- 1. Part (a) of Lemma B.3 is a standard Functional CLT result for Empirical Processes, see van der Vaart (1998), Theorem 19.5. In fact, the result holds jointly with the weak convergence in (28) for other empirical distributions involving the bids {Bil}.
Paper not yet in RePEc: Add citation now
- 2. The first claim in part (b) of the lemma follows from part (a) by the FDM, see van der Vaart (1998), Lemma 20.10 and Lemma 21.3 for quantile functions. Note that Lemma B.2 implies that 0 i is a bounded function. The α-Hölder continuity result holds by (i) the α-Hölder continuity for of 0 i with α = 1/2 shown in Lemma B.2, and (ii) because the sample paths of Gi and G0 i are α-Hölder continuous with probability one for any α < 1/2, see for example Revuz and Yor (1999), Theorem 2.2. 3. Part (c) uses a point-wise approximation of empirical processes by Gaussian processes, see van der Vaart (1998), page 268, and Hölder continuity of 0 i in Lemma B.2. Proof of Lemma B.3. To simplify the notation, we omit bidder’s index i in whenever there is no risk of confusion.
Paper not yet in RePEc: Add citation now
- A Appendix: Extended Functional Delta Method The following lemma is an extension of the FDM (van der Vaart, 1998, Theorem 20.8) and allows for functionals that depend on the sample size L. This includes functionals with sample-size-dependent trimming. Lemma A.1 (Extended Functional Delta Method). Let D and E be normed linear spaces. Suppose that: (i) rLkÆL(F) − Æ(F)k → 0, where rL → ∞ as L → ∞, and ÆL, Æ : D → E. (ii) There is a continuous linear map Æ0 F,L : D → E such that, for every compact D ∈ D0 ⊂ D, sup h∈D:F+h/rL∈D ÆL(F + h/rL) − ÆL(F) 1/rL − Æ0 F,L(h) → 0. (iii) kÆ0 F,L(hL) − Æ0 F (h)k → 0 for all hL such that khL − hk → 0 with h ∈ D0, where Æ0 F : D0 → E is a continuous linear map. (iv) GL = rL(FL − F) G, where P(G ∈ D0) = 1. Then, rL(ÆL(FL) − Æ(F)) Æ0 F (G).
Paper not yet in RePEc: Add citation now
- A. Aradillas-López, A. Gandhi, and D. Quint. Identification and inference in ascending auctions with correlated private values. Working Paper. University of Wisconsin, Madison, 2011.
Paper not yet in RePEc: Add citation now
- A. W. van der Vaart. Asymptotic Statistics. Cambridge University Press, Cambridge, 1998.
Paper not yet in RePEc: Add citation now
Artyom Shneyerov. An empirical study of auction revenue rankings: The case of municipal bonds. RAND Journal of Economics, 37(4):1005–1022, 2006.
Bjarne Brendstrup and Harry J. Paarsch. Semiparametric identification and estimation in multi-object, english auctions. Journal of Econometrics, 141(1):84 – 108, 2007. K. Chen and S. H. Lo. On a mapping approach to investigating the bootstrap accuracy.
- By adapting the proof of Lemma 21.3 in van der Vaart (1998) and as in the proof of Lemma B.3(b), we can write √ L(̂ − ) = √ L Ĝ0 (G−1 ) − G0 (G−1 ) − g0 (G−1 ) g(G−1) √ L Ĝ(G−1 ) − Ä +op √ L Ĝ0 (G−1 ) − G0 (G−1 ) + √ L Ĝ(G−1 ) − Ä , (65) √ L(̂†− ) = √ L Ĝ0,†(G−1 ) − G0 (G−1 ) − g0 (G−1 ) g(G−1) √ L Ĝ†(G−1 ) − Ä +op √ L Ĝ0,†(G−1 ) − G0 (G−1 ) + √ L Ĝ†(G−1 ) − Ä , where the op term is uniform in Ä, and therefore, √ L(̂†− ̂) = √ L Ĝ0,†(G−1 ) − Ĝ0 (G−1 ) − g0 (G−1 ) g(G−1) √ L Ĝ†(G−1 ) − Ĝ(G−1 ) +op √ L Ĝ0,†(G−1 ) − Ĝ0 (G−1 ) + √ L
Paper not yet in RePEc: Add citation now
Daniel A Graham and Robert C Marshall. Collusive bidder behavior at single-object second-price and english auctions. Journal of Political Economy, 95(6):1217–1239, 1987.
- Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1999.
Paper not yet in RePEc: Add citation now
Dominic Coey, Bradley Larsen, and Kane Sweeney. The bidder exclusion effect. NBER Working Paper 20523, 2014.
E. Guerre, I. Perrigne, and Q. Vuong. Optimal nonparametric estimation of first-price auctions. Econometrica, 68(3):525–74, 2000. P. A. Haile and E. Tamer. Inference with an incomplete model of english auctions.
G. Hanoch and H. Levy. The efficiency analysis of choices involving risk. Review of Economic Studies, 36(3):335–346, 1969.
Gaurab Aryal and Maria F Gabrielli. Testing for collusion in asymmetric first-price auctions. International Journal of Industrial Organization, 31(1):26–35, 2013.
