Unit Root Tests For Heavy-tailed Time Series And Its Applications

Since heavy tailed data widely exist in real life,it is necessary to add the premise assumption of heavy tailed distribution in modeling to ensure that its statistical properties can be accurately described.On the other hand,it is of positive practical significance to study the unit root test method,because the stability of the sequence is the prerequisite for modeling most time series models,which represents that the statistical properties of random variables do not change with time.This paper firstly reviews the development process of the unit root test method for heavy-tailed data,and summarizes the defects of the existing method,including the test depends on the Alpha stable distribution,does not involve the case of α≤ 1,and the Bootstrap method is too complicated.Therefore,the author constructs a new empirical likelihood test statistic and deduces its large sample theorem in this essay.In the unit root test of the AR(1)model,the theoretical process shows that the condition of belongs to the domain of attraction of a distribution can be replaced,and it is also applicable to the situation where random errorterms are symmetric martingale difference sequences.In the following content,the statistic is extended to the adjusted empirical likelihood statistic,and a statistic suitable for AR(p)model is constructed by adding the condition of the Alpha stable distribution.Next,the heavy tailed unit root test methods are compared by Monte Carlo.The results show that AR(p)statistic is effective in AR(1)and AR(p)models with independent identically distributed or autocorrelation or conditional heteroscedasticity of errorterms.In contrast,AR(1)statistic has a smaller scope of application and is only applicable to AR(1)models whose errorterms are martingale difference sequence.Further,compared with other statistics,the new statistic has high efficiency in the case of heavy tail,but low efficiency in the case of light tail,which is the same as Cauchy m-estimation statistic.In the next section,this paper determines the optimal value of the hyperparameter through simulation.At the last part of the empirical analysis,the new empirical likelihood statistics are used in daily data,5-minute and 15-minute data of the stock market.The conclusion shows that China’s stock market has not reached weak form efficiency,and the efficiency of the current stock market is still at a low level and can’t converge to the equilibrium price in a short time.