Extreme Treatment Effect: Extrapolating Dose-Response Function into Extreme Treatment Domain

Abstract

The potential outcomes framework serves as a fundamental tool for quantifying causal effects. The average dose–response function $\mu \left(t\right)$ (also called the effect curve) is typically of interest when dealing with a continuous treatment variable (exposure). The focus of this work is to determine the impact of an extreme level of treatment, potentially beyond the range of observed values—that is, estimating $\mu \left(t\right)$ for very large t. Our approach is grounded in the field of statistics known as extreme value theory. We outline key assumptions for the identifiability of the extreme treatment effect. Additionally, we present a novel and consistent estimation procedure that can potentially reduce the dimension of the confounders to at most 3. This is a significant result since typically, the estimation of $\mu \left(t\right)$ is very challenging due to high-dimensional confounders. In practical applications, our framework proves valuable when assessing the effects of scenarios such as drug overdoses, extreme river discharges, or extremely high temperatures on a variable of interest.

1. Introduction

Quantifying causal effects is a fundamental problem in many diverse fields of research [1,2,3,4]. Some prevalent examples include the impact of smoking on developing cancer [5], the influence of education on increased wages [6], the effects of various meteorological factors on precipitation [7] or the effect of policy design on various economy factors [8].

The potential outcomes framework [9] has been the fundamental language to express the notion of the causal effect. The crux of this framework lies in acknowledging that, in any given scenario, multiple potential outcomes exist based on different interventions or exposures [10]. This perspective challenges researchers to consider not only the observed outcome but also the unobserved outcomes that could materialize under alternative conditions. The typical focus in causal inference is on the binary treatment variable (exposure). However, binary treatment is unable to differentiate between different levels of the treatment variable. This issue can be partially solved by assuming a continuous treatment. For example, Refs. [11,12] proposed an estimator based on local linear smoothing. Ref. [13] discussed the combination of parametric and non-parametric models for the effect curve estimation. Refs. [14,15] utilized marginal structural causal models framework. Refs. [16,17] applied neural networks for the effect estimation. However, typical methods that work with a continuous treatment variable are not well suited for the inference that goes beyond the observed range of the data.

In this paper, we are interested in the extreme treatment effect, that is, the quantity of interest is the effect of an extreme level of treatment, outside of the observed range. Consider the following example from medicine: Assume that the data of a study (either randomized or observational) are available to us, with the health status (Y) of patients and the corresponding dose of a medicine administrated (T). The data available only depict Y when $T\le 20$ mg. What if then, we would like to know the change in Y when the dose is increased to $T=25$ mg? Answering this inquiry is hard, since we have zero data to answer it (this might be considered unethical to give such a dose to a patient), and we must rely on strong unverifiable assumptions and extrapolation. Additionally, in the case of observational studies, high-dimensional confounders pose yet another significant challenge.

The connection between causal inference and extreme value theory has been receiving increasing interest. Refs. [18,19] analyze the Extreme Quantile Treatment Effect (EQTE) of a binary treatment on a continuous, heavy-tailed outcome. The paper authored by [20] develops a method to estimate the EQTE and the Extreme Average Treatment Effect (EATE) for continuous treatment. Ref. [21] developed a framework for Granger-type causality in extremes. Some other approaches for causal discovery using the extreme values include [22,23,24,25]. Ref. [26] propose graphical models in the context of extremes. Ref. [27] analyzed the the effect of climate change on weather extremes. Ref. [28] proposed a framework for extreme event propagation. Ref. [29] study probabilities of necessary and sufficient causation as defined in the counterfactual theory using the multivariate generalized Pareto distributions. We contribute to this growing literature and provide a theoretically well-founded approach for estimation and inference of the extremal treatment effect.

Recent advancements in machine learning research have spotlighted the extrapolation capabilities of different models [30,31,32,33]. For instance, ‘engression’, as proposed by [34], presents a framework that serves as an extrapolating alternative to regression-based neural networks. Similarly, Ref. [35] explored extrapolation of conditional expectations by assuming that the maximum derivative occurs within the observed range of support, and while these approaches are not inherently causal, they can be construed as such under certain assumptions. Despite achieving cutting-edge performance, these methods encounter two primary limitations: difficulty handling multiple confounders and reliance on often uninterpretable extrapolation assumptions. In contrast, our framework focuses on the causal aspect of the extrapolation and can handle a large number of confounders. This is achieved under weak assumptions commonly embraced in the extreme value theory. Moreover, our framework relies on strong yet more interpretable extrapolation assumptions; while our primary focus is on linear regression, our approach has the flexibility to integrate with various machine learning methodologies, potentially improving the overall performance. However, this integration may come at the expense of losing interpretability for certain assumptions.

As for the application in this work, we consider a dataset describing extreme precipitation and river discharge levels in Switzerland. A historical record indicates a maximum precipitation level near Zurich’s meteo-station on 6 June 2002, reaching an extreme of 111 $\frac{\mathrm{m}\mathrm{m}}{{\mathrm{m}}^2}$. This event coincided with the river Reuss (near Zurich) nearly breaching its banks, causing damage to adjacent settlements. We focus on the following question: how would the river discharge alter if the precipitation on that day were to reach 120 $\frac{\mathrm{m}\mathrm{m}}{{\mathrm{m}}^2}$? Would the river breach its banks under such circumstances? We anticipate that the effect of precipitation on river discharge may vary between the body of the distribution and its tail. This anticipation stems from several factors: During periods of light to moderate precipitation, the ground absorbs a significant portion of the rainfall, reducing its contribution to the river flow. In contrast, during severe rainstorms, a larger proportion of the precipitation directly contributes to the river flow, potentially resulting in a more pronounced impact on discharge levels. Therefore, we expect to observe differing, potentially more severe, impacts of extreme precipitation on discharge levels compared to moderate events, highlighting the importance of understanding such dynamics across varying levels of precipitation intensity.

The structure of this paper is as follows: we introduce the notation and preliminaries on causal inference and extreme value theory in Section 2. In Section 3, we present the main assumptions along with some simple theoretical implications. In Section 4, we introduce a practical statistical methodology for estimating an extreme treatment effect from the data. Section 5 explains our methodology using a simple simulated example and discusses simulation results. In Section 6, we explore the application of inferring the effect of extreme precipitation on river discharge levels.

This manuscript includes five appendices: Appendix A introduces a second real-world application regarding the compressive strength of concrete, which has been relocated to the appendix for the sake of brevity. Appendix B contains a detailed simulation study, exploring the methodology under various conditions, including (1) a varying dimension $dim\left(\mathbf{X}\right)$, (2) comparison with classical methods from the literature, (3) a hidden confounder, (4) different dependence structures, and (5) varying dose–response functions. Appendix C contains a detailed inference process for the river application described in Section 6. Appendix D provides a more detailed explanation of the bootstrap algorithm used in the inference process and presents the theory behind the consistency result. Finally, proofs can be found in Appendix E.

