Symmetric thermal optimal path and time-dependent lead-lag relationship: novel statistical tests and application to UK and US real-estate and monetary policies

We present the symmetric thermal optimal path (TOPS) method to determine the time-dependent lead-lag relationship between two stochastic time series. This novel version of the previously introduced thermal optimal path (TOP) method alleviates some inconsistencies by imposing that the lead-lag relationship should be invariant with respect to a time reversal of the time series after a change of sign. This means that, if ‘X comes before Y’, this transforms into ‘Y comes before X’ under a time reversal. We show that a previously proposed bootstrap test lacks power and leads too often to a lack of rejection of the null that there is no lead-lag correlation when it is present. We introduce instead two novel tests. The first criterion, based on the free energy p-value ρ$ \rho $, quantifies the probability that a given lead-lag structure could be obtained from random time series with similar characteristics except for the lead-lag information. The second self-consistent test embodies the idea that, for the lead-lag path to be significant, synchronizing the two time series using the time varying lead-lag path should lead to a statistically significant correlation. We perform intensive synthetic tests to demonstrate their performance and limitations. Finally, we apply the TOPS method with the two new tests to the time-dependent lead-lag structures of house price and monetary policy of the United Kingdom (UK) and United States (US) from 1991 to 2011. We find that, for both countries, the TOPS paths indicate that interest rate changes were lagging behind house price index changes until the crisis in 2006–2007. The TOPS paths also suggest a catch up of the UK central bank and of the Federal Reserve still not being on top of the game during the crisis itself, as diagnosed by again the significant negative values of TOPS paths until 2008. Only later did the central banks interest rates as well as longer maturity rates lead the house price indices, confirming the occurrence of the transition to an era where the central bank is ‘causally’ influencing the housing markets more than the reverse. The TOPS approach stresses the importance of accounting for change of regimes, so that similar pieces of information or policies may have drastically different impacts and developments, conditional on the economic, financial and geopolitical conditions. This study reinforces the view that the hypothesis of statistical stationarity in economics is highly questionable.