value either u^^^^ or F~^{p;9), the m a x o p e r a t i o n .  -, 2  Ws{m:i9)  a n d n o longer involves  Therefore, q is twice differentiable e v e r y w h e r e o n 0 \  p r o v i d e d t h a t Fy a n d fy are differentiable. W e t h e n get a p x p m a t r i x for  V^q{9)  =  {b-Yi9)yY{9)  J2  +  ib-X{9))VY{e)-{-  [(n(^) - y ( ^ ) ) ' n ( e ) + iXsi9)  5(5-1)  -  Xi9))VYsie)]iBA)  s=l  where VYs{9) Denote dF{9)/d9j.  = dYs{9)/d9  and V y ( ^ ) = l / 5 X ^ f ^ i Vys(6l).  t h e j - t h element i n t h e vector 9.  fyj = dfy{9)/d9j.  Fj =  Ysj is t h e j-th element i n Ys- T h e b o u n d e d H e s s i a n o f / „ i m p l i e s  t h a t t h e g r a d i e n t o f fy is b o u n d e d b y a constant vector as w e l l . W e use t o represent s u c h a b o u n d i n g vector a n d its j - t h element is Zj. define t h e b o u n d i n g vector for t h e gradient o f F as  S i m i l a r l y , we  a n d i t s j - t h element  T o show t h a t V^q is b o u n d e d , we notice t h e f o l l o w i n g :  \Ysm  <  < <  u'f, V]  fyj{9)F-Hp,9)-Fj{9) 9{u)  9{u) 9{u) 9{u)  \fvj  J  + +  1 9{u)  IF,  1 9{u)  N o w consider t h e j A ; - t h element of V l ^ ( & ) .  D]  •.= Mb,u)  Denote Z the bounding  m a t r i x for V ^ / „ a n d F t h e b o u n d i n g m a t r i x for V ^ F . W e use s u b s c r i p t jk  for the jk-th element i n the c o r r e s p o n d i n g m a t r i x .  <  9{u)  + < <  1  + v  +v 9{u)  1  îv{e)9{u)  9{u)  9{u)  z]\\Dl\  \Zik\ +  +  1 9[u)  3k  Djk\ ••= 4>i{b,u)  Furthermore,  9W  (B.4) suggests t h a t 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 m u s t exist a c o m p o s e d f u n c t i o n (j){b, u) := (j>{(j)\{b, u), 4>2{b, u), cj)2,{b, u)) such t h a t |V2<7(6,u,6l)| <  Since i is chosen a r b i t r a r i l y a n d (i){h,u)  does not d e p e n d o n i , the c o n t i n u i t y of q over 0 (j) : JR-^+i  i m p l i e s the existence of  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 n e x t e s t a b l i s h the result t h a t r is a L i p s c h i t z f u n c t i o n .  Pick 6 e  0\{0o}. T h e n we have t h a t  Vr{b,u,e)  =  Vg(b,u,g)-Vg(b,u,go) ll^-^oll  q{b, u, 9) - q{b, u,9o)-  V'qjb, u, 9o)id - OQ)  Note that  \\o-eo\\ ^q{b,u,9)-Vqib,u,9o)  <4>{b,u).  A l s o , note t h a t for some 9 o n t h e Hne segment j o i n i n g 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)\\ <(j>{b,u) \\9-e4  \\9-9o\\  \\9-94 (B.5)  <4>{b,u) \\9-9of. It follows t h a t q{b, u, 9) - q{b, u, 9o) - Vq{b, u, 9Q){9 - OQ) \\e-9or T h u s , we have t h a t ||Vr(6,«,^)||<20(fe,w). Because  {b,u) G V b x V g ' ^ a n d 9 G 6\{6'o}, i t follows  {b,u) G V b X V g ' ^ a n d 9Q G 9 , r ( 6 , u , ) : 6 ^  t h a t for each  M is a L i p s c h i t z func-  t i o n o n 6\{^o} w i t h t h e L i p s c h i t z constant set equal t o 2 0 ( 6 , u ) .  F u r t h e r , i t follows f r o m (B.5) t h a t for each 9 G 0\{^o} a n d each {b,u) G V b X Vg-^ that \r{h,u,9)-r{b,uM\  = \r{h,u,9)\ _  =  1  q{b, u, 9) - q{b, u, 9o) - V ' g ( 6 , u, 9o) {9 - 9o) \  cj>{b,u)\\9-94<24>{b,u)\\9-94.  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 t o 20(6, u ) .  T h e L e m m a 2.13 i n P a k e s a n d  P o l l a r d (1989)[46] i m p l i e s t h a t R is E u c l i d e a n for t h e envelope n{b,u) := 2 y ^ s u p 0 ||^-^o||0(6, u ) . Since 9 is c o m p a c t , t h e n supe ||^-^o|| < 0 0 . M o r e over, t h e finiteness of 0(6, u) is i m p l i e d b y t h e f o r m u l a t i o n of 0 i , 02 a n d 0 3 . A l l these facts together ensure t h a t ETZ < 0 0 , w h i c h f u r t h e r i m p l i e s t h a t  (B.3) h o l d s b y the L e m m a 2.16 i n P a k e s a n d P o l l a r d (1989)[46].  F u r t h e r m o r e , (B.2) a n d (B.3) together i m p l y t h a t sup|i;Tr(.,^)|-^0  (B.6)  UT  for each sequence of b a l l s {UT}  t h a t center at 9o a n d s h r i n k d o w n to 0. B y  n o t i n g t h a t (B.6) is a stronger c o n d i t i o n t h a n ( B . l ) , we have s h o w n t h a t the r e m a i n d e r f u n c t i o n r{9)  satisfies the stochastic d i f f e r e n t i a b i l i t y c o n d i t i o n as  desired. N e x t , we show the d i f f e r e n t i a b i l i t y 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. H o w e v e r , notice  t h a t E[ç] c a n be w r i t t e n as S folds of integrals, each of w h i c h represents the a v e r a g i n g over the r a n d o m v a r i a b l e u^^^y  T h e each of s u c h integrals  c a n be s e p a r a t e d i n t o t a k i n g averaging over two sets: a n d tt*2) < F~^{P;9Q),  u^2) >  because the event U(2) — F~^{p;9o)  F~^{P\9Q)  happens w i t h  p r o b a b i l i t y zero as u is c o n t i n u o u s l y d i s t r i b u t e d w i t h d e n s i t y g.  