2. Problem Statement, Notation and Preliminaries

Following [36], we define dose–response functions in the potential outcomes framework. We consider the triplet of $(\mathbf{X},T,Y)$, where $\mathbf{X}\in \mathcal{X}\subseteq {\mathbb{R}}^d$, $T\in \mathcal{T}\subseteq \mathbb{R}$, and $Y\in \mathcal{Y}\subseteq \mathbb{R}$ denote the confounders, treatment, and response variable, respectively, in an observational causal study. We assume a continuous treatment setting, where $\mathcal{T}=({\tau }_L,{\tau }_R)$ for some ${\tau }_L,{\tau }_R\in \overline{\mathbb{R}}:=\mathbb{R}\cup \{-\infty \}\cup \{\infty \}$. Here, ${\tau }_R\in \mathbb{R}\cup \{\infty \}$ is the right endpoint of the support of T. For simplicity, assume that the right endpoint of the support of $T\mid \mathbf{X}=\mathbf{x}$ is equal to ${\tau }_R$ for all $\mathbf{x}\in \mathcal{X}$. Let $Y\left(t\right)$ be a set of potential outcomes corresponding to the hypothetical world in which $T=t$ is set deterministically. The fundamental problem of causal inference arise, since in the real world, each individual can only receive one treatment level T, and we only observe the corresponding outcome $Y=Y\left(T\right)$.

We observe a random sample ${\{{\mathbf{X}}_i,T_i,Y_i\}}_{i=1}^n$ of size $n\in \mathbb{N}$. It follows from our setting that, given the observed covariates, the distribution of the potential outcome for one unit is assumed to be unaffected by the particular treatment assignment of another unit (Stable Unit Treatment Value Assumption). We utilize the letter H for a (possible) hidden confounder. We denote vectors by bold letters. For any pair of continuous random variables Z and $Z^{\prime }$, we denote its probability distribution function $P_Z(·)$, density function $p_Z(·)$ and a conditional density $p_{Z\mid Z^{\prime }}(·\mid ·)$.

The average dose–response function and patient-specific dose–response function are defined as

$$\mu \left(t\right)=\mathbb{E}\left[Y\left(t\right)\right],{\mu }_{\mathbf{x}}\left(t\right)=\mathbb{E}[Y\left(t\right)\mid \mathbf{X}=\mathbf{x}],$$

respectively. Although the term “dose” is typically associated with the medical domain, we adopt here the term dose–response learning in its more general setup: estimating the causal effect of a treatment on an outcome across different (continuous) levels of treatment. Our objective is to learn the behavior of $\mu \left(t\right)$ or ${\mu }_{\mathbf{x}}\left(t\right)$ for $t\approx {\tau }_R$.

For a pair of real functions $f_1,f_2$, we employ the following notation: $f_1\left(t\right)\sim f_2\left(t\right)$ for $t\to {\tau }_R$, if ${lim}_{t\to {\tau }_R}\frac{f_1\left(t\right)}{f_2\left(t\right)}=1$. The sequence of random variables approaches the same distribution as the sample size grows. In the remainder of the paper, we assume that $\mu \left(t\right),{\mu }_{\mathbf{x}}\left(t\right)$ are continuous on some neighborhood of ${\tau }_R$ for all $\mathbf{x}$.

2.1. Classical Assumptions

Two classical assumptions in the literature for identifying the average dose–response function are as follows:

• Unconfoundedness: Given the observed covariates, the distribution of treatment is independent of the potential outcome. Formally, we have $T\perp \perp Y\left(t\right)|\mathbf{X},\forall t\in \mathcal{T}$, where $\perp \perp$ denotes the independence of random variables.
• Positivity: $p_{T\mid \mathbf{X}}(t\mid \mathbf{x})>0$ for all $t,$$\mathbf{x}$, where $p_{T\mid \mathbf{X}}$ represents the conditional density function of the treatment given the covariates.

Under these assumptions, [36] showed the identifiability of the dose–response function via

$${\mu }_{\mathbf{x}}\left(t\right)=\mathbb{E}\left[Y\right|T=t,\mathbf{X}=\mathbf{x}],\mathrm{and}\mu \left(t\right)=\mathbb{E}\left[{\mu }_{\mathbf{X}}\left(t\right)\right]=\mathbb{E}\left[\mathbb{E}\left[Y\right|T=t,\mathbf{X}=\mathbf{x}]\right],$$

(1)

where the inner expectation is taken over Y, and the outer expectation is taken over $\mathbf{X}$.

Even if we are not willing to rely on the unconfoundedness assumption, it may often still be of interest to estimate the function $t\to \mathbb{E}\left[\mathbb{E}\right[Y|T=t,\mathbf{X}=\mathbf{x}]]$ as an adjusted measure of association, defined purely in terms of observed data. It may be interpreted as the average value of Y in a population with exposure fixed at $T=t$, but it is otherwise characteristic of the study population with respect to $\mathbf{X}$ [11,12,37].

When the positivity assumption is violated, a different type of extrapolation arises [38], which is different from the one considered in this work. This scenario occurs when the distributional support of variable T varies across different levels of confounding variables $\mathbf{X}$. Various approaches have been devised to confront this challenge, including propensity thresholding [39].

Several algorithms were proposed to estimate the function $\mu \left(t\right)$ in the body of the distribution of T. State-of-the-art methods estimate $\mu \left(t\right)$ via ${\sum }_{i:T_i\approx t}{w}_i{Y}_i$ for appropriate weights $w_i$ what serve to “erase” the confounding effect of $\mathbf{X}$ [11,40,41,42,43]. Typically, the estimation of $\mu \left(t\right)$ involves a two-step procedure [5,36,44]. In the first step, we model the distribution $T\mid \mathbf{X}$, also known as the propensity. In the second step, we model the distribution of $Y\mid T$, suitably adjusted by the propensity, with the aim of mitigating the confounding effect of $\mathbf{X}$.

Ref. [36] introduced a generalized propensity score (GPS) defined as $e(t,\mathbf{x}):=p_{T\mid \mathbf{X}}(t\mid \mathbf{x})$. One common approach is to model $p_{T\mid \mathbf{X}}$ using a Gaussian model. In binary treatment cases (when $\mathcal{T}=\{0,1\}$), the propensity score is a probability denoted as $e(1,\mathbf{X})=P(T=1\mid \mathbf{X})$ and is typically modeled using logistic regression. Subsequently, we define weights $w_i$ as $w_i:=\frac{1}{\widehat{e}(T_i,{\mathbf{X}}_i)}$ or stabilized weights $w_i:=\frac{{\widehat{p}}_T\left(T_i\right)}{e(T_i,{\mathbf{X}}_i)}$, where we additionally model and estimate the marginal distribution of T, denoted as ${\widehat{p}}_T$.

