(Fractional) Beta Convergence

Unit roots in output, an exponential 2% rate of convergence and no change in the underlying dynamics of output seem to be three stylized facts that cannot go together. This paper extends the Solow-Swan growth model allowing for cross-sectional heterogeneity. In this framework, aggregate shocks might vanish at a hyperbolic rather than at an exponential rate. This implies that the level of output can exhibit long memory and that standard tests fail to reject the null of a unit root despite mean reversion. Exploiting secular time series properties of GDP, we conclude that traditional approaches to test for uniform (conditional and unconditional) convergence suit first step approximation. We show both theoretically and empirically how the uniform 2% rate of convergence repeatedly found in the empirical literature is the outcome of an underlying parameter of fractional integration strictly between (1)/(2) and 1. This is consistent with both time series and cross-sectional evidence recently produced. Keyword(s): Growth model; Convergence; Long memory; Aggregation