Long†Run Covariability

We develop inference methods about long†run comovement of two time series. The parameters of interest are defined in terms of population second moments of low†frequency transformations (“low†pass†filtered versions) of the data. We numerically determine confidence sets that control coverage over a wide range of potential bivariate persistence patterns, which include arbitrary linear combinations of I(0), I(1), near unit roots, and fractionally integrated processes. In an application to U.S. economic data, we quantify the long†run covariability of a variety of series, such as those giving rise to balanced growth, nominal exchange rates and relative nominal prices, the unemployment rate and inflation, money growth and inflation, earnings and stock prices, etc.