In a similar vein, Ref. [5] introduced the concept of a “uniquely parameterized propensity function assumption,” which states that for every value of $\mathbf{X}$, there exists a unique finite-dimensional parameter $\theta \in \Theta$ such that $e(·\mid \mathbf{X})$ depends on $\mathbf{X}$ only through $\theta \left(\mathbf{X}\right)$. Since $\theta \left(\mathbf{X}\right)$ contains all information about the confounding, we only model $\mathbb{E}[Y\mid T=t,\theta (\mathbf{X})=\mathbf{s}]$ instead of $\mathbb{E}[Y\mid T=t,\mathbf{X}=\mathbf{x}]$ in Equation (1). In a vast majority of applications, $\theta \left(\mathbf{X}\right)$ corresponds to the parameters of a normal distribution; to the best of our knowledge, there has been no exploration of the extreme value distribution in this particular context.

2.2. Extreme Value Theory

When dealing with extreme values, it is easy to introduce a strong selection bias. A naive approach for estimating $\mu \left(t\right)$ for large values of t might involve only considering observations where t exceeds a certain threshold, denoted as $\tau$, and computing $\mu \left(t\right)$ using conventional techniques, while disregarding all values below this threshold. This is a typical approach of many classical algorithms, which estimate $\mu \left(t\right)$ by focusing solely on a local neighborhood of observations around t. However, this approach can introduce a significant selection bias. In its extreme manifestation, all observations where t exceeds $\tau$ might exclusively pertain to men, for instance. The selection bias arises if the effect of T on Y differs between men and women (see Figure 2 in Section 5.1 with $\tau =3$). We employ the Extreme Value Theory technique known as peaks-over-threshold to tackle this issue.

Extreme value theory is a sub-field of statistics that explores techniques for extrapolating the behavior (distribution) of T beyond the observed values. A limiting theory posits that the tail of T can be well approximated by the Generalized Pareto Distribution (GPD), as detailed in the following explanation.

Consider a sequence ${\left(T_i\right)}_{i\ge 1}$ of independent and identically distributed (iid) random variables with a common distribution F, and $M_n={max}_{i=1,\dots ,n}{T}_i$ represents the running maximum. It is well known [45] that if there exists a non-degenerate distribution G such that $\frac{M_n-b_n}{a_n}\stackrel{D}{\to }G$ as $n\to \infty$ for some sequences of constants ${\{a_n,b_n\}}_{n=1}^{\infty }\in {\mathbb{R}}_{+}^{\mathbb{N}}\times {\mathbb{R}}^{\mathbb{N}}$, then G falls within the Generalized Extreme Value (GEV) distribution family. This can equivalently be expressed using the following definition:

Definition 1

([46]). The distribution F is in the max domain of attraction of a generalized extreme value distribution (notation $F\in MDA\left(\gamma \right)$) if there exist $\gamma \in \mathbb{R}$ and sequences of constants $a_n>0,b_n\in \mathbb{R},n=1,2,\dots$ such that ${lim}_{n\to \infty }{F}^n(a_{n}x+b_n)=exp(-{(1+\gamma x)}^{-1/\gamma })$ for all x satisfying $1+\gamma x>0$. In case $\gamma =0$, the right side is interpreted as $exp(-e^{-x})$. The parameter γ is called the extreme value index (or shape index).

This condition is mild as it is satisfied for most standard distributions, for example, the normal, Student-t and beta distributions. The following crucial theorem states that the tail of T can be well approximated by GPD if the distribution of T belongs to $MDA\left(\gamma \right)$.

Theorem 1

(Theorem 4.1 in [47]). Let $T\sim F\in MDA\left(\gamma \right)$. Then, for large $\tau \approx {\tau }_R$, there exist $\sigma >0,\gamma \in \mathbb{R}$ such that the distribution of $T-\tau \mid T>\tau$ is approximately $GPD(0,\sigma ,\gamma )$.

GPD distribution has three parameters, namely a location $\tau \in \mathbb{R}$, scale $\sigma >0$ and a shape $\gamma \in \mathbb{R}$. Its distribution function takes the following form:

$$H\left(x\right)=\left\{\begin{array}{cc}\hfill & 1-{\left(1+\gamma \frac{x-\tau }{\sigma }\right)}^{-1/\gamma },\gamma \ne 0,\hfill \\ \hfill & 1-exp\left(-\frac{x-\tau }{\sigma }\right),\gamma =0,\hfill \end{array}\right.$$

defined on the support $[\tau -\sigma /\gamma ,\infty ),(-\infty ,\infty ),(-\infty ,\tau -\sigma /\gamma ]$ for cases $\gamma <0,\gamma =0,\gamma >0$, respectively. Cases when $\gamma >0,\gamma =0$, and $\gamma <0$ correspond to the Fréchet, Gumbel, and Weibull distributions, respectively, [45,48].

Note that when the distribution of $T-\tau$ given $T>\tau$ follows a GPD with parameters 0, $\sigma$, and $\gamma$, an equivalent assertion can be made that T given $T>\tau$ follows a GPD with parameters $\tau$, $\sigma$, and $\gamma$. We denote $\theta ={(\tau ,\sigma ,\gamma )}^{\top }$.

Assumption 1.

We assume that the distributions T and $T\mid \mathit{X}$ are in the max domain of attraction of a generalized extreme value distribution.

3. Our Tail Framework

We aim to model the effect of a treatment variable T in the context of extreme values of T. However, it is essential to approach the term ‘extreme’ with caution, considering the discrepancy between real-world implications and the interpretations within our model. Take, for instance, if T represents a drug dose in milligrams; our model operates under the assumption that T tends toward ${\tau }_R$, which can be potentially larger than several kilograms. While this mathematical abstraction lacks practical significance—given that administering several kilograms of a drug is physically implausible—the model does include values of T that are arbitrarily large. Of course, we do not claim that our model performs well when T equals several kilograms but only that it performs well for T in the ‘reasonable neighborhood’ of the observed values.

3.1. Assumptions

We are not aiming to estimate the complete $\mu \left(t\right)$ but rather only its values for large t. Therefore, we can relax the classical assumptions for the identification of $\mu \left(t\right)$; what we specifically need is their tail counterparts.

Assumption 2

(Unconfoundedness in tail). For all $\mathit{x}\in \mathcal{X}$, it holds that

$$\mathbb{E}[Y\left(t\right)\mid \mathit{X}=\mathit{x}]\sim \mathbb{E}[Y\mid \mathit{X}=\mathit{x},T=t]\mathit{as}t\to {\tau }_R.\left(\mathrm{Unconfoundedness}\mathrm{in}\mathrm{tail}\right)$$

We always assume the existence of the expected values.

