Non-Linear Time Series by Kamil Feridun Turkman Manuel González Scotto & Patrícia Zea Bermudez

Author:Kamil Feridun Turkman, Manuel González Scotto & Patrícia Zea Bermudez
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

4. Inference for Nonlinear Time Series Models

Kamil Feridun Turkman1 , Manuel González Scotto2 and Patrícia de Zea Bermudez1

(1)Departmento de Estatística e Investigação Operacional, Faculdade de Ciências Universidade de Lisboa, Lisboa, Portugal

(2)Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

4.1 Identification of Nonlinearity

Suppose we have an observed time series and want to know if a linear time series model is adequate for the data, or an alternative nonlinear model should be considered. Linear models are often taken as the null hypotheses against a nonlinear alternative due to the simplicity of inference. Often we know much about the underlying process which generate the data set. Therefore it is possible to decide if a linear model will be adequate and if not, what aspects of nonlinearity should be modeled as alternative. For example, if the data is on population dynamics, as explained before, limit cycle behavior can be expected, and an adequate model such as a self-exciting threshold model should seriously be considered as an alternative. On the other hand, bilinear models may be more adequate for telecommunication data which often exhibit heavy-tailed behavior. However, if we know little about the underlying process, we will have to rely only on the information contained in the data, and empirical methods for testing nonlinearity are needed. In this case, formulation of formal tests is quite difficult due to the fact that there are many different ways a process can be nonlinear, and often it is difficult to specify alternative hypotheses to a linear model. Naturally, the power of tests constructed will depend on how well the alternative hypotheses are constructed, and complicated tests of hypotheses may not worth the effort put into devising such tests. Therefore, quick graphical methods and portmanteau tests are often used for checking nonlinearity.

Subba Rao and Gabr (1980) were possibly the first to propose a linearity test, which is based on the characteristics of the third-order cumulant spectrum of a linear process, and later this test was improved by Hinich (1982). McLeod and Li (1983) proposed a portmanteau test and Keenan (1985) devised an easy to use test which is based on arranged auto-regressions. Petruccelli and Davies (1986) constructed a CUSUM test which is also based on the arranged auto-regression approach. Brock et al. (1996) (see also Luukkonen et al. 1988) proposed a test for linearity against a STAR model alternative. Tsay (1989) devised an F-type test for assessing a TAR/SETAR model alternative hypothesis that used the arranged auto-regression methodology. Tsay (1991) modified his previously proposed test in order to be both simple to use and general enough in order to be able to identify several types of nonlinearity, and consequently a wide range of possible nonlinear models such as Bilinear, STAR, EXPAR and SETAR were included as alternative hypotheses. These tests were also used as basis for model selection. Peat and Stevenson (1996) slightly modified the test proposed by Tsay (1991). Bera and Higgins (1997) proposed two stage tests for GARCH and bilinear type models. In the first stage, a joint test is devised to see if existing nonlinearity can be attributed to ARCH or bilinearity.

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