A note on testing for nonlinearity with partially observed time series

We have implemented a Lagrange multiplier test for the alternative hypothesis of a nonlinear continuous-time autoregressive model with the instantaneous mean having multiple degrees of nonlinearity. This test is an extension of a Lagrange multiplier test proposed by Tsai & Chan (2000), with the alternative model analogous to the model used in Tsayʹs (1986) discrete-time work.The performance of the test in the finite-sample case is compared with several existing tests for nonlinearity including Keenanʹs (1985) test, Petruccelli & Daviesʹ (1986) test, Tsayʹs (1986, 1989) tests and Tsai & Chanʹs (2000) test. The comparison is based on simulated data from some linear autoregressive models, self-exciting threshold autoregressive models, bilinear models and the nonlinear continuous-time autoregressive models for which the Lagrange multiplier test is designed. In general, the test is more powerful than all the other tests. The test is further illustrated with the annual sunspot data and the lynx data.