Inference on Common Stochastic Trends in Functional Time Series

Abstract

We provide statistical procedures to identify the number of common stochastic trends embedded in functional time series, which may be especially important in economic applications that are often accompanied by nonstationarity. Each of those is given by sequential applications of a proposed test based on a generalized eigenvalue problem associated with sample covariance operators. In particular, we provide a bottom-up procedure that determines the estimate by sequentially testing the null hypothesis that there are a specified number of common stochastic trends against the alternative that there are more. This is distinct from the existing top-down testing procedures and has theoretical advantages especially in an infinite-dimensional setting. Interestingly, the bottom-up procedure entails a top-down procedure as its reverse, and those may be viewed as two different consequences of a single asymptotic phenomenon. We also find some connections between the existing tests and ours: specifically, we provide tests that are asymptotically or exactly equivalent to the existing tests with slight modifications from the eigenvalue problem that our testing procedures are based on. Our Monte Carlo experiments suggest that the finite-sample performances of the proposed tests are satisfactory. We apply our methodology to two empirical examples: U.S. age-specific shares of full-time employment and hourly real wage densities.