Rather than simply writing $t\to {\tau }_R$, we frequently opt for the notation $t\left(\mathbf{x}\right)\to {\tau }_R$ to emphasize its dependence on the random variable $\mathbf{X}$. Note that Assumption 2 is strictly less restrictive than the Unconfoundedness assumption introduced in Section 2.1.

Remark 1.

To provide some intuition regarding the permissiveness of Assumption 2, we rephrase our framework in the language of structural causal models (SCM, [49]). Assume that the data-generating process of the output Y is as follows:

$$Y=f_Y(T,\mathit{X},H,\epsilon ),\epsilon \perp \perp (T,\mathit{X},H).$$

Here, H represents a (possible) latent confounder of T and Y. Then, the dose–response function has a form $\mu \left(t\right)=\mathbb{E}\left[f_Y(t,\mathit{X},H,\epsilon )\right]$ where the expectation is taken with respect to ${(\mathit{X},H,\epsilon )}^{\top }$.

Assumption 2 can be rephrased as follows: there exists a function ${\tilde{f}}_Y$ such that

$$f_Y(t,\mathit{x},h,e)\sim {\tilde{f}}_Y(t,\mathit{x},e)\mathit{as}t\to {\tau }_R,(\mathrm{Unconfoundedness}\mathrm{in}\mathrm{tail}\mathrm{in}\mathrm{SCM})$$

for all admissible values of $\mathit{x},h,e$. This assumption is valid for example in additive models; that is, when $f_Y(t,\mathit{x},h,e)={\tilde{f}}_Y(t,\mathit{x},e)+g\left(h\right)$ for some functions $\tilde{f},g$.

Additionally, we restate the positivity assumption in the context of its tail counterpart.

Assumption 3

(Positivity in tail). $p_{T\mid \mathit{X}}(t\mid \mathit{x})>0$ for all $\mathit{x}$ and all $t>t_0$ for some $t_0\in \mathcal{T}$, where $p_{T\mid \mathit{X}}$ represents the conditional density function of the treatment given the covariates.

Note that this assumption is weaker than Assumption 1.

3.2. Adjusting Only for $\theta \left(\mathit{X}\right)$

The following lemma serves as a tail counterpart of an identifiability for the classical framework. It states that, under Assumptions 2 and 3, the tail of the dose–response function is identifiable from the observational distribution via the propensity function ${\pi }_0(t,\mathbf{x})$.

Lemma 1

(Identifiability). Under Assumptions 2 and 3, it holds that

$$\mu \left(t\right)\sim \mathbb{E}\{{\pi }_0(T,\mathit{X})Y\mid T=t\},\mathit{as}t\to {\tau }_R,$$

(2)

where ${\pi }_0(t,x):=\frac{p_T\left(t\right)}{p_{T\mid \mathit{X}}(t\mid \mathit{x})}$ is the (stabilized) propensity function.

Recall that the distribution of $T\mid \mathbf{X}=\mathbf{x}$, conditioned on $T>\tau \left(\mathbf{x}\right)$ for large $\tau \left(\mathbf{x}\right)\approx {\tau }_R$, is approximately GPD with parameters $\theta \left(\mathbf{x}\right)=\left(\tau \right(\mathbf{x}),\sigma (\mathbf{x}),\gamma (\mathbf{x}\left)\right)$. The following result suggests that instead of conditioning on (potentially high-dimensional) covariates $\mathbf{X}$, we only need to condition on $\theta \left(\mathbf{X}\right)$.

Lemma 2.

Under Assumptions 1 and 2, for all $\mathit{s}$ in the support of $\theta \left(\mathit{X}\right)$ holds

$$\mathbb{E}[Y\left(t\right)\mid \theta \left(\mathit{X}\right)=\mathit{s}]\sim \mathbb{E}[Y\mid T=t,\theta \left(\mathit{X}\right)=\mathit{s}]\mathit{for}t\to {\tau }_R.$$

(3)

Hence,

$$\mu \left(t\right)\sim \int \mathbb{E}[Y\mid T=t,\theta \left(\mathit{x}\right)]p_{\theta \left(\mathit{X}\right)}\left(\mathit{x}\right)d\mathit{x}\mathit{for}t\to {\tau }_R.$$

Lemma 2 suggests that it is sufficient to condition only on $\theta \left(\mathbf{X}\right)$ rather than on $\mathbf{X}$. This finding is pivotal for dimension reduction, effectively reducing the dimension from $dim\left(\mathbf{X}\right)$ to at most 3. Nonetheless, this is merely a limiting result, and it introduces an approximation error into the GPD approximation for finite samples.

3.3. Model for the Conditional Expectation of Y Given a T

Under Assumption 2, modeling ${\mu }_{\mathbf{X}}$ reduces to a statistical modeling of $\mathbb{E}[Y\mid T,\mathbf{X}]$. Furthermore, under Assumptions 1 and 2, it reduces to modeling $\mathbb{E}[Y\mid T,\theta (\mathbf{X}\left)\right]$. In principle, a wide range of models can be considered, ranging from simple linear models to non-parametric neural networks. The principle of Occam’s razor suggests that, especially when extrapolating beyond the range of observed values, simpler models often prove to be the most effective choices [50]. The extrapolation capabilities of various models have recently garnered attention in machine learning research. Ref. [34] introduced the ‘engression’ framework as an extrapolating counterpart to regression-based neural networks, and while we build our framework under a linear model for simplicity, it is possible to utilize different models, such as the engression-based ones. Figure 1 illustrates the extrapolating properties of various commonly used models.

A straightforward approach to model $\mathbb{E}[Y\mid T=t,\mathbf{X}=\mathbf{x}]$ under the assumption of linearity-in-the-tail would be assuming an existence of functions $\tilde{\alpha },\tilde{\beta }$ such that

$$\mathbb{E}[Y\mid T=t,\mathbf{X}=\mathbf{x}]\sim \tilde{\alpha }\left(\mathbf{x}\right)+\tilde{\beta }\left(\mathbf{x}\right)t,\mathrm{as}t\to {\tau }_R.$$

(4)

Following the notation in Remark 1, this corresponds to assuming

$$f_Y(t,\mathbf{x},h,e)\sim \tilde{\alpha }\left(\mathbf{x}\right)+\tilde{\beta }\left(\mathbf{x}\right)t\mathrm{as}t\to {\tau }_R,$$

for all admissible values of $h,e$. This assumption is valid for example in additive models where $f_Y(t,\mathbf{x},h,e)=\tilde{\alpha }\left(\mathbf{x}\right)+\tilde{\beta }\left(\mathbf{x}\right)t+g(\mathbf{x},h,e)$ for some function g. However, using the result from Lemma 2, it is sufficient to condition only on $\theta \left(\mathbf{X}\right)$ instead of potentially high-dimensional $\mathbf{X}$. Therefore, we introduce the following model assumption:

Assumption 4

(Conditional linearity of tail). There exist functions α and β such that for all $\mathit{s}$ in the support of $\theta \left(\mathit{X}\right)$, the following holds:

$$\mathbb{E}[Y\mid T=t,\theta \left(\mathit{X}\right)=\mathit{s}]\sim \alpha \left(\mathit{s}\right)+\beta \left(\mathit{s}\right)t,\mathit{as}t\to {\tau }_R.$$

(5)

Such an assumption was explored in various contexts (typically where $\theta \left(\mathbf{X}\right)$ represents parameters of a normal distribution, see [5] or Section 2.2.1 in [44]; to the best of our knowledge, the extreme case was not yet explored). We can construct our inference method by estimating $\alpha$ and $\beta$ using various machine-learning methodologies. This is discussed in Section 4.

