# Hypothesis Testing Fusion for Nonlinearity Detection in Hedge Fund Price Returns
## 4 Results
### Hedge Fund Nonlinearity
Four styles are in this category, Funds of Funds, Sector, Macro and Relative Values Arbitrage. As observed in [17], Macro has near normal statistics but are sometimes classified as nonlinear. 4. The last family is that classified as nonlinear in a base and linear in the other one. These styles are Convertible arbitrage, Merger Arbitrage and Equity Market Neutral. For this last family no conclusion can be drawn. For the first family of HFs, the factorial linear model is valid. Moreover, the evolution of the pertinent risk factors is also governed by a linear equation. For the second family, the linearity of the model is rejected. Two reasons can explain the nonlinearity in the HF price return. The first one is that the price return does not depend linearly on the risk factors. The second reason is that the risk factors are nonlinear TS. A first indication is given by observing that these three HF styles are negatively and strongly correlated with the exchange rate (see [64] chapter 11) with a coefficient correlation respectively equal to \(-0.14\), \(-0.17\), \(-0.15\), but the exchange rates are known to nonlinearly evolve [65]. Thus, the nonlinearity detected for the family of funds is due to the risk factor and not to a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**HF Style** & **Linear** & **Nonlinear** & **Linear** & **Nonlinear** \\ \hline Fund of Funds & 25 & 27 & 75 & 35 \\ \hline Event-Driven & 10 & 4 & 19 & 6 \\ \hline Emerging Market & 7 & 1 & 10 & 3 \\ \hline Convertible Arbitrage & 2 & 7 & 16 & 2 \\ \hline Equity Non-Hedge & 5 & 14 & 15 & 12 \\ \hline Fixed Income & 3 & 0 & 11 & 5 \\ \hline Managed Future & 18 & 18 & 31 & 29 \\ \hline Equity Hedge & 17 & 30 & 47 & 34 \\ \hline Merger Arbitrage & 7 & 5 & 3 & 7 \\ \hline Foreign Exchange & 4 & 5 & 8 & 1 \\ \hline Sector & 8 & 7 & 8 & 5 \\ \hline Equity Market Neutral & 4 & 11 & 11 & 3 \\ \hline Distressed Securities & 5 & 1 & 11 & 6 \\ \hline Macro & 4 & 6 & 13 & 7 \\ \hline Relative Value Arbitrage & 1 & 2 & 7 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Classification of Hedge funds according to the decision rule of nonlinearity for the TASS base (second and third column) and the S&P base (fourth and fifth column). nonlinearity in the factorial model. This point is confirmed by observing that the linear HFs do not depend on the exchange rate (correlation coefficients equal to \(-0.03\), \(0.03\), \(0.06\)). The exception is the Fixed Income style with a positive correlation coefficient of \(0.15\). In fact, Exchange rate has been pointed out as a leverage factor in some HFs (but not in all the cases, as detailed on the [PERSON] fund in [63]). Amazingly, the emerging market HFs are not classified as nonlinear, although the exchange rate is also assumed to be a key factor in these funds [63]. As also noted in this paper, this exposure depends on the level really invested in local currencies. The linearity of the emerging market HF price return is observed despite the turbulent times related to the multiple currency and financial crises (Mexican crisis of December 1994; Asian crisis since July 1997; Brazilian devaluation of January 1999; Turkish liquidity crises of 2000-2001...). The linearity is also declared for Exchange Rate HFs. A possible reason is that in these funds the nonlinearities between all the exchange rates can cancel each other out. The third family can be explained by observing that the funds are very diversified and thus there can be a specific risk factor (on a specific sector of the HF for instance) that explains why some HF are classified as nonlinear and the others as linear. A final and global conclusion of this study is that the linear factorial model is generally validated. When nonlinearities are observed, they can be explained by the nonlinear evolution of a special risk factor (the exchange rate).