Testing for Threshold Diffusion

The threshold diffusion model assumes a piecewise linear drift term and a piecewise smooth diffusion term, which constitutes a rich model for analyzing nonlinear continuous-time processes. We consider the problem of testing for threshold nonlinearity in the drift term. We do this by developing a quasi-likelihood test derived under the working assumption of a constant diffusion term, which circumvents the problem of generally unknown functional form for the diffusion term. The test is first developed for testing for one threshold at which the drift term breaks into two linear functions. We show that under some mild regularity conditions, the asymptotic null distribution of the proposed test statistic is given by the distribution of certain functional of some centered Gaussian process. We develop a computationally efficient method for calibrating the p-value of the test statistic by bootstrapping its asymptotic null distribution. The local power function is also derived, which establishes the consistency of the proposed test. The test is then extended to testing for multiple thresholds. We demonstrate the efficacy of the proposed test by simulations. Using the proposed test, we examine the evidence of nonlinearity in the term structure of a long time series of U.S. interest rates.