# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 2 Nonlinearity Detection Hypothesis Testing
### Tests Based on Parametric Linear and Nonlinear Models  
As stated in the introduction, the purpose of the parametric tests is first to identify a linear system and to test whether the residuals (denoted \(\hat{e}(n)\) that is the difference between the predicted values by the estimated AR model and the real values \(X(n)\) or in others words an estimate of model excitation, see Equations (4a)-(4c)) contain nonlinear terms (with the underlying assumption that the identified linear model captures only the linear part, this being not so obvious, see [24; 47] for a discussion). As usual, the aim of the tests is to verify that the residuals are white. For these tests, the AR model identification is assumed to be performed with a least squares solution (i.e., by solving the Yule-Walker equations). In many articles, the order of the AR model is assumed to be known. In practical situations of Sections 4.1 and 4.2, the Bayesian Information Criterion (BIC) is applied to determine the order of the AR model (see [48] for the BIC and [49] for an exhaustive review of the order selection criteria). Several tests over the residual whiteness are proposed. * Tsay index: In [29], Tsay improves a test proposed by [PERSON] [50] derived from the one-degree test (of additivity) by [PERSON] [51]. In this paper, we consider only the Tsay indices (F and C tests) since they provide slightly better results than [PERSON]'s indices. After the model identification and the residual estimation, we regress the TS squared samples over the TS samples at previous lags and we estimate the new residuals \(\zeta(n)\) (obviously this step involves Higher Order Statistics, HOS, as explicitly mentioned in [PERSON] [50] and Section 2.2.1). The last step involves finding a correlation between the residuals of the TS fitting and the residuals of the squared TS fitting. If the TS is linear, then the two kinds of residuals are uncorrelated. This property is verified by using the F-distribution test. We denote \(q_{5}\) this index in the rest of the paper. Due to bad results of a Volterra model, [PERSON] proposes a second test based on the correlation of the squared residuals with the TS samples. This test is the C test and is denoted \(q_{6}\) in the latter part of this paper. * [PERSON] et al. index. The test proposed by [PERSON] et al. is based on whether the TS can be modeled by a Smooth Transition AutoRegressive (STAR) model. By expanding the STAR model up to the third order, this model can be rewritten as a Nonlinear AutoRegressif model containing nonlinear terms \(X(n-j)\), \(X(n-i)X(n-j)\), \(X(n-i)X^{2}(n-j)\) and \(X(n-i)X^{3}(n-j)\). The HT of nonlinearity is based on the test of the nullity of the nonlinear kernel coefficients. As for the Tsay index, the first step involves calculating the residual after regression of \(X(n)\) over an AR model and we denote \(S_{0}\), the sum of the squared residuals. Finally, the residuals are regressed (using a mean least squares criterion) on \(X(n-j)\), \(X(n-i)X(n-j)\), \(X(n-i)X^{2}(n-j)\) and \(X(n-i)X^{3}(n-j)\), (then HOS are also involved in this step). The squared sum of these new residuals is denoted \(S_{1}\) leading to the nonlinearity index \(q_{7}=N(S_{0}-S_{1})/S_{0}\), \(N\) being the TS length. This procedure being fairly complicated due the large number of parameters to be estimated, the authors propose a simpler test based on a reduced QARMA model (4b). In this test, the index is then given by \(q_{8}=N(S_{0}-S_{2})/S_{0}\) where \(S_{2}\) is the sum of the squared residuals of the nonlinear regression of \(X(n)\) over \(X(n-i)X(n-j)\) and \(X^{3}(n-j)\) (see [42]). Other nonlinear models are difficult to identify, in particular Volterra model, except in simplified cases such a [PERSON] and [PERSON] second-order models [52; 53; 54]. * BDS test: The idea developed by [PERSON] et al. in [43], always verifies that the residuals are white, is close to Order Statistics (see [55]). They define the sequence \(u^{m}(n)=\big{(}\hat{e}(n),\cdots,\hat{e}(n+m)\big{)}\).