arXiv subjects

Matthieu Garcin

Publications and source records attributed to Matthieu Garcin.

16 recordsLinked to original sources

arXiv2026

Composite likelihood inference of fractional Gaussian processes with sequentially optimal subset selection

The composite likelihood method reduces the computational cost of parameter estimation in time series by considering several subsets of observations instead of all observations at once. The asymptotic properties of this method are related to the Godambe information, an extension of the Fisher information that accounts for the dependence between subsets of observations. We aim to apply this method to linear Gaussian models, in particular fractional Brownian motion and fractional Gaussian noise. We derive theoretical expressions for their Fisher information and their Godambe information and deduce a subset selection design that sequentially maximizes the Godambe information. The size of the subsets then allows us to control the trade-off between estimation accuracy and computational cost. Through simulations, we compare this method with the method of moments and maximum likelihood estimation, and we apply it to real data, namely volatility series of a stock index and a wind speed time series.

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arXiv2026

Directional Dependence of Extreme Events

This paper introduces a novel measure to quantify the directional dependence of extreme events between two variables. The proposed approach is designed to capture asymmetric tail dependence by studying conditional tail expectations of rank-transformed variables, thereby quantifying the behavior of one variable when the other takes extreme values. We investigate the theoretical asymptotic behavior of the associated estimator. The effectiveness of the approach is demonstrated through an extensive simulation study. In addition, we discuss the use of the proposed coefficient for the detection of causal effects in extreme events. Finally, we apply the method to an oceanographic dataset, where the results highlight the strong asymmetric nature of extreme events and identify the dominant directions of extremal influence among key oceanographic variables. As a directional measure of tail dependence, our approach provides a natural tool for exploring causal-effect relationships in extreme-value settings.

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arXiv2025

Asymptotic and finite-sample distributions of one- and two-sample empirical relative entropy

In the perspective of building statistical tests of divergence between two probability distributions, we study the distribution of empirical relative entropy and derive several types of approximations: concentration inequalities for finite samples, asymptotic distributions, and Berry-Esseen bounds in a pre-asymptotic regime. For the latter, we introduce a new approach to obtain Berry-Esseen inequalities for nonlinear functions of sum statistics under some convexity assumptions. Our theoretical contributions cover both one- and two-sample empirical relative entropies.

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arXiv2025

Prediction of linear fractional stable motions using codifference, with application to non-Gaussian rough volatility

The linear fractional stable motion (LFSM) extends the fractional Brownian motion (fBm) by considering $\alpha$-stable increments. We propose a method to forecast future increments of the LFSM from past discrete-time observations, using the conditional expectation when $\alpha>1$ or a semimetric projection otherwise. It relies on the codifference, which describes the serial dependence of the process, instead of the covariance. Indeed, covariance is commonly used for predicting an fBm but it is infinite when $\alpha<2$. Some theoretical properties of the method and of its accuracy are studied and both a simulation study and an application to real volatility data, with a comparison to the fBm and to the heterogeneous auto-regressive model, confirm the relevance of the approach. The LFSM-based method shows a promising performance in the forecast of time series of volatilities, decomposing properly, in the fractal dynamic of rough volatilities, the contribution of the kurtosis of the increments and the contribution of their serial dependence. Moreover, the analysis of hit ratios suggests that, beside independence, persistence, and antipersistence, a fourth regime of serial dependence exists for fractional processes, characterized by a selective memory controlled by a few large increments.

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arXiv2024

Market information of the fractional stochastic regularity model

The Fractional Stochastic Regularity Model (FSRM) is an extension of Black-Scholes model describing the multifractal nature of prices. It is based on a multifractional process with a random Hurst exponent $H_t$, driven by a fractional Ornstein-Uhlenbeck (fOU) process. When the regularity parameter $H_t$ is equal to $1/2$, the efficient market hypothesis holds, but when $H_t\neq 1/2$ past price returns contain some information on a future trend or mean-reversion of the log-price process. In this paper, we investigate some properties of the fOU process and, thanks to information theory and Shannon's entropy, we determine theoretically the serial information of the regularity process $H_t$ of the FSRM, giving some insight into one's ability to forecast future price increments and to build statistical arbitrages with this model.

