Testing For Multivariate White Noise Under Unknown Dependence Based On Random Weighting Bootstrap Method

Testing for white noise has been widely applied to many problems in statistics and econometrics,such as effectiveness testing of financial market and adequacy checking for ARMA model.From the perspective of theoretical research,testing for white noise has been a classic topic in statistic theory.Proposed by Box and Pierce(1970),portmanteau test has become the most frequently used method to test the white noise.Francq and Raissi(2007)pointed out that the traditional portmanteau test statistic for the multiple white noise under unknown dependence is no longer asymptotically distributed by null ?2-distribution,which leaves the traditional portmanteau test to lose asymptotic validity.To resolve this problem,a random weighting method employed by Zhu(2016)in the context of univariate time series is extended to multivariate case to bootstrap the asymptotically valid critical value for the traditional portmanteau test statistic.Based on the asymptotic analysis of random weighting method,a bunch of Monte Carlo experiments have been utilized to investigate the finite-sample properties of the portmanteau test based on random weighting method.Traditional method and random weighting method are both used simultaneously to test the sample series generated from independent and identically distributed series,martingale difference sequences and white noise series respectively,and it turns out that with effectively undoing the size distortion and over-rejection caused by traditional method,the random weighting method has better performance in size property.Then we use two methods above to test the VARMA models with different auto-correlation patterns and find that random weighting method still performs well in terms of power property.Finally,some additional simulations are offered to check the robustness of the finite-sample properties of random weighting method,which shows that the testing results are robust to the distribution of random weights in bootstrap steps while the finite-sample properties are relatively sensitive to the heavy-tail characteristics of the tested data,especially when the sample size is small.At last,random weighting method,as well as the traditional method,has been applied to the real data example,5-daily return observations of exchange rates of two Asian-Pacific currencies,namely the exchange rates of Malaysian ringgit-US dollar and Singapore dollar-US dollar,which has been analyzed by Tse(2000)and turns out to follow a CCC-GARCH(1,1)process.The result of random weighting method shows that tested multiple return series are white noise indeed while the result of traditional method goes opposite.Considering the previous theoretical analysis,simulation results and statistical characteristics of sample data,we tend to accept the result of random weighting method,which exactly coincides with the conclusion given by Tse(2000).