4. Inference and Estimation

Let ${({\mathbf{x}}_i,t_i,y_i)}_{i=1}^n$ be the observed data. In the following, we propose a methodology for the estimation of $\mu \left(t\right)$ for $t\approx {\tau }_R$ under Assumptions 1, 2, and 4.

Consider the following two-step procedure. In the first step, we approximate the tail of $T\mid \mathbf{X}$ using GPD (that is, we estimate the location, scale, and shape parameters $\theta \left(\mathbf{X}\right)=\left(\tau \right(\mathbf{X}),\sigma (\mathbf{X}),\xi (\mathbf{X}\left)\right)$). In the second step, we estimate the expectation of Y given a large T conditional on the estimated GPD parameters $\widehat{\theta }\left(\mathbf{X}\right)$.

• Estimate $\theta \left(\mathbf{x}\right)$:
  • Choose $q\in (0,1)$.
  • Estimate the covariant-dependent threshold $\tau \left(\mathbf{x}\right)$ using a quantile regression, that is, estimate q-quantile of $T\mid \mathbf{X}=\mathbf{x}$.
  • From now on, restrict our inference on the observations from $S:=\{i:t_i>\widehat{\tau }\left({\mathbf{x}}_i\right)\}$.
  • Estimate $\theta \left(\mathbf{x}\right)$ in the tail model, that is, estimate $(\sigma ,\xi )$ from the data points in S in the model, where

    $$T\mid T>\widehat{\tau }\left(\mathbf{x}\right),\mathbf{X}=\mathbf{x}\sim GPD(\widehat{\tau }\left(\mathbf{x}\right),\sigma \left(\mathbf{x}\right),\xi \left(\mathbf{x}\right)).$$
• Estimate $\mu \left(t\right)$ or ${\mu }_{{\mathbf{x}}^{★}}\left(t\right)$ using $\widehat{\theta }\left(\mathbf{x}\right)$:
  • Estimate $\alpha ,\beta$ in model (5) from the data points in S (that is, we only consider $t>\widehat{\tau }\left(\mathbf{x}\right)$).
  • Return $\widehat{\mu }\left(t\right):=\frac{1}{n}{\sum }_{i=1}^n\widehat{\alpha }\left[\widehat{\theta }\left({\mathbf{x}}_i\right)\right]+\widehat{\beta }\left[\widehat{\theta }\left({\mathbf{x}}_i\right)\right]t$ or ${\widehat{\mu }}_{{\mathbf{x}}^{★}}\left(t\right):=\widehat{\alpha }\left[\widehat{\theta }\left({\mathbf{x}}^{★}\right)\right]+\widehat{\beta }\left[\widehat{\theta }\left({\mathbf{x}}^{★}\right)\right]t$.

The first step is a very standard procedure in the extreme-value literature called ‘peak-over-threshold’ [47]; it is standard to assume a constant shape parameter $\gamma \left(\mathbf{x}\right)\equiv \gamma \in \mathbb{R}$ since in practice, it is untypical for the shape parameter to change with covariates [53,54]. For the estimation of $\tau \left(\mathbf{x}\right),\sigma \left(\mathbf{x}\right),\alpha \left(\mathbf{x}\right),\beta \left(\mathbf{x}\right)$, we use either linear parametrization (that is, $\tau \left(\mathbf{x}\right)={\tau }^{\top }\mathbf{x},\sigma \left(\mathbf{x}\right)={\sigma }^{\top }\mathbf{x},\alpha \left(\mathbf{s}\right)={\alpha }^{\top }\mathbf{s},\beta \left(\mathbf{s}\right)={\beta }^{\top }\mathbf{s}$ for some real coefficients $\tau ,\sigma ,\alpha ,\beta$ and their estimation is carried out via classical maximum likelihood) or non-parametric smooth estimation using splines (GAM, [51]), but any method can be used in practice. In case of a very small sample size, we can also assume a constant scale parameter $\sigma \left(\mathbf{x}\right)\equiv \sigma \in \mathbb{R}$ in order to reduce the dimension of the estimation.

The choice of q in the first step is a standard issue in extreme value theory [55,56,57]. For theoretical results, q should be growing with the sample size; that is, $q=q_n$ satisfying ${lim}_{n\to \infty }{q}_n=1$ and ${lim}_{n\to \infty }n(1-q_n)=\infty$. In practical terms, q should be set as high as possible while ensuring that a sufficient amount of data remains above the threshold to maintain good inferential properties. Classical choices include $q=0.9$, $q=0.95$, or $q=0.99$, depending on the size of our dataset.

We utilize the basic bootstrap technique (sometimes also called Efron’s percentile method, see Chapter 23 in [58]) to establish confidence intervals. This involves random sampling, with replacement, from our dataset to generate multiple bootstrap samples, each matching the size of our original dataset. For each bootstrap sample, we calculate the estimate of the statistic ${\widehat{\mu }}^{★}\left(t\right)$. Subsequently, we determine the $\alpha$-percentiles of the re-sampled statistics to derive the confidence intervals. Details can be found in Appendix D.1.

Remark 2.

One must be cautious when interpreting confidence intervals during extrapolation. Generally, estimation of $\mu \left(t\right)$ is subject to two primary sources of bias: (1) bias stemming from model misspecification, and (2) bias arising from estimation variance; while the former bias can be mitigated within the body of the distribution by comparing different models and employing cross-validation, AIC or BIC criteria to select the most suitable model, this approach becomes less reliable in the extremal region. Eliminating this bias necessitates observation of data within the region of interest. The latter bias stemming from estimation uncertainty can be addressed by computing confidence intervals (in our case, via bootstrapping). It is important to note that our bootstrap confidence intervals only account for the latter bias, and consequently, the first type of bias presents a greater challenge during extrapolation since it is, in principle, unquantifiable without additional data.

