Nonparametric expectile shortfall regression for functional data

1 Introduction

Risk analysis is a pivotal element for financial investors. Traditionally, value at risk (VaR) has been the standard function for modeling financial risk, a model officially endorsed by the Basel Committee in 1996 and 2006. Despite its popularity, VaR has notable weaknesses, particularly its insensitivity to outliers or extreme risks. To address these limitations, the Basel Committee in 2014 introduced the expected shortfall (ES) function [1], which controls the expected loss or gain beyond a given threshold, usually defined by VaR. In this study, we consider the ES function using the expectile model instead. The expectile model, introduced by Newey and Powell [2], is highly sensitive to outliers, enhancing its ability to fit financial risk better than the VaR function. This sensitivity has led to significant attention in risk analysis literature [3–6]. Additionally, the expectile function has been applied in heteroscedasticity analysis (e.g., [7,8]) and outlier detection [9]. Expectile regression has been explored using vectorial regressors (e.g., [10]) and more recently in the context of functional statistics [11–22]. The concept of shortfall risk was introduced by Artzner et al. [23]. Based on its coherency property, the ES function has gained popularity in the last decade. A comparative study between VaR and shortfall was performed by Artzner et al. [23], showing that VaR is inappropriate when profit-loss distributions are non-Gaussian. ES models can be estimated using parametric, semi-parametric, and nonparametric approaches (e.g., [24–26]). In this article, we focus on the nonparametric approach, building on the work of researchers like Scaillet [27], who constructed an estimator using the kernel method. Using similar techniques, Cai and Wang [28] established the asymptotic distribution of the constructed estimator, while [29] used the Bahadur representation to construct an estimator for the ES function. Regarding nonparametric functional data analysis (NFDA), the literature is limited. Only two works have addressed the nonparametric estimation of functional ES regression. The first contribution was developed by Ferraty and Quintela-del Río [30], who studied the functional version of the kernel estimator of ES regression under the mixing assumption. Using the same estimator, Ait-Hennani et al. [31] established almost complete consistency under quasi-associated dependency.

In this work, we aim to develop a new risk model that combines the advantages of both the expectile and ES models. Specifically, contrary to existing works, we define ES using the expectile threshold. This new risk model accumulates the advantages of both approaches. The expectile is coherent and elicitable and is very sensitive to the magnitude of the lower tail, unlike VaR, which is not influenced by this aspect. On the other hand, the ES model fulfills all conditions of spectral risk measures [32]. Thus, the ES model based on expectile substantially improves risk management in practice. A key challenge in ES estimation is the absence of an associated backtesting measure. Our contribution constructs a computational kernel estimator that reduces the computational time of this risk tool. Mathematically, we establish the asymptotic properties of the estimator, including the convergence rate in the almost complete mode for both pointwise and uniform versions. The practical implementation of this risk model is evaluated using empirical analysis. To our knowledge, no attempt has been made so far to estimate the functional ES regression based on expectile regression functions.

This article is organized as follows: we present our risk model and its estimator in Section 2. Section 3 introduces the necessary assumptions. The pointwise convergence of the constructed estimator is shown in Section 4, while uniform consistency is given in Section 5. Section 6 discusses the computational ability of the estimator over simulated and real data applications. Finally, the proofs of the auxiliary results are given in the Appendix.

2 Model and estimator

Consider n independent random pairs ( X 1 , Y 1 ) , … , ( X n , Y n ) residing in ℱ × R , each identically distributed as ( X , Y ) . Throughout this study, we posit that the functional space ℱ is a semi-metric space, endowed with the semi-metric d ( ⋅ , ⋅ ) . Furthermore, assume the existence of a regular version of the conditional distribution of Y given X . In probability theory, it remains an open problem whether a regular version of conditional probability exists in every topological setting. Nevertheless, such a version has been established in certain frameworks [33,34]. Generally, the validity of this assumption hinges on various topological properties, such as completeness, separability, and compactness. For the sake of generality, we therefore assume the existence of a regular version without imposing additional topological constraints.