Further  note t h a t a l l the terms i n E[g] t h a t involves 9 are t w i c e differentiable by a s s u m p t i o n s . T h e s e facts together ensure the twice d i f f e r e n t i a b i l i t y of E[g] at  w h i c h is d e n o t e d as J. T h e n o n s i n g u l a r i t y of J is i n the a s s u m p t i o n .  N o w we show t h a t E A = 0. E A = EWeq{9o) = V e E ç ( ^ o ) = ^eQ{9o)  = 0,  where t h e l a s t e q u a l i t y follows f r o m the i d e n t i f i a b i l i t y a s s u m p t i o n .  The  second e q u a l i t y follows f r o m the i n t e r c h a n g e a b i l i t y of e x p e c t a t i o n a n d diff e r e n t i a t i o n , w h i c h i n t u r n follows f r o m the fact t h a t d i f f e r e n t i a t i o n of Efg] w i t h respect to the 9 o n the l i m i t s of integrals is e q u a l to zero.  To complete  the p r o o f for the t h e o r e m , now it r e m a i n s to derive the  covariance m a t r i x , S .  B y the t h e o r e m due to P o l l a r d (1985)[51], S  =  J-iE(AA')J-^ W e first consider J . W e note t h a t A = dq{6)/d9o. o b t a i n J = - 2 ( J i + J2 + J3 + Ji),  D i f f e r e n t i a t i n g E A , we  where  Ji  =  J2  =  E[(6-Xo)^],  =  1 ^ E[-^^^r-^5^no(no-yo)'],  =  E [ ^ | : ( X . o - X o ) ^ ] .  -EIFOFQ],  N o w for a n y s £ { 1 , 2 ,  _  [Xsa^yso)  are i.i.d.  a n d i n d e p e n d e n t of b.  T h u s , u s i n g the fact t h a t E[6] = E[X3o] = l{0(i), we get Ji  J2  J3  =  -[Var(Fo)+EFoEy[)]  =  -(ivar(y,o) + Er,oEy;o),  =  E[a(0o)-Xo)^]  =  - - C o v ( X . o , ^ ) ,  =  -^Var(no),  C a n c e l i n g t e r m s a n d u s i n g the E y ^ o = dEXso/dO  = fiQ b y i n v e r t i n g differ-  e n t i a t i o n a n d e x p e c t a t i o n . Therefore, we have J = 2non'o. W e now t u r n to c o m p u t e E ( A A ' ) . D e n o t i n g /xxo = '(^0) W e have A = - 2 ( > l i - ^ 2 - ^ 3 + >l4),  a n d /xyo =  dl{9o)/9.  where Al  =  A2  =  A4  =  {b-fixo)Yo, (Xo  g  -  1 _  p.xo)lJ-Yo,  ^ YliXsO  -  fJ-xo){Yso -  Myo)-  Because E A = 0, E ( A A ' ) = V a r [ A ] . N o t e t h a t , for j = = E [ ( 6 - fixo)YoA'j]  Cov{Ai,Aj)  2,3,4,  = 0,  and  Cov[^2,^3]  =  1 ^ g2(g _ ^ ) m E [ ( ^ ( X , o  - Mxo)' s  +2 ^ ( X , o - M x o ) ( ^ t o - f^xo))iJ2(^so s<t S{S ^Cov[^2,^4]  =  s=l  1)  fJ-YOmXsQ-fiXO?{YsO-fJ-YoY],  1 ^ 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 ( V a r [ A i ] + Varf/la] + V a r [ ^ 3 -  ^4]).  -  MKo)')]  N o w i t r e m a i n s t o c o m p u t e t h e variance m a t r i c e s of Ai,A2  a n d {A3 — A4).  N o t e t h a t these vectors have zero e x p e c t a t i o n s . Since <JQ = E [ ( 6 — p-xo)'^]: we o b t a i n Var[Ai]  =  Var[yl2]  =  a ^ ( | v a r [ n o ] +/iyo/xVo), ^Var[Xso]fiYO(J-'Yù-  F o r t h e t h i r d v a r i a n c e 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[A4A^]. But,  s s<t  s=l  +2Y,{Yso-^iYo){Yto-^iYo)')] s<t  ^  '  s  s  s=\  s=l  + 4 ( ^ ( X , o - / x x o ) ( X t o - / i x o ) E(^-o  - /^yo)(Fto - /iyo)')]  1 =  g 2 ( g , i ) 2 E [ E ( ^ ^ Q - ^^0)^(^0 - Myo)(no - /iyo)' + "^{Xsù  - /xxo)^(l^to - Myo)(i'«o - M y o ) '  + 4 E("''-^o ~ AiA:o)(^to - A'xo)('5^so - Myo)(yto - /^yo)']  =  g ( g ^ 1 ) 2 ^ 1 ( ^ ^ 0 - / x x o ) ^ ( n o - M y o ) ( n o - /iyo)'] + ^ ^ ^ ^ [ V a r X , o V a r n o + 2Cov(X,ono)Cov(X,ono)'],  + ^{Xso  - Mxo)(^to - lJ-xo){Yso - ^Yo){Yto - HYO)']  g ( g ^ _ i ) 2 ^ [ ( ^ « o - M x o ) ' ( n o - M y o ) ( n o - Myo)']  =  ^~^^Cov{XsoYsu)Cow{XsM',  1 ^4]  =  g2(5 _  i)2^[(I](-^^Q  l^C^so - /ixo)(yto si^t =  ~ iJ^xaWsQ  - /xro) +  m))(E(^«o -  M x o ) ( n o - Myo)')]  s=i  1 S2(g _  ^  i)2^[(IZ(^«o  ~ /^xo)(>".o - /^yo))  s=l  =  E[A4A^].  T h e r e f o r e , we o b t a i n Var[^3-^4]  =  E[^3^^] - E[^4A^]  =  JiJZY)  [VarX,oVarno + Cov(X,oV;o)Cov(Xsono)'l  Thus,  Var[A]  =  4 { a g ( | v a r [ n o ] + HYOP'YO)  +  |var[Xso]/xyoMyo  + g^g^_^^[VarX,oVary,Q + Cov(X,oyso)Cov(X,ono)']} =  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 + Cov(X,ono)Cov(X,ono)']},  where t h e last e q u a l i t y follows b y o b s e r v i n g t h a t / i y o = dl{6o)/6 = /io-  •  Appendix C  Proofs for chapter 3 C.l  Details of the entry models  T h e f o l l o w i n g l e m m a is used for t h e p r o o f o f P r o p o s i t i o n 3.3. Lemma C . l Further,  The function  the function  Tl{s,s,N)  ]I{s,s,N)  s. For s G [ 0 , 1 ) , ll{s,s,N)  is a non-decreasing  is a continuous  is a decreasing  function  and increasing  function  of s.  function  of  of N, and constant  in  N ifs = 1.  