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arXiv2024

Estimation of bid-ask spreads in the presence of serial dependence

Starting from a basic model in which the dynamic of the transaction prices is a geometric Brownian motion disrupted by a microstructure white noise, corresponding to the random alternation of bids and asks, we propose moment-based estimators along with their statistical properties. We then make the model more realistic by considering serial dependence: we assume a geometric fractional Brownian motion for the price, then an Ornstein-Uhlenbeck process for the microstructure noise. In these two cases of serial dependence, we propose again consistent and asymptotically normal estimators. All our estimators are compared on simulated data with existing approaches, such as Roll, Corwin-Schultz, Abdi-Ranaldo, or Ardia-Guidotti-Kroencke estimators.

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arXiv2023

Fractal properties, information theory, and market efficiency

Considering that both the entropy-based market information and the Hurst exponent are useful tools for determining whether the efficient market hypothesis holds for a given asset, we study the link between the two approaches. We thus provide a theoretical expression for the market information when log-prices follow either a fractional Brownian motion or its stationary extension using the Lamperti transform. In the latter model, we show that a Hurst exponent close to 1/2 can lead to a very high informativeness of the time series, because of the stationarity mechanism. In addition, we introduce a multiscale method to get a deeper interpretation of the entropy and of the market information, depending on the size of the information set. Applications to Bitcoin, CAC 40 index, Nikkei 225 index, and EUR/USD FX rate, using daily or intraday data, illustrate the methodological content.

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arXiv2023

Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis

We are interested in the nonparametric estimation of the probability density of price returns, using the kernel approach. The output of the method heavily relies on the selection of a bandwidth parameter. Many selection methods have been proposed in the statistical literature. We put forward an alternative selection method based on a criterion coming from information theory and from the physics of complex systems: the bandwidth to be selected maximizes a new measure of complexity, with the aim of avoiding both overfitting and underfitting. We review existing methods of bandwidth selection and show that they lead to contradictory conclusions regarding the complexity of the probability distribution of price returns. This has also some striking consequences in the evaluation of the relevance of the efficient market hypothesis. We apply these methods to real financial data, focusing on the Bitcoin.

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arXiv2022

A statistical test of market efficiency based on information theory

We determine the amount of information contained in a time series of price returns at a given time scale, by using a widespread tool of the information theory, namely the Shannon entropy, applied to a symbolic representation of this time series. By deriving the exact and the asymptotic distribution of this market information indicator in the case where the efficient market hypothesis holds, we develop a statistical test of market efficiency. We apply it to a real dataset of stock indices, single stock, and cryptocurrency, for which we are able to determine at each date whether the efficient market hypothesis is to be rejected, with respect to a given confidence level.

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arXiv2021

Nonparametric estimator of the tail dependence coefficient: balancing bias and variance

A theoretical expression is derived for the mean squared error of a nonparametric estimator of the tail dependence coefficient, depending on a threshold that defines which rank delimits the tails of a distribution. We propose a new method to optimally select this threshold. It combines the theoretical mean squared error of the estimator with a parametric estimation of the copula linking observations in the tails. Using simulations, we compare this semiparametric method with other approaches proposed in the literature, including the plateau-finding algorithm.

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arXiv2021

Credit scoring using neural networks and SURE posterior probability calibration

In this article we compare the performances of a logistic regression and a feed forward neural network for credit scoring purposes. Our results show that the logistic regression gives quite good results on the dataset and the neural network can improve a little the performance. We also consider different sets of features in order to assess their importance in terms of prediction accuracy. We found that temporal features (i.e. repeated measures over time) can be an important source of information resulting in an increase in the overall model accuracy. Finally, we introduce a new technique for the calibration of predicted probabilities based on Stein's unbiased risk estimate (SURE). This calibration technique can be applied to very general calibration functions. In particular, we detail this method for the sigmoid function as well as for the Kumaraswamy function, which includes the identity as a particular case. We show that stacking the SURE calibration technique with the classical Platt method can improve the calibration of predicted probabilities.

