#  A Stochastic Climate Model - An approach to calibrate the Climate-Extended
Risk Model (CERM)

Jean-Baptiste Gaudemet  Jules Deschamps  Amalthea FS, https://amaltheafs.com/
Olivier Vinciguerra

###### Abstract

The initial Climate-Extended Risk Model (CERM) [1] addresses the estimate of
climate-related financial risk embedded within a bank loan portfolio, through
a climatic extension of the Basel II IRB model. It uses a Gaussian copula
model calibrated with non stationary macro-correlations in order to reflect
the future evolution of climate-related financial risks.

In this complementary article, we propose a stochastic forward-looking
methodology to calibrate climate macro-correlation evolution from scientific
climate data, for physical and transition efforts specifically. We assume a
global physical and transition risk, likened to persistent greenhouse gas
(GHG) concentration in the atmosphere. The economic risk is considered
stationary and can therefore be calibrated with a backward-looking
methodology. We present 4 key principles to model the GDP and we propose to
model the economic, physical and transition effort factors with three
interdependent stochastic processes allowing for a calibration with seven
precisely defined parameters. These parameters can be calibrated using public
data. This new approach means not only to evaluate climate risks without
picking any specific scenario but also allows to fill the gap between current
one year approach of regulatory and economic capital models and the
necessarily long-term view of climate risks by designing a framework to
evaluate the resulting credit loss on each step (typically yearly) of the
transition path.

This new approach could prove instrumental in the 2022 context of central
banks weighing the pros and cons of a climate capital charge.

Keywords: climate model, stochastic model, climate risk, net-zero transition,
GDP.

## Introduction

Since the very first version of this new climate-related financial risk
framework in June 2020 [2], the ecosystem has moved forward a lot. Central
Banks have progressed in their analysis of climate change impact on financial
stability by running multiple stress-tests exercises. In UK, the Bank of
England in 2021 (due in May 2022), French ACPR, ended in 2021. The Network for
Greening the Financial System (NGFS) has also grown tremendously, gathering
now 105 central banks including the Federal Reserve Bank which joined in
January 2021 thanks to the new administration involvement in USA. NGFS refined
last year the climate scenari taken in consideration and describes now 6
scenari [3]. Finally, commercial banks have also picked up their games,
quickly drafting Net Zero plans and forming coalitions to support the energy
transition plan, the latest being 19 banks in North America forming a
consortium with the Risk Management Association.

This paper is the result of the continuous forward thinking work Green RWA is
fostering, keeping the momentum the CERM has sparkled in its community. We
describe here a calibration methodology allowing to take into account a
distribution of scenari (as opposed to scenario picking in the current stress-
test approach) through a diffusion model relying on a limited number of well
defined parameters. This calibration methodology paves the way for the
calculation of credit loss provisions on the net-zero pathway and potentially
for a single upfront capital coverage by discounting this curve.
In the first section, we will introduce four key principles we will use in the
second session to define a stochastic diffusion model of climate change. It
will allow to derive the CERM macro-correlation parameters, in the third
section. As we will see, our calibration methodology is slightly modifying the
mathematics of the original CERM as we now need auto-correlated risk factors
and a dynamic correlation matrix. We will discuss these model adjustments in
the fourth section of the paper. In the fifth section we will study the
resulting distribution of the GDP growth curve, which will allow to calculate
in the sixth section the asymptotic probability to reach a net-zero economic
growth.

The seventh and final section of the paper is a call for a numerical
application of the calibration.

## About

### Green RWA

Green RWA is a unique NGO rooted in the belief that financial institutions
have a key role to play in the energy transition. Green RWA supports banks by
developing innovative climate financial risk models. It now counts 84 members,
bankers, insurers and fintech specialists, and 10 corporates, typically
FinTech and Consultancies.

### Amalthea FS

Amalthea is a start-up Fintech developing SaaS technology for climate risk and
reporting. Amalthea believes banks will need new technology to help achieve
the challenges of new regulation and risk frameworks that energy transition
and regulation will impose, launching a necessary evolution of their
technological infrastructure.

## Acknowledgments

We want to thank specifically Josselin Garnier and Anne Gruz for the high
quality interactions we regularly have on the financial climate risk and its
modelling. We are also grateful to the central banks and international
institutions we met including ECB, FED, European Commission and IMF for their
openness and interest in our approach and this calibration methodology. Their
expertise and in-depth understanding of the matter was instrumental to
narrowing down on this approach.

##  1 Key principles

The CERM is calibrated through an estimation of the borrower’s assets’
sensitivity to the global economic and climate-related financial risks
(typically transition and physical risks). In this paper, we suggest to model
the joint dynamic of the Global Domestic Product (GDP) between economic
cycles, physical climate change damages, and transition costs, with a system
of equations derived from the following four key principles:

  * •

P1: Climate risks are additive to the other economic risks observed
historically.

  * •

P2: Physical risk associated with the global rise of temperature adds up with
a growing incertitude over time as the climate reacts to the accumulation in
the atmosphere of greenhouse gas (GHG) emissions generated by economic
activity [4]. This assumption is a general consensus amongst climate
specialists, as it is underlying in most climate models.

  * •

P3: Accumulated transition efforts reduce future climate change damage by
adapting the economy and limiting new greenhouse gas emissions. This typically
is represented by net-zero emission plans, aggregated both at state and
corporate levels.

  * •

P4: Transition efforts stem from the political pressure and ecological
awareness generated by climate change. Extreme weather events, flooding and
large fires around the world typically encourage citizens to accept increasing
transition efforts and urge decision-makers to enforce climate adaptation
investments and GHG mitigation policies.

##  2 Modeling climate change

###  2.1 GDP model

We define a yearly time axis $t\in\mathbb{Z}$. We choose to set $t=0$ as the
current year and $t=-t_{0}$ as the year from which GHG emissions have started
to modify climate. We will model the growth of the economy from time $-t_{0}$
to time $t$ as the additive growth of the log GDP during this climate change
period. We introduce three stochastic cumulative growth factors from $-t_{0}$
to $t$:

  * •

$\tilde{Y}_{E}^{t}$, the hypothetical climate-free growth of the economy
(i.e., if climate change did not exist) since $-t_{0}$.

  * •

$\tilde{Y}_{P}^{t}$, the cumulative physical damage to the economy due to
climate change since $-t_{0}$;

  * •

$\tilde{Y}_{T}^{t}$, the cumulative economic cost due to transition efforts
since $-t_{0}$.

By definition of $-t_{0}$ we have the initial conditions:
$\tilde{Y}_{E}^{-t_{0}}=\tilde{Y}_{P}^{-t_{0}}=\tilde{Y}_{T}^{-t_{0}}=0$. As
per the first principle P1, those three random variables add up to model the
growth of the economy.

$\forall t\in\mathbb{Z},t\geq-
t_{0},GDP^{t}=GDP^{-t_{0}}\exp{\left[\tilde{Y}_{E}^{t}-\tilde{Y}_{T}^{t}-\tilde{Y}_{P}^{t}\right]}$
(1)

The variables $\tilde{Y}_{E}^{t}$, $\tilde{Y}_{P}^{t}$, $\tilde{Y}_{T}^{t}$
are fixed for all past (and present) dates $t\in\\{-t_{0},...,0\\}$. In the
next section we will introduce a forward-looking stochastic diffusion model
for $t>0$.

###  2.2 Forward-looking climate factors model

In this section we propose to model the dynamic of the three stochastic growth
factors defined above. Since we model the log GDP, we have a year-on-year
additive model:

$\forall
t\in\mathbb{N},\begin{cases}\tilde{Y}_{E}^{t+1}=\tilde{Y}_{E}^{t}+R+e\epsilon_{E}^{t+1}\,\,(\tilde{L}_{1})\\\
\tilde{Y}_{P}^{t+1}=\tilde{Y}_{P}^{t}+\tilde{\gamma}(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-\tilde{\alpha}\tilde{Y}_{T}^{t+1}+\tilde{p}W_{P}^{t+1}\,\,(\tilde{L}_{2})\\\
\tilde{Y}_{T}^{t+1}=\tilde{Y}_{T}^{t}+\beta(\tilde{Y}_{P}^{t}-\tilde{Y}_{P}^{t-1})+\theta\epsilon_{\theta}^{t+1}\,\,(\tilde{L}_{3})\end{cases},$
(2)

where $R$, $e$, $\tilde{p}$, $\theta$, $\tilde{\alpha}$, $\beta$
$\tilde{\gamma}$ are all positive coefficients to be calibrated, and all
$(\epsilon_{r}^{t})_{r\in\\{E,P,\theta\\}}\sim\mathcal{N}(0,1)$ are i.i.d.

As per the IRB regulatory principle, we first model in equation
$(\tilde{L}_{1})$ a stationary climate-free economic dynamic $\tilde{Y}_{E}$
with a stochastic variable $e\epsilon_{E}^{t}$ of constant variance, to which
we add a constant growth factor R. Note it is possible without changing the
formalism of the model to use a deterministic growth factor $R^{t}$.

In the second equation $(\tilde{L}_{2})$, we increase the physical damage
$\tilde{Y}_{P}$ by the incremental impact on climate change
$\tilde{\gamma}(\tilde{Y}_{E}-\tilde{Y}_{P}-\tilde{Y}_{T})$ resulting from
marginal GHG emissions generated by the economy (as per the second principle
P2). To it is subtracted the benefit $\tilde{\alpha}\tilde{Y}_{T}$ of the
cumulative transition effort (as per the third principle P3). This allows to
see $\tilde{\gamma}$ as the climate change intensity of the pre-transition
economic activity, and $\tilde{\alpha}$ as the transition efficiency. In
addition, we integrate the growing climate change incertitude by adding the
independent Wiener process $\tilde{p}W_{P}$. Note it possible without changing
the formalism of the model to add an acceleration factor to the climate damage
$\tilde{Y}_{P}^{t}$ in order to capture secondary climate change effects.

The third equation $(\tilde{L}_{3})$ formalizes the forth principle P4 by a
linear regression between the incremental physical damage to the economy and
the additional transition effort made by the economy. We note $\beta$ the
intercept term (transition efforts reactivity) and $\theta$ the error term
(idiosyncratic transition efforts).

The next section will derive the CERM parameters from the forward-looking GDP
model presented.

###  2.3 Glossary

variable | interpretation
---|---
$\tilde{Y}_{E}$ | cumulative hypothetical climate-free economic growth
$\tilde{Y}_{P}$ | cumulative economic cost due to climate change
$\tilde{Y}_{T}$ | cumulative economic cost due to transition efforts
$\tilde{Y}_{E}-\tilde{Y}_{P}-\tilde{Y}_{T}$ | cumulative economic growth

parameter | interpretation
---|---
$\tilde{\gamma}$ | climate change intensity of the economic activity
$\tilde{p}$ | idiosyncratic physical risk (standard deviation rate)
$\tilde{\alpha}$ | transition efficiency
$\beta$ | transition effort reactivity to climate change
$\theta$ | independent transition effort (standard deviation)
$R$ | hypothetical climate-free average growth rate
$e$ | idiosyncratic economic risk (standard deviation)

##  3 CERM parameters

###  3.1 Risk model

We introduce the risk factors $Y_{E}^{t}$, $Y_{P}^{t}$, $Y_{T}^{t}$, as the
random parts of the incremental growth factors of the log GDP between time
$t-1$ and $t$. We have:

$\forall
t\in\mathbb{N},\begin{cases}Y_{E}^{t+1}=\tilde{Y}_{E}^{t+1}-\tilde{Y}_{E}^{t}-\mathbb{E}(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{E}^{t})\\\
Y_{P}^{t+1}=\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t}-\mathbb{E}(\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t})\\\
Y_{T}^{t+1}=\tilde{Y}_{T}^{t+1}-\tilde{Y}_{T}^{t}-\mathbb{E}(\tilde{Y}_{T}^{t+1}-\tilde{Y}_{T}^{t})\end{cases},$
(3)

with $Y_{E}^{t}=Y_{P}^{t}=Y_{T}^{t}=0$ for $t\leq 0$ since historical growth
factors are known. By construction, for $t\geq 0$ all
$(Y_{r}^{t})_{r\in\\{E,P,\theta\\}}$ are centered.

