Detecting Co‐Movements in Non‐Causal Time Series

This paper introduces the notion of common non-causal features and proposes tools to detect them in multivariate time series models. We argue that the existence of co-movements might not be detected using the conventional stationary vector autoregressive (VAR) model as the common dynamics are present in the non-causal (i.e. forward-looking) component of the series. We show that the presence of a reduced rank structure allows to identify purely causal and non-causal VAR processes of order P>1 even in the Gaussian likelihood framework. Hence, usual test statistics and canonical correlation analysis can be applied, where either lags or leads are used as instruments to determine whether the common features are present in either the backward- or forward-looking dynamics of the series. The proposed definitions of co-movements are also valid for the mixed causal—non-causal VAR, with the exception that a non-Gaussian maximum likelihood estimator is necessary. This means however that one loses the benefits of the simple tools proposed. An empirical analysis on Brent and West Texas Intermediate oil prices illustrates the findings. No short run co-movements are found in a conventional causal VAR, but they are detected when considering a purely non-causal VAR.