PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract

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 GHG 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.
Is climate change time-reversible?
Francesco Giancaterini 1a , Alain Hecqa, Claudio Moranab
aMaastricht University
bUniversity of Milano-Bicocca
Center for European Studies -Milan
RCEA, RCEA-Europe ETS
CeRP - Collegio Carlo Alberto
Abstract
This paper proposes strategies to detect time reversibility in stationary stochas-
tic 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 envi-
ronmental tipping point to the statistical property of time irreversibility and
assesses fourteen climate indicators. We find evidence of time irreversibility in
GHG emissions, global temperature, global sea levels, sea ice area, and some
natural oscillation indices. While not conclusive, our findings urge the im-
plementation 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.
Keywords: mixed causal and noncausal models, time reversibility,
Hodrick-Prescott filter, climate change, global warming, environmental tipping
points.
JEL: C22
1. Introduction
According to the most recent International Panel on Climate Change report,
humanity is unlikely to prevent global warming by 1.5above pre-industrial lev-
els. Still, aggressive curbing of greenhouse-gas emissions and carbon extraction
from the atmosphere could limit its rise and even bring it back down (IPCC
(2022)). But this window is rapidly closing, and, above the 1.5threshold, the
chances of tipping points, extreme weather, and ecosystem collapse will become
even more sizeable.
1Corresponding author: Francesco Giancaterini, Maastricht University, School of Business
and Economics, Department of Quantitative Economics, P.O.box 616, 6200 MD, Maastricht,
The Netherlands.
Email: f.giancaterini@maastrichtuniversity.nl.
November 2022
An environmental tipping point is when small climatic changes might trig-
ger large, abrupt, and irreversible environmental changes and lead to cascading
effects. Recent IPCC assessments suggest that tipping points might arise be-
tween 1and 2warming, and likely to manifest at current emissions levels if
they have not already occurred. Well-known tipping points concern the Green-
land and the West Antarctic ice sheets, the Atlantic Meridional Overturning
Circulation (AMOC), thawing permafrost, ENSO, and the Amazon rainfor-
est. Recent evidence suggests that melting ice sheets is accelerating because of
warming air and ocean temperatures and less snowfall. Some studies indicate
that the irreversible disintegration of the Greenland ice sheet could occur at 0.8
and 3.2warming (Wunderling et al. (2021)). An unstoppable ice sheet melting
in Antarctica would manifest at 2warming (DeConto et al. (2021)). Ice sheets
melting adds fresh water to the North Atlantic, weakening the AMOC, one of
the main global ocean currents, which is already in its weakest state in 1,000
years (Caesar et al. (2021)). Its shutdown would cause significant cooling along
the US east coast and Western Europe, alter rainfall and cause more drying. At
the current global warming pace, a 50% weakening of AMOC is expected by
2100, and a tipping point between 3and 5.5warming. Moreover, the Arctic
is warming twice as faster as the planet on average, and it has already warmed
2, causing permafrost thawing, which releases CO2 and methane into the at-
mosphere. Available estimates point to 1400 billion tons of carbon frozen in the
Arctic’s permafrost, twice as much carbon already in the atmosphere, and a 2
warming could even cause the thawing of 40% of the world’s permafrost. The El
Ni˜no-Southern Oscillation or EN S O cycle is an oscillating warming and cooling
pattern affecting rainfall intensity and temperatures in tropical regions. It can
strongly influence weather in many parts of the globe. El Ni˜no and La Ni˜na are
the warm and cool phases of the EN S O cycle, respectively. Oceans warming
can trigger a tipping point in the EN SO cycle, increasing its variability and
intensity and shifting its teleconnection eastward (Cai et al. (2021)). Extreme
rainfalls and droughts will no longer occur in tropical regions but throughout
the earth due to the destabilization of these natural oscillations. The Amazon
rainforest has already lost about 17% of its tree cover. At the current rate of
deforestation, the loss could reach 27% by 2030. Lovejoy and Nobre (2018) esti-
mate the dieback of the Amazon Forest at 20%-25%; beyond this deforestation
threshold, the rainforest would transform into a savannah, potentially releasing
up to 90 gigatons of CO2. Some climate models already indicate that the Ama-
zon will be a net generator of C02 by 2035, setting the dieback threshold at 3
warming.
Further uncertainty on the compound effect of the above phenomena arises
from their potential interaction, allowing tipping points to occur even below 2
warming. Overall, greenhouse gases generated by human activity over the last
two centuries have driven the global trend temperature up. This temperature
warming has widely impacted the natural environment and has raised the risk of
irreversible changes of state with catastrophic consequences (see also Schellnhu-
ber (2008)) and Solomon et al. (2009)).
In this paper, we link the concept of an environmental tipping point to the
2
statistical property of time irreversibility. A stationary process {Yt}T
t=1 is said
to be time-reversible if its statistical properties are independent of the direction
of time. In other words, the vectors (Y1, Y2, . . . , YT) have identical joint distri-
butions as (YT, Y(T1), . . . , Y1) for every integer T. Hence, a time-reversible
process (TR) exhibits a temporal symmetry in its probabilistic structure. In
the alternative circumstance, we have time irreversibility when the stochastic
process behaves differently according to the direction of time considered. TR
has been under investigation in various fields over the years, for instance, in the
different branches of physics, where researchers have been investigating whether
time has some preferred direction in explaining physical phenomena (see Wald
(1980), Levesque and Verlet (1993), Holster (2003)). This univocity along the
time direction appears to be a tipping point property, as once a tipping point
is reached, the system undergoes an irreversible state change.
This paper aims to investigate whether TR has the potential to offer insight
into the process of climate change and its implications for the natural envi-
ronment. Studying TR in the context of climate change is motivated by the
possibility of answering the following questions: are there divergences between
the forward-time and backward-time joint probability distributions for the pro-
cess of climate change and global warming? Are these processes symmetric
over time? Is this property similarly present in natural oscillations that tem-
perature warming might have permanently impacted, inducing changes in their
frequencies and intensity of occurrence? Irreversibility in this context might
carry insights into the event of state changes.
This paper then introduces new strategies to detect whether a stochastic
process is time-reversible. There are already several tests for TR in the econo-
metric literature. See, for instance, Ramsey and Rothman (1996), Hinich and
Rothman (1998), Chen et al. (2000), Belaire-Franch and Contreras (2003), and
Proietti (2020). The shortcoming of many of these approaches is that they usu-
ally impose strong restrictions on the model or are not trivial to apply. Our new
strategies are grounded on mixed causal and noncausal models (see Gourieroux
and Jasiak (2016)). Unlike causal models, which only consider the relationship
between present and lagged values, mixed casual and noncausal models also
compute the relationship between present and future values. This framework
leads to nonlinear conditional expectations (e.g., Gourieroux and Jasiak (2022)).
The connection between these models and TR gives rise to our testing strategies.
Furthermore, similarly to Proietti (2020), we can test for TR on non-stationary
time series using a novel approach. We extract the trend component using the
Hodrick-Prescott (HP) filter imparted in a time-reversible closed-form solution.
Then, the cyclical component, which records the process’s oscillations around its
trend, is responsible for the potential time-irreversibility feature of the stochas-
tic process.
The rest of the paper is as follows. Section 2 summarizes the properties of
time-reversible processes and reviews the existing methods to detect TR. Section
3 introduces our new TR strategies. Namely, we show how our new approaches
exploit the properties of mixed causal and noncausal models. We then evaluate
their performance through Monte Carlo experiments. Section 4 extends our
3
framework to non-stationary time series, and Section 5 presents the empirical
assessment of some relevant climate variables. Finally, Section 6 concludes.
2. Time reversibility
Weiss (1975) shows that if a Gaussian error term characterizes an ARMA
model, then the process is time-reversible. Indeed, Gaussian processes are en-
tirely defined by their second-order moments, which have the property of being
time symmetrical.
