Volume 4, Issue 3

When the hypothesis of linearity of a univariate time series model is tested using a battery of tests for neglected nonlinearity, the probability of one or more tests' leading to a false rejection increases with the number of tests being performed. This paper discusses how this undesirable effect of multiple testing may be controlled by means of some simple and easily implemented procedures. Monte Carlo experiments are used to demonstrate the finite-sample effectiveness of the various methods, and an analysis of the nonlinearity properties of GDP data from five OECD countries is presented as an illustration.

The search for deterministic chaos in economic and financial time series has attracted much interest over the past decade. Evidence of chaotic structures is usually blurred, however, by large random components in the time series. In the first part of this paper, a sophisticated algorithm for estimating the largest Lyapunov exponent with confidence intervals is applied to artificially generated and real-world time series. Although the possibility of testing empirically for positivity of the estimated largest Lyapunov exponent is an advantage over other existing methods, the interpretability of the obtained results remains problematic. For instance, it is practically impossible to distinguish chaotic and periodic dynamics in the presence of dynamical noise even for simple dynamical systems. We conclude that the notion of sensitive dependence on initial conditions, as it has been developed for deterministic dynamics, can hardly be transferred into a stochastic context. Therefore, the second part of the paper aims to measure the dependencies of stochastic dynamics on the basis of a distributional characterization of the dynamics. For instance, the dynamics of financial return series are essentially captured by heteroskedastic models. We adopt a sensitivity measure proposed in literature and derive analytical expressions for the most important classes of stochastic dynamics. In practice, the sensitivity measure for the a priori unknown dynamics of a system can be calculated after estimating the conditional density of the system's state variable.

This paper examines possible nonlinearities in growth rates of nine U.K. macroeconomic time series, namely gross domestic product, price, consumption, retail sales, personal disposable income, savings, investment, industrial production and unemployment, chosen as representative of series typically used to investigate business cycle fluctuations. By basing analysis on the class of smooth-transition autoregressive (STAR) models, it is assumed that the economy can be in one of two states with distinct dynamics or in transition between these states. Except for consumption, industrial production, and unemployment, I successfully estimate STAR models that pass a set of mis-specification tests. For the former three variables, the results indicate that two-threshold (three-regime) STAR models may be needed for a better description of their dynamics. The comparison of nonlinear models with their linear counterparts shows that although in most of the cases estimated nonlinear models yield lower residual variances and lower root-mean-square errors (RMSEs) in some cases, there is essentially no evidence of nonlinearity according to Diebold and Mariano's (1995) test of equal forecast accuracy. It is worthy of note that my modeling procedure with and without dummy variables introduced to account for abnormal observations suggests that these observations should not be overlooked within the context of STAR-type nonlinear modeling.

Following the framework in Oliva et al. 1987, GEMCAT II implements a flexible method to test catastrophe models containing multivariate (i.e., latent) variables while allowing for a priori variable specifications. The system uses an efficient hybrid minimization algorithm combining the Downhill Simplex and Powell's Conjugate Gradient method. GEMCAT II is compiled in Delphi V3.0 and is sufficiently fast to allow for the use of resampling methods (bootstrap as well as jackknife) to determine the statistical significance of latent variables' indicator weights. In addition, a Pseudo-R2 index of model fit is provided, together with a test of significance, and options are included to facilitate competitive model tests of nested and non-nested catastrophe models as well as linear models. Two simulation studies are reported. Based on 61,250 simulated data sets of varying sizes, the first study addressed the effects of indicator reliability on the quality of the weight estimations, and the second dealt with the problem of false positives in model identification. The results strongly support the viability of the GEMCAT II approach over a wide range of reasonable indicator reliabilities and sample sizes. Moreover, it proved possible to distinguish reliably between cusp catastrophes and linear models based on the Pseudo-R2 values. Finally, GEMCAT II is applied to actual market data in order to demonstrate its use in an economic context. Using 34 quarters of panel data, we examine the fit of a cusp catastrophe model of organizational product adoption as applied to competing software standards in the presence of network externalities. The results are consistent with economic theory and published work on network externalities. This example also illustrates GEMCAT II's bootstrap tests for indicator weights and its options for competitive model testing.