###  3.2 Risk covariance

In order to calculate the correlation of the CERM economic and climate
factors, we must derive from (2) the covariance matrix of the risk factors
$Y_{E}$, $Y_{P}$ and $Y_{T}$. The difference in equation $(2)$ between times
$t+1$ and $t$ reads:

$\forall t\in\mathbb{N},\begin{cases}Y_{E}^{t+1}=e\epsilon_{E}^{t+1}\\\
(1+\tilde{\gamma})Y_{P}^{t+1}=Y_{P}^{t}+\tilde{\gamma}Y_{E}^{t+1}-(\tilde{\alpha}+\tilde{\gamma})Y_{T}^{t+1}+\tilde{p}\epsilon_{P}^{t+1}\\\
Y_{T}^{t+1}=\beta Y_{P}^{t}+\theta\epsilon_{\theta}^{t+1}\\\ \end{cases},$

with $\epsilon_{P}^{t+1}=W_{P}^{t+1}-W_{P}^{t}\sim\mathcal{N}(0,1)$ (by
definition of a Wiener process). Then, by substituting $Y_{E}^{t+1}$ and
$Y_{T}^{t+1}$ from the first and the third equations in the second one, we
get:

$\forall t\in\mathbb{N},\begin{cases}Y_{E}^{t+1}=e\epsilon_{E}^{t+1}\\\
(1+\tilde{\gamma})Y_{P}^{t+1}=(1-(\tilde{\alpha}+\tilde{\gamma})\beta)Y_{P}^{t}-(\tilde{\alpha}+\tilde{\gamma})\theta\epsilon_{\theta}^{t+1}+e\tilde{\gamma}\epsilon_{E}^{t+1}+\tilde{p}\epsilon_{P}^{t+1}\\\
Y_{T}^{t+1}=\beta Y_{P}^{t}+\theta\epsilon_{\theta}^{t+1}\end{cases}$

To simplify the system we introduce the reduced parameters:

$\begin{cases}\alpha=\frac{\tilde{\alpha}}{1+\tilde{\gamma}},\,\,\gamma=\frac{\tilde{\gamma}}{1+\tilde{\gamma}},\,\,p=\frac{\tilde{p}}{1+\tilde{\gamma}}\\\
q=\frac{1-(\tilde{\alpha}+\tilde{\gamma})\beta}{1+\tilde{\gamma}}=1-\alpha\beta-(1+\beta)\gamma=(1-\gamma)-(\alpha+\gamma)\beta\,\,\text{(is
assumed to be $<1$)}.\end{cases}$ (4)

Substituting gives the following equations:

$\forall
t\in\mathbb{N},\begin{cases}Y_{E}^{t+1}=e\epsilon_{E}^{t+1}\,(L_{1})\\\
Y_{P}^{t+1}=qY_{P}^{t}-(\alpha+\gamma)\theta\epsilon_{\theta}^{+1}+\gamma
e\epsilon_{E}^{t+1}+p\epsilon_{P}^{t+1}\,(L_{2})\\\ Y_{T}^{t+1}=\beta
Y_{P}^{t}+\theta\epsilon_{\theta}^{t+1}\,(L_{3})\end{cases}.$ (5)

For $t\geq 0$, writing the vector
$\mathbf{Y}^{t}=(Y_{E}^{t}\,\,Y_{P}^{t}\,\,Y_{T}^{t})^{T}$ gives:

$\forall t\in\mathbb{N},\,\mathbf{Y^{t+1}}=\left(\begin{array}[]{ccc}0&0&0\\\
0&q&0\\\
0&\beta&0\end{array}\right)\mathbf{Y^{t}}+\left(\begin{array}[]{c}e\epsilon_{E}^{t+1}\\\
-(\alpha+\gamma)\theta\epsilon_{\theta}^{t+1}+e\gamma\epsilon_{E}^{t+1}+p\epsilon_{P}^{t+1}\\\
\theta\epsilon_{\theta}^{t+1}\end{array}\right).$

For the following, we will note:

$\mathbf{A}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&q&0\\\
0&\beta&0\end{array}\right),\,\;\text{and: }\forall
t>0,\,\mathbf{E_{t}}=\left(\begin{array}[]{c}e\epsilon_{E}^{t}\\\
-(\alpha+\gamma)\theta\epsilon_{\theta}^{t}+e\gamma\epsilon_{E}^{t}+p\epsilon_{P}^{t}\\\
\theta\epsilon_{\theta}^{t}\end{array}\right).$

By induction (initiated with $\mathbf{Y}^{0}=0$): $\forall t\geq
1,\mathbf{Y^{t}}=\sum_{k=0}^{t-1}\mathbf{A}^{k}\mathbf{E_{t-k}}\text{, with
}\mathbf{A}^{0}=\mathbf{I_{3}}$.

As all $(\mathbf{E_{k}})_{k}$ are independent, the variance is:

$\forall t\geq
1,\,\mathbb{V}[\mathbf{Y^{t}}]=\mathbb{V}[\sum_{k=0}^{t-1}\mathbf{A}^{k}\mathbf{E_{t-k}}]=\sum_{k=0}^{t-1}\mathbb{V}[\mathbf{A}^{k}\mathbf{E_{t-k}}]=\sum_{k=0}^{t-1}\mathbf{A}^{k}\mathbf{V}(\mathbf{A}^{k})^{T},$
(6)

with

$\mathbf{V}=\mathbb{V}[\mathbf{E_{t}}]=\left(\begin{array}[]{ccc}e^{2}&\gamma
e^{2}&0\\\ \gamma
e^{2}&(\alpha+\gamma)^{2}\theta^{2}+e^{2}\gamma^{2}+p^{2}&-(\alpha+\gamma)\theta^{2}\\\
0&-(\alpha+\gamma)\theta^{2}&\theta^{2}\end{array}\right).$ (7)

We introduce the characteristic standard deviation
$\sigma=\sqrt{(\alpha+\gamma)^{2}\theta^{2}+e^{2}\gamma^{2}+p^{2}}$ to
simplify future calculations. Having $\left|q\right|<1$ also allows to
introduce the sum $c_{t}=\sum_{k=0}^{t-1}q^{2k}=\frac{1-q^{2t}}{1-q^{2}}$ for
$t\geq 1$ with $c_{0}=0$.

We know by induction that:

$\forall t\geq 1,\,\mathbf{A^{t}}=\left(\begin{array}[]{ccc}0&0&0\\\
0&q^{t}&0\\\ 0&\beta q^{t-1}&0\end{array}\right),$ (8)

and from (6) and (7), we draw:

$\forall t\geq
1,\,\mathbb{V}[\mathbf{Y^{t}}]=\left(\begin{array}[]{ccc}e^{2}&\gamma
e^{2}&0\\\ \gamma
e^{2}&\sigma^{2}c_{t}&-(\alpha+\gamma)\theta^{2}+\sigma^{2}\beta qc_{t-1}\\\
0&-(\alpha+\gamma)\theta^{2}+\sigma^{2}\beta
qc_{t-1}&\theta^{2}+\beta^{2}\sigma^{2}c_{t-1}\end{array}\right)$ (9)

###  3.3 CERM macro-correlations

In the CERM, the macro-correlations at time $t+1$ are defined as the
respective sensitivities to the economic and climate risk factors between time
$t$ and $t+1$. We can now derive these quantities as the standard deviations
of the incremental growth factors:

$\forall
t\in\mathbb{N+},\begin{cases}\xi_{E}^{t}=\sqrt{\mathbb{V}[Y_{E}^{t}]}=e\\\
\xi_{P}^{t}=\sqrt{\mathbb{V}[Y_{P}^{t}]}=\sigma\sqrt{c}_{t}\\\
\xi_{T}^{t}=\sqrt{\mathbb{V}[Y_{T}^{t}]}=\sqrt{\theta^{2}+\beta^{2}\sigma^{2}c_{t-1}}\end{cases}$
(10)

We can now calculate the correlation matrix for CERM risk factors
$\mathbf{Z^{t}}\sim\mathcal{N}(0,C_{t})$ at time $t\geq 1$ as:
$C_{t}=Corr[Y_{E}^{t},-Y_{P}^{t},-Y_{T}^{t}]$. Using equations (7) and (9), we
get:

$C_{t}=\left(\begin{array}[]{ccc}1&-\frac{\gamma
e^{2}}{\xi_{E}^{t}\xi_{P}^{t}}&0\\\ -\frac{\gamma
e^{2}}{\xi_{E}^{t}\xi_{P}^{t}}&1&\frac{\sigma^{2}\beta
qc_{t-1}-(\alpha+\gamma)\theta^{2}}{\xi_{P}^{t}\xi_{T}^{t}}\\\ \
0&\frac{\sigma^{2}\beta
qc_{t-1}-(\alpha+\gamma)\theta^{2}}{\xi_{P}^{t}\xi_{T}^{t}}&1\end{array}\right)$
(11)

It is noteworthy that the current version of the CERM cannot integrate time-
dependent correlation matrices. In the next section we will see how to adapt
the CERM accordingly.

####  3.3.1 Asymptotic study

The macro-correlations of equation (8) converge to:

$\begin{cases}\xi_{E}^{\infty}=e\\\ \xi_{P}^{\infty}=\sigma/\sqrt{1-q^{2}}\\\
\xi_{T}^{\infty}=\sqrt{\frac{\beta^{2}\sigma^{2}}{1-q^{2}}+\theta^{2}}\
\end{cases},$ (12)

from which we can calculate the limit $C_{\infty}$ of the correlation matrix.
With $c=\beta^{2}+\frac{\theta^{2}(1-q^{2})}{\sigma^{2}}$, we have:

$C_{\infty}=\left(\begin{array}[]{ccc}1&-\frac{\gamma
e\sqrt{1-q^{2}}}{\sigma}&0\\\ -\frac{\gamma
e\sqrt{1-q^{2}}}{\sigma}&1&\frac{(1-\gamma)\beta}{\sqrt{c}}-(\alpha+\gamma)\sqrt{c}\\\
0&\frac{(1-\gamma)\beta}{\sqrt{c}}-(\alpha+\gamma)\sqrt{c}&1\end{array}\right)$
(13)

As expected, we note that the asymptotic correlation between physical and
transition risk decreases with the idiosyncratic transition effort parameter
$\theta$ and is negative in the absence of reactivity of transition efforts to
climate change ($\beta=0$).