C.l.  P r o o f for L e m m a  S u b s t i t u t i n g P ( s ) a n d F*{v\s)  from  (C.2) a n d ( C . 3 ) r e s p e c t i v e l y i n t o ( C . 6 ) gives after some m a n i p u l a t i o n t h e f o l l o w i n g e x p r e s s i o n for  j\\  \l{s,s,N):  - F{v\s)){\ - Pr{Si  >-s,Vi>r)  + Pr{Si  > s,  S [r,  V\)f-^dv. (C.l)  N o t e t h a t b y t h e affiliation o f Vi a n d Si, F{v\s) is n o n - i n c r e a s i n g i n s ( T h e o r e m 23 i n M i l g r o m a n d W e b e r (1982)[38]), so 1 - F{v\s) is n o n - d e c r e a s i n g i n s a n d c o n s e q u e n t l y n ( s , s, N) is non-decreasing i n s. A l s o , t h e t e r m w i t h i n the second parentheses i n ( C . l ) is i n c r e a s i n g i n s w h e n v €  i^Pr{Si  >s,Vi£  [r, V]} - Pr{Si  (^j\nv\s)  - F(r|5)]^-  =  ^  =  F{r\s) - F(v\s) +  =  1 - F{v\s) > 0,  F{r\s)]  >s,Vi>  {v,v):  r}]  F{r\s)di^  because F{v\s) 6 (0,1) i f u S {v, v). It follows t h a t II(s, s, N) is a continuous a n d i n c r e a s i n g f u n c t i o n of s on [0,1). A l s o observe t h a t for a n y s G [0,1), the t e r m w i t h i n t h e second parentheses 1 - Pr{S^ so t h a t II(s, s, N)  >s,Vi>r}  + Pr{Si  > 5, K € [r, V]}  is a decreasing f u n c t i o n of N.  € [0,1),  •  F i x b i d d e r i a n d assume t h a t his  P r o o f for P r o p o s i t i o n 3.3.  r i v a l s follow t h e i r e q u i l i b r i u m strategies represented by a cutoff s.  From  b i d d e r i ' t h v i e w p o i n t , c o n d i t i o n a l o n e n t r y he is p a r t i c i p a t i n g i n a first-price a u c t i o n w i t h a r a n d o m n u m b e r of bidders.  Specifically, the n u m b e r of his  rivals follows a b i n o m i a l d i s t r i b u t i o n w i t h parameters A'^ a n d the p r o b a b i l i t y of b i d d i n g e q u a l to P{s)  (C.2)  = Pr{Si>s,Vi>r}.  (we suppress t h e dependence of s on i V for n o w ) , a n d the r i v a l s have  iid  d i s t r i b u t e d v a l u a t i o n s according to F*{v\s) = Pr{Vi N o t e t h a t i f s = 1, t h e n P{s)  < v\Si >s,Vi>  r}.  = 0 a n d F*{v\s) is not defined.  (C.3) Assuming  existence of a b i d d i n g e q u i l i b r i u m i n w h i c h a l l bidders use the same b i d d i n g s t r a t e g y B : [v,v]  —* R+,  a n increasing f u n c t i o n , a b i d d e r w i n s against a  g i v e n p o t e n t i a l r i v a l either if the r i v a l does not b i d , or b i d s b u t his v a l u a t i o n c o n d i t i o n a l o n b i d d i n g is less t h a n v. T h i s p r o b a b i l i t y is e q u a l to 1 — P{s) P{s)F*(v\s).  +  B y independence, the p r o b a b i l i t y of w i n n i n g the a u c t i o n for a  b i d d e r w i t h v a l u a t i o n v is (1 - Pis)  + P{s)F*{v\s))'^-K  (C.4)  S t a n d a r d e n v e l o p e - t h e o r e m a r g u m e n t i m p l i e s t h a t z ' t h profit c o n d i t i o n a l o n r e c e i v i n g t h e s i g n a l Si = s at t h e b i d d i n g stage is e q u a l t o J\l-Pis)+Pis)F*{v\s)f-'d^, A s s u m i n g p r o v i s i o n a l l y t h a t a n e q u i l i b r i u m given b y a cutoff s exists, t h e b i d d i n g s t r a t e g y B{v) c a n b e f o u n d f r o m t h e a l t e r n a t i v e expression for t h i s profit, {v - B{v)){l  - Pis) +  P{s)F*iv\s))''-\  w h i c h gives t h e 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 a r g u m e n t s (e.g. M i l g r o m (2004)[37]) i m p l y t h a t B{-) is a n increasi n g f u n c t i o n , a n d is i n d e e d a best response.  T h e e x p e c t e d profit a t t h e e n t r y stage  n(s,s,N)  =  £  =  J\l  f{v\s) J\l-P{s) - Fiv\s))il  +  P(s)F*{v\s))^-^éiidv-k  - P{s) + P{s)F*iv\s))^-'^dv  - k,  where t h e last line follows b y i n t e g r a t i o n b y p a r t s .  