Applying spline-based phase analysis to macroeconomic dynamics

The article uses spline-based phase analysis to study the dynamics of a time series of low-frequency data on the values of a certain economic indicator. The approach includes two stages. At the first stage, the original series is approximated by a smooth twice-differentiable function. Natural cubic splines are used as an approximating function y y . Such splines have the smallest curvature over the observation interval compared to other possible functions that satisfy the choice criterion. At the second stage, a phase trajectory is constructed in ( t , y , y ′ ) \left(t,y,y^{\prime} ) -space, corresponding to the original time series, and a phase shadow as a projection of the phase trajectory onto the ( y , y ′ ) (y,y^{\prime} ) -plane. The approach is applied to the values of GDP indicators for the G7 countries. The interrelation between phase shadow loops and cycles of economic indicators evolution is shown. The study also discusses the features, limitations and prospects for the use of spline-based phase analysis.