###  3.4 Auto-correlations

In this section we analyze how the risk factors previously presented are auto-
correlated. We introduce the delay $\tau>0$, giving for $t\geq 0$:
$\mathbf{Y^{t+\tau}}=A^{\tau}\mathbf{Y^{t}}+\sum_{k=0}^{\tau-1}A^{k}\mathbf{E_{t+\tau-k}}$.
Having this formula for $\mathbf{Y^{t+\tau}}$, we can derive:

$Cov[\mathbf{Y^{t+\tau}},\mathbf{Y^{t}}]=A^{\tau}\mathbb{E}[\mathbf{Y^{t}}\mathbf{Y^{t}}^{T}]+\sum_{k=0}^{\tau-1}A^{k}\mathbb{E}[\mathbf{E_{t+\tau-k}}\mathbf{Y^{t}}^{T}]=A^{\tau}\mathbb{E}[\mathbf{Y^{t}}\mathbf{Y^{t}}^{T}]=A^{\tau}\mathbb{V}[\mathbf{Y^{t}}]$
(14)

because $\mathbf{Y^{t}}$ is centered and independent from
$\mathbf{E_{t+\tau-k}}$ for $k\in\\{0,...,\tau-1\\}$.

Here, we have:

$Cov[\mathbf{Y^{t+\tau}},\mathbf{Y^{t}}]=\left(\begin{array}[]{ccc}0&0&0\\\
\gamma
e^{2}q^{\tau}&\sigma^{2}q^{\tau}c_{t}&-(\alpha+\gamma)\theta^{2}q^{\tau}+\sigma^{2}\beta
q^{\tau+1}c_{t-1}\\\ \beta\gamma e^{2}q^{\tau-1}&\sigma^{2}\beta
q^{\tau-1}c_{t}&-(\alpha+\gamma)\beta\theta^{2}q^{\tau-1}+\sigma^{2}\beta^{2}q^{\tau}c_{t-1}\end{array}\right),$
(15)

giving:

$Cov[\mathbf{Y^{t+\tau}},\mathbf{Y^{t}}]\xrightarrow{t\rightarrow\infty}\left(\begin{array}[]{ccc}0&0&0\\\
\gamma
e^{2}q^{\tau}&\frac{\sigma^{2}q^{\tau}}{1-q^{2}}&-(\alpha+\gamma)\theta^{2}q^{\tau}+\frac{\sigma^{2}\beta
q^{\tau+1}}{1-q^{2}}\\\ \beta\gamma e^{2}q^{\tau-1}&\frac{\sigma^{2}\beta
q^{\tau-1}}{1-q^{2}}&-(\alpha+\gamma)\beta\theta^{2}q^{\tau-1}+\frac{\sigma^{2}\beta^{2}q^{\tau}}{1-q^{2}}\end{array}\right).$

And then, as $\boldsymbol{\xi}^{t}$ is the standard deviation vector of
$\mathbf{Y^{t}}$ (meaning, the square root of the diagonal of its co-variance
matrix), we note $\frac{1}{\boldsymbol{\xi}^{t}}$ the vector whose coefficient
are the inverses of the coefficients of $\boldsymbol{\xi}^{t}$. We have:

$\frac{1}{\boldsymbol{\xi}^{t}}\left(\frac{1}{\boldsymbol{\xi}^{t+\tau}}\right)^{T}=\begin{matrix}((\frac{1}{\sigma(\mathbf{Y^{t}})_{i}\sigma(\mathbf{Y^{t+\tau}})_{j}})_{i,j})\end{matrix}$

and then ($*$ is coefficient-by-coefficient product):

$Corr(\mathbf{Y^{t+\tau}},\mathbf{Y^{t}})=\frac{1}{\boldsymbol{\xi}^{t}}\left(\frac{1}{\boldsymbol{\xi}^{t+\tau}}\right)^{T}*Cov[\mathbf{Y^{t+\tau}},\mathbf{Y^{t}}].$
(16)

From this we can draw:

$\lim\limits_{t\to\infty}Corr(\mathbf{Y^{t+\tau}},\mathbf{Y^{t}})=\left(\begin{array}[]{ccc}0&0&0\\\
\frac{\gamma e^{2}q^{\tau}\sqrt{1-q^{2}}}{\sigma}&q^{\tau}&\frac{\beta
q^{t+1}}{\sqrt{c}}-\frac{(\alpha+\gamma)\theta^{2}q^{\tau}(1-q^{2})}{\sigma^{2}\sqrt{c}}\\\
\frac{\beta\gamma eq^{\tau-1}\sqrt{1-q^{2}}}{\sigma\sqrt{c}}&\frac{\beta
q^{\tau-1}}{\sqrt{c}}&\frac{\beta^{2}q^{\tau}}{c}-\frac{\beta(\alpha+\gamma)\theta^{2}q^{\tau-1}(1-q^{2})}{\sigma^{2}c}.\end{array}\right)$
(17)

We notice that physical and transition risks are auto- and cross-correlated.
It is quite a fundamental difference with the original CERM as it was
considering risk factors as independently distributed. We believe this model
better reflects the reality of climate change which is obviously a persistent
phenomenon. The economic risk stays without auto-correlation in order to model
punctual economic crisis consistently with the regulatory IRB model.

##  4 Adaptation of the CERM

The following approach is based on the third approach developed in the
original CERM [1]. For coherence purposes, it adopts the same template of
presentation. As mentioned previously, the time-dependent model for auto-
correlations calls for an adaptation of the original CERM paper.

In the following, as is in the original CERM paper, $\alpha_{g,i,t,j}$ are the
micro-correlation adjustment parameters (each borrower in group $g$ with
rating $i$ at time $t$ has a micro-correlation $\alpha_{g,i,t,j}$ with risk
$j$), and $\xi_{t,j}$ are the macro-correlation parameters, as presented
earlier on. The correlation coefficient $R_{g,i,t}$ is the proportion of the
variance of the normalized log asset value that is due to systematic risks.
The corresponding loading factors are noted $a_{g,i,t}$. Also, $M_{g,t}$ is
the migration matrix for borrowers in group $g$ at time $t$: a borrower in
group $g$ with rating $i$ at time $t$ will move to a new rating $j$ with
corresponding probability $M_{g,t,i,j}$. The $z_{g,t,i,j}$ coefficients are
the corresponding thresholds: the same borrower will move to that same new
rating if their log asset value falls behind it.

Like in the original CERM model, we decide here that:

  * •

the time unit is one year (horizon of the migration matrices);

  * •

at time 1 the migration matrix $M_{g,1}$ is equal to $M_{g}^{reg}$ and the
correlation $R_{g,i,1}$ is determined by the regulator’s formula;

  * •

the migration matrices and the regulator’s formula for the correlation are
updated at time $t\geq 2$ because, contrary to the economic and idiosyncratic
risks, which are stationary, the physical and transition efforts evolve with
time;

  * •

the factor loadings $a_{g,i,t,j}$ are proportional to the product of the
macro-correlation and micro-correlation adjustment parameters.

We note:

$M_{g,1}=M_{g}^{reg},$
$R_{g,i,1}=R_{g,i}^{reg},\,\,R_{g,i}^{reg}=\mathbf{R}((M_{g}^{reg})_{iK})\text{
(the }\mathbf{R}\text{ correlation function is given in the original paper)},$
$a_{g,i,1}=a_{g,i}^{reg}=\sqrt{R_{g,i}^{reg}}\frac{\tilde{a}_{g,i,1}}{\sqrt{\tilde{a}_{g,i,1}\cdot
C_{1}\tilde{a}_{g,i,1}}}$

Similarly to the approach 3, we define (all products and quotients are
coefficient-by-coefficient)

$\tilde{a}_{g,i,t}=\alpha_{g,i,t}*\xi_{t},\,\,c_{g,i,t}=\frac{a_{g,i,1}*\tilde{a}_{g,i,t}}{\tilde{a}_{g,i,1}}.$

At time $t\geq 1$, we have:

$(M_{g,t})_{ij}=\begin{cases}1-\Phi(z_{g,t,i2})&j=1,\\\
\Phi(z_{g,t,ij})-\Phi(z_{g,t,ij+1})&2\leq j\leq K-1,\\\
\Phi(z_{g,t,iK})&j=K,\end{cases}$

with

$z_{g,t,ij}=\frac{z_{g,ij}^{reg}}{\sqrt{1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}}},$
$z_{g,ij}^{reg}=\Phi^{-1}(\sum_{j^{\prime}=j}^{K}(M_{g}^{reg})_{ij^{\prime}}),$

and we also have

$R_{g,i,t}=\frac{c_{g,i,t}\cdot C_{t}c_{g,i,t}}{1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}}$
$a_{g,i,t}=\frac{c_{g,i,t}}{\sqrt{1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}}}$

Both formulas for the loading factors and for the correlations coincide, as
proven hereunder.

###  4.1 Proof

At time 1, the normalized log asset value is given by

$X_{1}^{(q)}=a_{g,i}^{reg}\cdot Z_{1}+\sqrt{1-a_{g,i}^{reg}\cdot
C_{1}a_{g,i}^{reg}}\epsilon_{1}^{(q)}.$

In this approach, the idiosyncratic risk is stationary while the micro-
correlation and macro-correlation parameters evolve in time. This means that
we have in fact

$\bar{X}_{t}^{(q)}=c_{g,i,t}\cdot Z_{t}+\sqrt{1-a_{g,i}^{reg}\cdot
C_{1}a_{g,i}^{reg}}\epsilon_{t}^{(q)},$

which is a Gaussian variable with mean zero and variance $1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}.$ As a consequence, the
probabilities of rating changes are

$(M_{g,t})_{ij}=\mathbb{P}(\bar{X}_{t}^{(q)}\in[z_{g,ij+1}^{reg},z_{g,ij}^{reg}]),$

where the $z_{g,ij}^{reg}$s are the threshold values associated with the given
unconditional matrix $M_{g}^{reg}$. Furthermore, after normalization, the log
asset value (i.e. $X_{t}^{(q)}=\bar{X}_{t}^{(q)}/{\sqrt{1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}}}$) has now the form:

$X_{t}^{(q)}=a_{g,i,t}\cdot Z_{t}+\sqrt{1-a_{g,i,t}\cdot
C_{t}a_{g,i,t}}\epsilon_{t}^{(q)},$

with $a_{g,i,t}$ defined earlier on.

This normalized formula which is a function of the risk factor correlation
$C_{t}$ at time t only, results from the following calculation. Using the
definition of the correlation with systemic risk factors:

$R_{g,i,t}=\frac{c_{g,i,t}\cdot C_{t}c_{g,i,t}}{1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}},$

we get

$\sqrt{1+c_{g,i,t}\cdot C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot
C_{1}a_{g,i}^{reg}}=\sqrt{\frac{1-R_{g,i}^{reg}}{1-R_{g,i,t}}}$

and that is why

$X_{t}^{(q)}=\frac{\bar{X}_{t}^{(q)}}{\sqrt{1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}}}=\frac{c_{g,i,t}\cdot
Z_{t}+\sqrt{1-a_{g,i}^{reg}\cdot
C_{1}a_{g,i}^{reg}}\epsilon_{t}^{(q)}}{\sqrt{1+c_{g,i,t}\cdot
C_{t}c_{g,i,t}-a_{g,i}^{reg}\cdot C_{1}a_{g,i}^{reg}}}$ $=a_{g,i,t}\cdot
Z_{t}+\frac{\sqrt{1-R_{g,i}^{reg}}}{\sqrt{\frac{1-R_{g,i}^{reg}}{1-R_{g,i,t}}}}\epsilon_{t}^{(q)}=a_{g,i,t}\cdot
Z_{t}+\sqrt{1-R_{g,i,t}}\epsilon_{t}^{(q)}$ $=a_{g,i,t}\cdot
Z_{t}+\sqrt{1-a_{g,i,t}\cdot C_{t}a_{g,i,t}}\epsilon_{t}^{(q)}$

##  5 The GDP distribution

In this section, we want to calculate the distribution of the GDP as a
function of time. The equation (1) models the GDP as a stochastic process
$GDP^{t}$ following at each time t a log-normal distribution law of
parameters:

