Statistical inference of nonlinear Granger causality: a semiparametric time series regression analysis.

Abstract

Since the seminal work of Granger (1969), Granger causality has become a useful concept and tool in the study of the dynamic linkages between economic variables and to explore whether or not an economic variable helps forecast another one. Researchers have suggested a variety of methods to test the existence of Grangercausality in the literature. In particular, linear Granger causality testing has been remarkably developed; (see, for example, Toda & Philips (1993), Sims, Stock & Watson (1990), Geweke (1982), Hosoya (1991) and Hidalgo (2000)). However, in practice, the real economic relationship between different variables may often be nonlinear. Hiemstra & Jones (1994) and Nishiyama, Hitomi, Kawasaki & Jeong (2011) recently proposed different methods to test the existence of any non-linear Granger causality between a pair of economic variables under a α-mixing framework of data generating process. Their methods are general with nonparametric features, which however suffer from curse of dimensionality when high lag orders need to be taken into consideration in applications. In this thesis, the main objective is to develop a class of semiparametric time series regression models that are of partially linear structures, with statistical theory established under a more general framework of near epoch dependent (NED) data generating processes, which can be easily used in exploring nonlinear Granger causality. This general NED framework, which takes the α-mixing one as a very special case, is popular in nonlinear econometric analysis. The reasons why we will adopt such a semi-parametric model structure for time series regression under the general NED framework are not only because it is a natural extension of the structure of linear Granger causality analysis but also because it enables us to estimate and hence understand the likely structure of the non-linear Granger-causality if it exists. To our best knowledge, in the literature, this is still an early effort that seeks the causality structure in nonlinear Granger-causality beyond the testing of its existence. Furthermore, semiparametric structures help to reduce the curse of dimensionality that purely nonparametric methods suffer from in Granger-causality testing. We study the semiparametric regression models under a more general framework of NED time series processes, which include, for example, the popular ARMA(p,q)-GARCH(r,m) model in financial econometrics which is hard to be shown to be α-mixing except in some very special cases. By using the idea of Robinson (1988) and the theory developed in Lu & Linton (2007) under NED, we can construct the estimators of both the parameters and the unknown function in our model. We have also established asymptotic theory for these estimators under NED. The estimated unknown functional part and its confidence interval can tell us useful information on whether or not there exists a nonlinear Granger causality between the variables, and moreover, we can find the functional form of the nonlinear Granger causality. In order to examine the finite-sample performance of the proposed estimators for our model under study, besides the developed large-sample theory, we have also conducted Monte Carlo simulation studies. The simulation results clearly demonstrate that we can estimate both the parameters and the unknown function rather accurately under moderate sample sizes. Finally, we have empirically applied the proposed methodology to examining the existence or effects of nonlinear Granger-causality between each pair of the financial markets involving Australia and the USA, UK and China which are closely related to Australia in economy. Weekly return data are used to avoid the possible market micro structure differences. Interestingly, we have found that there is a strongly nonlinear Granger causality from the UK FTSE100 to the Australian ASX200, and some linear or nonlinear Granger causality from the USA S&P500 to the Australian ASX200, as well as some negatively linear Granger causality from the China SSE index to the Australian ASX200. From these results, we can fairly say that the USA, UK and Chinese stock markets have close linkages with and impacts on the Australian market, with the stock prices from these foreign countries 1 week ago can help to predict current Australian stock market behavior. We have also found that there is some linear Granger causality from the UK FTSE100 and the USA S&P500 to the China SSE index, respectively, but we cannot see the Granger-causality from the Australian ASX200 to the China SSE. We either can not find apparent evidence of the Granger causalities from the other countries to the UK or the USA. In addition, we have examined the lag 2 Granger-causality effect between each pair of these markets, and only find that the Chinese stock price 2 weeks ago appears to help predict this week's Australian stock price. In summary, the main contributions of this thesis are as follows: • We have suggested semiparametric time series regression models of partially linear structures to examine possibly nonlinear Granger causality, with methods to estimate both the parameters and the unknown nonparametric function proposed. • We have established the consistency and asymptotic normality for the estimators under a more general, popular data generating framework of near epoch dependence. • Monte Carlo simulation studies reveal that the proposed methodology works well for both linear and nonlinear functional forms of Granger causality in finite samples. • Interesting empirical applications find that there are clear dynamic linkages of the Australian stock market impacted by the USA, UK and Chinese markets.