Wavelet Methods And Application For Time Series Econometrics

Time series econometric analysis originated from the cognition of people that economic activities are carried out in a certain time and space conditions, and are influenced by social,economic, cultural and other factors together, so that the outside economic phenomenon usually will show the autocorrelation between before and after in the time dimension. They constitute the core goal for time series econometric analysis to scientifically characterize or approximate the dynamic relations or laws of economic phenomena in time dimension, and to establish a model to meet the needs of other management practices such as economic forecasting and decision making. Since the publication of "Time series analysis: Forecasting and control" in 1970, the theory and application of time series econometrics have been developed progressively.It is particularly noteworthy that some theories, such as non-stationary unit root processes,cointegration systems, heterogeneity and random heteroskedasticity models, have been rising since the early 1980s and have largely changed the analysis theory and method of the traditional time series econometrics. Stationary time series is no longer the only object of econometric research, and non-stationary time series is no longer an area that can not be involved. Especially the process and cointegration process have become the main object of research,have been widely used in the economy and finance field.However, most of the current time series econometric practices are carried out in the time domain, including the construction, estimation and testing of the econometric models. In addition, time domain analysis and frequency domain analysis seem to be cut off, and they walked in two independent tracks respectively. Usually the stage results obtained from time domain and frequency domain did not been combined organically, thus the integrity of the results been greatly reduced. In fact, it has been proved that many valuable statistics can be constructed in the frequency domain for estimating or testing econometric models, and maybe possess better statistical properties over time domain analysis. Wavelet analysis, as a new mathematical theory and method, has good resolution in time domain and frequency domain。Its application can make time series analysis to achieve a good localized effect both in the time domain and frequency domain,and provides a new perspective and tools to insight into the dynamic of a time series.On the basis of deep understanding of wavelet analysis, time series analysis theory and summarizing previous research experience, this dissertation synthesizes the knowledge of statistics, probability theory and mathematical statistics, and financial economics, embedded the advantages of wavelet analysis in time-frequency domain in the existing time series analysis theory. The theoretical research and application of the non-stationary time series analysis are extended and enriched. Specifically, the contribution of this dissertation mainly includes the following three major aspects.(1) Methodological aspects of unit root testing. Firstly, based on the keen sense of difference between the null hypothesis (unit root) the and the alternative assumption (absence of the unit root) for coupling mechanism, which exists in the sample variance of the original sequence, that of the series of wavelet coefficient sequence and that of scale coefficients sequence, A new test statistic TXL1 is constructed in the wavelet domain to test the unit root process with drift items. Fan and Gencay (2010) is extended in construction strategy of the test statistic and the scope of the test object. Secondly, the large-sample properties of the test statistic TXL1 under the original hypothesis and the alternative hypothesis are fully proved. The results show that the statistical distribution of the test statistic TXL1 converges to the random functional of the two independent standard Wiener process under the null hypothesis, while the test statistics TXL1 converges to 0 point by probability under the alternative hypothesis. This excellent nature is very useful for ensuring the high test power of the test statistic. In addition,taking the reality into account that the limited sample data can be obtained in the application practice, some Monte Carlo experiment is used to study the test power and the test level of the test statistic TXL1 under small sample condition. Monte Carlo experimental results show that the test statistic TXL1 has a certain level of test size distortion under small-sample condition,but it has very high test power under the same conditions. Finally, in the process of constructing the test statistic TXL1 and proving its the large sample properties, the other new convergence properties of a random walk are expanded and saved in two lemmas whose proof is given at the same time. These properties can provide the valuable help for the other studies of non-stationary time series.(2) Methodological aspects of cointegration testing. First, it is demonstrated sufficiently that if the cointegration test in wavelet domain is to be considered in reference to the EG two-step frame, the potential cointegration model under the initial estimation should select securely the regression model with the intercept term, and in testing the stability of residual from potential cointegration regression, autoregressive model of random walk without drift should be considered. Then, in order to realize the cointegration test in the wavelet domain, another new test statistic TXL1* is developed for verifying the unit root test without drift, and its large sample properties are deduced. Secondly, we learn from the direct simulation strategy in Dickey and Fuller (1979) and Phillips and Perron (1988), and give the critical value of the test statistic TXL1* when applied to the pseudo-cointegration regression test by a large number of stochastic simulations. In addition,six random experiments were designed to study the specific performance of the test statistic TXL1* in cointegration test. The results show that the test size has very weak distortion under various conditions, and test power become high when the sample capacity exceeds 500. Finally, through the study of an actual case, the validity of the test statistic TXL1* in cointegration test is verified, and the empirical evidence is provided for the long-term equilibrium relationship between the Chinese gold market and the international gold market.(3)Application of wavelet domain hidden Markov model. According to these facts:Following banker is generally phenomenon in stock market. That means that short-term trader’soperation depends on long-term traders’. Long or short-term transaction is denominated in a certain time scale for the conditions. The traditional time series analysis, focusing on time series at a given scale,lacks the abilities to explore the links between transaction behaviors at different scales. So In this dissertation, we study the statistical properties of the information flow between China’s stock market volatilities along time scales,using high-frequency data and wavelet domain hidden Markov model. The results show that there is a significant asymmetry during the volatilities conduction. The asymmetry senses that a low volatility state at a long time horizons is most likely followed by low volatility states at shorter time horizons, while a high volatility state at long time horizon does not necessarily imply a high volatility state at a shorter time horizons. Finally, the policy implications of the result are explained.