Long-range dependent curve time series

Abstract

We introduce methods and theory for functional or curve time series with long- range dependence. The temporal sum of the curve process is shown to be asymp- totically normally distributed, the conditions for this covering a functional version of fractionally integrated autoregressive moving averages. We also construct an estimate of the long-run covariance function, which we use, via functional principal component analysis, in estimating the orthonormal functions spanning the dominant sub-space of the curves. In a semiparametric context, we propose an estimate of the memory parameter, and derive its consistency result. A Monte-Carlo study of finite-sample performance is included, along with two empirical applications. The first of these finds a degree of stability and persistence in intra-day stock returns. The second finds similarity in the extent of long memory in age-specific fertility rates across some developed nations.