Testing for non-linearity and asymmetry in time series

The Ph.D. thesis, called Testing for Non-linearity and Asymmetry in Time Series, focuses on various issues related to testing for non-linearity and marginal asymmetry of economic time series. This is an important issue since testing for non-linearity and/or asymmetry represents an early, yet crucial, step in the whole process of time series modelling. A mistake in this preliminary step may lead to model misspecification, and, subsequently, to a sequence of related issues throughout all the modelling steps (i.e. identification, estimation, and forecasting). As a result, this type of mistakes is very likely to result in wrong business or economic policy decisions. The thesis is divided into six chapters. The first chapter explains the motivation for the thesis. The second chapter, called Robustness of the Power of Non-linearity Tests, examines the statistical properties of the selected univariate non-linearity tests under different conditions. In particular, special attention is paid to the robustness of the power properties of the tests against moment condition failure of innovations, asymmetry of innovations, and the parameter configuration of data generating processes. Since analytical results are available only for a very limited number of the test statistics, an extensive Monte Carlo approach is implemented instead. The Monte Carlo results reveal that the power of the selected non-linearity tests is statistically significantly inflated under asymmetry of innovations and moment condition failure. In the third chapter, called Testing for Non-linearity Using a Modified Q Test, a new version of the portmanteau Q test, based on auto- and cross-correlations, is developed. The main task of this chapter is to propose a new type of the Q test in order to bypass some of the shortcomings of the McLeod and Li Q test discovered in Chapter 2. Our results, based on extensive Monte Carlo experiments, suggest the proposed Q test significantly improves the power against some non-linear time series models (e.g. threshold autoregressive and moving average models) and is capable to detect some interesting non-linear processes (e.g. non-linear moving average models), for which the standard Mcleod and Li Q test completely fails. In the fourth chapter, called Testing for Marginal Asymmetry in Time Series, a modified test for symmetry of the marginal law of weakly dependent processes is proposed. The test statistic is based on sample quantiles. It is shown that the test has an intuitive interpretation, it is easy and fast to calculate, it it follows a standard limiting distribution, and much more importantly, it is robust against weak dependence of observations. Especially the last feature makes the test very attractive for the use in applied economics since it minimizes inferential errors due to the incorrect configuration of the test. The finite sample properties of the test are examined via Monte Carlo experiments. The results suggest that the quantile-based test of symmetry performs very well. In the fifth chapter, called Testing for Non-linearity in Multivariate Time Series, two new principal component-based multivariate non-linearity tests are considered. The main goal of this chapter is to modify two well known multivariate test statistics which suffer from the curse of dimensionality. It is shown that a dimensionality problem can be easily bypassed by means of a principal component analysis. Our results, based on extensive Monte Carlo experiments, suggest that a principal component analysis reduces the dimensionality problem very efficiently without any systematic power distortion. The results also reveal that the BIC stopping rule performs best in determining the number of components for the selected multivariate non-linearity tests. The last chapter summarizes the results of this thesis and discusses directions for further research.