Hallin et al. (1988) consider two-sided linear models of the form:
Yt=
X
k=−∞
θktk,(1)
where the stationary condition P
k=−∞ |θk|<is satisfied. They claim that
if {Yt}T
t=1 is time-reversible, then either tis a Gaussian white noise, or there
exists a kand s {0,1}such that θ2k+j= (1)sθ2kj. However, thas to be a
sequence of i.i.d. zero-mean random variables with finite moments of all orders.
It is an unrealistic assumption for non-Gaussian processes and many time series.
Breidt and Davis (1992) extend Weiss’s results to non-Gaussian processes
assuming milder conditions than Hallin et al. (1988). They take the following
ARMA(p, q) process into account:
φ(L)Yt=θ(L)t,(2)
where Lindicates the backshift operator, φ(z) has rroots outside and sroots
inside the unit circle (r+s=p), and thas a finite variance. For simplicity, we
set the polynomial θ(L) = 1, such that (2) can be rewritten as:
φ+(L)φ(L)Yt=t,(3)
where φ+(L) has rroots outside the unit circle while φ(L) has sroots inside.
It is well known that (3) has a unique stationary solution given by a two-sided
moving average representation, as expressed in (1). Breidt and Davis (1992)
claim that if φ(z) and φ(z1) have different roots, then Ytis reversible if and
only if the error term is Gaussian. In the other case, that is when the two
polynomials φ(z) and φ(z1) have the same roots, (1) (or equivalently (3)) is
time-reversible regardless of the distribution of t. Indeed, if p > 0 and φ(z)
and φ(z1) have the same roots, 1(z) has the Laurent expansion
1
φ(z)=
X
−∞
θjzj,(4)
with θp/2j=θp/2+j, for j= 0,1, . . . (see Breidt and Davis (1992)). This
implies that the result of Hallin et al. (1988) is a consequence of the conclusion
that the two polynomials φ(z) and φ(z1) have the same roots. Moreover, un-
like Hallin et al. (1988), Breidt and Davis (1992), only assume that the error
4
term must have finite variance.
Ramsey and Rothman (1996) define the stationary stochastic process {Yt}T
t=1
is time-reversible only if:
γi,j =E[Yi
tYj
tk]E[Yj
tYi
tk] = 0 (5)
for all i, j, k N+. This is a sufficient condition for TR, but not a necessary one
since it only considers a proper subset of moments from the joint distributions
of {Yt}. Since it is impractical to show that (5) holds for any i,j, and k, they
adopt a restricted definition of TR by imposing i+kmand kK. In
particular, they restrict m= 3 so that the symmetric-bicovariance function is
given by:
γ2,1=E[Y2
tYtk]E[YtY2
tk]=0,(6)
for all integer values of k. Ramsey and Rothman (1996) claim that i+j= 3 is
sufficient to provide a valid indication of time irreversibility.
Ramsey and Rothman (1996) also introduced a new procedure to test TR
that became a standard approach to investigating business cycle properties such
as asymmetry. It amounts to a TR test statistic distributed as a standard normal
distribution: T[bγ2,1γ2,1]
pV ar(bγ2,1)
d
N(0,1),(7)
with:
bγ2,1=b
B2,1(k)b
B1,2(k),
and:
b
B2,1= (Tk)1
T
X
t=K+1
Y2
tYtk;b
B1,2= (Tk)1
T
X
t=K+1
YtY2
tk,
for various integer values of k. Under the null hypothesis, we have a time-
reversible process. The pre-requisite of the test is that the data must possess
finite first sixth moment. If the distribution lacks this property, the test size
can be seriously distorted (see Belaire-Franch and Contreras (2003)).
Chen et al. (2000) propose a new class of TR tests, which, unlike Ramsey
and Rothman (1996), does not require any moment restrictions. This class of
tests relies on the fact that if {Yt}T
t=1 is a time-reversible process, then for every
k= 1,2, . . . , the distribution of Xt,k =YtYtkis symmetric about the origin.
The drawback of this approach is that it allows for testing the symmetry of
Xt,k for each value of k, but not jointly for a collection of kvalues, which would
require a portmanteau test.2Moreover, its implementation is not trivial.
Similar reasoning is followed by Proietti (2020) since also his test is based
on the idea that Xt,k has to be symmetric for every k > 0. However, Proietti
(2020) uses a weaker definition of TR as {Yt}T
t=1 can also be non-stationary.
2Chen et al. (2000) state that to jointly test Xt,k for a collection of kvalues, a portmanteau
test is required.
5
3. New strategies to detect time reversibility on stationary time series
This Section introduces new strategies to assess TR in stationary stochas-
tic processes, exploiting the properties of mixed causal and noncausal models.
Breidt et al. (1991) introduce mixed causal and noncausal models as expressed
in equation (3). They define the polynomial φ(z) as noncausal and the poly-
nomial φ+(z) as causal. A required condition for identifying the causal from
the noncausal component is the non-Gaussianity of the innovation term.
Lanne and Saikkonen (2011), rewriting the noncausal polynomial in (3) as a
lead polynomial, start with a mixed causal and noncausal model expressed as:
φ(L)ϕ(L1)Yt=t,(8)
where L1produces lead such that L1Yt=Yt+1. A mixed causal and non-
causal model represented in this way is denoted as MAR(r,s), where ϕ(L1) is
the noncausal polynomial of order sand φ(L) is the causal polynomial of order
r. Exactly as representation (3), r+s=pis true even in this case. Purely causal
and purely noncausal models are obtained setting respectively ϕ(L1) = 1 and
φ(L) = 1 (see Gouri´eroux et al. (2013), Hencic and Gouri´eroux (2015), Hecq
et al. (2016), Fries and Zakoian (2019), Hecq and Voisin (2021), Giancaterini and
Hecq (2022), and Fries (2021)). In (8), both causal and noncausal polynomials
have their roots outside the unit circle, such that:
φ(z)6= 0 and ϕ(z)6= 0 for |z| 1.(9)
The tests for TR that we propose have the common feature of extending the
results obtained by Breidt and Davis (1992) to the MAR(r,s) representation
(8). This is possible if and only if Condition 3.1 is true.
Condition 3.1 A stochastic process that can be expressed as a MAR model
is time-reversible if and only if φ(z)ϕ(z1) have the same roots as φ(z1)ϕ(z).
Namely, when:
r=s and φi=ϕi, for i = 1, . . . , s.
This implies that MARs are time-reversible if and only if the causal polyno-
mial has the same order and the same coefficients as the noncausal polynomial
and vice versa. Remember that it is impossible to identify a MAR model un-
der the Gaussianity of the innovation term. Hence, in that case, we have a
time-reversible process (see Weiss (1975)).
3.1. Strategy 1: For detecting time reversibility
The first strategy aims to evaluate whether a stochastic process meets Condition 3.1.
In particular, it uses a procedure similar to the one used to identify MAR mod-
els (see Lanne and Saikkonen (2011) and Hecq et al. (2016)). The procedure is
as follows:
6
1. We estimate a conventional autoregressive process (also called pseudo-
causal model) by OLS, and the lag order pis selected using information
criteria (for instance, AIC or BIC).
2. We test the normality in the residuals of the AR(p). If the null hypothesis
of Gaussianity is not rejected, we cannot identify a MAR(r,s) model, and
for the reasons above, we have a time-reversible process. Moreover, if
the null hypothesis of normality is rejected and the estimated pis an
odd number, the condition r=scan never be satisfied. According to
Condition 3.1, this result would allow us to identify our process as time-
irreversible. However, the selection of pmight not be univocal and depend
on the information criterion employed. As such, to have more robust
results before proceeding to the next step, we increase pby one unit so
that r=sis still possible. In the alternative case that pis an even number,
we directly proceed to the next step.
3. We select a model among all MAR(r,s) specifications with r+s=pif
pis an even number; otherwise r+s=p+ 1. This step is performed
using a maximum likelihood approach (see Giancaterini and Hecq (2022)
and references therein). In the selection procedure, we also include the
model given by the restricted likelihood that imposes commonalities in
causal and noncausal parameters (the model with the same restrictions as
in Condition 3.1). Note that when we compute the information criteria of
the model with restricted likelihood, instead of estimating pparameters
(or p+ 1 if pis an odd number), we estimate p/2 of them (or (p+ 1)/2),
implying a smaller penalty term. Finally, we choose the model with the
smallest information criteria.