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arXiv2021

Forecasting with fractional Brownian motion: a financial perspective

The fractional Brownian motion (fBm) extends the standard Brownian motion by introducing some dependence between non-overlapping increments. Consequently, if one considers for example that log-prices follow an fBm, one can exploit the non-Markovian nature of the fBm to forecast future states of the process and make statistical arbitrages. We provide new insights into forecasting an fBm, by proposing theoretical formulas for accuracy metrics relevant to a systematic trader, from the hit ratio to the expected gain and risk of a simple strategy. In addition, we answer some key questions about optimizing trading strategies in the fBm framework: Which lagged increments of the fBm, observed in discrete time, are to be considered? If the predicted increment is close to zero, up to which threshold is it more profitable not to invest? We also propose empirical applications on high-frequency FX rates, as well as on realized volatility series, exploring the rough volatility concept in a forecasting perspective.

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arXiv2020

Long vs Short Time Scales: the Rough Dilemma and Beyond

Using a large dataset on major FX rates, we test the robustness of the rough fractional volatility model over different time scales, by including smoothing and measurement errors into the analysis. Our findings lead to new stylized facts in the log-log plots of the second moments of realized variance increments against lag which exhibit some convexity in addition to the roughness and stationarity of the volatility. The very low perceived Hurst exponents at small scales is consistent with the rough framework, while the higher perceived Hurst exponents for larger scales leads to a nonlinear behavior of the log-log plot that has not been described by models introduced so far.

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arXiv2020

Efficiency of the financial markets during the COVID-19 crisis: time-varying parameters of fractional stable dynamics

This paper investigates the impact of COVID-19 on financial markets. It focuses on the evolution of the market efficiency, using two efficiency indicators: the Hurst exponent and the memory parameter of a fractional L\'evy-stable motion. The second approach combines, in the same model of dynamic, an alpha-stable distribution and a dependence structure between price returns. We provide a dynamic estimation method for the two efficiency indicators. This method introduces a free parameter, the discount factor, which we select so as to get the best alpha-stable density forecasts for observed price returns. The application to stock indices during the COVID-19 crisis shows a strong loss of efficiency for US indices. On the opposite, Asian and Australian indices seem less affected and the inefficiency of these markets during the COVID-19 crisis is even questionable.

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arXiv2020

Estimation of time-varying kernel densities and chronology of the impact of COVID-19 on financial markets

The time-varying kernel density estimation relies on two free parameters: the bandwidth and the discount factor. We propose to select these parameters so as to minimize a criterion consistent with the traditional requirements of the validation of a probability density forecast. These requirements are both the uniformity and the independence of the so-called probability integral transforms, which are the forecast time-varying cumulated distributions applied to the observations. We thus build a new numerical criterion incorporating both the uniformity and independence properties by the mean of an adapted Kolmogorov-Smirnov statistic. We apply this method to financial markets during the COVID-19 crisis. We determine the time-varying density of daily price returns of several stock indices and, using various divergence statistics, we are able to describe the chronology of the crisis as well as regional disparities. For instance, we observe a more limited impact of COVID-19 on financial markets in China, a strong impact in the US, and a slow recovery in Europe.

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arXiv2020

A comparison of maximum likelihood and absolute moments for the estimation of Hurst exponents in a stationary framework

The absolute-moment method is widespread for estimating the Hurst exponent of a fractional Brownian motion $X$. But this method is biased when applied to a stationary version of $X$, in particular an inverse Lamperti transform of $X$, with a linear time contraction of parameter $\theta$. We present an adaptation of the absolute-moment method to this framework and we compare it to the maximum likelihood method, with simulations and an application to a financial time series. While it appears that the maximum-likelihood method is more accurate than the adapted absolute-moment estimation, this last method is not uninteresting for two reasons: it makes it possible to confirm visually that the model is well specified and it is computationally more performing.

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