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A semiparametric varying coefficient model of monotone auction bidding processes

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Abstract

We propose an econometric model of the price processes of online auctions. Since auction bids monotonically increase within individual auctions but can differ considerably across auctions, we construct a monotone series estimator with a common relative price growth curve and auction-specific slopes. Furthermore, because the impacts of auction-specific attributes may evolve along the course of auctions, we employ a varying coefficient approach to accommodate these possibly time-varying effects. We present an algorithm to solve the proposed partially linear panel model with a nonparametric monotone component. We apply the proposed model to eBay auctions of a Palm personal digital assistant. The estimates capture closely the overall pattern of online auction price processes, in particular, the bidding drought midway through auctions and the bid acceleration associated with bid sniping (last-minute bidding) at the end of auctions.

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Notes

  1. See Lucking-Reiley (2000) for some early adopters of the second price auctions, and a fascinating account of the history of the second price auction.

  2. eBay also uses some variations of the hard-closed auction. For example, some auctions offer a “Buy It Now” price. In this kind of auction, bidders have the option to purchase the auctioned item at a posted “Buy It Now” price, which terminates the auction.

  3. To recover the underlying common continuous price processes, researchers often rely on a large number of similar auctions with usually small number of bids, which results in an unbalanced panel of bids. The consequences of this particular nature of pooled bid histories on model construction and estimation are discussed below.

  4. See Ockenfels and Roth (2002), Bajari and Hortaçsu (2003, 2004) and Ockenfels and Roth (2006) for plausible explanations of bid snipping.

  5. Instead of submitting their maximum willingness to pay, many bids submit their current willingness to pay and increase their bids as the auction progresses as in ascending price auctions.

  6. For example, consider an auction with a reserve price of $10 and a minimum increment of $1. Suppose bidder A is the first bidder and his proxy bid is $20. Then, eBay displays $10 as the current highest bid. Next, bidder B enters the auction with a proxy bid of $15. eBay automatically raises the current bid to $16, which is one bid increment above the second highest bid. If a third bidder C submits a proxy bid of $23, he will become the highest bidder while the displayed highest bid will be raised to $21. The process continues until the auction ends.

  7. Model (4) is a simplification of Ramsay (1998)’s original monotone series estimator, which takes the form

    $$\begin{aligned} m^*\left( t\right) =\int _{0}^{t}\exp \left( \int _{0}^{u}w^{*}\left( v\right) dv\right) du. \end{aligned}$$

  8. Note that in \(w(\cdot )\), we do not have the usual constant term \(\phi _0\). This is because that \(\phi _0\) and \(\beta \) cannot be identified separately.

  9. See Chapter 11 of Wooldridge (2002) and Wooldridge (2005) for a detailed treatment of panel models with individual slopes.

  10. As suggested by an associate editor, another possibility to model the individual effects is to allow them to depend on the day of the auctions. This approach is not considered in this study due to the fact that we model the effects of auction specific attributes as smooth functions of time. Further incorporating time dependent individual effects may introduce the problem of multicollinearity.

  11. On eBay, a buyer can rate a seller by giving him a positive (+1), neutral (0), or negative (-1) score, along with a text comment. eBay records and displays all of these comments, including the ID of the person making the comment. eBay also displays some summary statistics of user feedback.

  12. Higher order terms for the time-varying coefficients are not statistically significant.

  13. Almost all of the feedback on eBay is positive. One interpretation can be that users are hesitant to leave negative feedback for fear of retaliation. Another factor limiting the potential usefulness of feedback, reported by Resnick et al. (2006), is that feedback provision is an arguably costly activity that is completely voluntary, and that not all buyers (52.1 %) actually provide reviews about their sellers.

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Authors and Affiliations

  1. International School of Economics and Management, Capital University of Economics and Business, Beijing, 100070, People’s Republic of China

    Wenchuan Liu

  2. Department of Agricultural Economics, Texas A&M University, College Station, TX, 77843, USA

    Yu Yvette Zhang

  3. Department of Economics, Texas A&M University, College Station, TX, 77843, USA

    Qi Li

  4. ISEM, Capital University of Economics and Business, Beijing, 100070, China

    Qi Li

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  1. Wenchuan Liu
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  2. Yu Yvette Zhang
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Correspondence to Qi Li.

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Liu, W., Zhang, Y.Y. & Li, Q. A semiparametric varying coefficient model of monotone auction bidding processes. Empir Econ 48, 313–335 (2015). https://doi.org/10.1007/s00181-014-0857-z

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