Consider a short example to illustrate how the strategy works. We suppose
that we estimate a conventional AR model by OLS, and we reject the Gaussian
hypothesis of the residuals, for instance, using the Jarque-Bera test. Further-
more, we assume we select the number of lags pequal to 2. To analyze whether
our process is time-reversible, we then compute the log-likelihoods and then
the information criteria of the following four models: MAR(2,0), MAR(0,2),
MAR(1,1) as well as the MAR(1,1) with the restriction φ=ϕ. If the model
with the smallest information criteria is the one with the restriction, we have a
time-reversible process. We have a time-irreversible process in the alternative
case where another model is selected. This approach allows knowing with a
limited number of steps whether the process is time-reversible. Its shortcoming
is that information criteria are very sensitive to the sample size, and model se-
lection might not be robust to sample update or trimming. Moreover, model
selection can depend on the information criterion employed, i.e., AIC rather
than BIC, HQ, or others. Finally, even for the same information criterion, the
value used for model selection can only slightly differ from values shown by
either lower or higher-order alternative models.
7
3.2. Strategy 2: For detecting time reversibility
The second strategy we introduce is more robust concerning the sample and
slight differences in the value of information criteria when models are compared.
However, more steps are required to identify the TR of the process than for the
previous approach. It requires the following steps: steps 1 and 2 are identical
to what we described in 3.1;
3. We select a model among all MAR(r,s) specifications with r+s=pif p
is an even number (otherwise r+s=p+ 1). Then, we choose the one
with the largest likelihood (since we are considering models with the same
number of parameters).
4. If the selected model is the one with r=s(in our previous example,
it was the MAR(1,1)), we compute a likelihood ratio test, taking into
account the same restrictions as in Condition 3.1. If we do not reject the
test’s null hypothesis, we have TR. On the other hand, if we reject the null
hypothesis, we identify the process under investigation as time-irreversible.
3.3. Simulation study
We now analyze the performance of these two strategies using Monte Carlo
experiments. We take into account data-generating processes (dgp) defined by
an error term with a skewed Student’s-tdistribution, generated by joining two
scaled halves of the Student’s-t distribution (see Fern´andez and Steel (1998)):
f() = 2
γ+1
γ(g
γ!I() + gγI()),(10)
where I() and I() stand for the indicator function:
I() = (1, 0
0, < 0,
g() stands for the density function of a symmetric Student’s-t, and γR+.
In case γ= 1, we have f() = g(), hence (10) is a symmetric Student’s-
twith νdegrees of freedom. The assumption that the error term follows a
Student’s-tis not a particularly strong hypothesis. It is a distribution that
offers a good summary of the features of other (non-Gaussian) fat-tailed and
symmetric distributions. Furthermore, our Monte Carlo experiments consider
N= 1000 replications, four different sample sizes, T= (100,200,500,1000),
and the following combinations of causal and noncausal coefficients:
MAR(1,1) :φ0= 0.8, ϕ0= 0.8; time-reversible process;
MAR(1,1) :φ0= 0.8, ϕ0= 0.5; time-irreversible process;
MAR(1,1) :φ0= 0.8, ϕ0= 0.1; time-irreversible process;
MAR(1,0) :φ0= 0.8; time-irreversible process.
8
In our Monte Carlo study, we also include results obtained by Ramsey and
Rotham’s test, setting k= 2.
Table 1 shows the frequencies with which the two new strategies and the
test proposed by Ramsey and Rotham detect the processes as time-irreversible
when γ= 1, pis known, and rand sare unknown. In particular, columns
Strategy 1 and Strategy 2 indicate the percentage of times the stochastic pro-
cesses are identified as time-irreversible when the strategies from Sections 3.1
and 3.2 are implemented. The last column, RR (1996), indicates how often we
reject the null hypothesis of TR when the methodology proposed by Ramsey
and Rothman (1996) is used. The Bayesian Information Criteria (BIC) is used
in Strategy 1. The results exhibit that Strategy 1 detects TR with greater pre-
cision, but is ”undersized” for large T. This is because the penalty terms can
differ from a tiny number in a large sample. On the other hand, Strategy 2
looks consistent and performs better when the processes under investigation are
time-irreversible (frequencies are not size-adjusted, though, which makes the
results of Strategies 1 and 2 not easy to compare). Finally, the test proposed by
Ramsey and Rothman (1996) clearly shows size distortion problems. This is not
an unexpected result since, as previously stated, the test can show a seriously
distorted size if the distribution lacks a finite sixth moment. The Student’s-t
distribution has a finite sixth moment for ν > 6. As a consequence, the power
of the test also performs poorly for RR (1996).
Our simulation studies also consider cases where the error term is charac-
terized by γ6= 1. In these scenarios, we simulate a process with a skewed error
term and proceed as if γ= 1: Strategies 1 and 2 are followed assuming a sym-
metric Student’s-tdistributed error term. The results obtained under these new
circumstances are similar to those in Table 1. This suggests that the test size
and power are not sensitive to the eventual asymmetry of the error term. The
outcomes are available upon request.
Table 2 shows different results when pis assumed unknown. In this case,
before implementing our strategies, we estimate a pseudo-causal model in each
replica of our simulation study to capture the dynamics p. Since there is more
uncertainty under these new conditions, the results are less precise with small
sample sizes (T= (100,200)). However, the table displays that the outcomes
align with Table 1 for large values of T. The percentages displayed in the col-
umn RR(1996) of Table 2 are unchanged from those shown in Table 1 since the
same method is applied.
9
MAR(1,1); φ0= 0.8, ϕ0= 0.8, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 7.1% 16.4% 8.1%
T=200 3.1% 7.5% 11.5%
T=500 1.4% 5.0% 11%
T=1000 0.8% 4.5% 13.6 %
MAR(1,1); φ0= 0.8, ϕ0= 0.5, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 51.4% 63.7% 20.7%
T=200 77.9% 84.8% 29.4%
T=500 99.0% 99.5% 40.7%
T=1000 100% 100% 51.8 %
MAR(1,1); φ0= 0.8, ϕ0= 0.1, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 87.4% 93.2% 33.2%
T=200 99.6% 99.9% 43.2%
T=500 100% 100% 57.5%
T=1000 100% 100% 68.4%
MAR(1,0); φ0= 0.8, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 91.5% 93.2% 34.2%
T=200 99.6% 99.9% 43.8%
T=500 100% 100% 58.9%
T=1000 100% 100% 70.1%
Table 1: Frequencies with which time irreversibility is detected when the error term has a
symmetric Student’s-tdistribution (γ=1) and ν0= 3. Finally, rand sare assumed as
unknown and pas known.
MAR(1,1); φ0= 0.8, ϕ0= 0.8, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 20.9% 21.6% 8.1%
T=200 9.5% 12.6% 11.5%
T=500 3.5% 7.1% 11%
T=1000 4.3% 7.8% 13.6 %
MAR(1,1); φ0= 0.8, ϕ0= 0.5, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 61.6% 67.1% 20.7%
T=200 79.0% 85.2% 29.4%
T=500 99.0% 99.5% 40.7%
T=1000 100% 100% 51.8 %
MAR(1,1); φ0= 0.8, ϕ0= 0.1, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 92.2% 93.8% 33.2%
T=200 99.8% 99.9% 43.2%
T=500 100% 100% 57.5%
T=1000 100% 100% 68.4%
MAR(1,1); φ0= 0.8, ν0= 3, γ= 1
Strategy 1 Strategy 2 RR (1996)
T=100 95% 95.7% 34.2%
T=200 99.9% 99.9% 43.8%
T=500 100% 100% 58.9%
T=1000 100% 100% 70.1%
Table 2: Frequencies with which time irreversibility is detected when the error term has
a symmetric Student’s-t distribution (γ=1) with ν0= 3 and p,r, and sare assumed as
unknown.
