Mixed Causal–Noncausal Autoregressions: Bimodality Issues in Estimation and Unit Root Testing1

Abstract

This paper stresses the bimodality of the likelihood function of the Mixed causal-noncausal AutoRegressions (MAR), and it is shown that the bimodality issue becomes more salient as the causal root approaches unity from below. The consequences are important as the roots of the local maxima are typically interchanged, attributing the noncausal component to the causal one and vice-versa. This severely changes the interpretation of the results, and the properties of unit root tests of the backward root are adversely affected. To circumvent the bimodality issue, this paper proposes an estimation strategy which (i) increases noticeably the probability of attaining the global MLE; and (ii) selects carefully the maximum used for the unit root test against a MAR stationary alternative.

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

“…This can be seen in the magnitude of the lag and lead coefficients. The bimodality of the coefficient distribution in the estimation can lead, in the optimization of the likelihood function, to a local maximum (Bec, Bohn Nielsen, and Saïdi, 2019). This phenomenon is subject to initial values and can induce a switch between the lag and lead coefficients.…”

A short term credibility index for central banks under inflation targeting: an application to Brazil

“…(and not only weak white noise) non-Gaussian to ensure the identifiability of the causal and the noncausal part (Breidt et al, 1991). There is a increasing literature making use of MAR models; see among others Karapanagiotidis (2014), Hencic and Gouriéroux (2015), Gouriéroux and Jasiak (2016), Lof and Nyberg (2017), Hecq and Sun (2021), Bec et al (2020), Gourieroux et al (2021b), Gourieroux et al (2021a).…”

Convolution‐based filtering and forecasting: An application to WTI crude oil prices

“…This section recalls the causal-noncausal convolution model and provides the joint characteristic function that 1 The bimodality in the distribution of monthly WTI crude oil prices evidenced in this paper has been often disregarded in the existing literature (see, e.g., Bec et al, 2020).…”

“…Many financial time series display local explosions occurring as spikes and bubbles, which are short-lasting local trends followed by a sudden or gradual burst. These patterns characterize the series of commodity prices (Bec, Nielsen, & Saidi, 2020;Voisin & Hecq, 2020), cryptocurrency exchange rates (see, e.g., Cavaliere, Nielsen, & Rahbek, 2018;Gourieroux & Hencic, 2015;Hou, Wang, Chen, & Härdle, 2020) and other financial and macro-economic time series (see, e.g., Fries & Zakoian, 2019;Gourieroux, Hencic, & Jasiak, 2020;Hecq, Issler, & Telg, 2020;Hecq & Sun, 2020). Over the past 10 years, univariate locally explosive time series have been commonly modelled and estimated from the causal-noncausal mixed autoregressive (MAR) models (Lanne & Saikkonen, 2011).…”