Monotonicity-constrained nonparametric estimation and inference for first-price auctions

Abstract

In the independent private values framework for first-price auctions, we propose a new nonparametric estimator of the probability density of latent valuations that imposes the monotonicity constraint on the estimated inverse bidding strategy. We show that our estimator has a smaller asymptotic variance than that of Guerre, Perrigne and Vuong’s estimator. In addition to establishing pointwise asymptotic normality of our estimator, we provide a bootstrap-based approach to constructing uniform confidence bands for the density function.

Keywords:

• Asymptotic normality
• bootstrap
• firstprice auctions
• monotonicity
• nonparametric estimation
• uniform confidence band

JEL Classification:

• C14
• C57

Acknowledgment

We thank two anonymous referees and Emmanuel Guerre for their helpful comments, and Christopher Parmeter for sharing his Gauss code. Xinrui Zhou provided excellent research assistance.

Notes

1 See Henderson and Parmeter (2009) for a comprehensive survey of the literature on monotone nonparametric regression.

2 See Henderson and Parmeter (2015, chapter 12) for discussion on rearrangement and constrained weighted bootstrapping. See Dette and Pilz (2006) for simulation studies that compare the rearrangement and reweighting approaches to monotone regression. Dette and Pilz (2006) also noticed that the rearrangement approach has computational advantage since the reweighting approach requires solving constrained optimization.

3 When implemented in Julia on a MacBook Pro with a 2.9 GHz Intel Core i7 processor, the elapsed time for computing pseudo valuations for 1500 bids using our rearrangement approach is approximately 4.6 s. We also implemented the approach of Henderson et al. (2012) in Julia using the NLopt package and the SLSQP algorithm (a sequential quadratic programming method). With monotonicity constraints imposed at 121 points and a sample of 1500 bids, the elapsed time for their method is approximately 66 s.

4 See Hickman and Hubbard (2015) for more discussion on why we should avoid trimming when estimating the auction model using GPV approach.

5 See remark 2.3 of MMS.

6 See Jales et al. (2017) and Ma et al. (2020) for more recent applications of MCE in econometrics.

7 Note that the MCE requires knowledge of the locations of the endpoints $\underset{¯}{b}$ and $\overline{b}.$ Since the estimators $\widehat{\underset{¯}{b}}$ and $\widehat{\overline{b}}$ are super-consistent: $\widehat{\underset{¯}{b}}=\underset{¯}{b}+O_p(\hspace{0.17em}\text{log}\hspace{0.17em}(L)/L)$ and $\widehat{\overline{b}}=\overline{b}+O_p(\hspace{0.17em}\text{log}\hspace{0.17em}(L)/L),$ we can replace the unknown endpoints in the MCE with these estimators without affecting the validity of the asymptotic results.

8 See Chernozhukov et al. (2009, proposition 1).

9 In this article, the theory is done for deterministic sequences of bandwidths as opposed to random sequences. This article focuses on the effect of smooth rearrangement and highlights the comparison of the results with those of MMS. However, an extension of Theorem 1 which allows for random sequences of bandwidths is possible but very much involved. Suppose that the bandwidths are ${\widehat{h}}_f={\widehat{\lambda }}_{f}h,$ ${\widehat{h}}_g={\widehat{\lambda }}_{g}h$ and ${\widehat{h}}_r={\widehat{\lambda }}_{r}h$ with random scaling constants $({\widehat{\lambda }}_f,{\widehat{\lambda }}_g,{\widehat{\lambda }}_r)$ and $({\widehat{\lambda }}_f,{\widehat{\lambda }}_g,{\widehat{\lambda }}_r){\to }_p({\lambda }_f,{\lambda }_g,{\lambda }_r)$ with a fast enough rate of convergence for some scaling constants $({\lambda }_f,{\lambda }_g,{\lambda }_r).$ Then, the maximal inequalities for empirical processes (see the inequalities in Chernozhukov et al., 2014a and Chen and Kato, 2019) can be applied to show that the RGPV estimator using $({\widehat{h}}_f,{\widehat{h}}_g,{\widehat{h}}_r)$ is asymptotically equivalent to the RGPV estimator using $(h_f,h_g,h_r)$ and the result stated in Theorem 1 also holds for the estimator using $({\widehat{h}}_f,{\widehat{h}}_g,{\widehat{h}}_r).$

10 To the best of our knowledge, such a result is new in the literature of rearrangement-based shape-constrained estimators of nonparametric curves. See e.g., the main results in Dette et al. (2006), Birke and Dette (2007) and Dette and Pilz (2009). We are not aware of any existing paper that showed that an estimator of any nonparametric curve with a shape constraint imposed by rearrangement or any other method could have a smaller variance asymptotically, compared with that of a standard unconstrained estimator.

11 Also see Birke and Dette (2007) and Dette and Pilz (2009).

12 See e.g., Arai and Ichimura (2018). In our context, a simultaneously selected AMSE-minimizing bandwidth h_g will be different from the AMSE-minimizing bandwidth selected for the GPV estimator.

13 For a test of this assumption, see Liu and Luo (2017).

14 Constrained weighted bootstrapping of Hall and Huang (2001) is easily generalized to multiple covariates. See Du et al. (2013). The homogenization approach is used extensively in the literature, e.g., Athey et al. (2011), Liu and Luo (2017), Luo and Wan (2018) and many others.

15 For example, ${\mathit{x}}_0$ can be taken to be the sample mean $L^{-1}{\sum }_{l=1}^L{\mathit{X}}_l.$ See Haile et al. (2003).

16 Alternatively, one can bootstrap at the auction level re-sampling entire auctions as subsection 6.2 of MMS. However, in that case one must apply the homogenization procedure to every bootstrap sample.

17 We also tried a finer grid $\left[v_l:0.0001:v_u\right],$ which produced similar results.

18 It is easy to adapt the proofs to show that the remainder terms are indeed uniform in $v\in I.$ See the proof of Theorem 2.1 of MMS.

19 See the proof of Lemma B.2 of MMS for more details.

Additional information

Funding

We gratefully acknowledge the financial support from the National Natural Science Foundation of China10.13039/501100001809 under grant 71903190 (Ma), the Social Sciences and Humanities Research Council of Canada10.13039/501100000155 under grant 435-2017-0329 (Marmer), and the General Research Fund by HK Research Grant Council (Xu). Marmer and Xu also appreciate the financial support by HKU Business School10.13039/501100003803 under the visiting scholar scheme.