Quasi-likelihood estimation of a threshold diffusion process

The threshold diffusion process, first introduced by Tong (1990), is a continuous-time process satisfying a stochastic differential equation with a piecewise linear drift term and a piecewise smooth diffusion term, e.g., a piecewise constant function or a piecewise power function. We consider the problem of estimating the (drift) parameters indexing the drift term of a threshold diffusion process with continuous-time observations. Maximum likelihood estimation of the drift parameters requires prior knowledge of the functional form of the diffusion term, which is, however, often unavailable. We propose a quasi-likelihood approach for estimating the drift parameters of a two-regime threshold diffusion process that does not require prior knowledge about the functional form of the diffusion term. We show that, under mild regularity conditions, the quasi-likelihood estimators of the drift parameters are consistent. Moreover, the estimator of the threshold parameter is super consistent and weakly converges to some non-Gaussian continuous distribution. Also, the estimators of the autoregressive parameters in the drift term are jointly asymptotically normal with distribution the same as that when the threshold parameter is known. The empirical properties of the quasi-likelihood estimator are studied by simulation. We apply the threshold model to estimate the term structure of a long time series of US interest rates. The proposed approach and asymptotic results can be readily lifted to the case of a multi-regime threshold diffusion process.