On a Class of Estimation and Test for Long Memory

This paper advances a new analysis technology path of estimation and test for long memory time series. I propose the definitions of time scale series, strong variance scale exponent and weak variance scale exponent, and prove the strict mathematical equations that strong and weak variance scale exponent can accurately identify the time series of white noise, short memory and long memory, especially derive the equation relationships between weak variance scale exponent and long memory parameters. I also construct two statistics which SLmemory statistic tests for long memory properties. The paper further displays Monte Carlo performance for MSE of weak variance scale exponent estimator and the empirical size and power of SLmemory statistic, giving practical recommendations of finite-sample, and also provides brief empirical examples of logarithmic return rate series data for Sino-US stock markets.