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Peaks, Gaps, and Time Reversibility of Economic Time Series
... It is shown in Proietti (2023) that, under the stated assumptions, s + it and s − it are homogeneous first order Markov chain processes with ergodic probabilities π + ik and π − ik , and transition probabilities p + ...
... The gap process is stationary with zero mean; analytic expressions for the autocovariance function of the gap process are given in Proietti (2023) , along with a proof of its absolute summability. The above stochastic processes are related as follows: ...
... Notable differences arise with respect to Hamilton's output gap estimate, which shows a larger amplitude. This is characteristic of the method, as it was already pointed out by Proietti (2023) , whereas the turning points of the deviation cycle are in broad agreement. ...
... However, this assumption may be too restrictive in some cases, and alternative distributions may provide a better fit. For example, as shown in Gouriéroux and Zakoïan (2017), the Cauchy distribution can more accurately capture extreme events, such as market crashes, while the Skewed t distribution is better suited to model asymmetric patterns in financial time series, as discussed in Proietti (2023). Since the distribution of the error term is usually unknown to the researcher, we propose to exploit the adaptability of SMC to distributions and introduce an identification methodology, based on the MDD and the Bayesian Information Criterion (BIC), that not only determines the causal and noncausal polynomial orders but also identifies the error term distribution that best fits. ...
This paper proposes a Sequential Monte Carlo approach for the Bayesian estimation of mixed causal and noncausal models. Unlike previous Bayesian estimation methods developed for these models, Sequential Monte Carlo offers extensive parallelization opportunities, significantly reducing estimation time and mitigating the risk of becoming trapped in local minima, a common issue in noncausal processes. Simulation studies demonstrate the strong ability of the algorithm to produce accurate estimates and correctly identify the process. In particular, we propose a novel identification methodology that leverages the Marginal Data Density and the Bayesian Information Criterion. Unlike previous studies, this methodology determines not only the causal and noncausal polynomial orders but also the error term distribution that best fits the data. Finally, Sequential Monte Carlo is applied to a bivariate process containing SP Europe 350 ESG Index and Brent crude oil prices.
... Chen et al. (2000), Chen and Kuan (2002), Paparoditis and Politis (2002), Chen (2003), Racine and Maasoumi (2007), while Psaradakis (2008) proposes tests based on the fact that, for any time-reversible {X t : t ∈ Z}, the process {Y t,k := X t − X t−k : t ∈ Z} is symmetric about the origin for all k ∈ N . Proietti (2023) constructs a test based on the transition probability of the process associated with a maximum process. Diks et al. (1995) and Beare and Seo (2014) propose a test based on joint distribution functions and Darolles et al. (2004) a test based on canonical directions. ...
Time-reversibility is a crucial feature of many time series models, while time-irreversibility is the rule rather than the exception in real-life data. Testing the null hypothesis of time-reversibilty, therefore, should be an important step preliminary to the identification and estimation of most traditional time-series models. Existing procedures, however, mostly consist of testing necessary but not sufficient conditions, leading to under-rejection, or sufficient but non-necessary ones, which leads to over-rejection. Moreover, they generally are model-besed. In contrast, the copula spectrum studied by Goto et al. ( 2022, : 3563--3591) allows for a model-free necessary and sufficient time-reversibility condition. A test based on this copula-spectrum-based characterization has been proposed by authors. This paper illustrates the performance of this test, with an illustration in the analysis of climatic data.
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.
The properties of Maximum Likelihood estimator in mixed causal and noncausal models with a generalized Student’s t error process are reviewed. Several known existing methods are typically not applicable in the heavy-tailed framework. To this end, a new approach to make inference on causal and noncausal parameters in finite sample sizes is proposed. It exploits the empirical variance of the generalized Student’s t, without the existence of population variance. Monte Carlo simulations show a good performance of the new variance construction for fat tail series. Finally, different existing approaches are compared using three empirical applications: the variation of daily COVID-19 deaths in Belgium, the monthly wheat prices, and the monthly inflation rate in Brazil.
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