| A Lagrange multiplier (LM) approach for testing for nonlinearity with irregularly sampled data is developed. We have implemented a Lagrange multiplier (LM) test specifically for the alternative of a nonlinear continuous-time autoregressive (NLCAR) model with the instantaneous mean having one-degree of nonlinearity. The test is then extended to testing for the alternative of general NLCAR models with multiple degrees of nonlinearity. The performance of the LM test in the finite-sample case is compared with several existing tests for nonlinearity including Keenan's (1985) test, Petruccelli and Davies' (PD) (1986) test, and Tsay's (1986, 1989) tests. The comparison is based on simulated data from some linear AR models, self-exciting threshold autoregressive (SETAR) models, bilinear models and the NLCAR models for which the LM test is some linear AR models, self-exciting threshold autoregressive (SETAR) models, bilinear models and the NLCAR models for which the LM test is designed to detect. The LM test outperforms the other tests in detecting the NLCAR model for which it is designed. Compared with the other tests, this LM test has excellent power in detecting bilinear models, but seems less powerful in detecting SETAR nonlinearity. This LM test is further illustrated with the Hong Kong beach water quality data. |