Inference of Bivariate Long-memory Aggregate Time Series

We study the autocorrelation structure of aggregates from a bivariate continuous-time process. The underlying model is assumed to be a stationary bivariate Continuous-time Auto-Regressive Fractionally Integrated MovingAverage (CARFIMA) process driven by two independent standard fractional Brownian motions with two Hurst parameters. The limiting model of the aggregates, as the extent of aggregation increases to infinity, is shown to be an instantaneous linear transformation of two independent fractional Gaussian noises with the same Hurst parameters of the continuous-time process from which the aggregates arederived. We derive the likelihood ratio test for testing the equality of the two Hurst parameters, within the framework of Whittle likelihood. The limiting properties of the proposed test statistic and Whittle likelihood estimation are derived, and their finite sample properties are studied by simulation. The efficacy of the proposed approach is demonstrated with a data analysis.