Threshold Regression with a Nearly Integrated Threshold Variable

This paper considers threshold regression with an integrated or local-to-unity threshold variable and a threshold fixed in levels. We study the least squares threshold estimator and establish its asymptotic theory under both fixed and diminishing threshold effects. Under a fixed effect, the threshold estimator has a compound Cox argmin limit, whereas under a diminishing effect, its limiting distribution is characterized by a two-sided Brownian argmax with a random scale. We also develop likelihood ratio inference, which is asymptotically pivotal under a diminishing effect and conservative under Gaussian errors for a fixed effect, and show that conventional $t$-ratios for the slope coefficients remain asymptotically standard normal. A sup-Wald test is proposed to detect threshold effects that may be too small for consistent threshold localization. We further extend the framework to allow for integrated slope regressors and for contemporaneous endogeneity of the threshold variable, showing that the leading endogenous contribution cancels over shrinking threshold windows. Monte Carlo simulations examine the finite-sample performance of the proposed methods. An illustration using public debt and economic growth highlights the relevance of allowing for a persistent threshold variable.