Now, the conventional ES regression is defined for s ∈ ℱ as

where CVaR α denotes the conditional VaR at level 1 − α . In this article, we introduce an alternative ES-regression framework that employs the expectile instead of VaR α . Specifically, we define the ES-regression as

where 1 I A denotes the indicator function of the set A , and CExP p is the expectile regression defined by

Replacing CVaR α with CExP p enhances the regression’s sensitivity to extreme values, addressing the inherent limitation of VaR α in capturing tail risks. This improvement is particularly crucial in practical applications where catastrophic losses or extreme events are of primary concern.

Throughout this article, we assume that F ( ⋅ ) is a known measurable function and that a = a n is a positive sequence of real numbers converging to zero as n approaches infinity. Under these assumptions, the ES-regression estimator is given by

where CExP ^ p ( s ) is the kernel estimator of CExP p ( s ) , defined as the solution to

with

where

This formulation ensures that the ES-regression is both theoretically robust and practically sensitive to extreme outcomes, thereby enhancing its applicability in risk-sensitive analyses.

3 Pointwise convergence

Before stating the asymptotic properties of the estimator CEX p ^ , we need to introduce some notations and assumptions. First, we set by C s or C ′ some strictly positive generic constants, N s is a given neighborhood of s , and, for all t ∈ R , we define

To present our main findings, we now introduce and rely on the following hypotheses:

1. P ( X ∈ B ( s , r ) ) = φ ( s , r ) > 0 ,

   where

   B ( s , h ) = { x ′ ∈ ℱ : d ( x ′ , s ) < h } .

2. There exists δ > 0 such that for all

   ( t 1 , t 2 ) ∈ [ CExP p ( s ) − δ , CExP p ( s ) + δ ] , and for all ( s 1 , s 2 ) ∈ N x 2 ,

   we have

   ∣ ES ( t 1 , s 1 ) − ES ( t 2 , s 2 ) ∣ ≤ C x ( d b ( s 1 , s 2 ) b + ∣ t 1 − t 2 ∣ ) , b > 0 .

3. For all m > 2 ,

   E [ ∣ Y ∣ m ∣ X = s ] ≤ C ′ < ∞ , a.s.

4. The kernel function F ( ⋅ ) is supported on (0, 1) such that

   C 1 I [ 0,1 ] ( t ) < F ( t ) < C ′ 1 I [ 0,1 ] ( t ) .

5. The parameter a satisfies

   lim n → ∞ n φ ( s , a ) ln n = ∞ .

4 Uniform consistency

To enhance the mathematical support of ES ^ ( t , s ) , we derive its uniform almost complete convergence over a subset S ℱ in ℱ . To accomplish this, we denote some generic constants in R * + by C or C ′ .

This concept was introduced by Kolmogorov and Tikhomirov [49]. It represents a measure of the complexity of a set, in the sense that high entropy indicates that a significant amount of information is required to describe an element with an accuracy of ε . Therefore, the choice of the topological structure (in other words, the choice of the semi-metric) plays a crucial role when examining uniform asymptotic results over S ℱ . For more examples, refer to [50,51].

In our analysis, we will impose the following conditions:

1. For all x ∈ S ℱ , we have

   0 < C φ ( a ) ≤ P ( X ∈ B ( s , a ) ) ≤ C ′ φ ( a ) < ∞ .

2. There exists δ > 0 such that for all ( t 1 , t 2 ) ∈ R 2 and all ( x 1 , x 2 ) ∈ S ℱ 2 ,

   ∣ ES ( t 1 , x 1 ) − ES ( t 2 , x 2 ) ∣ ≤ C ( d ( x 1 , x 2 ) b + ∣ t 1 − t 2 ∣ ) , b > 0 .

3. For all m > 2 ,

   E [ ∣ Y ∣ m ∣ X ] ≤ C m < ∞ , a.s.

4. The kernel function F ( ⋅ ) is an a -Lipschitz continuous function on its support [0, 1] and is itself supported in (0, 1). Moreover,

   C 1 I [ 0,1 ] ( t ) < F ( t ) < C ′ 1 I [ 0,1 ] ( t ) .

5. There exists α > 0 such that

   lim n → ∞ ( n α a φ ( a ) ) = ∞ and lim n → ∞ ψ S ℱ ( n − α − 1 2 ) n φ ( a ) ,

   where ψ S ℱ is the Kolmogorov ε -entropy of S ℱ .