Theorem 2

(Idea: Precise statements can be found in Appendix D). Assuming that the conditions outlined in either Theorem A1 or Theorem A2 are met, our procedure is consistent. Furthermore, under the assumptions detailed in Theorem A3, the bootstrap confidence intervals are asymptotically consistent at a correct level.

In our procedure, we adopt a practice common in extreme value theory, where we concentrate solely on the extreme observations (set S) while discarding all non-extreme values. This approach stems from the rationale that extreme observations provide the most valuable insights into out-of-support behavior. Utilizing data within the body of the distribution may introduce bias, as these values may exhibit different behavioral patterns. Mathematically, this rationale can be expressed through an examination of the precision of the GPD approximation. This approximation shows high precision exclusively in extreme values while displaying bias and low precision for non-extreme values.

5. Illustration and Experiments

In this section, we assess the performance of our methodology using both a simple illustrative example and experimental data. A comprehensive simulation study is provided in Appendix B.

The quantity of interest in the application presented in Section 6 is the difference $\mu \left(t_1\right)-\mu \left(t_2\right)$ for $t_1,t_2<{\tau }_R$. Hence, in the simulations, we focus on estimating

$${\omega }_{\mathbf{x}}:=\underset{t\to \infty }{lim}[{\mu }_{\mathbf{x}}(t+1)-{\mu }_{\mathbf{x}}\left(t\right)]\mathrm{or}\omega :=\underset{t\to \infty }{lim}[\mu (t+1)-\mu \left(t\right)],$$

assuming ${\tau }_R=\infty$ and that the corresponding limits exist. Note that under the linear model (5), the limit exists and corresponds to the parameter ${\omega }_{\mathbf{x}}=\beta \left[\theta \left(\mathbf{x}\right)\right]$. Here, ${\omega }_{\mathbf{x}}$ can be regarded as the tail counterpart of a coefficient ${\beta }_{\mathbf{x}}$ in a linear model $Y={\alpha }_{\mathbf{x}}+{\beta }_{\mathbf{x}}T+\epsilon$, where ${\alpha }_{\mathbf{x}}$ and ${\beta }_{\mathbf{x}}$ are real coefficients, possibly dependent on $\mathbf{X}$.

5.1. Simple Example

The subsequent illustrative example outlines our methodology and the main ideas. Consider a single confounder $X=X_1\sim \mathrm{Bernoulli}\left(0.75\right)$ (where $X_1=1$ denotes men and $X_1=0$ denotes women, for instance). Define $T=X_1+{\epsilon }_T$, where ${\epsilon }_T\sim \mathcal{N}(0,1)$ (indicating that T generally tends to be larger for men than for women). Let

$$Y=\left\{\begin{array}{cc}\hfill & T+\epsilon ,\mathrm{when}{X}_1=1,T>1,\hfill \\ \hfill & 2T+\epsilon ,\mathrm{when}{X}_1=0,T>1,\hfill \\ \hfill & 2-T+\epsilon ,\mathrm{when}T\le 1,\hfill \end{array}\right.$$

where $\epsilon \sim \mathcal{N}(0,1)$.

Simple computation gives us $\mu \left(t\right)=0.75t+(1-0.75)2t=1.25t$ for any $t>1$, while $\mu \left(t\right)=-t+2$ for $t\le 1$. Consequently, our primary interest lies in estimating the slope

$$\omega =\mu (t+1)-\mu \left(t\right)=1.25\mathrm{for}t>1.$$

We generate data as specified with a sample size of $n=500$. Setting the threshold at $q=0.9$, we employ the methodology outlined in Section 4 to estimate $\omega$. This process is repeated 100 times, yielding a mean and $0.95$ quantile of

$$\widehat{\omega }=1.26\pm 0.39.$$

Additionally, we employ the bootstrap technique to calculate confidence intervals, and we obtain a confidence interval of the form $\omega \in (0.72,1.87)$ on average. We see from other simulations that these confidence intervals are slightly conservative for $n\le 1000$ but work well for larger sample sizes.

In Figure 2, we present one generated dataset with a sample size of $n=500$, showcasing various methodologies from the existing literature and their extrapolation efficacy. Notably, classical techniques often exhibit a tendency to underestimate $\mu \left(t\right)$ for t large, primarily due to the fact that only the ‘men’ category ($X=1$) received $T>2$.

We conclude with an important remark regarding the sample size: a substantial amount of valuable information is lost when we discard $90\%$ of the data by focusing solely on the data in the set S (data above the threshold $\tau \left(\mathbf{x}\right)$). This is the primary reason behind the considerably large confidence intervals and the heightened variability in our estimates. We encounter the inevitable bias-variance trade-off; the inclusion of more data introduces a potential bias, given that the behavior of $\mu \left(t\right)$ differs in the body and in the tail.

5.2. Simulations

We provide a comprehensive discussion of all simulations in detail in Appendix B. In our study, we devised several simulation setups to model diverse scenarios and explore them thoroughly. Specifically, we focused on the following five key scenarios:

• Investigating how our method scales with respect to the dimension of the confounders $d=dim\left(\mathbf{X}\right)$.
• Comparing our method with classical methods from the literature.
• Expanding upon the simple example introduced in Section 5.1, wherein we evaluated performance across various dependence structures (employing different copulas), sample sizes, and a spectrum of causal effects.
• Examining the presence of a hidden confounder affecting both T and Y.
• Focusing on variations in the function $\mu \left(t\right)$.

In this section, we present two key findings from our simulation study.

Table 1. Estimates of $\omega =-1$ with varying dimensions of the confounders $d=dim\left(X\right)$ and with different distributions of the noise of T. The sample size is $n=5000$. The full simulations setup can be found in Appendix B.1.

True $\mathit{\omega }=-1$	Gaussian ${\mathit{\epsilon }}_{\mathit{T}}$	Exponential ${\mathit{\epsilon }}_{\mathit{T}}$	Pareto ${\mathit{\epsilon }}_{\mathit{T}}$
$d=5$	$\widehat{\omega }=-1.0\pm 0.03$	$\widehat{\omega }=-1.0\pm 0.01$	$\widehat{\omega }=-1.0\pm 0.001$
$d=25$	$\widehat{\omega }=-0.96\pm 0.08$	$\widehat{\omega }=-0.99\pm 0.01$	$\widehat{\omega }=-1.0\pm 0.001$
$d=50$	$\widehat{\omega }=-0.75\pm 0.28$	$\widehat{\omega }=-0.97\pm 0.09$	$\widehat{\omega }=-0.99\pm 0.01$
$d=200$	$\widehat{\omega }=-0.44\pm 0.37$	$\widehat{\omega }=-0.53\pm 0.42$	$\widehat{\omega }=-0.91\pm 0.68$

illustrates how our methodology scales with varying dimensions of the confounders $d=dim\left(X\right)$. Additionally,

Table 2. Comparing the extrapolation performance of various models using the average Absolute Relative Error (ARE) across 100 simulations with varying dimension of the confounders $d=dim\left(\mathbf{X}\right)$. The interpretation of the values is as follows: if the true value of $\mu \left(\tilde{t}\right)$ is 1, an ARE of 0.17 indicates an approximate typical error of $|\widehat{\mu }\left(\tilde{t}\right)-\mu \left(\tilde{t}\right)|\approx 0.17$. Bold values correspond to the algorithm with the best performance.

