# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 2 Nonlinearity Detection Hypothesis Testing
### Tests Based on Parametric Linear and Nonlinear Models
The distance between two sequences is defined as \(||u^{m}(i)-u^{m}(j)||=\max\limits_{0\leq n<m}\big{(}|\hat{e}(i+n)-\hat{e}(j+n )|\big{)}\). The index is based on the number of sequences whose distance is below a given threshold \(e\). Index \(q_{9}\) is defined as the difference between the number of sequences of length \(m\) (with a distance below the threshold) with the number of sequences of length \(1\) powered at \(m\) (with a distance also below the threshold). This difference is normalized in order that the index has standard normal statistics. When the model is well fitted, the noise is iid and the index is close to 0. In this index, two parameters have to be fixed. The first is the sequence length \(m\) (if we choose \(m\) too small, then \(q_{9}\) will be always close to 0 whatever the model is) and the second is the value of \(\epsilon\) (in fact, if \(\epsilon\) is too small, then the number of close sequences will not be large enough for a robust estimate of \(q_{9}\)). In Section 4, we test the values \(m=4\), \(m=8\) and \(\epsilon=std/2\), \(\epsilon=std\), \(\epsilon=2 std\), with \(std\) being the standard deviation of the residuals. * [PERSON] and [PERSON] test. [PERSON] and [PERSON] in [44] propose a test based on the logarithm of the determinant of the correlation matrix of the squared residuals. In other words, we have \[q_{10}=-\frac{N}{m+1}\log|R_{m}|,\] (8) where \(R_{m}\) is the normalized correlation matrix of the squared residuals (the normalizing coefficient is \(\dfrac{N+2}{N-k}\) and the size of this square matrix of dimension \(m\), \(k\) being the lag of the correlation coefficient). The idea behind this test is to verify that the correlation matrix is diagonal (i.e., the residuals are white and then the model is well fitted). In this test, the parameter \(m\) has to be chosen. As proposed by the authors, \(m\) is determined by the greatest correlation length in the residual. This estimation of the correlation is performed as the MA order estimation described below. * [PERSON] and [PERSON] test on residuals. Always with the idea of comparing the best predictor and the linear predictor (see Section 2.2.2), [PERSON] and [PERSON] compare the two predictors for the residuals (always under an AR model fitting). The idea behind this calculation is to find nonlinear terms in the residuals, as stated in the introduction to this section. In this case, the index \(q_{11}\) is also defined as the mean square error between the two predictors. * The main drawback of the previously presented approaches is to be based on an AR model and then they assumed that the process is infinite memory. In [45], we propose a HT based on an MA model (i.e., with finite ACF), that is of particular interest in our case since HF returns are assumed to verify this property of finite memory. The order of the MA process is to detect a lag above which the autocorrelation function is null. Such an order estimate can be performed either by a Ljung-Box test or with a \(T^{2}\) Hottleing by estimating the correlation sequence over subsignals (in the results of Section 4, we use four sub signals). We estimate the MA parameters by a Giannakis' formula since we have \(b_{i}/b_{0}=M_{3}^{X}(i,q)/M_{3}^{X}(0,q)\) (see [56; 57]). Unlike the AR model, MA are not necessarily invertible (the transfer function can vanish). Since the residual are not available as for the previous indices, we propose to verify that the estimated coefficients agree with the estimated ACF as: \[q_{12}=\sum_{n=0}^{q_{M}}\left(\hat{M}_{2}^{X}(n)-\phi^{2}\sum_{i=0}^{q_{M}-k }\hat{b}_{i}\hat{b}_{i+n}\right)^{2}\] (9) where \(q_{M}\) is the estimated MA order, \(\hat{M}_{2}^{X}(n)\) is the estimated second-order moment, \(\hat{b}_{i}\) is the MA coefficient estimated by a Giannakis' formula and \(\phi^{2}\) is estimated to minimize \(q_{12}\). When the TS is the output of a linear MA model, then \(q_{12}\) is theoretically null, but non-null under \(\mathcal{H}_{1}\), (see [45], for practical implementation).