Essays on testing for nonlinearity in time series : issues in nonlinear cointegration, structural breaks and changes in persistence

Abstract

Besides obvious nonlinear relations like a nonlinear error correction model are nonlinearities in time series closely related to structural breaks and changes in persistence. Since both kinds of changes can induce regime-switching, they qualify well to capture the characteristic of time-variability. On the contrary, linear models are often an insufficient simplification of the real underlying DGP because they fail ro reproduce trends, shocks like finance crises and stylized facts such as long-range dependencies as well as volatility clustering. This is why testing for the presence of these nonlinear properties as the first step of any statistical analysis is very crucial, especially with regard to effective model building. Nonlinear Cointegration In the first chapter, a nonlinear cointegration test is proposed which builds on Kapetanios et al. (2006) who where the first who addressed cointegration in a nonlinear error correction framework under the alternative. The switch between regimes is modeled to follow a second order logistic smooth transition (D-LSTR) function and a null hypothesis of no cointegration is tested against globally stationary D-LSTR cointegration. From the nonlinear error correction regression, t-type and F-type statistics are derived and finite-sample investigations are conducted. The results of the modified nonlinear cointegration test are compared to a comparable linear cointegration test, namely the test proposed by Johansen (1991). The D-LSTR function qualifies well as an overall-generalization of transition functions and it is found that the D-LSTR error correction model has power against both alternatives, D-LSTR as well as 3-regime TAR nonlinearity which is nested for large gamma in the D-LSTR function. Structural breaks The topic of the second paper is to survey the most frequently applied volatility break tests when they are employed to a broad range of different DGPs. Within a simulation study, the break tests are applied to DGPs which can exhibit either single- or double-shifting or the process can experience a smooth increase in the magnitude of the volatility break. The surveyed tests are a CUSUM test in a version proposed by Deng and Perron (2008) and conventional Wald and LM tests. Besides size and power comparisons the break tests are empirically validated and it is found that more breaks are found in equity series than in exchange rate series. One main finding is that huge outliers in the data can impact the long-run variance of the squared return process to be no longer finite which renders non-monotonic power functions. Changes in Persistence Chapter three addresses the specific question whether either structural breaks or nonstationarity in the conditional volatility affect the testing decision of the R test proposed by Leybourne et al. (2007). The additional structural breaks in the conditional volatility process are not specified under the null hypothesis and may therefore lead to a non-pivotal limiting distribution. Hence, heteroskedasticity of an unknown form is encountered and in order to potentially robustify the testing procedure, a wild-bootstrapped version of Leybourne et al. (2007)'s R test is suggested. Within a simulation study, size and power of the originally proposed test and the wild-bootstrap analogue are compared for various constellations of simultaneous breaks in the AR parameter as well as the GARCH parameter. It is found that the Leybourne et al. (2007) test seems heavily impacted by additional structural breaks in the conditional volatility, especially in very finite sample sizes. In an empirical application the two testing procedures are applied and evaluated.