Mixed Causal–Noncausal Autoregressions: Bimodality Issues in Estimation and Unit Root Testing1
| Author | Frédérique Bec,Sarra Saïdi,Heino Bohn Nielsen |
| Date | 01 December 2021 |
| DOI | http://doi.org/10.1111/obes.12372 |
| Published date | 01 December 2021 |
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©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 82, 6 (2020) 0305–9049
doi: 10.1111/obes.12372
Mixed Causal–Noncausal Autoregressions:Bimodality
Issues in Estimation and Unit Root Testing*
Fr´
ed´
erique Bec,†Heino Bohn Nielsen,‡Sarra Sa¨
idi§
†CNRS, THEMA and CREST, CY Cergy Paris Universit´e, Cergy-Pontoise, France (e-mail:
fbec@cyu.fr)
‡Department of Economics, University of Copenhagen, Copenhagen, Denmark (e-mail:
heino.bohn.nielsen@econ.ku.dk)
§CNRS, THEMA, CY Cergy Paris Universit´e, Cergy-Pontoise, France (e-mail:
sarra.saidi@cyu.fr)
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.
I. Introduction
As emphasized by Hanfelt (2000) in his comment on Small, Wang and Yang (2000), most
statisticians doubt the proposition that multiple roots pose a serious problem in data anal-
yses. Also, very few published data analyses in scientific journals mention the existence of
local maxima or describe what methods are used to select among them. In this paper, we
investigate the multiple roots issue in the case of mixed causal–noncausal autoregressive
(MAR) models, and as a by-product we propose an improvement to the test for a unit root
against MAR stationary alternatives. More precisely, a maximum likelihood estimation
strategy is proposed to select the relevant maximum for the topic at hand.
JEL Classification numbers: C12, C22.
*This work has benefited from discussions with Christian Francq,Alain Guay, Alain Hecq, Rasmus Søndergaard
Pedersen, Anders Rahbek, and Jean-Michel Zakoian and from remarks by participants at the annual congress of
the Canadian Society for Economic Science, Quebec 2019, and Quantitative Finance and Financial Econometrics
conference, Marseille 2019. In addition, constructivecomments from the editor, Jonathan Temple,and two anonymous
referees are gratefully acknowledged. F. Bec and S. Sa¨ıdi acknowledge financial support from the Labex MME-DII
(ANR-11-LBX-0023-01). H. B. Nielsen acknowledges financial support from the Institute for AdvancedStudies at
the CY Cergy Paris Universit´e and the Danish Council for Independent Research (DSF Grant 7015-00028).
1414 Bulletin
Introduced decades ago in the statistics literature (see for instance Weiss (1975), Findley
(1986), Lawrance (1991), Breidt and Davis (1991), Breidt et al. (1991), Breidt, Davis
and Dunsmuir (1992), Rosenblatt (1993), Cambanis and Fakhre-Zakeri (1994), (1996)
or Rosenblatt (2000)), mixed causal–noncausal autoregressions have recently known a
revival of interest among researchers in economics and econometrics (see, e.g., Lanne
and Saikkonen (2011), Lanne, Luoma and Luoto (2012), Lanne and Saikkonen (2013),
Hencic and Gouri´eroux (2015), Gouri´eroux and Zakoian (2015), Gouri´eroux and Jasiak
(2016), Gouri´eroux and Zakoian (2017), Cavaliere,Nielsen and Rahbek (2020) or Fries and
Zakoian (2019)). A noncausal component might be interpreted as capturing the epochs of
bubble build-up and burst, as well as non-fundamentalness of shocks.The latter can in turn
be seen as evidence that the econometrician uses a smaller information set than economic
agents do.
To fix ideas, let us introduce the univariate MAR(r,s) model as formulated by, for
example, Lanne and Saikkonen (2011),
(B)(B−1)yt=t,(1)
where B is the backward shift operator (Bkyt=yt−kfor k=0, ±1,…) and (B)=1−1B−
…−rBr,(B−1)=1−1B−1−…−sB−s. Finally, tis a sequence of non-Gaussian
independent, identically distributed random variables with mean zero, density function
f(t|), where is a set of parameters to be specialized later, and E(2
t)<∞unless otherwise
mentioned. Indeed, if the error terms were Gaussian distributed, the model could be written
equivalently as a backward or a forward autoregression, as these two representations are
observationally equivalent asymptotically in this case, see, for example, Cambanis and
Fakhre-Zakeri (1996).
Under the assumption that the polynomials (z) and (z)(z∈C) have roots outside
the unit circle, it is well-known that ythas a stationary solution in terms of the two-sided
moving average representation:
yt=∞
j=−∞
jt−j.(2)
As can be seen from equation (2), MAR models allow for dependence on both the past and
the future, in contrast with the well-known backward-looking autoregression which rules
out dependence on future observations. If r>0 and s=0, the process defined in equation
(1) becomes a purely causal process, namely the MAR(r,0) or AR(r):
(B)yt=t.(3)
If s>0 and r=0 in equation (1), one obtains the following purely noncausal process
MAR(0,s):
(B−1)yt=t.(4)
Papers by Hencic and Gouri´eroux (2015), Gouri´erouxand Zakoian (2015), Gouri ´eroux
and Jasiak (2016), Gouri´eroux and Zakoian (2017) and Fries and Zakoian (2019), assume
Cauchy distributed disturbances in (1), that is, very fat-tailed distributions needed to capture
bubble-like dynamics. Forother macroeconomic variables such as inflation or interest rates,
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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