of τTis likely to be well approximated by the Student’s t-distribution under
H 0 if the distribution of ∆p(xi) in RT is approximately Gaussian. In unre-
ported simulations, working with the log transformation of the estimated
density improved the homogeneity of variance (of ∆log{p(x)}) across typical
neighborhoods R—thus, all the tests that we report in the text are based on
∆log{p(x)} rather than on ∆p(x), though the inferences appear similar. In
any case, we do not solely rely on normality. Instead we compare the τTto
other τ-values computed for nearby points.
We examine the rank of τTrelative to the other τ’s as well as its relative
magnitudes to assess whether a discontinuity at Tcan be established. For-
tunately, clear, unambiguous results obtain: using the full sample, the τT
values always prove to be the largest when compared to the other τvalues.^43
2.ElaborationsThe basic test sketched above is satisfactory as long as the point at which
the density being examined for a discontinuity (T) falls significantly on one
side of the peak of the probability density distribution. Denote the peak by
P. Now consider the case when the symmetric construction of RTsketched
above would include P. Since points on different sides of Pare likely to
have slopes of the density function of opposite signs, the symmetric RTwill
no longer be composed of similar points in the sense of similar slopes.
Case A1. Symmetric neighborhood around Twould include peak, P,
though T≠P. For this case, we construct an asymmetric neighborhood RT
around T. When T<P(T>P), construct RTto be the most symmetric re-
gion possible around Tof 2r+1 points such that all the points lie at or
below (above) P. The intuition for this construction is that by selecting
points on the same side of Pwe obtain a neighborhood with points that
have similar slopes of the (log) density function. Given such an RT, we com-
pute τTas in the basic test approach above.
Case A2. Suspected threshold coincides with the peak, that is, T=P. Con-
sider the case of the analysts’ forecast as the threshold, T. In this case, the dis-
tribution of reported earnings is likely centered at this Tif analysts forecast
the mode or if the latent distribution of earnings is nearly symmetric and
forecasters minimize the mean squared forecast error or the mean absolute
error. Now, we identify an earnings management effect by testing whether the
EARNINGS MANAGEMENT 663(^43) Given the 10 neighborhood values to which we compare tT(see n. 3 above), the likeli-
hood of obtaining tTas the largest value by chance is slightly less than 10 percent. Looking at
the magnitudes themselves, the neighborhood tvalues interestingly always compute to less
than 2 while the tTvalues always exceed 2.