Finally, to analyze the result sensitivity to the persistence level, we imple-
10
ment new Monte Carlo experiments considering as dgp a MAR(1,1) with the
following combinations of causal and noncausal coefficients:
MAR(1,1) :φ0= 0.95, ϕ0= 0.95; time-reversible process;
MAR(1,1) :φ0= 0.95, ϕ0= 0.5; time-irreversible process;
MAR(1,1) :φ0= 0.95, ϕ0= 0.1; time-irreversible process;
MAR(1,0) :φ0= 0.95; time-irreversible process.
Even in this case, the outcomes are similar to those displayed in Tables 1 and 2
and available upon request.
4. Testing for time reversibility on non-stationary processes
The goal of this section is to detect TR in non-stationary processes {Yt}T
t=1
that can be expressed as:
Yt=fY
t+ccY
t,(11)
where fYis a generic trend function, and ccYis a stationary process that cap-
tures the cyclical fluctuations of Yaround fY. We show that whenever the
trend component is computed using the HP filter, then fYcan be expressed as
a time-reversible process. As a consequence, the potential time irreversibility
of process Ywould be captured by its cyclical component ccY. In other words,
whenever fYis estimated by using the HP filter, model (11) is time-irreversible
(or reversible) if and only if its cyclical component is irreversible (or reversible).
The HP filter estimates the trend component through the following mini-
mization problem (see Hecq and Voisin (2021)):
min
{fY
t}T
t=1 PT
t=1 y2
t+λPT
t=1(ftft1)(ft1ft2)2.(12)
According to De Jong and Sakarya (2016), the optimization problem (11)
has the following closed-form solution:
fY
t=λL24λL1+ (1 + 6λ)4λL +λ21Yt,(13)
for t= 3, . . . , T 2. The λparameter penalizes the filtered trend’s variability;
therefore, the higher its value, the smoother the trend component:
λ=number of observations per year
4i×1600,
with either i= 2 (see Backus and Kehoe (1992)) or i= 4 (Ravn and Uhlig
(2002)). It can be shown that (12) can be rewritten as:
fY
t="1ψ1(λ)Lψ2(λ)L21ψ1(λ)L1ψ2(λ)L2+
ψ2
1(λ) + ψ2
2(λ)+6ψ2(λ)#1
Yt,
(14)
11
where ψ1(λ) = 4λ
λ+1 , and ψ2(λ) = λ. For instance, for annual data, we can
adopt λ= 6.25, implying
fY
t="1100
29 L+ 6.25L21100
29 L1+ 6.25L213.456#1
Yt.
The results underline that the filter of the trend component is given by a time-
reversible MAR(2,2) polynomial minus a constant value. Since the latter does
not affect the symmetry over time of our process, and our goal is to investi-
gate the time reversibility of fY, we do not consider the constant term in our
investigation. As a consequence, we can approximate fYas follows:
fY
t"1100
29 L+ 6.25L21100
29 L1+ 6.25L2#1
Yt.(15)
Using the Laurent expansion as in (4), we have:
fY
t
+
X
j=−∞
δjYtj,(16)
where because of the identity of the lead and lag polynomials, δis symmetric
over time. Hence, even if fYis a non-stationary process, we can apply a weaker
definition of TR and define it as time-reversible. This result implies that the
potential time irreversibility (or reversibility) lies with the cyclical component
of Y.
To illustrate how our new strategies perform under the new conditions, we
implement new Monte Carlo experiments where fYis a random walk with drift:
Xt=Xt1+δ+ηt,(17)
with ηN(0,1), and ccYas a MAR(1,1):
(1 + φL)(1 + ϕL1)˜
Yt=εt.(18)
Finally, the process {Y}T
t=1 is obtained by the sum of the two processes, that
is:
Yt=Xt+˜
Yt.(19)
Alternatively, we could have considered the following process as dgp:
Yt=Yt1+δ+˜
YtYt=δ+˜
Yt.(20)
However, the reason not to consider such a process is that, as expressed in (20),
the resulting dgp implies that the first difference process (∆Yt) is a MAR(1,1).
This is not a realistic assumption because MARs are typically used to capture
explosive bubbles, and the first difference operation eliminates most locally ex-
plosive behaviors (see Hecq and Voisin (2021)).
12
In each replica of our Monte Carlo experiment, we simulate the non-stationary
process {Yt}T
t=1, remove the trend component using the HP filter, and then ap-
ply our strategies on ccY. The coefficients used for the cyclical component ccY
are the same as in the previous section. Table 3 displays the results.
ccY:MAR(1,1); φ0= 0.8, ϕ0= 0.8; ν0= 3, γ= 1
Strategy 1 Strategy 2
T=100 15.1% 33.3%
T=200 6.7% 8.1%
T=500 1.4% 5.0%
T=1000 0.8% 4.5%
ccY:MAR(1,1); φ0= 0.8, ϕ0= 0.5; ν0= 3, γ= 1
Strategy 1 Strategy 2
T=100 40.6% 59.5%
T=200 58.0% 69.7%
T=500 86.7% 94.8%
T=1000 99.2% 100%
ccY:MAR(1,1); φ0= 0.8, ϕ0= 0.1; ν0= 3, γ= 1
Strategy 1 Strategy 2
T=100 71.9% 89.2%
T=200 92.0% 97.3%
T=500 99.6% 100%
T=1000 100% 100%
ccY:MAR(1,0); φ0= 0.8; ν0= 3, γ= 1
Strategy 1 Strategy 2
T=100 75.1% 91.1%
T=200 93.4% 98.8%
T=500 100% 100%
T=1000 100% 100%
Table 3: Frequencies with which time irreversibility is detected on non-stationary time series;
rand sare assumed as unknown and pas known. Data are considered to have quarterly
frequency (λ= 1600).
The results are similar to those displayed in Tables 1 and 2, with the differ-
ence that the power of the strategies is less accurate under these new conditions,
especially when the sample size considered is small (T= (100,200)).
5. Is climate change time-reversible?
In our empirical investigation, we analyze annual data for the global land
and ocean temperature anomaly (GLO), the global land temperature anomaly
(GL), the global ocean temperature anomaly (GO), solar activity (SA), emis-
sions of greenhouses gas (GHG), emissions of nitrous oxide (N2O). When avail-
able, we also use monthly data to control for potential small sample distortions
in our statistics, as revealed by the simulation results. In particular, we con-
sider the following monthly series: the Southern Oscillation Index (SOI ), the
North Atlantic Oscillation Index (NAO), the Pacific Decadal Oscillation Index
(P DO), the global mean sea level (GM SL), the Northern Hemisphere sea ice
13
area (NH), the Southern Hemisphere sea ice area (SH ), the global component
of climate at a glance (GCAG), and, finally, the global surface temperature
change (GIS T EM P ).3
GLO,GL,GO,GC AG, and GI ST E MP measure global warming. They
provide the difference between the current temperature from a standard bench-
mark value. Positive anomalies show that the observed temperature is warmer
than the benchmark value, and negative temperatures show that the observed
temperature is colder than the benchmark value. In particular, GCAG provides
global-scale temperature information using data from NOAA’s Merged Land
Ocean Global Surface Temperature Analysis (NOAAGlobalTemp), which uses
comprehensive data collections of increased global coverage over land (Global
Historical Climatology Network-Monthly) and ocean (Extended Reconstructed
Sea Surface Temperature) surfaces.
