Integrating the Structural Auction Approach and Traditional Measures of Market Power



number of bidders on a given lot of cattle (Nj). Following Guerre et al. (2000), the
estimates of bid cumulative distribution and density functions are obtained via the
empirical distribution
F(pjf ) and kernel density estimator f (pjf), respectively as:

(2.14)


F(pf )=MJ∑ ∑ (pf ≤pf ),


(2.15)


^    S

f(. pf ) =


ɪ M J K
MJh i=1    j=1


where h is a bandwidth defining the size of the “neighborhood” around and arbitrary bid

p f, pijf is the jth bid in the interval (p f - h, p f + h), J is the total number of cattle lots, and

K() is the kernel density function, which assigns weights to every bid in the
neighborhood of
p f.

The kernel density function defined by equation (2.15), is estimated assuming a

Gaussian kernel function as:

(2.16)


K(u) =


1              12

. exp(—u ), where u =
2π     2


f
pf


Previous studies indicate that while the choice of the form of the kernel functional form
does not affect results in practice, the choice of the bandwidth (
h) may affect results
(DiNardo and Tobias, 2001; Hardle et al., 2004). Sheather (2004, p.596) recommends
the Sheather-Jones plug-in method (SJPI) due to good performance. The SJPI is defined
as:

(2.17)


h = σ(4Z3J )1/5,

where σ is sample standard deviation of the bids and J is the number of bids in the

sample. The kernel density function is estimated using the KDE Procedure in SAS 9.1.

22



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