# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
###### Abstract  
In this paper, we present the results of nonlinearity detection in Hedge Fund price returns. The main challenge is induced by the small length of the time series, since the return of this kind of asset is updated once a month. As usual, the nonlinearity of the return time series is a key point to accurately assess the risk of an asset, since the normality assumption is barely encountered in financial data. The basic idea to overcome the hypothesis testing lack of robustness on small time series is to merge several hypothesis tests to improve the final decision (i.e., the return time series is linear or not). Several aspects on the index/decision fusion, such as the fusion topology, as well as the shared information by several hypothesis tests, have to be carefully investigated to design a robust decision process. This designed decision rule is applied to two databases of Hedge Fund price return (TASS and SP). In particular, the linearity assumption is generally accepted for the factorial model. However, funds having detected nonlinearity in their returns are generally correlated with exchange rates. Since exchange rates nonlinearly evolve, the nonlinearity is explained by this risk factor and not by a nonlinear dependence on the risk factors. n  
Lefefing Fusion for Nonlinearity Detection in Hedge Fund Price Returns. 15 June 202  
performance (either mutual or hedge), for deriving the skill of the fund manager (i.e., \(\alpha_{m}\)) or to estimate the exposure to a given risk factor. The risk factors are generally a priori identified and depend on the authors and their number varying from 7 (the well admitted proposal of [2]) to 31 (the return explanation by a given risk factor is generally not verified through conditional independence testing [13] or Bayesian network [14] for instance). The idea of this paper is to propose a method to directly classify HFs according to their price return (i.e., their performance) model. When observing a time series (TS, in our case, the HF price returns), a natural idea is to determine the equations governing this TS (quantitative model). [PERSON] and [PERSON] [15], [PERSON] [16], [PERSON] et al. [4], [PERSON] [10], and [PERSON] [17] have noticed that the HF price returns exhibit non-Gaussian skewness and kurtosis. Usually, non-Gaussian statistics are thought to be generated by some nonlinearities [10]. * For instance, [PERSON] [16] proposes a price return model based on the nonlinear payoff model developed by [PERSON] [18; 19] but without statistical validation of this model. * [PERSON] proposed in [9] an extension of Equation (1) by adding powered risk factors. * [PERSON] and [PERSON] [8] and [PERSON] [10] proposed a model with a loading factor having different values depending on the risk factor (i.e., \(\beta_{i}(F_{i}(n))\cdot F_{i}(n)\) and, thus nonlinearly, depends on \(F_{i}(n)\)), the simplest loading function being a threshold model. This threshold model possibly conveys the less or more liquid periods [5]. * Similarly, [PERSON] [10] proposed to add switching the random variable to the factorial model of Equation (1), this switching variable explains the phase locking behavior observed in crisis periods. The idea of including nonlinear terms in Equation (1) is supported by the fact that usual risk factors have a normal distribution [17]. Thus, if the risk factors are normal (right side of Equation (1)) and the returns are non-normal (left side of Equation (1)), then a straightforward explanation is some nonlinearities. The main goal of this paper is to provide an algorithm to validate the linearity of the HF return model. The nonlinearity detection problem in an observed TS has led to the definition of several Hypothesis Testing (HT). In the signal processing community, tests based on the bispectrum have generated some interest (see [20; 21; 22; 23]), and a comparison of the non parametric tests is given in [24]). In the econometric community, nonlinearity tests are generally based on linear/nonlinear parametric models.