  We propose a novel two-regime regression model where regime switching is
driven by a vector of possibly unobservable factors. When the factors are
latent, we estimate them by the principal component analysis of a panel data
set. We show that the optimization problem can be reformulated as mixed integer
optimization, and we present two alternative computational algorithms. We
derive the asymptotic distribution of the resulting estimator under the scheme
that the threshold effect shrinks to zero. In particular, we establish a phase
transition that describes the effect of first-stage factor estimation as the
cross-sectional dimension of panel data increases relative to the time-series
dimension. Moreover, we develop bootstrap inference and illustrate our methods
via numerical studies.