If t h e reserve p r i c e is b i n d i n g , t h e n t h e b i d d e r w i t h t h e lowest  active  t y p e is t h e one w i t h t h e lowest t y p e possible, i.e. v. U n l i k e i n t h e b i n d i n g reserve p r i c e case, t h i s b i d d e r n o w makes a p o s i t i v e profit a n d w i n n i n g t h e a u c t i o n i n t h e event w h e n no one else chooses t o enter, i.e. w i t h p r o b a b i l i t y (1 - P{s))^~^{v U{s,s,N)  - r). A p p h c a t i o n of the E n v e l o p e T h e o r e m n o w results i n  =  {l-P{s))'^-\v-r) + J\l  - F ( i ; | s ) ) ( l - P ( s ) -t- P ( s ) P * ( ï ï | s ) ) ^ - i d ï ï - k,  a n d the b i d d i n g s t r a t e g y m u s t be m o d i f i e d accordingly. T h e c r u c i a l q u a n t i t y w i l l be the m a r g i n a l b i d d e r ' s profit n ( s , s, TV), i.e. w h e n b i d d e r i has a s i g n a l e q u a l to the e q u i l i b r i u m c u t o f i ' s . Pi{s,  s, N)  Note that  defined even i f s = 1 (no rivals enter), i n w h i c h case i t does not  d e p e n d o n A^: n ( l , 1, N) = j \ l  - F{v\s))éS  -  k.  N e x t , note t h a t i n v i e w of L e m m a C . l , for a g i v e n N >2,  either n ( 0 , 0 , TV) >  0, so t h a t a n e q u i l i b r i u m w i t h c u t o f f s = 0 exists, or 11(1,1, TV) < 0 so t h a t a n e q u i l i b r i u m w i t h cutoff s = 1 (no entry) exists, or a n e q u i l i b r i u m such t h a t a b i d d e r w i t h s G (0,1) t h a t solves the indifl?erence e q u a t i o n n ( s , s, TV) = 0 exists. T h i s i m p l i c i t l y defines the e q u i l i b r i u m cutoff s as a f u n c t i o n of TV, s(TV). Since n ( s , s , TV) is a n increasing continuous f u n c t i o n of s a n d a decreasing f u n c t i o n of TV, t a k i n g o n the same value for a l l TV if s = 1, it follows t h a t if s(TV) G ( 0 , 1 ) , t h e n also s(TV) G (0,1) for N' > TV, a n d s{N')  > s(TV).  •  C o n s i d e r the o n l y i f p a r t first. T h e r e  P r o o f for P r o p o s i t i o n 3.4. are two cases to consider.  F i r s t , suppose t h a t p{N)  < ... < p{N).  Denote  for future reference w i t h i n this p r o o f  T(TV) = J\l  - F*{v))(l-p{N)  +p{N)F*{v)f-^dv.  W e k n o w t h a t for TV > TV, s (TV) G (0,1) a n d the m a r g i n a l b i d d e r is indiflï'erent between e n t e r i n g or n o t , fc = [1 - F ( r ) ] r ( A ^ )  V TV > TV,  (C.6)  F o r TV = TV, i t m a y be either t h a t s(TV) = 0, so t h a t b i d d e r s enter w i t h p r o b a b i l i t y 1, or s(TV) G ( 0 , 1 ) . 1 - F{r)  T h e key o b s e r v a t i o n is t h a t the q u a n t i t y  is not i d e n t i f i e d , so t h a t (C.6) cannot be used to i d e n t i f y k.  We  only know that  w i t h s t r i c t e q u a l i t y o n l y i f s ( i V ) = 0, a n d the weak i n e q u a l i t y iis{N_) N o w , suppose t h a t p{If) 1 - F{r)  = p{N)  I n t h i s case, s{N)  = ... = p(N).  V N eAf,  € (0,1).  = 0 so t h a t  (in p a r t i c u l a r , 1 - F ( r ) is i d e n t i f i e d ) . W e can  o n l y p u t a n u p p e r b o u n d o n k: k  <  [l-F{r)]TiN)  <  [l-F{r)]T(N),  where the last i n e q u a l i t y follows f r o m the fact t h a t [1 - F{r)]T{N)  is the  profit i f a p o t e n t i a l b i d d e r w h e n each r i v a l enters w i t h p r o b a b i l i t y 1. W e n o w prove the " i f p a r t . If H < i n case 2 above, 1 - F(r)  < N, t h e n b y t h e same logic as  is identified since  w h i l e b y the same logic as i n case 1 above, k is i d e n t i f i e d f r o m ( C . 6 ) since for AT = A''» + 1, b i d d e r s enter w i t h p r o b a b i l i t y s{N) are indifferent between e n t e r i n g or not.  C.2  £ (0,1) a n d therefore  •  Details of the estimation method  W e m a k e the f o l l o w i n g a s s u m p t i o n s concerning the d a t a g e n e r a t i n g p r o cess. A s s u m p t i o n C . l (a) (b)  The marginal pact support  PDF X  bounded partial  {{Ni,xi)  : I = 1,...,L}  of xi,ip , is strictly C R**, and admits  derivatives  on Interior  are  i.i.d.  positive, at least R (X).  continuous >  on its com-  2 continuous  and  (c)  The {N,  distribution ...,N}  of Ni  conditional  on xi,  Tr{N\x),  has support N  2.  for allx€X,N>  (d)  Vil o,nd Ni are independent  (^)  {Vit '• i = ^,---,Ni;l  (f)  For ail X £ X,  conditional  = l,...,L}  on  are i.i.d.  the density of valuations  xi.  conditional  on  {Ni,xi)  f{-\-) is strictly  positive  bounded away from zero on its support, a compact interval R + , and admits its (g)  (h)  at least R continuous  The  and  [v{x),v{x)\  and bounded partial  C  derivatives  interior.  7r(iV|-) admit at least R>2 (X)  =  for all N  conditional  all N & M  continuous  bounded derivatives  on  Interior  eAf.  entry probability  tive for  continuous  derivatives  X + e Interior{X)  on {N,x),  and x G X,  is strictly  and p{N, •) admits  bounded away from  and all N  p{N,x),  zero on  posi-  at least R > 2 an open  subset  eAf.  