Volume 12, Issue 1 - Nonlinear Dynamical Methods and Time Series Analysis

In this paper we introduce an autoregressive model with a deterministically shifting intercept. This implies that the model has a shifting mean and is thus nonstationary but stationary around a nonlinear deterministic component. The shifting intercept is defined as a linear combination of logistic transition functions with time as the transition variables. The number of transition functions is determined by selecting the appropriate functions from a possibly large set of alternatives using a sequence of specification tests. This selection procedure is a modification of a similar technique developed for neural network modelling by White (2006). A Monte Carlo experiment is conducted to show how the proposed modelling procedure and some of its variants work in practice. The paper contains two applications in which the results are compared with what is obtained by assuming that the time series used as examples may contain structural breaks instead of smooth transitions and selecting the number of breaks following the technique of Bai and Perron (1998).

In nonparametric tests for serial independence the marginal distribution of the data acts as an infinite dimensional nuisance parameter. The decomposition of joint distributions in terms of a copula density and marginal densities shows that in general empirical marginals carry no information on dependence. It follows that the order of ranks is sufficient for inference, which motivates transforming the data to a pre-specified marginal distribution prior to testing. As a test statistic we use an estimator of the marginal redundancy. We numerically study the finite sample properties of the tests obtained when the data are transformed to uniform as well as normal marginals. For comparison purposes we also derive a rank-based test against local ARCH alternatives. The performance of the new tests is compared with a modified version of the BDS test and with the Ljung-Box test.

This paper revisits the well known Feldstein-Horioka saving-investment correlation puzzle from a time series perspective using a sample of 21 OECD countries. We argue that the strong positive correlation between saving and investment as originally identified by Feldstein and Horioka (1980) arises due to the neglect of the nonstationary properties of the variables as well as the failure to account for potential instabilities in the long run relationship between them. Our methodology is based on instability tests recently proposed in Kejriwal and Perron (2006a) as well as the cointegration test in Arai and Kurozumi (2005) extended to allow for multiple breaks under the null hypothesis of cointegration. Our empirical results show that for all countries except Mexico and the U.K., the cointegrating relationship has changed over time; in most cases, the change being towards a lower saving-investment correlation regime. This is perfectly consistent with the recent evidence on international diversification and integration of world capital markets. Finally, we find that while the saving-investment link bears a close relationship with the degree of openness of the country, there seems to be very little evidence in favour of the commonly held view that the correlation varies with the size of the country.

The validity of any test for nonlinearity based on resampling techniques depends heavily on the consistency of the generated resampled data to the null hypothesis of linear stochastic process. The surrogate data generating algorithms AAFT, IAAFT and STAP, as well as a residual-based bootstrap algorithm, all used for the randomization or bootstrap test for nonlinearity, are reviewed and their performance is compared using different nonlinear statistics for the test. The simulations on linear and nonlinear stochastic systems, as well as chaotic systems, reveals a variation in the test outcome with the algorithm and statistic. Overall, the bootstrap algorithm led to smallest test power whereas the STAP algorithm gave consistently good results in terms of size and power of the test. The performance of the nonlinearity test with the resampling techniques is evaluated on volume and return time series of international stock exchange indices.

It has been shown in the literature that the task of estimating the parameters of nonlinear models may be tackled with optimization heuristics. Thus, we attempt to carry these intuitions over to the estimation procedure of smooth transition autoregressive (STAR, Teräsvirta, 1994) models by introducing the following three stochastic optimization algorithms: Simulated Annealing, (Kirkpatrick, Gelatt, and Vecchi, 1983), Threshold Accepting (Dueck and Scheuer, 1990) and Differential Evolution (Storn and Price, 1995, 1997). Besides considering the performance of these heuristics in estimating STAR model parameters, our paper additionally picks up the problem of identifying redundant parameters which, according to our view, has not been addressed in a satisfactory way by now. The resulting findings of our simulation studies seem to argue for an implementation of heuristic approaches within the STAR modeling cycle. In particular for the case of STAR model specification, an application of these heuristics might offer valuable information to empirical researchers.

We develop a representation of nonlinear integrated vector processes based on the martingale representation theorem of Hall and Heyde (1980). In the representation, linear combinations of the components of the vector process may be stationary, so the system may be linearly cointegrated, yet exhibit nonlinear stationary, or short-run, dynamics. We test for linear cointegration relations with nonlinear dynamics in weekly U.S. interest rates. We find that the individual rates are I(1) and that the system is linearly cointegrated. Furthermore, both cointegration relations exhibit nonlinear dynamics so the the system's short-run dynamics are nonlinear.