Non-parametric determination of real-time lag structure between two time series: The “optimal thermal causal path” method with applications to economic data

We introduce a novel non-parametric methodology to test for the dynamical time evolution of the lag-lead structure between two arbitrary time series. The method consists in constructing a distance matrix based on the matching of all sample data pairs between the two time series. Then, the lag-lead structure is searched as the optimal path in the distance matrix landscape that minimizes the total mismatch between the two time series, and that obeys a one-to-one causal matching condition. We apply our method to the question of the causality between the US stock market and the treasury bond yields and confirm earlier results on a causal arrow of the stock markets preceding the Federal Reserve Funds adjustments as well as the yield rates at short maturities in the period 2000-2003. The application to inflation, inflation change, GDP growth rate and unemployment rate unearths non-trivial causal relationships: the GDP changes lead inflation especially since the 1980s, inflation changes lead GDP only in the 1980 decade, and inflation leads unemployment rates since the 1970s. In addition, we detect multiple competing causality paths in which one can have inflation leading GDP with a certain lag time and GDP feeding back/leading inflation with another lag time.