A s s u m p t i o n C . l (a) is the u s u a l iid a s s u m p t i o n o n the d a t a g e n e r a t i n g process for t h e covariates.  Assumptions C.l(b),  (f), a n d smoothness  of  functions i n (g) a n d (h) are s t a n d a r d i n the n o n p a r a m e t r i c a u c t i o n s l i t e r a t u r e (see, for e x a m p l e , G P V ) . A s s u m p t i o n C . l ( c ) defines t h e s u p p o r t of the d i s t r i b u t i o n of TVj c o n d i t i o n a l o n the covariates. A s s u m p t i o n C . l ( d ) is one of the m o s t i m p o r t a n t a s s u m p t i o n s ; it asserts t h a t i n the n u m b e r of p o t e n t i a l b i d d e r s N is exogenous c o n d i t i o n a l on xi = x, w h i c h allows us t o use t h e v a r i a t i o n i n A^^ for the purpose of t e s t i n g . I n S e c t i o n 3.6, we e x p l a i n w h y t h i s a s s u m p t i o n is p l a u s i b l e i n the context of our e m p i r i c a l a p p h c a t i o n . A s s u m p t i o n C . l ( e ) is the I P V a s s u m p t i o n . F o r k e r n e l e s t i m a t i o n , we use k e r n e l functions K s a t i s f y i n g the following s t a n d a r d a s s u m p t i o n (see, for e x a m p l e , N e w e y (1994)). A s s u m p t i o n C.2 derivatives  The kernel K has at least R>2  continuous  and bounded  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 n o n p a r a m e t r i c regression a r g u m e n t s i m p l y t h a t the e s t i m a t o r o f e n t r y p r o b a b i l i t i e s p{N, x) is a s y m p t o t i c a l l y n o r m a l as well (see, for e x a m p l e , P a g a n a n d U l l a h (1999)[45], T h e o r e m 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 o o ; Lh^ —» oo and \/U?h^  isfies as L  3 and 4, VLhM{p{N,x) and  —p{N,x))  that the bandwidth  —» 0. Then,  is asymptotically  under  normal  h sat-  Assumptions  with mean zero  variance  Moreover, distinct  the estimators  p{N,x)  are asymptotically  independent  for any  N, N' G {TV, ...N} and x, x' G Ml  S i n c e t h e d i s t r i b u t i o n of values a n d , consequently, b i d s have c o m p a c t  t h e d i s t r i b u t i o n of  s u p p o r t s , t h e e s t i m a t o r o f t h e d e n s i t y g* is a s y m p -  t o t i c a l l y b i a s e d near t h e b o u n d a r i e s .  O u r q u a n t i l e a p p r o a c h allows one  to avoid the problem b y considering only inner intervals of the supports. Specifically, l e t [v{N,x),v{N,x)] let A b e some c o m p a c t \v{N,x),v{N,x)]. b y Ti{N,x)  denote t h e s u p p o r t of F*{v\N,x),  i n n e r i n t e r v a l , A{N,x)  =  [vi(N,x),V2{N,x)]  and C  T h e q u a n t i l e orders c o r r e s p o n d i n g t o vi a n d V2 are g i v e n for i = 1,2. H e n c e , we consider q u a n t i l e o r -  = F*{vi{N,x)\N,x)  ders i n T ( A ' ' , x) = [TI{N,  X), T2{N, X)]. N e x t , the c o r r e s p o n d i n g i n n e r i n t e r v a l  of the s u p p o r t of G* is g i v e n b y the values between the n a n d T2 q u a n t i l e s : e ( 7 V , x ) = \bi{N,x),b2(N,x)],  where bi{N,x)  = q*{Ti{N,x)\N,x),  i = 1,2.  S i m i l a r l y , w e define the i n t e r v a l of q u a n t i l e orders for t r a n s f o r m e d q u a n t i l e s : T'^iN,x)  = [T^{N,x),T^iN,x)]  such t h a t r f ( 7 V , x ) = { i n f T|/3(r, A^, x ) >  r i ( i V , x ) , r G [0,1]} a n d 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  N€^f, (a)  ^(x)-^(x)=Op((j^)-i/2+  (b)  7r(7V|x) - 7r(7V|x) = Op{{{^)-'/'  + h^).  G Interior{X)  and  (c) piN,x)  - p{N,x)  = Op{i{^)-'/^  +  h^).  \G*{b\N,x) - G*{b\N,x)\ = O p ( ( | ^ ) - i / 2 +  (d) ^^PbmN,.),m.)] (e)  snp,^r(N,.)  (f)  s u p , , e ( ; v , . ) \rm,x)  (g)  snp,^riN,x)  ITirlN,  x) - q*{T\N, x)\ = Op{{{^r'l^  +  h^).  h^).  - 9*{b\N,x)\ = 0 , ( ( ^ ) - V 2 + ^ f l ) .  \Q*{r\N,x)  - Q*(r|iV,x)| = O p ( ( M f l i ) - i / 2 +  (h) s u p , e [ o , i ] l / S ( r , 7 V , a ; ) - / ? ( r , i V , : r ) | = 0 p ( ( j g ) - V 2 + / , f l ) . (i) s u p , e T ^ ( ; v , x ) I Q * ( - 3 ( r , i V , x ) | i V , a ; ) - Q * ( / 3 ( r , i V , a ; ) | 7 V , a ; ) |  P r o o f of L e m m a  C.3.  -  P a r t s (a)-(c) of the l e m m a follow f r o m  L e m m a B . 3 of N e w e y (1994)[43].  