White Noise Test Of High Dimensional Time Series

Testing for white noise has been widely applied to many problems in statistics and econometrics,such as the test of the efficiency hypothesis of financial market and the goodness of fit for ARMA model.From the perspective of theoretical research,testing for white noise has been a classical topic in statistical theory.This paper mainly studies how to extend the white noise test of U statistics in one-dimensional samples to the white noise test of multidimensional samples.The high-dimensional hypothesis testing problem is also known as a global test,such as the test of the mean vector or covariance matrix of the population.Since the high-dimensional white noise is the time series whose auto-covariance matrix with arbitrary delay is the zero matrix,the white noise test of this paper mainly tests the auto-covariance matrix of the given samples.Firstly,the auto-covariance matrix with arbitrary delay is transformed into a vector whose form is θ={el:l θ L),where el is the element we are interested in.To achieve overall inspection,the vector is converted into different norms,and we define ||θ||aa=Σl∈L ela which is the a norm of θ to the a power,and a is any positive integer.We show that under the null hypothesis,the U-statistics of different finite orders are independent and asymptotically normally distributed.We illustrate the accuracy and the power of the proposed test by simulation studies,which also show that the new test outperforms several commonly used methods,and the test is stable especially when the dimension of time series is large.In addition,the assumption of the specified sample sequences is no longer limited to the strong condition of independent identical distribution in this paper and the application range of the high-dimensional white noise test can be extended.