$\mu^{t}=\mathbb{E}(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})\text{
and
}(s^{t})^{2}=\mathbb{V}(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t}).$

As per the classic log-normal distribution formulas, we therefore have at each
time $t$:

$med(GDP^{t})=GDP^{-t_{0}}\exp{(\mu^{t})},\,\,\mathbb{E}({GDP^{t}})=GDP^{-t_{0}}\exp(\mu^{t}+(s^{t})^{2}/2),$
$\text{ and
}\mathbb{V}({GDP^{t}})=(GDP^{-t_{0}})^{2}(\exp{((s^{t})^{2}})-1)\exp{(2\mu^{t}+(s^{t})^{2})}.$

###  5.1 Expected cumulative risk factors

With similar calculations as in 3.2, we get:

$\forall
t\in\mathbb{N},\begin{cases}\mathbb{E}(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{E}^{t})=R\\\
\mathbb{E}(\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t})=q\mathbb{E}(\tilde{Y}_{P}^{t}-\tilde{Y}_{P}^{t-1})+\gamma
R\\\
\mathbb{E}(\tilde{Y}_{T}^{t+1}-\tilde{Y}_{T}^{t})=\beta\mathbb{E}(\tilde{Y}_{P}^{t}-\tilde{Y}_{P}^{t-1})\
\end{cases}$ (18)

The second equation is that of an arithmetico-geometric sequence. Since
$\tilde{\mathbf{Y}}^{k}$ are known for $k\leq 0$, this gives:

$\forall
t\in\mathbb{N},\begin{cases}\mathbb{E}(\tilde{Y}_{E}^{t+1})=\tilde{Y}_{E}^{0}+(t+1)R\\\
\mathbb{E}(\tilde{Y}_{P}^{t+1})=\tilde{Y}_{P}^{0}+\frac{\gamma
R}{1-q}(t+1)+\left((\tilde{Y}_{P}^{0}-\tilde{Y}_{P}^{-1})-\frac{\gamma
R}{1-q}\right)q\frac{1-q^{t+1}}{1-q}\\\
\mathbb{E}(\tilde{Y}_{T}^{t+1})=\tilde{Y}_{T}^{0}+\beta(\tilde{Y}_{P}^{0}-\tilde{Y}_{P}^{-1})+\beta\left(\frac{\gamma
R}{1-q}t+\left((\tilde{Y}_{P}^{0}-\tilde{Y}_{P}^{-1})-\frac{\gamma
R}{1-q}\right)q\frac{1-q^{t}}{1-q}\right)\ \end{cases}$ (19)

###  5.2 Asymptotic study

####  5.2.1 Log-GDP asymptotic median growth

As a result, the median growth of the economy is asymptotically linear:

$\mu^{t}=r_{\mu}^{\infty}t+\mu_{H}+o(1),$ (20)

with

$r_{\mu}^{\infty}=\frac{\alpha\beta R}{\alpha\beta+(1+\beta)\gamma}$ (21)

and

$\mu_{H}=\mu^{0}+\left((\tilde{Y}_{P}^{0}-\tilde{Y}_{P}^{-1})-\frac{\gamma
R}{1-q}\right)\times\left[1-\frac{1-\beta}{\alpha\beta+(1+\beta)\gamma}\right]$
(22)

From this, we draw the conclusion that although the climate-related factors
cannot reverse the median economic growth, they do asymptotically curb it, as
$\frac{\alpha\beta R}{\alpha\beta+(1+\beta)\gamma}\leq R$. The more efficient
the transition is in mitigating and adapting the economy to carbon emissions
(parameter $\alpha$, in comparison with $\gamma$), the closest the median
economic growth is to the climate-free growth ($R$).

####  5.2.2 Log-GDP asymptotic variance

We introduce the vector
$\mathbf{\tilde{Y}}=(\tilde{Y}_{E}\,\,\tilde{Y}_{P}\,\,\tilde{Y}_{T})^{T}$. We
first want to calculate $\mathbb{V}[\mathbf{\tilde{Y}}^{t}]$ for $t\geq 0$.
$\mathbf{\tilde{Y}}^{0}$ being known (as in constant),
$\mathbb{V}[\mathbf{\tilde{Y}}^{t}]=\mathbb{V}[\mathbf{\tilde{Y}}^{T}-\mathbf{\tilde{Y}}^{0}]$.
From equation (3) we derive:

$\mathbf{\tilde{Y}}^{t}-\mathbf{\tilde{Y}}^{0}=\sum_{k=1}^{t}\mathbf{Y^{k}}+C=\sum_{k=1}^{t}\sum_{i=0}^{k-1}\mathbf{A}^{i}\mathbf{E_{k-i}}+C\textrm{
with }\mathbf{A}^{0}=\mathbf{id}\textrm{ and where }C\textrm{ is a constant.}$

From this we draw:

$\mathbb{V}[\mathbf{\tilde{Y}^{t}}]=\mathbb{V}\left[\sum_{k=1}^{t}\mathbf{Y^{k}}\right]=\mathbb{V}\left[\sum_{k=1}^{t}\sum_{i=0}^{k-1}\mathbf{A}^{i}\mathbf{E_{k-i}}\right]$

With a classic manipulation of the double sum, we get:

$\mathbb{V}\left[\sum_{k=1}^{t}\mathbf{Y^{k}}\right]=\mathbb{V}\left[\sum_{k=1}^{t}\left(\sum_{i=0}^{t-k}\mathbf{A}^{i}\right)\mathbf{E_{k}}\right]$
(23)

And since the random vectors $\mathbf{E_{k}}$ are i.i.d., we have:

$\mathbb{V}[\mathbf{\tilde{Y}^{t}}]=\sum_{k=1}^{t}\mathbb{V}\left[\left(\sum_{i=0}^{t-k}\mathbf{A}^{i}\right)\mathbf{E_{k}}\right]=\sum_{k=1}^{t}\left[\left(\sum_{i=0}^{t-k}\mathbf{A}^{i}\right)\mathbb{V}(\mathbf{E_{k}})\left(\sum_{i=0}^{t-k}\mathbf{A}^{i}\right)^{T}\right]=\sum_{k=1}^{t}\left[\left(\sum_{i=0}^{t-k}\mathbf{A}^{i}\right)\mathbf{V}\left(\sum_{i=0}^{t-k}\mathbf{A}^{i}\right)^{T}\right]$
(24)

using the same notations as $(6)$, which gives in turn:

$\mathbb{V}[\mathbf{\tilde{Y}^{t}}]=\sum_{k=1}^{t}\left[\left(\sum_{i=0}^{k-1}\left[\mathbf{A}^{i}\right]\right)\mathbf{V}\left(\sum_{i=0}^{k-1}\left[(\mathbf{A}^{i})\right]\right)^{T}\right].$
(25)

Introducing the sum $b_{k}=\sum_{i=0}^{k-1}q^{i}=\frac{1-q^{k}}{1-q}$ for
$t\geq 1$ with $s_{0}=0$, we get:

$\forall
k\in\mathbb{N^{*}},\,\sum_{i=0}^{k-1}\left[\mathbf{A^{i}}\right]=\left(\begin{array}[]{ccc}1&0&0\\\
0&b_{k}&0\\\ 0&\beta b_{k-1}&1\end{array}\right).$

Hence the covariance matrix of the GDP is:

$\mathbb{V}[\mathbf{\tilde{Y}}^{t}]=\sum_{k=1}^{t}\left(\begin{array}[]{ccc}e^{2}&\gamma
e^{2}b_{k}&\beta\gamma e^{2}b_{k-1}\\\ \gamma
e^{2}b_{k}&\sigma^{2}b_{k}^{2}&\beta\sigma^{2}b_{k}b_{k-1}-(\alpha+\gamma)\theta^{2}b_{k}\\\
\ \beta\gamma
e^{2}b_{k-1}&\beta\sigma^{2}b_{k}b_{k-1}-(\alpha+\gamma)\theta^{2}b_{k}&\theta^{2}+\beta^{2}\sigma^{2}b_{k-1}^{2}-2\beta\alpha\theta^{2}b_{k-1}\end{array}\right).$
(26)

This allows to compute $s^{t}$ and $\mathbb{V}[GDP^{t}]$, as
${(s^{t})}^{2}=\mathbb{V}[\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}]=(1\,-1\,-1)\mathbb{V}[\mathbf{\tilde{Y}}^{t}]\left(\begin{array}[]{c}1\\\
-1\\\ -1\end{array}\right).$

More specifically, asymptotically we have:

$\left(s^{t}\right)^{2}=r_{s^{2}}^{\infty}t+s_{H}^{2}+o(1),$ (27)

with:

$r_{s^{2}}^{\infty}=\left[e^{2}+\theta^{2}-2\frac{\gamma\theta^{2}+(1+\beta)(\alpha\theta^{2}+\gamma
e^{2})}{1-q}+\frac{\sigma^{2}(1+\beta)^{2}}{(1-q)^{2}}\right].$ (28)

###  5.3 GDP statistics

Having asymptotic equivalents of $\mu^{t}$ and ${(s^{t})}^{2}$ draws:

  * •

$\log(med(GDP^{t}))\sim r_{\mu}^{\infty}t$;

  * •

$\log(\mathbb{E}[GDP^{t}])\sim\left(\frac{r_{s^{2}}^{\infty}}{2}+r_{\mu}^{\infty}\right)t$;

  * •

$\log(\mathbb{V}[GDP^{t}])\sim
2\left(r_{s^{2}}^{\infty}+r_{\mu}^{\infty}\right)t.$

##  6 The probability of a net-zero transition

The following sections give several distinct approaches to assess the
probability of asymptotic net-zero transition, i.e. the event that in the long
run physical risks will be impeded enough so that economic development is
sustainable. To that end, we propose 3 approaches, related to the asymptotic
behavior of the physical cost factor in regards to the growth factor of the
economy. They do not assess the exact same quantities:

  * •

the first approach computes the unconditional probability of net-zero
transition;

  * •

the second one assesses the probability of having no degradation of climate
conditionally to the median economic growth;

  * •

the last one reflects the probability of having no degradation of climate
conditionally to asymptotic positive economic growth.

To simplify the following calculations, we introduce some notations to
describe the distributions of the incremental growth factors of physical risks
and economic development, for $t\in\mathbb{N}$.