F o r p a r t (d), define a f u n c t i o n G*o{b,N,x)  =  Np{N,x)7r{N\x)G*{b\N,xMx),  a n d its e s t i m a t o r as L  G*o{b,N,x)  N,  = -rjjr^^yuHNi 1=1  = i V } l { 6 , , < b}K,h{xi  -  a;),  i=l  where  (C.8)  (C.9)  Next, / EGl{h,N,x)  =  E  l{Ni  =  NE[l{Ni  =  NE{E{l{bu  JV, \ N)K,h{xi-x)Y,yul{hu<h} r=l /  =  \  =  = N}K,h{xi  - x)yal{hu  < h}\N,xuVii  < b})  = 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  =  JG*o{b,N,x  +  -  u)^{u)du  hu)Kdi^)du.  B y L e m m a 3.5, G*(6|7V, •) a d m i t s at least R+1  c o n t i n u o u s derivatives.  T h e n , as i n the p r o o f 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 t h a t there exists a constant c > 0 such t h a t | G 5 ( 6 , i V , a ; ) - S G 5 ( 6 , i V , 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 n o r m , a n d D^GQ denotes t h e R - t h p a r t i a l d e r i v a t i v e of G Q w i t h respect to a;. It follows t h e n t h a t ^^Pbem,-),mx)]\Goib,N,x)  - EGUb,N,x)\  =  (C.IO)  0{h\  N o w , we show t h a t  ^mmN,r)hN,x)]\Gl{b,N,x)  - EGl{b,N,x)\  =  M —  j (C.ll)  W e follow the a p p r o a c h of P o l l a r d (1984) [50]. C o n s i d e r , for g i v e n N € ^f a n d X € Interior(X),  a class of functions Z i n d e x e d b y h a n d b, w i t h a  representative f u n c t i o n  ziib,N,x)  = J2yiinNi i=l  = N}l{bu  < b]h''K,h{xi  -  x).  B y the result i n P o l l a r d (1984) [50] ( P r o b l e m 28), t h e class Z has p o l y n o mial discrimination.  T h e o r e m 37 i n P o l l a r d (1984) [50](see also E x a m p l e  38) i m p l i e s t h a t for any sequences 5L, ai  such t h a t L5\a\jlogL  —>• oo,  Ezf{b)<ôl  sup  al^l"  (C.12)  \-Y^zi{h,N,x)-Ezi{h,N,x)\^Q  b€[b(N,x),b(N,x)]  ^  1=1  a l m o s t surely. W e c l a i m t h a t t h i s implies / rud \  1/2  — \LogL,/ is b o u n d e d as L ^  sup beUN,x),b{i^,x)]  \GUb,N,x)-EGUb,N,x)\  oo almost surely. T h i s implies t h a t  sup \Gl{b,N,x)-EG*o{b,N,x)\ 6e[È(Ar,x),b(iv,x)] T h e p r o o f is b y c o n t r a d i c t i o n .  Suppose not.  = Op  (-—) \\i'Ogi^j  j  T h e n there exist a sequence  7 i —> oo a n d a subsequence of L such t h a t a l o n g t h i s subsequence  sup |GS(6,7V,x)-FGS(&,iV,3:)|>7L 6e[6(W,x),6(7V,x)] o n a set of events fi' C let &\ = / i * * a n d  7-7  w i t h a positive p r o b a b i l i t y measure.  = 1L^{J^Y^I'^-,  •  (C.13)  N o w if we  t h e n the d e f i n i t i o n of z i m p l i e s t h a t ,  a l o n g t h e subsequence, o n a set of events fi'.  aL^'5L^sup^e[6(7v,x),6(iV,x)] I i E t i zi{b,N,x)  \l0ghj /  -  r  ,d \  -  Ezi{b,N,x)\  i,6[b(iV,x),6(JV,i)] ^ /=i 1/2  \iogijj  be[b{N,x)fi{N,x)]  [logLj  ^"-yiogL)  1/2  =  It  -^00,  where t h e i n e q u a h t y follows by ( C . 1 3 ) , a c o n t r a d i c t i o n to ( C . 1 2 ) .  This  establishes ( C . l l ) , so t h a t ( C . I O ) , ( C . U ) a n d the t r i a n g l e i n e q u a l i t y together imply that  sup  = O p { { ^ \ + h ^ \  \Gl{b,N,x)-Gl{b,N,x)\  be\b(N,x),b{N,x)]  (C.14)  /  Wog^J  T o c o m p l e t e the proof, r e c a l l t h a t , f r o m the definitions of GQ{b,N,x)  and  G*oib,N,x),  G*ib\N,x)  G*{b\n,x)  =  =  G*o{b,N,x) p{N,xMN\xMxy G*o{b,N,x) p{N,x)Tf{N\x)(p{x)'  so t h a t b y the m e a n - v a l u e t h e o r e m . GUb,N,x)-G*a{b,N,x) \G*{b\N, x) - G*{b\N, x)\ < C{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 g i v e n by ' p{N,  X)T!-{N,  x)ip{x)  Gl{b,N,x)  ^ '  G*o{b,N,x)  p{N,x)  '  n{N,x)  Gl{b,N,x'^ '  ^{N,x)  a n d | | ( G 0 - G ' O , p - p , ^ - 7 r , ^ - < ^ ) | | < ||(G" - G ^ p - p , 7 f a l l {b,N,x).  J  T T , ^ - f o r  F u r t h e r , b y A s s u m p t i o n C . l ( b ) , (c) a n d (h), a n d the results i n  p a r t s (a)-(c) of the l e m m a , w i t h the p r o b a b i l i t y a p p r o a c h i n g one p,7f a n d axe b o u n d e d away f r o m zero. T h e desired result follows f r o m ( C . 1 4 ) , (C.15) a n d p a r t s (a)-(c) of the l e m m a .  F o r p a r t (e) of the l e m m a , since G*{-\N,x) is m o n o t o n e b y c o n s t r u c t i o n , P{q*{n{N,x)\N,x) =  p(^m{{b:G'ib\N,x)  = =  <b{N,x)) >TI{N,X)}  p(G'{b{N,x)\N,x)>n{N,x)) 0(1),  where t h e last e q u a l i t y is b y the result i n p a r t (d). PiriT2{N,x)\N,x)  > b{N,x))  =  H e n c e , for a l l x € Interior{X)  Similarly,  P{G{b{N,x)\N,x)  =  one, b{N,x)  <b{N,x)^  <  0(1).  a n d N G M, w i t h t h e p r o b a b i l i t y a p p r o a c h i n g  < q*{TiiN,x)\N,x)  < q*{T2{N,x)\N,x)  < b{N,x).  