Assumptions (U1–U5) are the uniform versions of (P1–P5). These assumptions allow us to express the uniform convergence rate in terms of the Kolmogorov entropy of the subset S ℱ . This consideration is important in the functional context, where compactness is not standard. It is well known that Kolmogorov entropy can be characterized in many standard functional settings. In particular, when the underlying functions are driven by a standard Brownian motion, the Kolmogorov ε -entropy of the unit ball in a Sobolev space is on the order of C ε − 1 . This finding was originally established in [52]. We return to this last reference for more examples Additionally, assumptions (U4) and (U5) are technical in nature and used to simplify the proofs. We point out that (U4) is satisfied for the Beta-kernel, while assumption (U5) can be verified for many classes of functions, such as the closed ball in the Sobolev space or the unit ball in the Cameron-Martin space.

We obtain the following result.

5 Empirical analysis

This section is devoted to examining the practical use of the model studied in this work. This section is divided into three sections. In the first subsection, we discuss the selection of the smoothing parameter, which is the pivotal parameter in our estimation. For this reason, the choice of this parameter is primordial for the computational aspect. After the smoothing parameter selection, we examine the usefulness of the estimator. This practical study is conducted using two examples. The first one concerns artificial data, and the second one treats financial real data coming from some popular index markets according to the Dow-Jones index.

5.2 Simulated data

In this section, we will examine the behavior of the ES-expectile function over finite sample observations. More precisely, we evaluate how the estimator performs under conditions that fully match the standard theoretical assumptions and then contrast these results with what happens when those assumptions are partially relaxed. In particular, we focus on three main aspects: the role of the independence condition, the selection of an appropriate smoothing parameter, and the choice of kernel. By comparing these scenarios, we illustrate how each factor influences the overall behavior of the estimator. For this study, we compare our model to the alternative financial models such as the VaR, and the standard ES based on the percentile regression. For the empirical analysis, we generate the functional variable X ( ⋅ ) using the R-package rockchalk and the code routine dgp.fiid. Such curves represent a continuous trajectory of a financial asset, and the response variable Y represents a future characteristic of this trajectory. For this reason, for all i , we assume that Y i is the initial value of X i + 1 . In this empirical analysis, we compare the independence to the dependence case. In Figure 1, we plot the curves of the independent case versus the dependent case.

Figure 1

Shape of the curves.

As discussed in the following, the principal object of this computational part is to conduct a comparison study between the ES-expectile regression CEX p ^ and the ES-based on the VaR regression defined by the percentile regression

V p ( z ) = inf { z ∈ R : F ( z : z ) ≥ p } ,

where F is the conditional cumulative function of Y given X . The latter is estimated by using the kernel-estimator of the function F ( ⋅ ) as

F ^ ( z : z ) = ∑ i = 1 n 1 I ( Y i − z ) < 0 K d ( z , X i ) h n ∑ i = 1 n K d ( z , X i ) h n .

Thereafter, we estimate the VaR function by

V ^ p ( z ) = inf { z ∈ R : F ^ ( z : z ) ≥ p } .

Recall that, in this context, the ES regression is expressed by

CES p ˜ ( s ) = ∑ i = 1 n F ( a − 1 ‖ x − X i ‖ ) ( a G ( a − 1 ( V p ^ ( s ) − Y i ) ) + Y i ( 1 − H ( a − 1 ( V p ^ ( s ) − Y i ) ) ) ) p ∑ i = 1 n K ( a − 1 ‖ x − X i ‖ ) ,

where

G ( s ) = ∫ s ∞ u F ( u ) , H ( s ) = ∫ − ∞ s F ( u ) d u .

The two estimators CEX p ^ and CES p ˜ are calculated using the β -kernel, the L 2 metric, and the smoothing parameter of the CV rule (7). In the sense that

a CV opt ( s ) = arg min a ∈ H n ( s ) ∑ i = 1 n ( Y i − CExP ^ 0.5 ( X i ) ) 2 ,

where H n ( s ) is the set of the positive real a ( s ) such that the ball centered at x with radius a ( s ) contains exactly k neighbors of x . This smoothing selection method is usual in nonparametric functional statistics (see Ferraty and Vieu [45] for a deeper discussion on the subject). In addition, we compare our CV procedure with the bootstrap algorithm presented in [16,55]. To illustrate the influence of the chosen kernel and the associated bandwidth, we compute our estimators using a Gaussian kernel and a bandwidth set to a = a CV opt a boot opt , where a boot opt is the optimal bandwidth determined by the bootstrap selector. We evaluate the performance of these estimators via the following backtesting metrics:

