Testing for nonlinearity with partially observed time series

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

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