d i s t r i b u t i o n G*{b\N,x) is continuous i n b, G*(q*{T\N,x)\N,x) f o r r 6 T(N,  x),  T2{N,X))  Since the =  r , and,  we c a n w r i t e the i d e n t i t y  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 v a n der V a a r t (1998)[58], a n d b y the d e f i n i t i o n of G*, 0<G*{q*(T\N,x)\N,x)-T  <  ^ p{N,x)n{N\x)(p{x)NLh'i'  a n d b y t h e results i n (a)-(c), (C.17)  = T + Oj, [{Lh'')-')  G*iq*iT\N,x)\N,x)  u n i f o r m l y over r . C o m b i n i n g (C.16) a n d ( C . 1 7 ) , a n d a p p l y i n g t h e m e a n value t h e o r e m t o t h e l e f t - h a n d side o f ( C . 1 6 ) , we o b t a i n  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). g*{b\N,x)  is b o u n d e d  N O W , b y L e m m a 3.5,  away f r o m zero, a n d t h e result i n p a r t (e) follows  f r o m ( C . 1 8 ) a n d p a r t (d) of t h e l e m m a .  T o p r o v e p a r t ( f ) , b y L e m m a 3.5, 5*(-|iV, •) a d m i t s at least iï-1-l c o n t i n u o u s b o u n d e d p a r t i a l derivatives. L e t (C.19)  g*o{b,N,x)=piN,x)7:iN\xMx)g*{b\N,x), roib, N, x) = p{N, x)niN\x)0{x)rm, B y L e m m a B . 3 o f N e w y (1994)[43], gQ{b,N,x)  (C.20)  x).  is u n i f o r m l y consistent over  beeiN,x):  sup 6€e(N,x)  \roib,N,x)-g'oib,N,x)\  = Op  //r^d+i\-i/2 {^-^] + VV  y  B y t h e results i n p a r t s (a)-(c), t h e e s t i m a t o r s p{N,x), converge a t t h e rate faster t h a n t h a t i n (C.21).  \ •  (C.21)  y  n{N\x)  a n d 0{x)  T h e desired r e s u l t follows  b y t h e s a m e a r g u m e n t as i n the p r o o f of p a r t (d), e q u a t i o n  (C.15).  N e x t , we prove p a r t (g). B y L e m m a 3.5, g*{b\N,x) \Q*iT\N,x)-Q*iT\N,x)\  > Cg > 0. T h e n  \rir\N,x)-q*{r\N,x)  <  , \r ( r iT\N,x)\N,x)  +2^  -  g'iq*{T\N,X)\N,X)\  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 a n event EL{N,X)  = {r{n{N,x)\N,x)  a n d let 0L = {^^T'^^^  > bxiN,x),rir2iN,x)\N,x)  + f'^^-  <  b^iN^x)},  the result i n p a r t (e), P{El{N,x))  =  o ( l ) . H e n c e , i t follows f r o m p a r t (e) of the l e m m a t h e e s t i m a t o r ^ ( g * ( T | A'', x) \N,x)  is b o u n d e d  away f r o m zero w i t h the p r o b a b i l i t y a p p r o a c h i n g one.  Consequently, i t follows b y L e m m a 3.5 a n d p a r t (e) of t h i s l e m m a t h a t the first s u m m a n d o n t h e r i g h t - h a n d side of (C.22) is Op(/3^^ u n i f o r m l y over Next,  r{N,x).  P (sup^^r^^^,^(iL\r{q*ir\N,x)\N,x)-g*{r{r\N,x)\N,x)\  <  p(  sup  \T€Ï(JV,I)  >M )  /3L\ririr\N,x)\N,x)-g'ir{T\N,x)\N,x)\>M,EUx)  /  +PiEi{x)) <  p(  sup  \bee(.N,x)  PL\g'{b\N,x)-g'ib\N,x)\>M]+o(\). )  (C.23)  T h e result of p a r t (g) follows f r o m parts (c) a n d (f ) of t h e l e m m a a n d (C.23).  F o r p a r t (h), b y A s s u m p t i o n C . l ( h ) a n d p a r t (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 p a r t (h) follows since P is linear  i n T (see A n d r e w s (1992)[1]; also T h e o r e m s 21.9 a n d 21.10 o n p p .  337-339  of D a v i d s o n (1994)[12]).  L a s t l y , we prove p a r t (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) <  sup  - q*{Pir, N, x)\N, x)  \q*{T\N,x)-q*{T\N,x)\+Op  Ter(N,x)  R  + h  V iogL  (C.24)  w h e r e t h e i n e q u a l i t y follows f r o m p a r t (h) of the l e m m a a n d L e m m a 3.5. T h e r e s u l t of p a r t (i) follows f r o m the d e f i n i t i o n of Q * i n (3.23) a n d ( C . 2 4 ) .  L e m m a C . 4 Let Q{N,x) C.l  and  C.2  hold,  Vlhd+T^h^-^0.  be as in Lemma  and that the bandwidth  C.3.  Suppose that  Assumptions  h is such that Lh'^+^ —> oo,  Then 7V(0, Vgib, N,  V l h ^ i r m , x) - g*ib\N. x)) for b € Q{N,x),  X e Interior{X),  and N e Af,  x))  where Vg{b,N,x)  is given  by Vg{N,b,x)^ Furthermore, allNi^N2,  g*{b\Ni,x)  9' Np(N. ,xMN\xMx) andg*{b\N2,x)  Ni,N2eAf.  \J are asymptotically  J independent  for  P r o o f for L e m m a C.4.  C o n s i d e r g^ib, n , x) a n d g^{b, n, x) defined  i n (C.19) a n d ( C . 2 0 ) respectively.  It follows f r o m p a r t s (a)-(c) of L e m m a  C.3,  -55(6,7V,x)) + Op(l).  (C.25)  F u r t h e r m o r e , as i n L e m m a B 2 of N e w y (1994), Eg^ib, N, x) — g^ib, N, x) uniformly in 6 €  € J\f.  T h u s , i t r e m a i n s to e s t a b l i s h a s y m p t o t i c n o r m a l i t y of wLhP^ig^(b,N,x)  —  Q{N,x)  for a l l x  =  and N  0{h^)  € Interior{X)  Eroib,N,x)). Define  (=1  where  i=l  is defined i n ( C . 9 ) . W i t h above definitions we have t h a t  VNLhd^iToib,N,  x) - EToib,N,x))  = ^(WL^N  - EWL,N).  (C.28)  T h e n , b y t h e L i a p u n o v C L T (see, for e x a m p l e , C o r o l l a r y 11.2.1 o n 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)  p r o v i d e d t h a t Ew^i ^ < oo, a n d for some S > 0,  lim  --^E\wii^N  - Ewii^N\^~^^ = 0-  T h e last c o n d i t i o n follows from the L i a p u n o v ' s c o n d i t i o n ( e q u a t i o n (11.12) o n 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 g i v e n b y  ^I^E{p(N,  =  Vh^+ï  J  xiMN\xi)  ! K { ^ ) g*{u\N,xi)duKa  (^)  + hy)Tr{N\x + hy)  Jp{N,x  K{u)g*{b  + hu\N, x + hy)Kd{y)<p{x  +  hy)dudy  0. F u r t h e r , Ewfij^  is g i v e n b y  T;è^!!p{N,y)n{N\y)K^  =  J  JP(^^  {^)  ^ + hy)iT{N\x  {^)  cp{y)dudy  + hy)  K\u)g*ib <  g*{u\N,y)Kl  + hu\N, X + hy)KJiy)ipix  +  hy)dudy  00.  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 b o u n d e d b y  wmrrm  =  ^  1  1  l  '  ^  '  9*{u\N, y) \Kd{^)f^'  h i ^ J Jmu)\'+'9*{b + Tm)72  '^^  + hy)\Kd{y)\''+'<p{x +  ^ " P \K{u)f''^'^^^+'hupv{x)  «€[-1,1]  h(d+l)S/2 •  hu\N,x  xex  ^{y)dudy  s u p g*{b\N,x)  beB{N,x)  hy)dudy  Lastly,  ^  0,  since Lh'^'^^ —> CXD b y the a s s u m p t i o n . follows n o w f r o m  (C.31)  T h e first result of t h e l e m m a  (C.25)-(C.31).  N e x t , n o t e t h a t the a s y m p t o t i c covariance of W L . M a p r o d u c t of t h e t w o i n d i c a t o r functions, l{Ni is zero for a l l Ni  = i V i } l { J V / = N2},  which  The joint asymptotic normality and asymptotic  ^ N^.  independence oig*{b\Ni,x) device.  a n d WL,iV2 involves  a n d g{b\N2, x) follows t h e n b y the C r a m e r - W o l d  •  P r o p o s i t i o n C . 5 Suppose bandwidth  h satisfies  under Assumptions  that T e (0,1)  and x s Xl  Assume  C.l  and  that the 0.  as L -* 00 : LW^^^ —> 0 0 and y/Th^h^  Then,  C.2,  VLh^{Q'{T\N,x)-Q'{T\N,X))  Af(0,Vg(7V,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 { i V , A ' " } , r , r ' e T , and x,x' totically  independent,  € X\  C.4.  Moreover,  the estimators  as well as the estimators  for any distinct Q*{T\N,X)  are  Q*{f3{T,N,x)\N,x).  N, N' £ asymp-  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) a n d  (f), a n d t h e m e a n - v a l u e t h e o r e m .  (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 m e a n - v a l u e between g* a n d g* for b = q*{T\N,x). follows t h e n b y L e m m a C . 4 .  as L  00, and Assumptions  C.l  Assume  that Lh"^ -> 0 0 and VLh^h^  and C.2 hold.  supP/f^^^(T^^^(a;)>c)  PH,S{T^''^{X)  _^  P ( r ^ ^ ^ ( a ; ) > c),  ^•'"''^ ^HC^S denotes probabilities  tions of AME  and equality  restrictions  0  Then  =  where PHAME  defined in  T h e result  •  Let x € X^.  Proposition C.6  ,  > c)  under the inequality  of LS respectively,  (C.32) (C.33) restric-  and T ' ^ ^ ^ ( a ; ) is  (3.27).  T h e result i n ( C . 3 2 ) follows by  P r o o f for P r o p o s i t i o n C . 6 .  L e m m a 8.2 o f P e r l m a n (1969)[47], I n order t o show ( C . 3 3 ) , consider first the case of Â; =  1.  B y the results i n C h a p t e r 21.3.3 of G o u r i e r o u x a n d  M o n f o r t (1995)[16], , T ' * ^ ^ ( x ) is a s y m p t o t i c a l l y e q u i v a l e n t t o f^^^(x)=  where VLÎ?+^^  min  L/i''+i||7-^||2,  A''(0, J ^ A T - I ) ; however.  '-^^^{x)  =  min  mm n^/^(T,x)fi<o  \ / L ^ 7 - / i  a n d the result follows by the C o n t i n u o u s M a p p i n g T h e o r e m .  E x t e n s i o n to  the case of fc > 1 is s t r a i g h t f o r w a r d since there axe no cross r restrictions i n (3.26), a n d t h e q u a n t i l e estimators are a s y m p t o t i c a l l y i n d e p e n d e n t 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 i n t 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 c u r r e n t version is a further development of a p a p e r c i r c u l a t e d earlier u n d e r the t i t l e "Selective Entry in First-Price Auctions." T h e a u t h o r t o o k c e n t r a l roles i n a l l stages of the research, i n c l u d i n g i d e n t i f i c a t i o n of t h e research questions, development of t h e s t a t i s t i c a l m e t h o d o l ogy, i n v e s t i g a t i o n of t h e finite sample performance of the p r o p o s e d s t a t i s t i c a l m e t h o d s t h r o u g h M o n t e C a r l o s i m u l a t i o n s , a c q u i s i t i o n of the d a t a set used i n the e m p i r i c a l a n a l y s i s , a n d design of the counter f a c t u a l e x p e r i m e n t s .  

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