Mse = 1 n ∑ i = 1 n ( Y i − ϑ ^ p ( ⋅ ) ) 2 I { Y i > CExP ^ p ( X i ) } , Msp = 1 n ∑ i = 1 n ( Y i − ϑ ^ p ( ⋅ ) ) 2 I { Y i > V ^ p ( X i ) } .

These error measures are examined for p = 0.9 , 0.5, 0.1, 0.05, and 0.01. The corresponding results are presented in the following tables.

Error results of the ES-expectile

Cases	Estimation	p = 0.9	p = 0.5	p = 0.1	p = 0.05	p = 0.01
Independent case	Estimation with CV-rule	0.16	0.14	0.13	0.098	0.096
	Estimation with bootstrap-rule	0.19	0.16	0.11	0.089	0.077
	Estimation with arbitrary choice	0.45	0.53	0.48	0.31	0.32
Dependent case	Estimation with CV-rule	0.21	0.29	0.22	0.24	0.27
	Estimation with bootstrap-rule	0.25	0.19	0.23	0.25	0.21
	Estimation with arbitrary choice	0.68	0.72	0.87	0.56	0.69

Error results of the ES based on VaR

Cases	Estimation	p = 0.9	p = 0.5	p = 0.1	p = 0.05	p = 0.01
Independent case	Estimation with CV-rule	0.32	0.23	0.37	0.34	0.29
	Estimation with bootstrap-rule	0.27	0.23	0.38	0.20	0.19
	Estimation with arbitrary choice	0.91	0.87	0.82	0.98	0.89
Dependent case	Estimation with CV-rule	0.61	0.59	0.42	0.74	0.57
	Estimation with bootstrap-rule	0.78	0.79	0.81	0.87	0.82
	Estimation with arbitrary choice	1.036	1.13	0.97	1.11	1.05

It is evident that both estimators are strongly influenced by the choice of backtesting measure and the parameters involved in their construction, including independence assumptions and the selection of smoothing parameters. However, it appears that the ES-expectile model is more efficient compared to the ES based on VaR. This last is more stable because of its insensitivity to the extreme values. Indeed, by a sample comparison of the variability between CEX p ^ and CES p ˜ , we observe that the error values of the ES-expectile are in the range (0.096, 0.29) versus (0.23, 0.74) for the ES based on VaR.

5.3 Real data application

The last paragraph of this empirical analysis concerns the applicability of our model in real data examples. Specifically, we evaluate the efficiency of the ES-expectile model over financial data associated with the Dow Jones index. The latter is an old market index. It dates to May 26, 1896. The index value in this market is calculated by dividing the sum of the stock prices of the 30 companies by some fixed factor. In this experiment, we study the period between 11/10/1971 and 12/10/2022. It constitutes more than 25,000 days. The parent data are displayed in Figure 2. They are available in https://fred.stlouisfed.org/series/DJIA (accessed on 24 April 2024).

Figure 2

Shape of the curves.

Now, to explore the functional path of the data, we cut the initial data by pieces of 30 days. These values of these pieces represent the functional regressors X i . They are plotted in Figure 3.

Figure 3

Initial data.

Next, similar to the artificial case, we choose the response variable Y as the index of the first day, 1 month ahead. In the sense that Y i = X i + 1 ( d 1 ) and to ensure the independent structure of our work, we select distanced observations. Once again, we compare the two estimators CEX p ^ and CES p ˜ using a real data ( X i , Y i ) i = 1 , … , 120 . Such estimators are computed using the same algorithm as the artificial data. In the sense that we use the same kernel, we select the smoothing parameters by rule (7) and we employ the L 2 metric obtained by the principal component analysis-metric. We refer to Ferraty and Vieu [45] for more details on the mathematical formulation of these metrics. The comparison results are given in Figures 4 and 5 where we plot the true ( Y i ) i = 1 , … , 400 versus the estimator CEX p ^ ( X i ) and CES p ˜ ( X i ) for two values of p = 0.1 and p = 0.95 . The graphs show the superiority of the ES-expectile regression over the ES-quantile e model. This superiority is confirmed by computing the expectile mean square (Mse) against percentile mean square (Msp) that are, respectively, 0.147 and 0.318, showing that the ES-expectile is more precedence than the ES based on quantile regression.