SOI is one of the most important atmospheric indices for determining the
strength of El Ni˜no and La Ni˜na events and their possible effects on weather
conditions in the tropics and various other geographical areas. El Ni˜no events
are characterized by sustained warming of the central and eastern tropical Pa-
cific, whereas La Ni˜na events show sustained cooling of the same areas. These
changes in the Pacific Ocean and its overlying atmosphere occur in a cycle known
as the El Ni˜n o–Southern Oscillation (EN SO). High values of SOI indicate
La Ni˜na events, whereas negative values indicate El Ni˜no events. The N AO
determines the westerly winds’ speed and direction across the North Atlantic
and the winter sea surface temperature. When the NAO index is far above
average, there is a greater likelihood that seasonal temperatures in northern
Europe, northern Asia, and South-East North America will be warmer than
usual. In contrast, seasonal temperatures in North Africa, North-East Canada,
and southern Greenland will be cooler than usual. The opposite is true when
NAO is far below average. P D O is a climatic cycle that describes anomalies
in sea surface temperature in the Northeast Pacific Ocean. The P DO has the
power to influence weather patterns all over North America. Finally, GMSL,
NH, and SH are climate indicators providing information on how much of the
ice land is melting, and their connection with global warming is straightforward.
Figure 1 presents the data. GLO,GL,GO,SA,GHG, and N2Orange from
1881 to 2014, SOI and NAO from January 1951 to December 2021, P DO from
January 1854 to December 2021, GCAG and GI ST E M P from January 1880
to December 2016, GMSL from January 1880 to December 2015, and, finally,
NH and SH from January 1979 to 2021.
3GLO,GL, and GO are obtained from https://www.ncdc.noaa.gov/cag/global/time-series. S OI and NAO are
obtained from https://www.cpc.ncep.noaa.gov/data/indices/soi and
https://www.cpc.ncep.noaa.gov/products/precip/CWlink/pna/norm.nao.monthly.b5001.current.ascii.table, respec-
tively. PDO is obtained from https://www.ncdc.noaa.gov/teleconnections/pdo/. For GHG and SA, the source
is Hansen et al. (2017). For N2O, we use the historical reconstruction computed in Meinshausen et al.
(2017) (data available at https://www.climatecollege.unimelb.edu.au/cmip). N H and SH are obtained from
https://psl.noaa.gov/data/timeseries/monthly, GCAG and GIST E M P from https://datahub.io/core/global-temp
and, finally, GMSL from https://datahub.io/core/sea-level-rise.
14
−0.5
0.0
0.5
1880 1920 1960 2000
Global land and ocean temperature
anomalies (GLO): 1881−2014
(a) Annual data for
global land and
ocean temperature
anomalies.
−1.0
−0.5
0.0
0.5
1.0
1880 1920 1960 2000
Global land temperature
anomalies (GL): 1881−2014
(b) Annual data for
global land
temperature
anomalies.
−0.50
−0.25
0.00
0.25
0.50
1880 1920 1960 2000
Global ocean temperature
anomalies (GO): 1881−2014
(c) Annual data for
global ocean
temperature
anomalies.
0.0
0.1
1880 1920 1960 2000
Solar activity (SA) : 1881−2014
(d) Annual data for
solar activity.
0
1
2
3
1880 1920 1960 2000
Time
Greenhouses gas (GHG): 1881−2014
(e) Annual data for
greenhouses gas.
0.05
0.10
0.15
0.20
1880 1920 1960 2000
Time
Nitrous oxide (N2O): 1881−2014
(f) Annual data for
nitrous oxide.
−0.4
0.0
0.4
0.8
1.2
1900 1950 2000
Global component of
climate at a glance (GCAG): 1880−2016
(g) Monthly data for
global component
of climate at
a glance.
−0.5
0.0
0.5
1.0
1900 1950 2000
Global surface
temperature change (GISTEMP): 1880−2016
(h) Monthly data
for global surface
temperature change.
−100
0
1900 1950 2000
Global mean sea level (GMSL): 1880−2015
(i) Monthly data
for global mean
sea level.
−6
−3
0
3
1960 1980 2000 2020
Southern Oscillation (SOI): 1951−2021
(j) Monthly data
for Southern
Oscillation index.
−2
0
2
1960 1980 2000 2020
North Atlantic Oscillation (NAO): 1951−2021
(k) Monthly data
for North Atlantic
Oscillation index.
−4
−2
0
2
4
1850 1900 1950 2000
Pacific Decadal Oscillation (PDO): 1854−2021
(l) Monthly data
for Pacific Decadal
Oscillation index.
5
10
1960 1980 2000 2020
North Hemisphere Sea−
ice area (NH): 1979−2021
(m) Monthly data
for Northern
Hemisphere
sea ice area.
4
8
12
16
1960 1980 2000 2020
Southern Hemisphere Sea−
ice area (SH): 1979−2021
(n) Monthly data
for Southern
Hemisphere
sea ice area.
Figure 1: Climate time series.
15
As shown in Figure 1, time series (a)-(i) are characterized by a positive
trend. Hence, according to the strategy introduced in Section 4, their potential
time-reversibility (or irreversibility) lies with their cyclical component. For this
reason, we can remove their trend and extract their cyclical fluctuations using
the HP filter. Figure 2 displays the detrended time series.
−0.2
−0.1
0.0
0.1
0.2
0.3
1880 1920 1960 2000
Cyclical component of GLO (cc−GLO);
1881−2014
(a) Annual data for
cyclical component
of global land and
ocean temperature
anomalies.
−0.8
−0.4
0.0
0.4
1880 1920 1960 2000
Cyclical component of GL (cc−GL)
1881−2014
(b) Annual data for
cyclical component
of global land
temperature
anomalies.
−0.2
0.0
0.2
1880 1920 1960 2000
Cyclical component of GO (cc−GO);
1881−2014
(c) Annual data for
cyclical component
of global ocean
temperature
anomalies.
−0.05
0.00
0.05
0.10
1880 1920 1960 2000
Cyclical component of SA (cc−SA);
1881−2014
(d) Annual data for
cyclical component
of solar activity.
−0.01
0.00
0.01
0.02
1880 1920 1960 2000
Time
Cyclical component of GHG (cc−GHG);
1881−2014
(e) Annual data for
cyclical component
of greenhouses gas.
−0.001
0.000
0.001
0.002
1880 1920 1960 2000
Time
Cyclical component of N2O (cc−N2O);
1881−2014
(f) Annual data for
cyclical component
of nitrous oxide.
Figure 2: Cyclical components of the detrended time series.
−0.4
−0.2
0.0
0.2
1900 1950 2000
Cyclical component of GCAG: 1880−2016
(a) Monthly data for
cyclical component of
global component
of climate at
a glance.
−0.50
−0.25
0.00
0.25
1900 1950 2000
Cyclical component of GISTEMP v4: 1880−2016
(b) Monthly data for
cyclical component
of global surface
temperature change.
−20
−10
0
10
20
1900 1950 2000
Cyclical component of GMSL: 1993−2015
(c) Monthly data for
cyclical component of
global mean
sea level.
Figure 3: Cyclical components of the detrended time series.
The goal is to investigate the time-reversibility of the variables displayed in
16
Figure 2 and the latter five in Figure 1.
We estimate autoregressive models (see Sections 3.1 and 3.2) for each time
series. We use the BIC to identify the number of lags (p). Next, we test the nor-
mality of the residuals of the nine AR(p) models. Since for ccGLO ,ccGL,ccGO ,
ccSA , and SH we do not reject the null hypothesis of normality (significance
level 0.05) of the Shapiro-Wilk test (p-values equal to 0.83, 0.59, 0,24, 0.08, and
0.45, respectively) and the Jarque-Bera test (p-values equal to 0.64, 0.25, 0.15,
0.13, and 0.35 respectively), we identify them as time-reversible processes. On
the other hand, in ccGHG ,ccN2O,ccGCAG ,ccGIS T EM P ,ccGM SL ,SOI,NAO,
P DO, and N H, we reject the null hypothesis of Gaussianity of both the Shapiro-
Wilk test (p-values are close to zero for ccGHG,ccN2O,ccGC AG,ccGI ST EM P ,
ccGMS L,P D O,NH and 0.0362, 0.0020 for SOI and NAO, respectively) and
the Jarque-Bera test (p-value equal to 0.0307 for SOI and close to zero for all
the other variables) at a significance level of 0.05. We can then fit MAR models
to our data, identifying ccGHG,ccGCAG,and ccGIS T EM P as MAR(2,0), ccN2O
as MAR(4,0), ccGM SL as MAR(6,2), P D O as MAR(0,4), SOI as MAR(2,2),
NAO as MAR(1,1), and N H as MAR(12,2) (Table 4). Since the condition r=s
is not met, GHG,N2O,GCAG,GIS T EM P ,GMSL, and P DO are time-
irreversible. However, TR is still a possible outcome for SOI and N AO; hence,
we implement the next steps of Strategies 1 and 2. Since the information criteria
of the restricted MAR(2,2) (BIC=2705.795) is larger than the one provided by
the MAR(2,2) with no restrictions (BIC=2659.974), Strategy 1 identifies SOI
as time-irreversible. The same result follows from Strategy 2: the null hypoth-
esis of TR is rejected since the estimated likelihood ratio test statistic equals
52.57. Contrastingly, NAO is identified as time-reversible from both strategies:
the information criteria of the restricted MAR(1,1) is lower (BIC=2450.267)
than the one provided by the unrestricted MAR(1,1) (BIC=2456.317), and the
estimated likelihood ratio test statistic is equal to 0.6985. Even if this last time
series rejects the null hypothesis of Gaussianity, it is very close to the Gaussian
case since the estimated degrees of freedom ν) equals 96.2. However, identi-
fying NAO as non-Gaussian does not affect our conclusions as both strategies
identify it as time-reversible.
