slope of the density function immediately to the left of T(=P) is significantly
different from the corresponding slope (adjusted for sign) to the immediate
right of Tafter allowing for any general local skew in the distribution.
Define ∇pj≡∆log{p(xT+j)}−(− 1 ×∆log{p(xT−j)}). As remarked before, log
transformations of the density appear to stabilize variance across nearby j’s
in simulations as well for our samples (not reported). The test for case B2
amounts to examining whether ∇p 1 is unusual. We use the observations ∇pj
from a small neighborhood R(j>1,) to compute an estimate of the mean
of ∇p 1 as well as its standard deviation.^44 As before, we compute a t-like
test statistic, say τT=P,to assess the “unusualness” of ∇p 1.
In simulations that mimic the statistical structure of our sample while as-
suming a Gaussian distribution for the latent earnings, the statistic τT=P
proves to be greater than 2.0 less than 5 percent of the time. Nonetheless,
since the real distribution is unlikely to be as well behaved as Gaussian in
the absence of any discontinuity at T=P, the comparison of τT=Pwith the
real samples to the reference level of 2.0 is only taken as suggestive of a dis-
continuity. Thus, we also examine the rank of ∇p 1 to the corresponding
values at nearby j’s: in our samples, when τT=Pproves to be larger than 2.0,
∇p 1 is always the largest in the neighborhood.
664 DEGEORGE, PATEL, ZECKHAUSER
(^44) In the tests reported in the main text of this work the Rfor computing τT=Pspans 10
nearby values, that is, j=2,3,... , 11. Similar results, not shown, obtain with fewer nearby
values.