We note:

  * •

$\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t}\sim\mathcal{N}(\mu_{1}^{t+1},(\sigma_{1}^{t+1})^{2})$;

  * •

$(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})\sim\mathcal{N}(\mu_{2}^{t+1},(\sigma_{2}^{t+1})^{2});$

  * •

$\rho^{t+1}=Corr\left(\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t},(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})\right)=Corr\left(Y_{P}^{t+1},Y_{E}^{t+1}-Y_{P}^{t+1}-Y_{T}^{t+1}\right),$

where:

  * •

$\mu_{1}^{t+1}=\mathbb{E}[\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t}]=\gamma
Rb_{t}+q^{t}\left(\tilde{Y}_{P}^{0}-\tilde{Y}_{P}^{-1}\right)\xrightarrow{t\rightarrow\infty}\mu_{1}=\frac{\gamma
R}{1-q};$

  * •

$\mu_{2}^{t+1}=\mathbb{E}[(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})]=R\left(1-\gamma(b_{t}+\beta
b_{t-1})\right)\xrightarrow{t\rightarrow\infty}\mu_{2}=r_{\mu}^{\infty}=\frac{\alpha\beta
R}{\alpha\beta+(1+\beta)\gamma};$

  * •

$({\sigma_{1}^{t+1}})^{2}=\sigma^{2}c_{t+1}\xrightarrow{t\rightarrow\infty}\sigma_{1}^{2}=\frac{\sigma^{2}}{1-q^{2}};$

  * •

$({\sigma_{2}^{t+1}})^{2}=(1-2\gamma)e^{2}+(1+2\alpha+2\gamma)\theta^{2}+\sigma^{2}\left(c_{t+1}+(\beta^{2}-2\beta
q)c_{t}\right)\xrightarrow{t\rightarrow\infty}\sigma_{2}^{2}=(1-2\gamma)e^{2}+(1+2\alpha+2\gamma)\theta^{2}+\frac{\sigma^{2}}{1-q^{2}}\left(1+\beta^{2}-2\beta
q\right);$

  * •

$\rho^{t+1}=\frac{\sigma^{2}c_{t+1}-\gamma
e^{2}+(\alpha+\gamma)\theta^{2}-\sigma^{2}\beta
qc_{t}}{\sqrt{\sigma^{2}c_{t+1}\times\left((1-2\gamma)e^{2}+\theta^{2}+2(\alpha+\gamma)\theta^{2}+\sigma^{2}c_{t+1}+\sigma^{2}(\beta^{2}-2\beta
q)c_{t}\right)}}\xrightarrow{t\rightarrow\infty}\rho=\frac{1-\beta
q+\left((\alpha+\gamma)\theta^{2}-\gamma
e^{2}\right)\frac{1-q^{2}}{\sigma^{2}}}{\sqrt{1+\beta^{2}-2\beta
q+\frac{1-q^{2}}{\sigma^{2}}\left((1-2\gamma)e^{2}+(1+2\alpha+2\gamma)\theta^{2}\right)}}.$

  * •

$\Phi$ is the cumulative distribution of the standard Gaussian distribution.

  * •

$\Phi_{2}(\cdot\,,\cdot\,;\rho)$ is the bivariate cumulative Gaussian
distribution with correlation $\rho$, as found in [1].

###  6.1 Unconditional net zero probability

We introduce $P_{NZ}^{1}$ the asymptotic unconditional probability to reach a
net-zero transition. It is worthwhile to note that the auto-correlation
analysed in section 3.4, guarantees to stay with a high probability close to
net-zero for a certain period of time once it has been reached. We define
$P_{NZ}^{1}$ as the limit of the probability to have no degradation of
climate, i.e.:

$P_{NZ}^{1}=\lim\limits_{t\to\infty}\mathbb{P}\left[\tilde{Y}_{P}^{t+1}-\tilde{Y_{P}^{t}}<0\right].$
(29)

Using the cumulative distribution function of the Gaussian distribution we
get:

$P_{NZ}^{1}=\Phi\left(-\frac{\gamma
R}{\sigma}\sqrt{\frac{1+q}{1-q}}\right)=\Phi\left(-\frac{\gamma
R}{\sigma}\sqrt{\frac{2-\alpha\beta-(1+\beta)\gamma}{\alpha\beta+(1+\beta)\gamma}}\right).$
(30)

The net-zero probability $P_{NZ}^{1}$ increases with the transition efficiency
$\alpha$ and the transition reactivity $\beta$ which was expected. In
addition, $P_{NZ}^{1}$ is naturally decreasing with the climate-free average
economic growth $R$, meaning we cannot grow the economy with total climate
impunity. However, we will see in the next two sections that our model makes
it possible to reach a net-zero growing economy.

###  6.2 Net zero probability conditional to the median economic growth

We now define $P_{NZ}^{2}(r)$ the asymptotic probability to reach a net-zero
transition conditionally to the median economic growth analysed in section
5.2.1. Here we note:

$P_{NZ}^{2}(r)=\lim\limits_{t\to\infty}\mathbb{P}\left(\left[\tilde{Y}_{P}^{t+1}-\tilde{Y_{P}^{t}}<0\,|\,(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})=r\right]\right).$
(31)

For $t\in\mathbb{N}$:

$\left(\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t}\,|\,(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})=r\right)\sim\mathcal{N}\left(\mu_{1}^{t+1}+\rho^{t+1}\frac{\sigma_{1}^{t+1}}{\sigma_{2}^{t+1}}(r-\mu_{2}^{t+1}),\,(1-(\rho^{t+1})^{2})(\sigma_{1}^{t+1})^{2}\right).$

Asymptotically, for $r=\mu_{2}=r_{\mu}^{\infty}$ (i.e. conditionally to a
growth equal to the asymptotic median growth of the economy in the long run):

$P_{NZ}^{2}=\lim_{t\rightarrow\infty}\mathbb{P}\left(\tilde{Y}_{P}^{t+1}-\tilde{Y}_{P}^{t}<0\,|\,(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})=r_{\infty}\right)=\Phi\left(-\frac{\mu_{1}}{\sigma_{1}\sqrt{1-\rho^{2}}}\right)$
(32) $=\Phi\left(-\frac{\gamma
R}{1-q}\sqrt{\frac{\frac{\sigma^{2}}{1-q^{2}}(1+\beta^{2}-2\beta
q)+(1-2\gamma)e^{2}+(1+2\alpha+2\gamma)\theta^{2}}{(\frac{\sigma^{2}}{1-q^{2}})^{2}\beta^{2}(1+q^{2})+\frac{\sigma^{2}}{1-q^{2}}[e^{2}(1+2\beta
q\gamma)+\theta^{2}(-2\beta
q(\alpha+\gamma))]-\gamma^{2}e^{4}+(\alpha+\gamma)^{2}\theta^{4}-2(\alpha+\gamma)\theta^{2}\gamma
e^{2}}}\right)$

.

###  6.3 Net-zero probability conditional to a positive economic growth

We now define $P_{NZ}^{3}$ as the asymptotic probability to reach a net-zero
transition conditionally to a positive economic growth:

$P_{NZ}^{3}=\lim\limits_{t\to\infty}\mathbb{P}\left(\left[\tilde{Y}_{P}^{t+1}-\tilde{Y_{P}^{t}}<0\,|\,(\tilde{Y}_{E}^{t+1}-\tilde{Y}_{P}^{t+1}-\tilde{Y}_{T}^{t+1})-(\tilde{Y}_{E}^{t}-\tilde{Y}_{P}^{t}-\tilde{Y}_{T}^{t})>0\right]\right).$
(33)

Which gives:

$P_{NZ}^{3}=\frac{\Phi(\frac{\mu_{1}}{\sigma_{1}})-\Phi_{2}(-\frac{\mu_{1}}{\sigma_{1}},-\frac{\mu_{2}}{\sigma_{2}};\rho)}{\Phi(\frac{\mu_{2}}{\sigma_{2}})}.$
(34)

####  6.3.1 Proof

We consider the random vector
$X=(X_{1},X_{2})^{T}\sim\mathcal{N}\left((\mu_{1},\mu_{2})^{T},\left(\begin{array}[]{cc}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\\
\rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{array}\right)\right)$ with
$0<\rho<1$, we have:

$\mathbb{P}(X_{1}<0\,|\,X_{2}>0)=\frac{\mathbb{P}(X_{1}<0\,\cap\,X_{2}>0)}{\mathbb{P}(X_{2}>0)}=\frac{\mathbb{P}(X_{1}<0)-\mathbb{P}(X_{1}<0\,\cap\,X_{2}\leq
0)}{\mathbb{P}(X_{2}>0)}=\frac{\Phi(\frac{\mu_{1}}{\sigma_{1}})-\Phi_{2}(-\frac{\mu_{1}}{\sigma_{1}},-\frac{\mu_{2}}{\sigma_{2}};\rho)}{\Phi(\frac{\mu_{2}}{\sigma_{2}})}.$

##  7 Call for numerical application

In this section, we present a methodology to calibrate the parameters of the
forward-looking diffusion model, from accessible historical data sets.

###  7.1 Historical dynamic

The historical worldwide GDP growth rate is very well documented in the
economic literature. It must be completed by an historical analysis of the
global economic cost of climate change, as well as an estimation of the global
transition effort made so far. This data set will allow to calibrate the
cumulative quantities $\tilde{Y}_{E}^{t}$, $\tilde{Y}_{P}^{t}$ and
$\tilde{Y}_{T}^{t}$ from $-t_{0}$. It also allows calculating the standard
deviation of the hypothetical climate-free economic factor
$\tilde{Y}_{E}^{t}$.

In the first version of the model (How Banks Can Save the Planet), we proposed
a calibration of against IMF data points. The World Economic Outlook database
gives access to the history of GDP at a country-granular level since 1980 [5].
World Bank also provides us with accurate GDP time series since 1961 [6].

###  7.2 Carbon intensity without transition

The historical anthropogenic emission of carbon dioxide at year $t$
$CO_{2}^{t}$ is well documented in the climatic literature. By definition
$-t_{0}$ is the year from which GHG emissions have started to modify climate
by exceeding the net-zero budget. We now introduce $-t_{1}$ as the year when
the transition effort started. The period $[-t_{0},-t_{1}]$ allows a
definition of the net carbon intensity of the economic activity in the absence
of transition effort:

$I=(CO_{2}^{-t_{1}}-CO_{2}^{-t_{0}})/(\ln{GDP^{-t_{1}}}-\ln{GDP^{-t_{0}}})$
(35)

The UNFCC provides a table of GHG emissions by country since 1990 [7] and the
World Resources Institutes references all kinds of GHG emissions sources
including PIK since 1850 and UNFCC data [8].

###  7.3 Transition efficiency

Several economic organizations such as the OECD have estimated the expected
economic cost $Y_{NZ}$ and duration $T_{NZ}$ to transition the global activity
to net-zero carbon emission. We also note $R_{NZ}$ the average economic growth
assumption made by the study. By taking the expected value of equation (2) we
get the following relationship between the climate change intensity of the
current economic activity $\tilde{\gamma}$ and the transition efficiency
$\tilde{\alpha}$:

$\tilde{\gamma}(\tilde{Y}_{E}^{0}-\tilde{Y}_{P}^{0}-\tilde{Y}_{T}^{0}+R_{NZ}T_{NZ})-\tilde{\alpha}Y_{NZ}=0$
(36)

We shall note $\tilde{\alpha}$ is constant, meaning it is assumed there is no
diminishing or increasing efficiency of transition effort. It is possible to
bypass this assumption at the cost of heavier formulas by making
$\tilde{\alpha}$ a function of $t$. However, it should not call into question
the computational implementation of our model, but only the formal
calculations. Another possibility is to stress this parameter in order to
assess the sensitivity of climate risk to the progress of green technologies.

###  7.4 Cost of climate change

Several economic organizations have documented the distribution of the yearly
economic cost of climate change by scenario of excess carbon concentration in
the atmosphere, with long maturity $T$ (typically $30$ years). The OECD
published in 2017 a study named Investing in Climate, Investing in Growth in
which the cost of the transition is very well described [9]. The climate
scenarios proposed by the NGFS have prompted the ECB to calculate their
respective associated GDP costs, both in terms of transition cost and climate
change costs [10]. It allows calibrating $a$ and $\tilde{p}$ in the equation
describing the economic cost of climate change at time T:
$\tilde{Y}_{P}^{T}-\tilde{Y}_{P}^{T-1}=a(CO_{2}^{T}-CO_{2}^{-t_{0}})+\tilde{p}W_{P}^{T}$.