In summary, our findings identify GLO,GL,GO,SA,NAO, and SH as
time-reversible and GHG,N2O,GC AG,GI ST E MP ,GM SL,SOI,P DO,
and NH as time-irreversible. The time irreversibility of ccGHG and ccN2Ois a
noticeable property of variables that account for the warming trend in global
temperatures (IPCC (2014), Morana and Sbrana (2019)). We expect time ir-
reversibility also to be present in other variables affected by greenhouse gas
emissions, as, statistically, a linear combination of time-irreversible and time-
reversible variables is also time-irreversible. This result can explain why GCAG,
GI ST EM P ,GM SL,SOI,P D O, and NH are time-irreversible. In particu-
lar, these results underline how global warming might have exerted feedback
effects on natural oscillations, temperatures, and the environment in general.
Among others, Morana and Sbrana (2019) show that GHG emissions are the
key determinant of the warming trend in global temperatures. The irreversibil-
ity of GCAG and GI ST E MP further corroborates these findings. Yet the
17
ccGHG ccN2OccGC AG ccGIST EM P ccGMS L SOI NAO PD O N H
ˆ
φ10.9620 0.9818 0.4417 0.4003 1.1233 -0.0933 -0.0966 / 0.1657
(0.0841) (0.0722) (0.0225) (0.0239) (0.0231) (0.0327) (0.00342) (0.0300)
ˆ
φ2-0.3230 -0.2413 0.1443 0.1245 -0.1169 -0.1315 / / -0.0071
(0.0822) (0.0980) (0.0225) (0.0239) (0.0347) (0.0326) (0.0306)
ˆ
φ3/ -0.0016 / / -0.6729 / / / -0.0824
(0.1030) (0.0336) (0.0304)
ˆ
φ4/ -0.2028 / / 0.3971 / / / 0.0025
(0.0753) (0.0335) (0.0303)
ˆ
φ5/ / / / 0.0898 / / / -0.0025
(0.0346) (0.0303)
ˆ
φ6/ / / / -0.1798 / / / -0.0390
(0.0229) (0.0303)
ˆ
φ7/ / / / / / / / 0.0058
(0.0303)
ˆ
φ8/ / / / / / / / -0.0136
(0.0303)
ˆ
φ9/ / / / / / / / -0.0926
(0.0303)
ˆ
φ10 / / / / / / / / 0.0496
(0.0304)
ˆ
φ11 / / / / / / / / 0.1039
(0.0305)
ˆ
φ12 / / / / / / / / 0.7015
(0.0300)
ˆϕ1/ / / / 0.0880 0.4951 0.2925 0.9183 0.7655
(0.0225) (0.0313) (0.0328) (0.0215) (0.0391)
ˆϕ2/ / / / 0.2709 0.3169 / -0.1365 -0.0512
(0.0225) (0.0313) (0.0291) (0.0391)
ˆϕ3/ / / / / / / 0.0063 /
(0.0291)
ˆϕ4/ / / / / / / 0.0664 /
(0.0214)
ˆν19.8 13.0 5.3 9.9 6.8 8.0 96.2 9.1 8.2
Table 4: Estimated coefficients of the time series identified as non-Gaussian. The figures in
parentheses are the standard errors computed by using the Hessian matrix.
18
evidence is inconclusive as the cyclical components of GO,GL, and GLO are
time-reversible. However, whether this latter result might be an artefact due
to their shorter sample is plausible. Morana and Sbrana (2019) also document
Atlantic hurricanes’ increasing natural disaster risk and destabilizing impact on
the EN SO cycle. Indeed, oceans warming can trigger a tipping point in the
EN SO cycle, increasing its variability and intensity and shifting its telecon-
nection eastward (Cai et al. (2021); see also Cai et al. (2014), and Cai et al.
(2015)). Global warming can also profoundly affect P DO, shortening its lifes-
pan and suppressing its amplitude (Li et al. (2020)). Moreover, the melting of
land ice and warming ocean waters cause rising sea levels affecting coastal shore-
lines. High-tide flooding is increasing in magnitude and frequency: minor floods
occur multiple times per year; major floods might occur even yearly. Even if
GHG emissions stopped, the sea level would continue to rise. Finally, the Arctic
is warming twice as faster as the planet on average, and finding irreversibility
in NH but not in SH , is interesting in this respect. Despite not being con-
clusive, the results might indicate that some irreversible environmental changes
are ongoing.
6. Conclusions
This paper links the concept of an environmental tipping point to the sta-
tistical concept of time irreversibility. A tipping point signals an environmental
change that is large, abrupt, and irreversible and generates cascading effects.
A tipping point is a point of no return, which we associate with a temporal
asymmetry in a phenomenon’s probabilistic structure, whereby it behaves dif-
ferently according to the direction of time considered. This univocity along the
time direction signals that the system has undergone an irreversible change.
Well-known tipping points concern the Greenland and the West Antarctic ice
sheets, the Atlantic Meridional Overturning Circulation (AMOC ), thawing per-
mafrost, EN SO, and the Amazon rainforest. Recent IP C C assessments suggest
that tipping points might occur even between 1Cand 2Cwarming relative
to pre-industrial temperature averages. Therefore, they are likely to arise at
current emissions levels if they have not already occurred.
We then introduce two new strategies, grounded on mixed causal and non-
causal models, to detect whether a stochastic process is time-reversible (TR).
Unlike existing approaches, our methods do not impose strong restrictions on
the model and are straightforward to implement. Moreover, similarly to Proi-
etti (2020), they can also be applied to non-stationary processes and, therefore,
useful to assess some key variables, such as temperature anomalies and GHG
emissions, which appear to exhibit this property. Our simulation studies show
that the strategies perform accurately and have a solid ability to detect TR.
In the empirical analysis, we have considered fourteen climate time series,
i.e., annual and monthly global temperature anomalies (GLO,GL,GO;GCAG,
GI ST EM P ), solar activity (SA), natural oscillations (N AO,SOI,P DO), the
global mean sea level (GMSL), the Northern (N H ) and Southern (SH) Hemi-
sphere sea ice areas, global sea levels, greenhouse gas emissions (GHG,N2O).
19
We detect time irreversibility in GHG and N2Oemissions, SOI and P DO,
GMLS,N H, and the monthly temperature anomaly series. Yet not in the an-
nual temperature series, SH , and N AO (and S A). The time irreversibility of
GHG emissions is a noticeable property of variables that are well-known causes
of global warming. It may then explain the time irreversibility of GMSL,NH,
global temperature, and some natural oscillation indices such as P DO and SOI,
and therefore signal that some potentially irreversible environmental changes
are ongoing. This evidence might not be apparent from annual temperature
data due to the relatively smaller sample size available for annual than monthly
data.
