Modification Of BDS Test And Its Application

For a long time,the independence test plays an important role in Statistics,E-conometrics and Finance.With the advent of the era of big data,the data becomes more and more complex.Therefore,before modeling the data,the independence of it must be initially detected to better guide the subsequent research analysis.In addi-tion,after modeling the data,diagnosis of the model can also be transformed into the independence test of the residual sequence.Therefore,driven by so many practical requirements,a variety of independence tests have emerged.Among them,the BDS test is widely used in Financial and Econometrics due to its commendable advantages.However,the BDS test method has an nonegligible shortcoming:it's a liberal test.This shortcoming will make the BDS test easy to mislead researchers when solving practical problems,which will affect subsequent research and analysis.The focus of our paper is to correct this shortcoming of the BDS test.Since the correlation inte-gral Cm,T(?)is the basis of the BDS test,we start with Cm,T(?),analyzed the reason that caused the BDS test liberal,and proposed two amendments.First,we remove all the non-independent terms in Cm,T(?)and re-prove the asymptotic normality of the retained part under the null hypothesis,and then get the first revised BDS test,record-ed as:Wm,T(?)test.Second,we precisely calculate the expectation of the correlation integral Cm,T(?)and give its consistent estimation,and obtain the second modified BD-S test by further calculations and proofs,which is recorded as Wm,T(?)test.What's more,we designed a series of simulation experiments.Through the simulation results,it's easy to find that these two revised tests we propose do not have the problem of over-rejecting the null hypothesis,which shows that the corrections we give are effec-tive.Besides,we give an empirical analysis to demonstrate the application of Wm,T(?)test and Wm,T(?)test in practical problems,and compare their performance with the BDS test.The results show that when the length of the actual data is less than 2000,two modified tests performs better that the BDS test.