	Our Method	Bia et al. [60]	Kennedy et al. [11]	HI with GAM [36]	IPTW [62]
$d=2$	0.18	0.68	0.64	0.42	3.89
$d=10$	0.48	0.81	0.65	0.67	5.69
$d=30$	0.79	0.92	could not handle	0.92	4.70

depicts the comparison of our method with other classical methods from the literature, where we estimated $\mu \left(\tilde{t}\right)$ for $\tilde{t}={max}_{i=1,\dots ,n}{t}_i+10$.

Table 1 suggests that the results are reasonably accurate as long as $d\le 25$. As discussed in the Appendix B, the reason for the bias observed in larger dimensions d is that Assumption 1 and Lemma 2 are only asymptotic results, and with higher dimensions d, we require more data for the asymptotic theory for $T\mid \mathbf{X}$ to take effect. It is well known that the convergence rate of the maxima of the Gaussian random sample to an extreme value distribution is very slow, whereas it is faster with Exponential or Pareto distribution [63].

Analysis of Table 2 reveals that our method achieves the smallest extrapolation error. Hirano and Imbens [36] method utilizing a GAM outcome model showed surprisingly reasonable performance, while the IPTW method [62] exhibits poor performance due to the quadratic nature of the extrapolating curve. Please note that our method assumes linearity in the tail. If this assumption is not met, our method may produce inferior results. In such instances, alternative regression techniques can be utilized in step 2 of our algorithm instead of linear regression. For example, neural networks, as demonstrated in [34], can be employed to address nonlinear behavior and enhance performance.

6. Application: River Discharge Dataset

Understanding the causal relationship between extreme precipitation and river discharge is crucial for effective water resource management. In this study, we examine how extreme precipitation events impact river discharge. By utilizing a comprehensive dataset spanning various hydro-logical conditions, our research seeks to provide insights into the critical nexus between extreme precipitation dynamics and extreme river discharge events. The data were collected by the Swiss Federal Office for the Environment (https://www.hydrodaten.admin.ch/ (accessed on 15 October 2023)) but were provided by the authors of [23,64], with some useful preliminary insights. We used precipitation data and other relevant measured variables from meteorological stations provided by Swiss Federal Office of Meteorology and Climatology, MeteoSwiss (https://gate.meteoswiss.ch/idaweb/login.do (accessed on 15 October 2023)).

We exclusively examine the discharge levels of the River Reuss, situated near Zurich in Switzerland (Figure 3). We selected this river due to the availability of excellent measurements of its discharge levels, complemented by well-documented weather conditions from nearby meteorological stations and diverse landscape. Our measurements include average daily discharges between January 1930 and December 2014 and daily precipitation in the nearby meteo-stations. To reduce any seasonal effects due to unobserved confounders, we only consider data during June, July and August, as the more extreme observations happen during this period when mountain rivers are less likely to be frozen.

We center our attention on addressing two distinct research questions: one characterized by a straightforward scenario where the ground truth is known, and another presenting a more intriguing challenge.

6.1. Known Ground Truth

We demonstrate our methodology using a straightforward example where the ground truth is known. Consider a pair of river stations, such as stations 2 and 1. Let T represent the water discharge at station 2, Y represent the water discharge at station 1, and $\mathbf{X}$ denote measurements taken at a nearby meteorological station (including precipitation, humidity, etc.; the full list of confounders can be found in Appendix C). Our objective is to investigate the impact of extreme discharge levels at station 2 on the water discharge observed at station 1. In mathematical terms, we seek to ascertain $\mu \left(t\right)$ or ${\mu }_{\mathbf{x}}\left(t\right)$ for large values of t. In this context, the ground truth is the following:

$$\mu (t_1+t_2)-\mu \left(t_2\right)={\mu }_{\mathbf{x}}(t_1+t_2)-{\mu }_{\mathbf{x}}\left(t_2\right)=t_1-t_2,\mathrm{for}\mathrm{all}{t}_1,t_2\ge 0\mathrm{and}\mathrm{all}\mathbf{x}\in \mathcal{X}.$$

This can also be explained in words as follows: if we pour $t_1$ liters of water into the river at station 2 (in causal terminology, we interpret this as an intervention $do(T=T+t_1)$), we expect the water discharge at station 1 (Y) to increase by exactly $t_1$. Hence, $\omega ={\omega }_{\mathbf{x}}=1$. As we will see below, our methodology consistently yields this expected outcome.

We follow the methodology introduced in Section 4 with $q=0.95$. Detailed steps, diagnostics and preliminary data analysis (just for a pair $2\to 1$) can be found in Appendix C. The resulting estimates can be found in

Table 3. Estimates $\widehat{\omega }$ between each pairs of the stations.

Truth: $\mathit{\omega }=1$	Stations $2\to 1$	Stations $3\to 2$	Stations $4\to 3$	Stations $5\to 3$
$\widehat{\omega }$	$1.03\pm 0.05$	$1.17\pm 0.24$	$1.21\pm 0.19$	$0.78\pm 0.41$

. The results are very similar with different choices of q (changing $\widehat{\omega }$ by not more than by $0.1$). We observe that our results align very well with the ground truth ($\omega =1$). However, there is a slight bias evident in the relationships between the pairs $5\to 3$ and $4\to 3$: this can be attributed to distinct geographical features. Notably, Lake Vierwaldstättersee lies between the pair 5 and 3, which diminishes the influence of 5 on 3. Additionally, a 3238 m Titlis mountain is situated between pair 4 and 3, amplifying the effect of 4 on 3 due to the melting glacier ice, acting as an unmeasured confounding factor. Our methodology relies on Assumptions 1–4, along with some continuity assumptions and the SUTVA assumption discussed in Section 2. Assumptions 1 and 3 are minor and are used frequently when dealing with these types of data. Assumption 2 is a common and challenging aspect of every causal inference methodology; while our assumption is weaker than the classical unconfoundness assumption (requiring no hidden confounder in the tail), complete rejection of the possibility of its violation is unattainable. However, we believe that the meteo-station between a pair of stations can capture the most significant confounders (with the exception of when the lake or mountains are present in between the river stations). Finally, Assumption 4 is a strong assumption that allows us to extrapolate observed values into the extremal region. However, this assumption (or at least some similar model assumptions) are necessary. In this case, the linear assumption is valid, since the underlying ground truth is known.

