Detecting Co‐Movements in Non‐Causal Time Series

Abstract

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. ). We thank the participants for helpful comments and suggestions. Additionally, we are greatly in debt to Lenard Lieb for fruitful discussions and two anonymous referees for valuable comments and suggestions.

Mixed causal–noncausal autoregressions with exogenous regressors

“…In the same spirit, Cubadda, Hecq, and Telg (2019) show that reduced rank restrictions allow identification of purely causal and noncausal VAR models in a Gaussian framework.…”

Predicting crashes in oil prices during the COVID-19 pandemic with mixed causal-noncausal models

“…However, the difference between estimated MAR parameters hints that the dynamics of those series differ. Cubadda et al (2019) extend the canonical correlation framework of Vahid and Engle (1993) from purely causal VARs to purely noncausal VARs. They show that more commonalities emerge when we also look at VARs in reverse time.…”

“…The requirement of y t being stationarity for both lag and lead polynomials gave rise to different strategies to transform nonstationary series to stationary ones. and Cubadda et al (2019) assume 3 that their commodity price series are I (1) and work with the returns ∆y t . However, this operation eliminates most of the locally explosive behaviors and the transformed series consist of many spikes instead.…”

“…ARMA models) cannot do so. MAR models have successfully been implemented on several commodity price series (see inter alia Hecq and Voisin, 2019, Hecq, Issler, and Telg, 2019, Fries and Zakoïan, 2019, Gouriéroux and Zakoïan, 2017, Cubadda, Hecq, and Telg, 2019, Lof and Nyberg, 2017, Karapanagiotidis, 2014. However, oil prices are challenging time series to be forecasted and modeled, and contrarily to many other commodities, they are clearly nonstationary (see Figure 5).…”

Modelling and forecasting economic time series with mixed causal-noncausal models

“…ARMA models) cannot do so. MAR models have successfully been implemented on several commodity price series (see inter alia Hecq and Voisin, 2021;Hecq, Issler, and Telg, 2020;Fries and Zakoïan, 2019a;Gouriéroux and Zakoïan, 2017;Cubadda, Hecq, and Telg, 2019;Lof and Nyberg, 2017;Karapanagiotidis, 2014). 1 Similarly to Gouriéroux and Zakoïan (2013), our goal when introducing a lead component in oil prices is not to provide an economic justification for the existence of a rational bubble.…”