Recent studies suggest global temperature has already warmed by 1.3C
and could cross the 1.5C threshold within a decade. 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.
Acknowledgment
The authors would like to thank the participants in the ECˆ2 2021 (Aarhus),
the CFE 2021 (London), the VI EMCC (Toulouse), the editor, and two anony-
mous referees for their valuable comments and suggestions. All remaining errors
are ours.
References
Backus, D.K., Kehoe, P.J., 1992. International evidence on the historical
properties of business cycles. The American Economic Review , 864–888.
Belaire-Franch, J., Contreras, D., 2003. Tests for time reversibility: a
complementarity analysis. Economics Letters 81, 187–195.
Breidt, F.J., Davis, R.A., 1992. Time-reversibility, identifiability and
independence of innovations for stationary time series. Journal of Time
Series Analysis 13, 377–390.
Breidt, F.J., Davis, R.A., Lh, K.S., Rosenblatt, M., 1991. Maximum likelihood
estimation for noncausal autoregressive processes. Journal of Multivariate
Analysis 36, 175–198.
Caesar, L., McCarthy, G., Thornalley, D., Cahill, N., Rahmstorf, S., 2021.
Current Atlantic Meridional Overturning Circulation weakest in last
millennium. Nature Geoscience 14, 118–120.
Cai, W., Borlace, S., Lengaigne, M., Van Rensch, P., Collins, M., Vecchi, G.,
Timmermann, A., Santoso, A., McPhaden, M.J., Wu, L., et al., 2014.
Increasing frequency of extreme el ni˜no events due to greenhouse warming.
Nature climate change 4, 111–116.
20
Cai, W., Santoso, A., Collins, M., Dewitte, B., Karamperidou, C., Kug, J.S.,
Lengaigne, M., McPhaden, M.J., Stuecker, M.F., Taschetto, A.S., et al.,
2021. Changing El Ni˜no–Southern Oscillation in a warming climate. Nature
Reviews Earth & Environment 2, 628–644.
Cai, W., Wang, G., Santoso, A., McPhaden, M.J., Wu, L., Jin, F.F.,
Timmermann, A., Collins, M., Vecchi, G., Lengaigne, M., et al., 2015.
Increased frequency of extreme la ni˜na events under greenhouse warming.
Nature Climate Change 5, 132–137.
Chen, Y.T., Chou, R.Y., Kuan, C.M., 2000. Testing time reversibility without
moment restrictions. Journal of Econometrics 95, 199–218.
De Jong, R.M., Sakarya, N., 2016. The econometrics of the hodrick-prescott
filter. Review of Economics and Statistics 98, 310–317.
DeConto, R.M., Pollard, D., Alley, R.B., et al., 2021. The Paris Climate
Agreement and future sea-level rise from Antarctica. Nature 593, 83–89.
Fern´andez, C., Steel, M.F., 1998. On bayesian modeling of fat tails and
skewness. Journal of the american statistical association 93, 359–371.
Fries, S., 2021. Conditional moments of noncausal alpha-stable processes and
the prediction of bubble crash odds. Journal of Business & Economic
Statistics , 1–37.
Fries, S., Zakoian, J.M., 2019. Mixed causal-noncausal ar processes and the
modelling of explosive bubbles. Econometric Theory 35, 1234–1270.
Giancaterini, F., Hecq, A., 2022. Inference in mixed causal and noncausal
models with generalized student’s t-distributions. Econometrics and
Statistics, https://doi.org/10.1016/j.ecosta.2021.11.007 .
Gourieroux, C., Jasiak, J., 2016. Filtering, prediction and simulation methods
for noncausal processes. Journal of Time Series Analysis 37, 405–430.
Gourieroux, C., Jasiak, J., 2022. Nonlinear forecasts and impulse responses for
causal-noncausal (s) var models. Manuscript, University of Toronto .
Gouri´eroux, C., Zakoian, J.M., et al., 2013. Explosive bubble modelling by
noncausal process. CREST.
Hallin, M., Lefevre, C., Puri, M.L., 1988. On time-reversibility and the
uniqueness of moving average representations for non-gaussian stationary
time series. Biometrika 75, 170–171.
Hansen, J., Sato, M., Kharecha, P., Von Schuckmann, K., Beerling, D.J., Cao,
J., Marcott, S., Masson-Delmotte, V., Prather, M.J., Rohling, E.J., et al.,
2017. Young people’s burden: requirement of negative co 2 emissions. Earth
System Dynamics 8, 577–616.
21
Hecq, A., Lieb, L., Telg, S., 2016. Identification of mixed causal-noncausal
models in finite samples. Annals of Economics and Statistics/Annales
d’´
Economie et de Statistique , 307–331.
Hecq, A., Voisin, E., 2021. Predicting bubble bursts in oil prices using mixed
causal-noncausal models. Forthcoming in Advances in Econometrics in
honor of Joon Y. Park .
Hencic, A., Gouri´eroux, C., 2015. Noncausal autoregressive model in
application to bitcoin/usd exchange rates, in: Econometrics of risk.
Springer, pp. 17–40.
Hinich, M.J., Rothman, P., 1998. Frequency-domain test of time reversibility.
Macroeconomic Dynamics 2, 72–88.
Holster, A., 2003. The criterion for time symmetry of probabilistic theories
and the reversibility of quantum mechanics. New Journal of Physics 5, 130.
IPCC, 2014. International panel on climate change fifth assessment report,
available at https://www.ipcc.ch/report/ar5/syr/ .
IPCC, 2022. International panel on climate change sixth assessment report,
available at https://www.ipcc.ch/report/sixth-assessment-report-cycle/ .
Lanne, M., Saikkonen, P., 2011. Noncausal autoregressions for economic time
series. Journal of Time Series Econometrics 3.
Levesque, D., Verlet, L., 1993. Molecular dynamics and time reversibility.
Journal of Statistical Physics 72, 519–537.
Li, S., Wu, L., Yang, Y., Geng, T., Cai, W., Gan, B., Chen, Z., Jing, Z.,
Wang, G., Ma, X., 2020. The pacific decadal oscillation less predictable
under greenhouse warming. Nature Climate Change 10, 30–34.
Lovejoy, T.E., Nobre, C., 2018. Amazon Tipping Point. Science Advances 4,
eaat2340.
Meinshausen, M., Vogel, E., Nauels, A., Lorbacher, K., Meinshausen, N.,
Etheridge, D.M., Fraser, P.J., Montzka, S.A., Rayner, P.J., Trudinger, C.M.,
et al., 2017. Historical greenhouse gas concentrations for climate modelling
(cmip6). Geoscientific Model Development 10, 2057–2116.
Morana, C., Sbrana, G., 2019. Some financial implications of global warming:
An empirical assessment. Economic Modelling 81, 274–294.
Proietti, T., 2020. Peaks, gaps, and time reversibility of economic time series.
WP Rome .
Ramsey, J.B., Rothman, P., 1996. Time irreversibility and business cycle
asymmetry. Journal of Money, Credit and Banking 28, 1–21.
22
Ravn, M.O., Uhlig, H., 2002. On adjusting the hodrick-prescott filter for the
frequency of observations. Review of economics and statistics 84, 371–376.
Schellnhuber, H.J., 2008. Global warming: Stop worrying, start panicking?
Proceedings of the National Academy of Sciences 105, 14239–14240.
Solomon, S., Plattner, G.K., Knutti, R., Friedlingstein, P., 2009. Irreversible
climate change due to carbon dioxide emissions. Proceedings of the national
academy of sciences 106, 1704–1709.
Wald, R.M., 1980. Quantum gravity and time reversibility. Physical Review D
21, 2742.
Weiss, G., 1975. Time-reversibility of linear stochastic processes. Journal of
Applied Probability 12, 831–836.
Wunderling, N., Donges, J.F., Kurths, J., Winkelmann, R., 2021. Interacting
tipping elements increase risk of climate domino effects under global
warming. Earth System Dynamics 12, 601–619.