By substituting with the carbon intensity in the absence of transition efforts
$I$ defined by (35), we get
$\tilde{Y}_{P}^{T}-\tilde{Y}_{P}^{T-1}=aI(\tilde{Y}_{E}^{T}-\tilde{Y}_{P}^{T}-\tilde{Y}_{T}^{T})-\tilde{\alpha}Y_{T}^{T}+\tilde{p}W_{P}^{T}$.
Hence, $\tilde{\gamma}=aI$ and we can deduct $\alpha$ from the equation (36).

###  7.5 Transition politics

The transition politics parameters $\beta$ and $\theta$ result from a linear
regression from $-t_{1}$ between the historical transition effort
$(\tilde{Y_{T}^{t+1}}-\tilde{Y_{T}^{t}})$ at time $t$ and the historical cost
of climate change $(\tilde{Y_{P}^{t}}-\tilde{Y_{P}^{t-1}})$ at time $t-1$.

## References

  * [1] Josselin Garnier  “The climate extended risk model (cerm)”, 2021  URL: https://arxiv.org/abs/2103.03275
  * [2] Green RWA  “How banks can save the planet – Part I”, 2020
  * [3] NGFS  “NGFS Climate Scenarios for central banks and supervisors”  URL: https://www.ngfs.net/sites/default/files/medias/documents/820184_ngfs_scenarios_final_version_v6.pdf
  * [4] Camilo Mora et al.  “Broad threat to humanity from cumulative climate hazards intensified by greenhouse gas emissions”  In _Nature Climate Change_ 8.12 Nature Publishing Group, 2018, pp. 1062–1071
  * [5] “Download entire World Economic Outlook database: October 2021”  URL: https://www.imf.org/en/Publications/WEO/weo-database/2021/October/download-entire-database
  * [6] “Worldbank GDP growth”  URL: https://data.worldbank.org/indicator/NY.GDP.MKTP.KD.ZG
  * [7] “GHG total without LULUCF (land-use, land-use change and forestry)”  URL: https://di.unfccc.int/time_series
  * [8] “The History of Carbon Dioxide Emissions”  URL: https://www.wri.org/insights/history-carbon-dioxide-emissions
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# How does the audience affect the way we express our gender roles?

MSMelody Sepahpour-Fard  MQMichael Quayle  MSMaria Schuld  TYTaha Yasseri  SFI
Centre for Research Training in Foundations of Data Science, Ireland
Department of Mathematics and Statistics, University of Limerick,
Düsternbrooker Weg 20, 24105 Castletroy, Limerick, Ireland  Centre for Social
Issues Research, University of Limerick, Castletroy, Limerick, Ireland
Department of Psychology, University of Limerick, Castletroy, Limerick,
Ireland  Department of Psychology, School of Applied Human Sciences,
University of KwaZulu-Natal, Durban, KwaZulu-Natal, South Africa  Department
of Psychology, University of Johannesburg, Johannesburg, South Africa  School
of Sociology, University College Dublin, Dublin, Ireland  Geary Institute for
Public Policy, University College Dublin, Dublin, Ireland

###### Abstract

In this paper, we explore the language of gender-specific groups of ”mothers”
and ”fathers” in comparison to the same individuals interacting just as
”parents.” Human beings adapt their language to the audience they interact
with. This may be for self-presentation reasons (to influence how they are
seen by others) or because being an authentic member of a group requires
alignment with its norms and culture. While audience effects have been studied
theoretically and in small-scale research designs, large-scale studies of
naturally-occurring audience effects are rare. To study the impact of audience
and gender in a natural setting, we choose a domain where gender plays a
particularly salient role: parenting. We look at the audience effects in
interaction with gendered contexts emphasizing different social identities
(i.e., mother, father, and parent). As a natural setting, we choose to study
interactions on Reddit. We collect posts from the three popular parenting
subreddits (i.e., topical communities on Reddit) r/Daddit, r/Mommit, and
r/Parenting. These three parenting subreddits gather different audiences,
respectively, self-identifying as fathers and mothers (ostensibly single-
gender), and parents (explicitly mixed-gender). By selecting a sample of users
who have published on both a single-gender and a mixed-gender subreddit, we
are able to explore both audience and gender effects. We analyse posts with
word embeddings by adding the user as a token in the corpus. This way, we are
able to compare user-tokens to word-tokens and measure their similarity.
Results show that mothers and fathers, when interacting in a mixed-gender
context, behave similarly and discuss a great diversity of topics while
focusing more around advising each other on educational and family matters. In
single-gender subreddits, mothers and fathers are more focused on specific
topics. Mothers in r/Mommit distinguish themselves from other groups by
discussing the most topics such as medical care, sleep and potty training, and
food. Both mothers and fathers celebrate parenting events and describe or
comment the physical appearance of their children with a single-gender
audience. In conclusion, this study shows how mothers and fathers express
different concerns and change their behaviour for different group-based
audiences. It also advocates for the use of Reddit and word embeddings to
better understand the dynamics of audience and gender in a natural setting.

gender stereotypes,

parenting,

audience effects,

word embeddings,

user embeddings,

Reddit,

natural language processing,

computational social science,

###### keywords:

## Content

Individuals identify with multiple social groups but only certain social
identities are contextually salient. The way we behave in a given context
depends on the audience in presence. The effect of an audience on speakers’
behaviour has been extensively documented since Goffman’s concept of
presentation of self or identity management goffman_presentation_1978 . He
uses the metaphor of a theatre with a backstage and a front stage, arguing
that individuals are like actors when interacting, performing to present a
desirable image of themselves goffman_presentation_1978 . Individuals are very
cautious about conforming to norms on the front stage, whereas they prepare
their presentation backstage goffman_presentation_1978 . The social identity
performance klein_social_2007  shows how the audience acts as a contextual
identity clue, making social identity salient and therefore making relevant
and appropriate a set of behaviours klein_social_2007 . These behaviours would
position individuals in relation to the social identity in question in a way
that allows them to achieve their social objectives klein_social_2007 . In
addition to influencing individual behaviour, the audience’s feedback shapes
the definition of social identity too klein_social_2007 . In contrast to the
concept of self-presentation, which focuses on being favourably seen as a
personal goal, social identity performance considers the situations where the
actor defines themselves as a group member and performs as such when they
think they are visible to an audience klein_social_2007 .

The present work studies audience effects in parenting communities on Reddit.
We use a novel method schuld_durrheim_mafunda_mazibuko_2022  of word embedding
using the agent as a token in the text corpus. This method allows not only to
locate agents from the way they talk but also to compare them to meaningful
word tokens. We use this method to focus on individuals and the influence
gender and audience play in their interactions. To do so, we analyse a dataset
of Reddit posts on the three popular parenting subreddits r/Parenting,
r/Daddit, and r/Mommit. We select a sample of Reddit users who have published
on these subreddits and study how the audience, either single-gender or mixed-
gender, can influence the expression of concerns related to parenting.

Given the salience of gender stereotypes in the parenting domain
wille19951990s ; jolly2014gender ; cinamon2002gender , the gender
configuration of the audience can push users towards specific behaviours, in
conformity or not with social expectations regarding gender norms
cinamon2002gender . The salience of social identities related to gender can
facilitate gender-stereotypical behaviours tajfel_social_1974 ;
tajfel_integrative_1979 ; turner_rediscovering_1987 ; ellemers2018gender .
According to gender stereotypes, women have communal characteristics (e.g.,
care and selflessness) while men have agentic characteristics (e.g.,
motivation to master and assertiveness) eagly_gender_1984 . In practice, these
stereotypes are reflected in gender roles as standards of ‘good’ behaviour for
men and women villicana_gender_2017 . These roles are associated with fields
of expected expertise such as work for men and family for women
ellemers2018gender . For instance, although fathers want to be more involved
in caring bulanda2004paternal , they are positioned as ”helpers” rather than
as primary caregivers cancian2000caring ; wall2007involved .

Audience and gender stereotypes are therefore frameworks that we will use to
understand behaviours. Do mothers and fathers adapt their expression of
concerns to the audience they interact with? On which topics specifically? Is
one gender more likely to change behaviour? These are questions we will
address in the present study.

Reddit has several very popular parenting communities that make possible the
study of those phenomena in a natural setting and at a large scale. We will
use Reddit as a means through which we can observe how the change in audience
can influence behaviour. Switching between different topical communities
(i.e., subreddits) and potentially social identities, with either a single or
mixed-gender audience, mothers and fathers may adapt their behaviour
accordingly. On Reddit, users’ interactions on parenting have been described
through several studies. Using a quantitative approach, Ammari et al.
ammari_pseudonymous_2018  looked at the three popular parenting subreddits
r/Parenting, r/Mommit, and r/Daddit with Latent Dirichlet Allocation topic
modelling blei_latent_2003 . They collected all the comments published on
these subreddits between 2008 and 2016, from which they were able to identify
47 topics. The authors found that users on r/Parenting talk more about
discipline and teen talk. r/Mommit users discussed the most topics such as
sleep training, breastfeeding, milestones, child weight gain, pregnancy
recovery, and housework. On r/Daddit, the most frequent topics were
congratulations, Neonatal Intensive Care Unit (NICU) experience, legal
questions for custody battles, and Halloween costumes. Therefore, mothers
discussed more issues related to health and raising children whereas fathers’
discussions focused more on events such as birth, Halloween, or divorce.

Using qualitative grounded theory, Feldman feldman__2021  analyzed posts
published on r/Daddit and r/Mommit. They found that while fathers merge work
and family and consider work as part of family duties, mothers see work and
family as two distinct domains and emphasise the double burden. Mothers’ posts
are also more centred around how to care for a child whereas fathers’ are
about whether they care feldman__2021 . Additionally, in family relationships,
fathers seek advice about how to provide support for the mother but mothers
focus on a dyadic mother-child relationship feldman__2021 .

Sepahpour-Fard and Quayle sepahpour2022how  showed that the audience might
play a role in the interactions parents have with each other. They selected a
set of Reddit users who have published on both a single-gender subreddit
(r/Mommit or r/Daddit) and a mixed-gender subreddit (r/Parenting) and looked
at how users change their expression of concerns depending on the audience.
They assumed, based on the subreddits’ descriptions, that r/Mommit is for
mothers, r/Daddit for fathers, and r/Parenting for parents. Using Latent
Dirichlet Allocation Topic Modelling blei_latent_2003 , they measured the
prevalence of topics in comments and averaged the prevalence by gender and
audience to isolate potential effects. Their results showed that mothers
generally express more concerns about the children’s basic needs (sleep, food,
and medical care) than fathers. Interestingly, they express those concerns
more between themselves (i.e., in r/Mommit) than with fathers (i.e., in
r/Parenting). In contrast, fathers focus on giving educational advice in
r/Parenting while celebrating events (e.g., pregnancies and births) and
sharing pictures with other fathers in r/Daddit. In addition to the conformity
of Reddit parents with gender roles regarding parenting, this study adds a
contextual approach which allows a better understanding of how users behave
with different audiences. However, results are aggregated by groups of users
and are therefore not showing what happens at the individual level. For
example, users who publish a lot will have a larger weight in the model and
results might be showing what they are concerned with, rather than what all
parents express.

In the present study, instead of looking at averages of posts such as in
previous studies, we use word embeddings (see MethodsMethods) and add the user
as a token in the text to locate each user from the way they speak
schuld_durrheim_mafunda_mazibuko_2022 . This way, we can give each user the
same weight and analyse the expression of concerns at the individual level in
parenting subreddits. From this embedding where users are located, forming a
speaker-landscape schuld_durrheim_mafunda_mazibuko_2022 , we can then see how
focused users are (i.e., discussing only specific topics), how users change
behaviour in front of different audiences, and how close they are to a
selection of topical keywords.

