Identification of causal relationships in non-stationary time series with an information measure: Evidence for simulated and financial data

The standard linear Granger causality test, based on the vector autoregressive model (VAR), requires stationarity of the time series. A VAR model is fitted to the first-differences of the time series, when they exhibit trends and are not co-integrated. In the case of co-integration, the vector error-correction model (VECM) is used instead. Alternatively, a nonlinear information causality measure is suggested, called partial transfer entropy on rank vectors (PTERV), which uses locally ranked observations. It is model-free and of a more general purpose, as it can be directly applied to the original time series without pre-testing for stationarity or co-integration. The significance test of the PTERV detects effectively the connectivity structure of complex multivariate systems. In particular, the size and power of this test are comparable to that of the standard linear Granger causality approach (VAR or VECM) when applied to systems with only linear causal effects, while the PTERV test outperforms the linear causality test when nonlinear causal effects exist, as long as the sample size is large enough. The application of PTERV to stock market data and interest rates illustrates that it can be a useful tool in the causality analysis of financial time series.