This paper investigates the effect of seasonal adjustment filters on the identification of mixed causal-noncausal autoregressive models. By means of Monte Carlo simulations, we find that standard seasonal filters induce spurious autoregressive dynamics on white noise series, a phenomenon already documented in the literature. Using a symmetric argument, we show that those filters also generate a spurious noncausal component in the seasonally adjusted series, but preserve (although amplify) the existence of causal and noncausal relationships. This result has has important implications for modelling economic time series driven by expectation relationships. We consider inflation data on the G7 countries to illustrate these results.

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.

This paper proposes concepts and methods to investigate whether the bubble patterns observed in individual time series are common among them. Having established the conditions under which common bubbles are present within the class of mixed causal–noncausal vector autoregressive models, we suggest statistical tools to detect the common locally explosive dynamics in a Student t-distribution maximum likelihood framework. The performances of both likelihood ratio tests and information criteria were investigated in a Monte Carlo study. Finally, we evaluated the practical value of our approach via an empirical application on three commodity prices.

This paper proposes strategies to detect time reversibility in stationary stochastic processes by using the properties of mixed causal and noncausal models. It shows that they can also be used for non-stationary processes when the trend component is computed with the Hodrick–Prescott filter rendering a time-reversible closed-form solution. This paper also links the concept of an environmental tipping point to the statistical property of time irreversibility and assesses fourteen climate indicators. We find evidence of time irreversibility in greenhouse gas emissions, global temperature, global sea levels, sea ice area, and some natural oscillation indices. While not conclusive, our findings urge the implementation of correction policies to avoid the worst consequences of climate change and not miss the opportunity window, which might still be available, despite closing quickly.

Financial assets often exhibit explosive price surges followed by abrupt collapses, alongside persistent volatility clustering. Motivated by these features, we introduce a mixed causal–noncausal invertible–noninvertible autoregressive moving average generalized autoregressive conditional heteroskedasticity (MARMA–GARCH) model. Unlike standard ARMA processes, our model admits roots inside the unit disk, capturing bubble-like episodes and speculative feedback, while the GARCH component explains time-varying volatility. We propose two estimation approaches: (i) Whittle-based frequency-domain methods, which are asymptotically equivalent to Gaussian likelihood under stationarity and finite variance, and (ii) time-domain maximum likelihood, which proves to be more robust to heavy tails and skewness—common in financial returns. To identify causal vs. noncausal structures, we develop a higher-order diagnostics procedure using spectral densities and residual-based tests. Simulation results reveal that overlooking noncausality biases GARCH parameters, downplaying short-run volatility reactions to news ($\alpha$) while overstating volatility persistence ($\beta$). Our empirical application to Bitcoin and Ethereum enhances these insights: we find significant noncausal dynamics in the mean, paired with pronounced GARCH effects in the variance. Imposing a purely causal ARMA specification leads to systematically misspecified volatility estimates, potentially underestimating market risks. Our results emphasize the importance of relaxing the usual causality and invertibility assumption for assets prone to extreme price movements, ultimately improving risk metrics and expanding our understanding of financial market dynamics.