From comments published in 2020 on the three most popular parenting subreddits
(i.e. r/Daddit, r/Mommit, and r/Parenting), we selected only users who have
published on both a single-gender subreddit and a mixed-gender subreddit,
identifying mothers (i.e., users who publish on r/Mommit and r/Parenting) and
fathers (users who publish on r/Daddit and r/Parenting). Their gender is
inferred based on the subreddits’ descriptions (see MethodsMethods for more
detail). This setting allows us to capture both gender differences and
audience effects. We can see if mothers and fathers differently adjust their
social identity performance when interacting with different audiences and if
the location of users in the embedding shows significant insight regarding the
change of behaviour.

## Results

### Speaker-landscape

We use the term ‘speaker-landscape’ to refer to the embedding created by users
from their linguistic use. We begin by visualising the vector representation
of Reddit users and qualitatively study the structure of the landscape. In
Figure 1, we can see the two-dimensional representation of users coloured by
their self-identified status as mothers or fathers, according to the subreddit
on which they posted. Each user has two data points representing them, one
when interacting with a single-gender audience (posting on r/Mommit or
r/Daddit) and one when interacting with a mixed-gender audience (posting on
r/Parenting).

Figure 1: Speaker-landscape of mothers and fathers interacting with different
audiences. Each dot represent an author posting on one subreddit (either
r/Daddit, r/Mommit, or r/Parenting).

Fathers on r/Daddit are clustered at the right of the landscape. In contrast,
mothers on r/Mommit are mostly on the bottom half of the landscape and are
more spread out. In the presence of a mixed-gender audience (r/Parenting),
both fathers and mothers seem to discuss a wide range of topics as their data
points are spread out around the landscape.

To quantify how spatially concentrated or dispersed each group is in the
embedding space, we measure the average cosine similarity between the vector
representations of each pair of users in a group (mothers on r/Mommit, mothers
on r/Parenting, fathers on r/Daddit, and fathers on r/Parenting). We then
measure the differences between the concentration of groups using T-tests and
Kolmogorov-Smirnov tests (see MethodsMethods for more detail). Results show
that all the differences between the different groups are significant (see
Additional FilesAdditional Files, Table 2). Fathers, when interacting with
other fathers are the most concentrated group, meaning they tend to gather
more around specific topics than other groups. Mothers in r/Mommit are also
more concentrated than users on r/Parenting. When in a mixed-gender context on
r/Parenting, parents interact about a greater set of topics than when in
single-gender contexts and are therefore less spatially concentrated. Looking
at mothers and fathers within the context of r/Parenting, Mothers are the most
spatially spread group. These results confirm the qualitative analysis of the
speaker-landscape with fathers in r/Daddit being the most concentrated group
followed by mothers in r/Mommit, fathers in r/Parenting, and mothers in
r/Parenting (Figure 1).

Figure 2 shows the probability density function of the cosine similarity. In
addition to all differences appearing to be significant (see Additional
FilesAdditional Files, Table 2), we can see two distinct types of behaviour
among both mothers and fathers: they are more focused on specific topics of
discussion when interacting with a single-gender audience while they diversify
topics when gathered in a mixed-gender environment.

Figure 2: Distribution of similarity between users within each group of gender
and audience. The higher the cosine similarity value, the more similar agents
are to each other.

### Similarity between two versions of the same author

Are there gender differences between mothers and fathers? i.e., Does one
gender change their behaviour more when switching from one subreddit to the
other? We use the cosine similarity between two vectors (associated with the
two data points representing each user in the two subreddits) to measure the
extent to which users change their discourse when interacting with different
audiences (Additional FilesAdditional Files, Figure 5). The Kolmogorov-Smirnov
test comparing two distributions shows that the difference between mothers and
fathers is not significant. Therefore, there is a similar tendency for mothers
and fathers to be more focused on specific topics within gender-specified
contexts, but more diverse when gathered in the more pluralistic r/Parenting
subreddit.

### Topics of discussion

Previous work by Sepahpour-Fard and Quayle sepahpour2022how  found, using
Latent Dirichlet Allocation Topic Modelling, twelve topics in users’ discourse
on r/Mommit, r/Daddit, and r/Parenting: Food, Sleep training, Medical care,
Change/Potty training, Birth/Pregnancy, Thank you/Appreciation, Physical
appearance/Picture, (Education/Family advice, Work/Raise children,
School/Teaching, Leisure activities, and Furniture/Design. Each topic was
associated with ten keywords representing them (more details available in
sepahpour2022how ). To validate these keywords and topics, we first visualise
the topic modelling keywords found in this study sepahpour2022how  (Figure 3).
The keywords are coloured by the topic they are the most associated with. As
expected, and validating our speaker-landscape, each topic’s keywords are
clustered together in the embedding.

Figure 3: Topic Modelling keywords Each color represents a topic found through
LDA topic modelling on the same dataset.

To identify which topics are the most discussed by each group of users, we
measure the distance between topics and users interacting in the different
subreddits by first calculating the centroid of each topic and the centroid of
each group, and then measuring the cosine similarity between the centroids of
topics and groups. Figure 4 shows the cosine similarity between the centroids
of each set of users and the centroids of each set of keywords in a topic.
Results show that for some of the topics such as Education/Family advise,
School/Teaching and Physical appearance/Picture, there seems to be a clear
distinction between users’ behaviour in single-gender subreddits and mixed-
gender subreddits. Users discuss more Education/Family advise, Work/Raise
children, and School/Teaching in r/Parenting than in r/Daddit or r/Mommit
while the topic Physical appearance/Picture is the most prevalent among
parents interacting in single-gender subreddits. Several topics are discussed
the most by mothers on r/Mommit: Birth/Pregnancy, Change/Potty training, Food,
and Sleep training. The topic Medical Care and Furniture/Design seem to
distinguish mothers from fathers, regardless of the audience.

Figure 4: Topic Modelling keywords Each color represents a topic found through
LDA topic modelling on the same dataset.

## Discussion

We explored the effects of audience and gender on interactions in parenting
subreddits. We validated the speaker landscape as a method for the study of
individuals, groups, and group interactions. The results are consistent with
previous results, which analyzed the same dataset using topic modelling
sepahpour2022how  and therefore gave us additional confidence about the
position of users. However, going beyond topic modelling results, the speaker
landscape approach could capture how individual users change their behaviour
when switching subreddits without having to rely on averages biased towards
those who post a lot.

Coherently with previous works on the effect of an audience on behaviour
klein_social_2007 ; goffman_presentation_1978 ; turner_rediscovering_1987 , we
found that both mothers and fathers change their topics of discussion when
interacting with different audiences. We looked at the differences between
groups (i.e., mothers on r/Mommit, fathers on r/Daddit, mothers on
r/Parenting, and fathers on r/Parenting), and saw two clusters of behaviour
appearing with users in a single-gender subreddit being more focused (i.e.,
higher cosine similarity value) than users in a mixed-gender subreddit.
Results also showed that parents, when interacting in a mixed-gender
subreddit, were more diverse linguistically. In contrast, in a single-gender
subreddit, mothers and fathers were more clustered in the landscape which
indicates fewer topics of discussion. Previous research hilte2022linguistic
showed that teenage girls and boys (aged 13 to 20) tend to converge to more
similar behaviour when in a mixed-gender group, girls and boys adopting less
gender-specific writing styles. We observe the same behaviour, with mothers
and fathers tending to be less ‘specialized’ in specific topics (more spread
in the landscape) when gathered in a mixed-gender subreddit. This phenomenon
is also explained by the Communication Accommodation Theory
giles1979accommodation , which describes how individuals adjust their language
to each other to get approval, communicate more efficiently, and maintain a
positive social identity. The latter might explain why users differ more when
in mixed-gender groups than when gathered in a single-gender group. However,
at an even more abstract level, groups and social identities, are founded in
language; and language is a primary way for people to position themselves as
group members (especially in online contexts where it is the only medium for
social interaction and group definition).

Moreover, for mothers specifically, we see that the topics they discuss the
most overall (Food, Sleep training, Change/Potty training, and Physical
appearance/Picture) are topics that they discuss less in a mixed-gender
setting than in a single-gender setting. These are topics particularly
associated with the gender role of mothers ellemers2018gender  as they are
about the basic needs of children and the practical aspects of raising a
child. By discussing them the most, they position themselves as more concerned
by these areas. In a single-gender subreddit, fathers discussed more topics
related to announcing births and pregnancies, appreciation, shared pictures
and commented on physical appearance. In a mixed-gender environment, parents
interacted more about general educational or family advice. Previous studies
on the same parenting subreddits ammari_pseudonymous_2018 ; feldman__2021 ;
sepahpour2022how  found similar results with users discussing more sleep
training, breastfeeding, and potty training on r/Mommit, congratulations on
r/Daddit and, discipline on r/Parenting.

Although allowing a deep analysis of parents’ interactions on Reddit, our
analysis has some limitations related to the method used. Future work should
focus on untangling the potential biases caused by the method. As word
embeddings are initially made for representing words as vectors and
highlighting relationships between words, adding a non-word token (i.e., the
user token) might misrepresent other word tokens. For instance, an important
part of the word tokens is at the periphery of the landscape. This might be
explained by the fact that ngrams surrounding user tokens might be more
diverse than those surrounding other word tokens, therefore making user tokens
more central in the landscape. In other words, if the word ”bed” will be most
of the time surrounded by words such as ”night” or ”sleep”, a user token can
once be surrounded by words such as ”night” or ”sleep” when talking about
sleep training but by ”school” and ”teaching” when discussing education. More
generally, the speaker-landscape distorts language by adding a non-word token
in the text. The consequences of the latter should be studied to prevent
biases in the representations of users and words. In our work, however, the
speaker-landscape seems to make intuitive sense and is validated by the
clustered topics’ keywords in the landscape.

Additionally, although we improved previous research by looking at individual
differences and giving the same weight to each user, we kept looking at the
global structure and averaged measures for the different groups. However,
there might be some differences within the groups that should be more
carefully analysed, e.g, there seems to be a cluster of fathers in r/Daddit
gathered on a corner of the landscape. Future research should look closer at
the landscape and analyse the dynamics within each group. Considering audience
effects, there might be some users with large differences in behaviour between
subreddits, but some others not changing much at all.

Using word embeddings, we represented users spatially to analyse audience
effects in different parenting subreddits. Our results show that mothers and
fathers are specialized on specific topics when in r/Mommit and r/Daddit but
converge into more similar topics when gathered in r/Parenting. In a single-
gender, mothers preferentially discuss matters regarding the basic needs of
children (food, sleep, health) between themselves while fathers interact the
most about physical appearance and photos of children. Our study demonstrates
how Reddit can be used as a means through which researchers can study audience
effects and behaviours at a large scale while taking into account individual
behaviour.