6.2. Effect of Precipitation on River Discharge

We employ our methodology to address a more complex inquiry where the ground truth is not known. Let us consider water discharge at station 3 (Y), and let T denote the precipitation measured in meteo-station M2. Our focus lies in understanding the impact of extreme precipitation events (T) on the water discharge (Y). As mentioned in the introduction, on 6 June 2002, we recorded a historical maximum precipitation level of $T_{\max }=111$$\frac{\mathrm{m}\mathrm{m}}{{\mathrm{m}}^2}$, coinciding with the scenario when the river nearly breached its banks. Our inquiry centers on the question: how would the river discharge Y alter if T were to reach 120 $\frac{\mathrm{m}\mathrm{m}}{{\mathrm{m}}^2}$? In mathematical terminology, we are interested in estimating $\mu \left(120\right)-\mu \left(111\right)$ or possibly ${\mu }_{{\mathbf{x}}^{★}}\left(120\right)-{\mu }_{{\mathbf{x}}^{★}}\left(111\right)$, where ${\mathbf{x}}^{★}$ are other covariates corresponding to that event. Addressing this question is challenging as we lack data within this extreme regime, necessitating reliance on extrapolation. This task is especially challenging, since we anticipate that the effect of precipitation on river discharge may vary between the body of the distribution and its tail, since the ground absorbs a significant portion of the rainfall during a light rain.

We follow the methodology introduced in Section 4. A straightforward approach would be to define T as precipitation and Y the water discharge on the same day, while choosing appropriate confounders $\mathbf{X}$ from some measurements at M2. Then, we can use the classical approach for estimating $\mu \left(t\right)$ in the body and our approach to estimate it in the tail. However, some problematic issues arise in this application:

• Time issue: $T_{monday}\to Y_{monday}$ but also $T_{monday}\to Y_{tuesday}$ since it takes time for the rain water to reach the river and rain tends to be more frequent around midnight. In fact, correlation (and extreme correlation coefficient as well, see Figure A11) is much higher for a pair ($T_{monday},Y_{tuesday}$) than for ($T_{monday},Y_{monday}$). The extreme storm on 6 June 2002 corresponded to extremely high river discharge on 7 June 2002 (where Y was about five times larger than on 6 June 2002). Hence, our interest lies in the effect $T_{monday}\to Y_{tuesday}$ (that is, we consider $t_i$ as precipitation on day i while $y_i$ is the discharge on day $i+1$). Additionally, the presence of time introduces an auto-correlation issue. This can be handled by taking for example weekly maxima or discarding consecutive observations within a certain time frame to reduce the auto-correlation effect. Alternatively, applying techniques like time series decomposition, differencing, or using autoregressive models can also mitigate the issue of auto-correlation in the data analysis process. We leave the data unchanged since the temporal dependence is primarily local, spanning only a few days, and does not introduce a substantial bias.
• Variable selection issue: choosing appropriate confounders $\mathbf{X}$ that act as confounders of Y and T. It is not clear which variables can be safely considered as confounders: if a variable X lie on a path $T\to X\to Y$, adjusting for X would lead to so-called path-canceling causal effect [65]. Here, we are interested in the so-called total causal effect, so we need to be cautious of which covariates to adjust for. However, not adjusting for a common cause leads to a bias. Moreover, there is often a feedback loop: $X_i\leftrightarrow Precipitation$ for $X_i$ for example humidity or temperature. However, some of the variables can be safely considered as common causes: for example, the temperature on Sunday (the day before measuring precipitation). There is a huge amount of literature for such a variable selection, and we do not aim to comment on this research area—we only provide a full list of chosen confounders in Appendix C.

We estimate two values: $\widehat{\omega }$ which is the tail quantity defined as the difference between $\mu (t+1)-\mu \left(t\right)$ for large t: in our case, how would Y change if it was raining by 1 $\frac{\mathrm{m}\mathrm{m}}{{\mathrm{m}}^2}$ more on 6 June 2002? Next, we also estimate $\widehat{\beta }=\mu (t+1)-\mu \left(t\right)$ corresponding to the body of the distribution (see Appendix C.2.2 for details on its computation). The resulting estimates can be found in

Table 4. Estimates $\widehat{\beta }$ and $\widehat{\omega }$ represent the estimation of the effect of T on Y in the body and in the tail, respectively. $\widehat{\beta }$ is computed using standard regression, while $\widehat{\omega }$ is the tail counterpart computed using steps introduced in Section 4. * Note that Station 5 is in different altitude and relatively far from meteo-station M2 with a lake in between; hence, there is a bias due to data collection problems.

Truth Unknown	Station 1	Station 2	Station 3	Station 4	Station 5
$\widehat{\beta }$	$2.4\pm 0.1$	$2.28\pm 0.1$	$1.44\pm 0.02$	$0.89\pm 0.02$	$0.38\pm 0.01$
$\widehat{\omega }$	$3.04\pm 0.95$	$2.61\pm 0.67$	$1.62\pm 0.35$	$0.99\pm 0.32$	$0.36\pm 0.13$ *

and visualization of the $\mu \left(t\right)$ can be found in Figure 4. We observe that the effect of T on Y is larger in the extreme region than in the body of the distribution by a factor of $\frac{3.04}{2.4}\approx 1.25$.

As for the answer to our question ‘how would the river discharge Y alter in station 3 if T were to reach 120 $\frac{\mathrm{m}\mathrm{m}}{{\mathrm{m}}^2}$ on 6 June 2002’, our results suggest that the water discharge would be larger by about $9\times 1.62=14.5$$\frac{{\mathrm{m}}^3}{\mathrm{s}}$ (note that median of Y is $11.2$, and the $95\%$ quantile of Y is $51.2$). Would this result in the river overflowing its banks? We cannot definitively say, as we lack the necessary data regarding the volume and contours of the river banks. Moreover, Y represents the daily average of the water discharge; while in order to answer this question, the daily maximum is a better suited variable for answering the question. Nonetheless, this advances us towards a more accurate understanding of the effects and impacts of extreme precipitation events and potentially enhancing statistical inference for hydroelectric power stations located along this river.

7. Conclusions and Future Work

Analyzing the impact of extreme levels of a treatment variable (exposure) is essential for comprehending its effects on diverse systems and populations. In this paper, we introduced a novel framework aimed at estimating the causal effect of extreme treatment values. Leveraging insights from extreme value theory, we enhanced the estimation of the extreme treatment effect. Our framework can handle a substantial number of confounders. Nonetheless, our methodology relies on extrapolation, presenting inherent challenges even in the absence of confounding variables, where the bias stemming from a model misspecification is impossible to quantify. Our framework holds promise for initial assessments of the impact of extreme environmental events, such as the effects of severe storms or droughts on economic damages. Future work may explore the application of our extreme value theory approach to address time-varying effects, a prevalent issue in environmental research.