23
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Originating in the equatorial Pacific, the El Niño–Southern Oscillation (ENSO) has highly consequential global impacts, motivating the need to understand its responses to anthropogenic warming. In this Review, we synthesize advances in observed and projected changes of multiple aspects of ENSO, including the processes behind such changes. As in previous syntheses, there is an inter-model consensus of an increase in future ENSO rainfall variability. Now, however, it is apparent that models that best capture key ENSO dynamics also tend to project an increase in future ENSO sea surface temperature variability and, thereby, ENSO magnitude under greenhouse warming, as well as an eastward shift and intensification of ENSO-related atmospheric teleconnections — the Pacific–North American and Pacific–South American patterns. Such projected changes are consistent with palaeoclimate evidence of stronger ENSO variability since the 1950s compared with past centuries. The increase in ENSO variability, though underpinned by increased equatorial Pacific upper-ocean stratification, is strongly influenced by internal variability, raising issues about its quantifiability and detectability. Yet, ongoing coordinated community efforts and computational advances are enabling long-simulation, large-ensemble experiments and high-resolution modelling, offering encouraging prospects for alleviating model biases, incorporating fundamental dynamical processes and reducing uncertainties in projections.
Article
Full-text available
Noncausal, or anticipative, heavy-tailed processes generate trajectories featuring locally explosive episodes akin to speculative bubbles in financial time series data. For (Xt) a two-sided infinite α-stable moving average (MA), conditional moments up to integer order four are shown to exist provided (Xt) is anticipative enough, despite the process featuring infinite marginal variance. Formulae of these moments at any forecast horizon under any admissible parameterisation are provided. Under the assumption of errors with regularly varying tails, closed-form formulae of the predictive distribution during explosive bubble episodes are obtained and expressions of the ex ante crash odds at any horizon are available. It is found that the noncausal autoregression of order 1 (AR(1)) with AR coefficient ρ and tail exponent α generates bubbles whose survival distributions are geometric with parameter ρα. This property extends to bubbles with arbitrarily-shaped collapse after the peak, provided the inflation phase is noncausal AR(1)-like. It appears that mixed causal-noncausal processes generate explosive episodes with dynamics à la Blanchard and Watson (1982) which could reconcile rational bubbles with tail exponents greater than 1.
Article
Full-text available
With progressing global warming, there is an increased risk that one or several tipping elements in the climate system might cross a critical threshold, resulting in severe consequences for the global climate, ecosystems and human societies. While the underlying processes are fairly well-understood, it is unclear how their interactions might impact the overall stability of the Earth's climate system. As of yet, this cannot be fully analysed with state-of-the-art Earth system models due to computational constraints as well as some missing and uncertain process representations of certain tipping elements. Here, we explicitly study the effects of known physical interactions among the Greenland and West Antarctic ice sheets, the Atlantic Meridional Overturning Circulation (AMOC) and the Amazon rainforest using a conceptual network approach. We analyse the risk of domino effects being triggered by each of the individual tipping elements under global warming in equilibrium experiments. In these experiments, we propagate the uncertainties in critical temperature thresholds, interaction strengths and interaction structure via large ensembles of simulations in a Monte Carlo approach. Overall, we find that the interactions tend to destabilise the network of tipping elements. Furthermore, our analysis reveals the qualitative role of each of the four tipping elements within the network, showing that the polar ice sheets on Greenland and West Antarctica are oftentimes the initiators of tipping cascades, while the AMOC acts as a mediator transmitting cascades. This indicates that the ice sheets, which are already at risk of transgressing their temperature thresholds within the Paris range of 1.5 to 2 ∘C, are of particular importance for the stability of the climate system as a whole.
Article
Full-text available
The Paris Agreement aims to limit global mean warming in the twenty-first century to less than 2 degrees Celsius above preindustrial levels, and to promote further efforts to limit warming to 1.5 degrees Celsius¹. The amount of greenhouse gas emissions in coming decades will be consequential for global mean sea level (GMSL) on century and longer timescales through a combination of ocean thermal expansion and loss of land ice². The Antarctic Ice Sheet (AIS) is Earth’s largest land ice reservoir (equivalent to 57.9 metres of GMSL)³, and its ice loss is accelerating⁴. Extensive regions of the AIS are grounded below sea level and susceptible to dynamical instabilities5–8 that are capable of producing very rapid retreat⁸. Yet the potential for the implementation of the Paris Agreement temperature targets to slow or stop the onset of these instabilities has not been directly tested with physics-based models. Here we use an observationally calibrated ice sheet–shelf model to show that with global warming limited to 2 degrees Celsius or less, Antarctic ice loss will continue at a pace similar to today’s throughout the twenty-first century. However, scenarios more consistent with current policies (allowing 3 degrees Celsius of warming) give an abrupt jump in the pace of Antarctic ice loss after around 2060, contributing about 0.5 centimetres GMSL rise per year by 2100—an order of magnitude faster than today⁴. More fossil-fuel-intensive scenarios⁹ result in even greater acceleration. Ice-sheet retreat initiated by the thinning and loss of buttressing ice shelves continues for centuries, regardless of bedrock and sea-level feedback mechanisms10–12 or geoengineered carbon dioxide reduction. These results demonstrate the possibility that rapid and unstoppable sea-level rise from Antarctica will be triggered if Paris Agreement targets are exceeded.
Article
Full-text available
The Atlantic Meridional Overturning Circulation (AMOC)—one of Earth’s major ocean circulation systems—redistributes heat on our planet and has a major impact on climate. Here, we compare a variety of published proxy records to reconstruct the evolution of the AMOC since about ad 400. A fairly consistent picture of the AMOC emerges: after a long and relatively stable period, there was an initial weakening starting in the nineteenth century, followed by a second, more rapid, decline in the mid-twentieth century, leading to the weakest state of the AMOC occurring in recent decades. The Atlantic Meridional Overturning Circulation (AMOC) is currently distinctly weaker than it has been for the last millennium, according to a synthesis of proxy records derived from a range of techniques.
Article
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.
Article
Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.
Article
Atmospheric greenhouse gas (GHG) concentrations are at unprecedented, record-high levels compared to the last 800 000 years. Those elevated GHG concentrations warm the planet and - partially offset by net cooling effects by aerosols - are largely responsible for the observed warming over the past 150 years. An accurate representation of GHG concentrations is hence important to understand and model recent climate change. So far, community efforts to create composite datasets of GHG concentrations with seasonal and latitudinal information have focused on marine boundary layer conditions and recent trends since the 1980s. Here, we provide consolidated datasets of historical atmospheric concentrations (mole fractions) of 43 GHGs to be used in the Climate Model Intercomparison Project - Phase 6 (CMIP6) experiments. The presented datasets are based on AGAGE and NOAA networks, firn and ice core data, and archived air data, and a large set of published studies. In contrast to previous intercomparisons, the new datasets are latitudinally resolved and include seasonality. We focus on the period 1850-2014 for historical CMIP6 runs, but data are also provided for the last 2000 years. We provide consolidated datasets in various spatiotemporal resolutions for carbon dioxide (CO2), methane (CH4) and nitrous oxide (N2O), as well as 40 other GHGs, namely 17 ozone-depleting substances, 11 hydrofluorocarbons (HFCs), 9 perfluorocarbons (PFCs), sulfur hexafluoride (SF6), nitrogen trifluoride (NF3) and sulfuryl fluoride (SO2F2). In addition, we provide three equivalence species that aggregate concentrations of GHGs other than CO2, CH4 and N2O, weighted by their radiative forcing efficiencies. For the year 1850, which is used for pre-industrial control runs, we estimate annual global-mean surface concentrations of CO2 at 284.3 ppm, CH4 at 808.2 ppb and N2O at 273.0 ppb. The data are available at https://esgf-node.llnl.gov/search/input4mips/ and www.climatecollege.unimelb.edu.au/cmip6. While the minimum CMIP6 recommendation is to use the global- and annual-mean time series, modelling groups can also choose our monthly and latitudinally resolved concentrations, which imply a stronger radiative forcing in the Northern Hemisphere winter (due to the latitudinal gradient and seasonality)..