## Methods

### Data

We collect comments published in 2020 on r/Daddit, r/Mommit, and r/Parenting,
three communities on Reddit. Reddit is an American content aggregation
platform. Users from any country can participate although English-speaking
countries are largely over-represented alexa ; similarweb . Pseudonymous users
publish content (e.g., links, text, images, or videos) on topical communities
or ‘subreddits’ which are defined by descriptions and community rules, and
governed by volunteer moderators. Subreddits can gather users around a large
variety of topics such as politics, science, or video games. The broad theme
of parenting has three popular subreddits: r/Parenting with 4,431,185 members,
r/Mommit with 538,017 members, and r/Daddit with 389,765 members as of June
2022.

r/Parenting is a mixed-gender subreddit. It was created in April 2008, and its
full name is “Reddit Parenting - For those with kids of any age!”. It is
described as “the place to discuss the ins and out as well as ups and downs of
child-rearing. From the early stages of pregnancy to when your teenagers are
finally ready to leave the nest (even if they don’t want to), we’re here to
help you through this crazy thing called parenting. You can get advice on
potty training, talk about breastfeeding, discuss how to get your baby to
sleep or ask if that one weird thing your kid does is normal.”

r/Mommit is a mother-centric single-gender subreddit. It was created in
September 2010, and its full name is “Mommit - Come for the support, stay for
the details.”. It is described as “We are people. Mucking through the ickier
parts of child-raising. It may not always be pretty, fun and awesome, but we
do it. And we want to be here for others who are going through the same
experiences and offer a helping hand.” r/Mommit explicitly indicates that
posters should be mothers.

r/Daddit is a father-centric single-gender subreddit. It was created in August
2010. It is described as “This is a subreddit for Dads. Single Dads, new Dads,
Step-Dads, tall Dads, short Dads, and any other kind of Dad. If you’ve got
kids in your life that you love and provide for, come join us as we discuss
everything from birth announcements to code browns in the shower.”. Similarly
to r/Mommit, r/Daddit is explicitly described to be exclusively for fathers.

We collected comments using Pushshift Reddit API111Code in Python by Mikołaj
Biesaga, The Robert Zajonc Institute for Social Studies, University of Warsaw.
GitHub repository: https://github.com/MikoBie/reddit baumgartner2020pushshift
. Pushshift is a platform collecting social media data in real time. Then, we
selected only users who have published on both a single-gender subreddit
(r/Mommit or r/Daddit) and a mixed-gender subreddit (r/Parenting), on the
assumption that posting on ”mommit” or ”daddit” indicates identification as a
mother or father, in accordance with the subreddit descriptions. Before
further preprocessing, our dataset contains 8,361 unique Reddit users who
published 194,497 comments.

#### Preprocessing

We first remove from the dataset 502 authors who have published comments both
on r/Mommit and r/Daddit, as we cannot know whether they primarily identify as
“mother” or “father”. We remove comments written by the auto-moderator and
deleted comments. We tokenize the documents, create ngrams, and remove a list
of stop words, i.e., words occurring in abundance but providing little or no
information about the content of text documents, using Scikit-learn
pedregosa2011scikit  to which we add “ai”, “im”, “m”, “s”, “ve”, “w”, “d”,
“ive”, “id”, “itll”, “shes”, “hes”, “theyre”, “youre”, “dont”, “got”,
“havent”, “didn”. We finally lemmatize the text tokens and remove comments
that became empty because of the cleaning process. The dataset now includes
148,304 posts with 24,035 posts on r/Mommit, 16,388 posts on r/Daddit, 78,450
posts on r/Parenting by mothers, and 29,431 posts on r/Parenting by fathers.

Before running the word embedding algorithm, we added the user as a token in
the text. The position of the agent token was defined randomly, i.e., for each
comment, a random number is chosen within the range of its length (i.e.,
number of words) and the user token is added at that position. We believe this
position to be the closest to the natural behaviour of other tokens and
therefore to minimize the impact of adding a non-word token in the text.

### Word embeddings

#### Definition

A word embedding is a Natural Language Processing technique representing words
as vectors. Words which share common linguistic characteristics (e.g.,
semantics and syntax) are located close to each other spatially. In other
words, if two words having the same context (i.e., the same surrounding words)
are passed as inputs, the outputs for these two words should be similar.

To create the embedding, we use Word2Vec in the Gensim library rehurek_lrec .
Word2Vec is one of the algorithms which produces vectors for words in textual
documents. It can be used through two models: the Continuous Bag-Of-Words
model or the Skip-Gram model. While the Continuous Bag-Of-Words model predicts
a target word from its context (i.e, surrounding words), the Skip-Gram model
predicts the context from target words and is better for larger datasets.
Parameters include the choice of model (i.e., Continuous Bag-Of-Words or Skip-
Gram), subsampling (e.g., removing too frequent or rare words), dimensionality
(i.e., the size of the vector), and window size (i.e., number of tokens to be
considered as the context of the target word). To train the word embedding
algorithm, we used the Skip-Gram model, kept all the words (the stopwords
removal in the preprocessing stage had already removed the frequent and non-
content words), set the size of the vector to 200 dimensions, and a window of
15 tokens, as 77% of posts are less than 30-tokens long (a window of 15 means
the algorithm will look at the 15 words before and the 15 words after the
target word).

#### User embeddings and the concept of speaker-landscapes

User embeddings represent users as vectors in an embedding space. By spatially
projecting users, we can analyze the relationship between users, their
concentration or spread in the embedding, and group dynamics. User embeddings
can be used to detect sarcasm amir2016modelling , build recommendations
systems pal2020pinnersage ; yu-etal-2016-user , predict gender and depression
xiaodong2020author , etc. In the present study, we create a ‘speaker-
landscape’, a novel method developed by Schuld et al.
schuld_durrheim_mafunda_mazibuko_2022  where the username is added in the text
document as a token. As word embeddings create a vector representation for a
target word from its surrounding words, the target word vector represents its
context. When adding the user as a token in a document, the embedding
algorithm will locate the user spatially relying on the words used in the
document. To create a ‘speaker-landscape’
schuld_durrheim_mafunda_mazibuko_2022 , we add the username at a random
position in the Reddit post they published. This method allows to create a
vector representation of users from the words they use and, as the user is a
word token in the text, it can be directly compared to other word tokens (e.g.
how close r/Daddit fathers are to word such as ‘food’, ‘sleep’, or ‘health’?).

#### Visualizing the embedding

As the output of the embedding algorithm gives us 200-dimension vectors, we
need to reduce the vectors into two-dimension vectors to be able to visualise
them. For reducing vectors, we use UMAP
https://doi.org/10.48550/arxiv.1802.03426 . UMAP uses graph layout algorithms
to represent data as a high-dimensional graph from which it builds a low-
dimensional graph with a similar structure. Two important parameters,
n_neighbors and min_dist, are used to control the balance between local and
global structure. A low value of n_neighbors will make UMAP consider more the
local structure, focusing on the closest points to analyse the high-
dimensional data whereas a high value will represent more the bigger picture.
min_dist corresponds to the minimal distance between the points represented in
low-dimensional space, i.e., low values will make the embeddings appear
tightly packed together. It should be noted that, as UMAP has to warp the
high-dimensional space to project the data to lower dimensions, the axes or
distances in the low-dimensional representation are not directly
interpretable.

For visualising our results, we keep users who posted at least five comments
on each subreddit to have a more robust position of users in the word
embedding. This leaves us 2872 users to be visualized, 956 mothers and 480
fathers. Each user will be represented by two dots, each for an audience with
which they interact (either single-gender or mixed-gender).

Additionally, as users are tokens along with other word tokens in our corpus,
we can directly compare their position with the position of other words. To
have a meaningful set of word tokens, we use the results of topic modelling
from a previous study conducted on the same dataset sepahpour2022how .
sepahpour2022how  found twelve topics to best represent discussions in
r/Daddit, r/Mommit, and r/Parenting subreddits: Thank you/Appreciation (i.e.,
thanking other users for their advice), Medical care, Education/Family advice
(i.e., giving advice to other parents), Furniture/Design (i.e., discussing
furniture for children in the house), Birth/Pregnancy (i.e., announcements or
congratulations related to child arrival), Change/Potty training, Physical
appearance/Picture (i.e., commenting on the physical appearance of children
and sharing pictures), Work/Raise children (i.e., discussing the balance
between work and family), Food, Leisure activities, School/Teaching, and Sleep
training. For each topic, we select six keywords among the topics’ ten most
associated keywords and add them to the visualisation.

#### Measures used to analyze the output

To measure the relationship between two tokens, we measure the similarity
between their vector representation. For this, we use cosine similarity (i.e.,
the dot product of the vectors divided by the product of their length), which
is defined as follows:

$\cos(\theta)={\frac{\mathbf{A}\cdot\mathbf{B}}{\|\mathbf{A}\|\|\mathbf{B}\|}}=\frac{\sum_{i=1}^{n}{\mathbf{A}_{i}\mathbf{B}_{i}}}{\sqrt{\sum_{i=1}^{n}{\mathbf{A}_{i}^{2}}}\sqrt{\sum_{i=1}^{n}{\mathbf{B}_{i}^{2}}}}$
(1)

The cosine values are between -1 and 1 and the greater the cosine value, the
more similar the two tokens are. To compare the similarity measures between
two groups of users or the same users but in two different contexts, we used
the Kolmogorov-Smirnov test 10.2307/2280095  and Student’s t-test
10.1093/biomet/6.1.1 . The Kolmogorov-Smirnov test for two samples gives the
probability that the two samples were drawn from the same distribution. The
Student’s t-test is used to compare the means of two samples.

Additionally, to measure the spread or concentration of a group and the
distance between two groups, we used the centroid of the set of vectors. The
centroid corresponds to the arithmetic mean position of a set of vectors.

## Acknowledgements

This publication has emanated from research conducted with the financial
support of Science Foundation Ireland under Grant number 18/CRT/6049 and the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 802421).

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## Additional Files

### Similarity between Reddit users

Table 1 shows the average and standard deviation values of the cosine
similarity between users in each group. The figure reproduces the pattern
found in Figure 2 with r/Mommit and r/Daddit users being more tightly gathered
in the landscape while r/Parenting users being less similar and therefore more
spread out in the landscape.

Table 2 shows the test statistics and the significance of Kolmogorov-Smirnov
tests conducted to compare the distribution of cosine similarity values
between groups. All differences were significant.

Groups | Average similarity between agents | Standard Deviation
---|---|---
Fathers on r/Daddit | 0.2213 | 0.0843
Mothers on r/Mommit | 0.2101 | 0.0818
Fathers on r/Parenting | 0.1631 | 0.0707
Mothers on r/Parenting | 0.1647 | 0.0699
Table 1: Average similarity with centroid and standard deviation for each group. Groups | Fathers on r/Daddit | Mothers on r/Mommit | Fathers on r/Parenting
---|---|---|---
Fathers on r/Daddit | - | - | -
Mothers on r/Mommit | 0.0535*** | - | -
Fathers on r/Parenting | 0.2941*** | 0.2426*** | -
Mothers on r/Parenting | 0.2853*** | 0.2342*** | 0.0136***
Table 2: Kolmogorov-Smirnov test statistics. P-values less than 0.05 are
summarized with one asterisk (*) and P-values less than 0.001 are summarized
with three asterisks (***). No asterisk means that no significant difference
was found.

### Similarity between two versions of the same user

Figure 5: Distribution of similarity between two versions of the same user for
mothers and fathers The higher the cosine similarity, the less users change
behaviours between subreddits.

Figure 5 shows the distributions of cosine similarity values between two
versions of the same user, i.e., the cosine similarity between the same user
interacting on r/Mommit or r/Daddit (not both) and interacting on r/Parenting.
Users assumed to be fathers (i.e., posting on r/Daddit and r/Parenting) are
shown on the left and users assumed to be mothers (i.e., posting on r/Mommit
and r/Parenting) on the right. Although the Kolmogorov-Smirnov test and the
t-test did not show any significant difference between the two groups, it is
interesting how the distribution seem to qualitatively differ. Among fathers,
there seem to be two clusters of behaviour, a group of fathers who change more
from one subreddit to the other and another group who focuses on similar
topics. Among mothers, there seem less variation with most of the mothers
moderately changing their topics of discussion across subreddits.