  This paper develops a threshold regression model where an unknown
relationship between two variables nonparametrically determines the threshold.
We allow the observations to be cross-sectionally dependent so that the model
can be applied to determine an unknown spatial border for sample splitting over
a random field. We derive the uniform rate of convergence and the nonstandard
limiting distribution of the nonparametric threshold estimator. We also obtain
the root-n consistency and the asymptotic normality of the regression
coefficient estimator. Our model has broad empirical relevance as illustrated
by estimating the tipping point in social segregation problems as a function of
demographic characteristics; and determining metropolitan area boundaries using
nighttime light intensity collected from satellite imagery. We find that the
new empirical results are substantially different from those in the existing
studies.
