Testing for nonlinearity with partially observed time series

We have implemented a Lagrange multiplier test specifically for the alterna tive of a nonlinear continuous-time autoregressive model with the instantan eous mean having one degree of nonlinearity. The test is then extended to t esting for the alternative of general nonlinear continuous-time autoregress ive models with multiple degrees of nonlinearity. The performance of the te st in the finite-sample case is compared with several existing tests for no nlinearity including Keenan's (1985) test. Petruccelli & Davies' (1986) tes t and Tsay's (1986, 1989) tests. The comparison is based on simulated data from some linear autoregressive models, self-exciting threshold autoregress ive models, bilinear models and the nonlinear continuous-time autoregressiv e models which the Lagrange multiplier test is designed to detect. The Lagr ange multiplier test outperforms the other tests in detecting the model for which it is designed. Compared with the other tests, the test has excellen t power in detecting bilinear models, but seems less powerful in detecting self-exciting threshold autoregressive nonlinearity. The test is further il lustrated with the Hong Kong beach water quality data.