Combining Long and Short Memory in Time Series Models: the Role of Asymptotic Correlations of the MLEs

A major practical problem in the application of the long memory ARFIMA model has been distinguishing between the long memory and short memory components and subsequent estimation of the model. The asymptotic correlations between the Maximum Likelihood Estimators (MLE) of the long memory parameter, d, and the short memory parameters within an ARFIMA estimation are derived. The correlation in an ARFIMA(1,d,0) model can be as large as −0.95. However, MLE still works well in these high correlation cases; even for non stationary situations. Similarly, QMLE also performs well in simulations where the innovations have t or skewed t densities. MLE also performs very well for ARFIMA(3,d,0) models when there is a moderate level of persistence in the short memory part. However, MLE can perform extremely poorly when there is a combination of long memory and substantial persistence in the short memory component. Some of these points are illustrated in the analysis of some Realized Volatility time series which contain long memory. The correlation in “gap ARFIMA” models is comparatively low which indicates that MLE is quite stable in this case. Some suggestions and recommendations for applied work are provided.