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.

Basu and Bundick, 2017 showed an intertemporal preference volatility shock has meaningful effects on real activity in a New Keynesian model with Epstein and Zin, 1991 preferences. We show that when the distributional weights on current and future utility in the Epstein–Zin time aggregator do not sum to 1, there is an asymptote in the responses to such a shock with unit intertemporal elasticity of substitution. In the Basu–Bundick model, the intertemporal elasticity of substitution is set near unity and the preference shock only hits current utility, so the sum of the weights differs from 1. We show that when we restrict the weights to sum to 1, the asymptote disappears and preference volatility shocks no longer have large effects. We examine several different calibrations and preferences as potential resolutions with varying degrees of success.