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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 1.5◦above 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 1.5◦threshold, 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 1◦and 2◦warming, 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 3.2◦warming (Wunderling et al. (2021)). An unstoppable ice sheet melting

in Antarctica would manifest at 2◦warming (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 3◦and 5.5◦warming. 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 (Y−T, Y−(T−1), . . . , Y−1) 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=−∞

θkt−k,(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)sθ2k−j. 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(z−1) 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(z−1) 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(z−1) have the same roots, 1/φ(z) has the Laurent expansion

1

φ(z)=

∞

X

−∞

θjzj,(4)

with θ−p/2−j=θ−p/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(z−1) 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

t−k]−E[Yj

tYi

t−k] = 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+k≤mand k≤K. In

particular, they restrict m= 3 so that the symmetric-bicovariance function is

given by:

γ2,1=E[Y2

tYt−k]−E[YtY2

t−k]=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[bγ2,1−γ2,1]

pV ar(bγ2,1)

d

−→ N(0,1),(7)

with:

bγ2,1=b

B2,1(k)−b

B1,2(k),

and:

b

B2,1= (T−k)−1

T

X

t=K+1

Y2

tYt−k;b

B1,2= (T−k)−1

T

X

t=K+1

YtY2

t−k,

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 =Yt−Yt−kis 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(L−1)Yt=t,(8)

where L−1produces lead such that L−1Yt=Yt+1. A mixed causal and non-

causal model represented in this way is denoted as MAR(r,s), where ϕ(L−1) 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(L−1) = 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(z−1) have the same roots as φ(z−1)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() + gγI(−)),(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 γ∈R+.

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(ft−ft−1)−(ft−1−ft−2)2.(12)

According to De Jong and Sakarya (2016), the optimization problem (11)

has the following closed-form solution:

fY

t=λL−2−4λL−1+ (1 + 6λ)−4λL +λ2−1Yt,(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="1−ψ1(λ)L−ψ2(λ)L21−ψ1(λ)L−1−ψ2(λ)L−2+

−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="1−100

29 L+ 6.25L21−100

29 L−1+ 6.25L−2−13.456#−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≈"1−100

29 L+ 6.25L21−100

29 L−1+ 6.25L−2#−1

Yt.(15)

Using the Laurent expansion as in (4), we have:

fY

t≈

+∞

X

j=−∞

δjYt−j,(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=Xt−1+δ+ηt,(17)

with η∼N(0,1), and ccYas a MAR(1,1):

(1 + φL)(1 + ϕL−1)˜

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=Yt−1+δ+˜

Yt⇒∆Yt=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 (ˆν) 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 1◦Cand 2◦Cwarming 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 1.3◦C

and could cross the 1.5◦C 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.

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