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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