Figure 4

Comparison between ES-expectile and ES-quantile (VaR) for p = 0.1 .

Figure 5

Comparison between ES-expectile and ES-quantile (VaR) for p = 0.95 .

6 Conclusion and future works

In this work, we have focused on the estimation of the free parameters in the ES-expectile regression model. We developed an estimator using kernel smoothing and examined both the theoretical and practical aspects of this model within the context of functional data analysis. On the theoretical side, we established convergence results using the Borell-Cantelli lemma for both pointwise and uniform cases. These findings provide strong mathematical support for the proposed model. The asymptotic results were derived under standard conditions, with precise convergence rates specified. Practically, our estimator is straightforward to implement and has shown superior performance. Additionally, our contribution paves the way for future research in several directions.

One critical aspect that remains unexplored in this article is the optimal selection of smoothing parameters aimed at minimizing the MSEs of the proposed kernel estimators. This topic holds significant relevance within the broader context of kernel-based nonparametric estimation and warrants focused research efforts. Although this issue is essential for refining the proposed method, we have chosen to defer a comprehensive investigation to future work. In subsequent studies, we intend to thoroughly address the selection of smoothing parameters, potentially employing CV or other optimal selection techniques (for instance, see [48]). An area of particular interest is determining the uniform convergence of the estimator with respect to bandwidth selection, which is crucial for optimal smoothing parameter determination.

Several avenues are available for further development of our method. A central challenge in kernel smoothing is the presence of bias error, which remains the primary limitation of standard kernel estimators. Although local linear fitting provides a theoretical approach to mitigating this bias, the problem continues to demand further attention in the field of NFDA. Indeed, the bias issue is especially pertinent when dealing with complex functional data, where simple kernel smoothing methods may yield suboptimal results. The computational results presented in this article demonstrate that implementing a local linear bias correction (CB) substantially enhances the performance of the estimation method, reducing the overall error. However, the use of any CB process, particularly in a nonparametric setting, requires robust mathematical support to ensure control over its asymptotic behavior. This remains a significant gap in the current literature. To the best of our knowledge, no comprehensive analysis has been undertaken in NFDA regarding the asymptotics of CB methods. Therefore, we believe this represents an exciting and unexplored area for future research.

Furthermore, it would be highly beneficial to explore how the proposed method can be extended or combined with established CB algorithms. Notably, we see great potential in incorporating the well-known CB algorithms, such as those developed by Yao [56] and Karlsson [57], which have been shown to improve estimation accuracy in related contexts. By integrating local linear smoothing with these CB algorithms, we anticipate a significant improvement in estimation quality, particularly for complex functional data. This integration could offer a more robust approach, balancing both bias reduction and variance control, which is crucial in NFDA.

Another natural question that arises is how to adapt our results to other popular nonparametric estimation techniques, such as wavelet-based estimators [58–60], delta sequence estimators [61], k -nearest neighbor ( k NN) estimators [62], and other local linear estimators [63]. These methods, widely used in various fields of functional data analysis, could potentially benefit from the CB techniques presented in this article. By investigating the extension of our approach to these estimators, we may uncover new synergies between kernel-based methods and other estimation frameworks, thereby advancing the state of the art in NFDA.

Finally, as a future direction, we propose relaxing the stationarity assumption employed in the present study. While stationarity is a common assumption in many functional data models, it may not always be realistic, particularly in dynamic or time-varying settings [64–66]. Therefore, extending our work to accommodate locally stationary functional ergodic processes could yield significant insights. In this more general setting, the conditional quantile estimator would be allowed to vary smoothly over time, thus accommodating nonstationary or evolving data structures. Investigating the uniform limit theorems for such processes would be a natural and fruitful extension of the current work, contributing to a deeper understanding of the asymptotic properties of kernel estimators in more flexible, nonstationary contexts. Furthermore, we aim to explore alternative estimation approaches using single index structures or additive modeling.