Nonstationarity in time series of state densities

This paper proposes a new framework to analyze the nonstationarity in the time series of state densities, representing either cross-sectional or intra-period distributions of some underlying economic variables. We regard each state density as a realization of Hilbertian random variable, and use a functional time series model to fit a given time series of state densities. This allows us to explore various sources of the nonstationarity of such time series. The potential unit roots are identified through functional principal component analysis, and subsequently tested by the generalized eigenvalues of leading components of normalized estimated variance operator. The asymptotic null distribution of the test statistic is obtained and tabulated. We use the methodology developed in the paper to investigate the state densities given by the cross-sectional distributions of individual earnings and the intra-month distributions of stock returns. We find some clear evidence for the presence of strong persistency in their time series.