George J Mailath and Peter Zemsky. Collusion in second price auctions with heterogeneous bidders. Games and Economic Behavior, 3(4):467–486, 1991. Robert C Marshall and Leslie M Marx. The vulnerability of auctions to bidder collusion.
- Isaac Meilijson. Estimation of the lifetime distribution of the parts from the autopsy statistics of the machine. Journal of Applied Probability, 18(4):pp. 829–838, 1981.
Paper not yet in RePEc: Add citation now
John Asker. A study of the internal organization of a bidding cartel. American Economic Review, 100(3):724–762, 2010.
- Joseph E Harrington. Detecting cartels. In Paolo Buccirossi, editor, Handbook of Antitrust Economics, pages 213–258. MIT Press, Cambridge, MA, 2008. Ken Hendricks, Robert Porter, and Guofu Tan. Bidding rings and the winner’s curse.
Paper not yet in RePEc: Add citation now
- Journal of Political Economy, 101(3):518–538, 1993. Robert H Porter and J Douglas Zona. Ohio school milk markets: An analysis of bidding.
Paper not yet in RePEc: Add citation now
Kenneth Hendricks and Robert H Porter. Collusion in auctions. Annales d’Economie et de Statistique, 15–16:217–230, 1989.
Laura H. Baldwin, Robert C. Marshall, and Jean-Francois Richard. Bidder collusion at forest service timber sales. Journal of Political Economy, 105(4):657–699, 1997.
Martin Pesendorfer. A study of collusion in first-price auctions. Review of Economic Studies, 67(3):381–411, 2000. Robert H Porter and J Douglas Zona. Detection of bid rigging in procurement auctions.
- Odd Aalen. Nonparametric inference for a family of counting processes. Annals of Statistics, 6(4):701–726, 1978.
Paper not yet in RePEc: Add citation now
P. A. Haile, H. Hong, and M. Shum. Nonparametric tests for common values at first-price sealed-bid auctions. NBER Working Paper 10105, 2003.
Patrick Bajari and Lixin Ye. Deciding between competition and collusion. Review of Economics and Statistics, 85(4):971–989, 2003.
Patrick Bajari, Han Hong, and Stephen P Ryan. Identification and estimation of a discrete game of complete information. Econometrica, 78(5):1529–1568, 2010.
Patrick Bajari, Stephanie Houghton, and Steven Tadelis. Bidding for incomplete contracts: An empirical analysis of adaptation costs. American Economic Review, 104 (4):1288–1319, 2014.
Paul Klemperer. What really matters in auction design. Journal of Economic Perspectives, 16(1):169–189, 2002.
Paul Robert Milgrom. Putting Auction Theory to Work. Cambridge University Press, Cambridge, UK, 2004.
S. Athey and P. A. Haile. Identification of standard auction models. Econometrica, 70 (6):2107–2140, 2002.
- See Guillemin and Pollack (1974), p. 42. (c) Under Assumption 1, there exists a version of the Gaussian process Mi such that for any α < 1/2, lim sup L→∞ Lα/2 √ L(̂i,L − i) − Mi < ∞ a.s. Remark 5.
Paper not yet in RePEc: Add citation now
- Sture Holm. A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(2):65–70, 1979.
Paper not yet in RePEc: Add citation now
- T. Komarova. Nonparametric identification in asymmetric second-price auctions: a new approach. Quantitative Economics, 4(2):269–328, 2013.
Paper not yet in RePEc: Add citation now
- The result in (33) now follows by the FDM for the bootstrap (van der Vaart, 1998, Theorem 23.5) and the same arguments as in the proof of Proposition 4, since F̃†= S̃†(Ĝ†). The result in (34) holds by the bootstrap FDM, Proposition 3.1 in Chen and Lo, (27), and since the functional Ècol is Hadamard differentiable on [v0, v] ⊂ (0, v]. To show (35), write √ L( ˆ ∆†i (b) − ˆ ∆i(b)) = √ L(Ĝ†i (b) − Ĝi(b)) − √ L(Ĝpred,†i (b) − Ĝpred i (b)). The result in (35) follows by the bootstrap FDM and the previous results of the proposition as the functional Èi,pred defined in (30) is Hadamard differentiable.
Paper not yet in RePEc: Add citation now
- Wayne Nelson. Hazard plotting for incomplete failure data. Journal of Quality Technology, 1(1):27–52, 1969. Wayne Nelson. Theory and applications of hazard plotting for censored failure data.
Paper not yet in RePEc: Add citation now
- where the first inequality follows because G0 ∈ Hα for any α < 1/2 by Theorem 2.2 in Revuz and Yor (1999), and the second inequality holds because G−1 is continuously differentiable and, therefore, Lipschitz. By Lemma B.2, |G(G−1 (t))(0 (t + δ) − 0 (t))| ≤ C|δ|1/2 supv∈[0,v] |G(v)|.
Paper not yet in RePEc: Add citation now
William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16(1):8–37, 1961.
Xun Tang. Bounds on revenue distributions in counterfactual auctions with reserve prices. RAND Journal of Economics, 42(1):175–203, 2011.