Essays on nonlinear time series modelling och hypothesis testing

Abstract: There seems to be a common understanding nowadays that the economy is nonlinear. Economic theory suggests features that can not be incorporated into linear frameworks, and over the decades a solid body of empirical evidence of nonlinearities in economic time series has been gathered. This thesis consists of four essays that have to do with various forms of nonlinear statistical inference. In the first chapter the problem of determining the number regimes in a threshold autoregressive (TAR) model is considered. Typically, the number of regimes (or thresholds) is assumed unknown and has to be determined from the data. The solution provided in the chapter first uses the smooth transition autoregressive (STAR) model with a fixed and rapid transition to approximate the TAR model. The number of thresholds is then determined using sequential misspecification tests developed for the STAR model. The main characteristic of the proposed method is that only standard statistical inference is used, as opposed to non-standard inference or computation intensive bootstrap-based methods. In the second chapter a similar idea is employed and the structural break model is approximated with a smoothly time-varying autoregressive model. By making the smooth changes in parameters rapid, the model is able to closely approximate the corresponding model with breaks in the parameter structure. This approximation makes the misspecification tests developed for the STR modelling framework available and they can be used for sequentially determining the number of breaks. Again, the method is computationally simple as all tests rely on standard statistical inference. There exists literature suggesting that business cycle fluctuations affect the pattern of seasonality in macroeconomic series. A question asked in the third chapter is whether other factors such as changes in institutions or technological change may have this effect as well. The time-varying smooth transition autoregressive (TV- STAR) models that can incorporate both types of change are used to model the (possible) changes in seasonal patterns and shed light on the hypothesis that institutional and technological changes (proxied by time) may have a stronger effect on seasonal patterns than business cycle. The TV-STAR testing framework is applied to nine quarterly industrial production series from the G7 countries, Finland and Sweden. These series display strong seasonal patterns and also contain the business cycle fluctuations. The empirical results of the chapter suggest that seasonal patterns in these series have been changing over time and, furthermore, that the business cycle fluctuations do not seem to be the main cause for this change. The last chapter of the thesis considers the possibility of testing for Granger causality in bivariate nonlinear systems when the exact form of the nonlinear relationship between variables is not known. The idea is to linearize the testing problem by approximating the nonlinear system by its Taylor expansion. The expansion is linear in parameters and one gets round the difficulty caused by the unknown functional form of the relationship under investigation.