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 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 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, 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 selfconsistent 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.

We employ the thermal optimal path method to explore both the long-term and short-term interaction patterns between the onshore CNY and offshore CNH exchange rates (2012)(2013)(2014)(2015). For the daily data, the CNY and CNH exchange rates show a weak alternate lead-lag structure in most of the time periods. When CNY and CNH display a large disparity, the lead-lag relationship is uncertain and depends on the prevailing market factors. The minute-scale interaction pattern between the CNY and CNH exchange rates change over time according to different market situations. We find that US dollar appreciation is associated with a lead-lag relationship running from offshore to onshore, while a (contrarian) Renminbi appreciation is associated with a lead-lag relationship running from onshore to offshore. These results are robust with respect to different sub-sample analyses and variations of the key smoothing parameter of the TOP method.

The diagonal effect of orders is well documented in different markets, which states that the orders are more likely to be followed by the orders of the same aggressiveness and implies † † Corresponding author. The detrending moving average analysis shows that there are crossovers in the scaling behaviors of overall fluctuations and order aggressiveness exhibits linear long-term correlations. We design an objective procedure to determine the two Hurst indexes delimited by the crossover scale. We find no correlations in the short term and strong correlations in the long term for all stocks except for an outlier stock. The long-term correlation is found to depend on several firm specific characteristics. We also find that there are nonlinear long-term correlations in the order aggressiveness when we perform the multifractal detrending moving average analysis. 1750041-1

This study presents an agent-based computational cross-market model for Chinese equity market structure, which includes both stocks and CSI 300 index futures. In this model, we design several stocks and one index futures to simulate this structure. This model allows heterogeneous investors to make investment decisions with restrictions including wealth, market trading mechanism, and risk management. Investors' demands and order submissions are endogenously determined. Our model successfully reproduces several key features of the Chinese financial markets including spot-futures basis distribution, bid-ask spread distribution, volatility clustering and long memory in absolute returns. Our model can be applied in cross-market risk control, market mechanism design and arbitrage strategies analysis.

In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent τ(q) is related to the partition function and the multifractal spectrum f(α) can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional p-model, the two-dimensional p-model, and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum f(α) can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.