Research Highlights

This paper considers robust inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump, nesting previous works that assume either continuity or discontinuity at the threshold. We find that the parameter values under the kink restriction are irregular points of the Hessian matrix, destroying the asymptotic normality and inducing the cube-root convergence rate for the threshold estimate. However, we are able to obtain the same asymptotic distribution as Hansen (2000) for the quasi-likelihood ratio statistic for the unknown threshold. We propose to construct confidence intervals for the threshold by bootstrap test inversion. Finite sample performances of the proposed procedures are examined through Monte Carlo simulations and an economic empirical application is given.