AbstractCommodity futures risk premiums vary across commodities and over time depending on the level of physical inventories, as predicted by the Theory of Storage. The interaction of the factors related to the inventories and other financial sources result in a complex futures price behaviour especially in the case of agricultural products. This behaviour has been surveyed and described with non-linear testing and modelling, respectively, within the last decades. The same process has been used to describe the formation of commodity prices. In the present study a number of un ivariate tests confirm the existence of nonlinearity in the cotton futures price formation, while a Mackey GARCH model is used to describe the  returns’ behaviour. This model selection is based upon the consideration of the mean process as dynamic chaotic. The model can be a useful in making a short term forecasts.
Key words: GARCH, nonlinearities, futures prices, cotton.
                                  Introduction
A substantial body of research examines volatility patterns in various asset return data, including stock returns and exchange rates, but there exists a much smaller body of research examining the volatility of agricultural commodity futures prices 1. The value of agricultural commodities is formatted by numerous  unpredictable factors (biological, economic and weather factors). To be more specific long perennial gestation periods and/or inventory adjustment factors play a significant role. The present study surveys the behaviour of cotton futures prices, a material extensively used in human clothing necessitating its extensive cultivation. A long frost-free period, plenty of sunshine, and a moderate level of precipitations constitute a satisfactory combination of climatic conditions required by the particular cultivation. The extensive use of cotton as a raw material in the production of different types of clothing like  terrycloth, denim, chambray, corduroy, seersucker, cotton twill and others makes the study of the cotton futures prices important. Furthermore, China, India, the United States and Pakistan are considered as the most important players in the world cotton market while, extensive cotton growing is also carried on in Brazil, Uzbekistan, Australia and Turkey.
　　The New York Board of Trade (NYBOT) and the New York Mercantile Exchange (NYMEX) are the trading centre of cotton futures contracts. The contract deliveries take place every year in March, May, July, September and December. What must also be mentioned involves that the volatility of futures prices increases as the delivery date approaches 2. This hypothesis has been empirically investigated by a great number of researchers. In another study 3 the variance of demand and supply shocks, as well as the correlation of supply shocks across producers are considered as sources of the volatility in the futures prices, giving the impression that Samuelson’s result is a special case in which uncertainty is resolved near maturity. That is because in the case of agricultural products a large supply shock positively correlated across producers (e.g. a weather-related shock) may lead to high volatility of the future prices even if the time to maturity is long. The aforementioned result is also confirmed by other studies 4-6despite that they are more volatile near maturity.
　　Futures increase the depth and the informative power of a market7 while an elegant explanation is provided by another study 8;to be more specific the derivatives’ market provides a mechanism for traders of insurance to hedge themselves against undesirable price changes. Actually, the risk is distributed to a number of investors and is finally being transferred through hedging spot positions to professional speculators who are capable and probably willing of taking it. Thus, this transfer may reduce the need to embody risk premium in spot market transactions to compensate for the risk variations resulting in a significant improvement of the spot market operation. Furthermore, it is argued 9 that well informed speculators buy mainly at low prices, pushing prices up and sell at high prices, thus forcing prices to fall, a fact that stabilizes the market and consequently reduces volatility, while the heterogeneity among traders regarding the responses to new information plays an important role to the spot prices’ volatility10-12. The efficiency of a spot market may be improved due to the creation of new information channels and their rapid evaluation and dissemination. Finally, it is known that derivatives expand investment opportunities, facilitate hedging and improve daily operation leading to a more mature and less uncertain market 13.    The commodity futures prices has been a subject of survey within      552  Journal of Food, Agriculture & Environment, Vol.9 (2), April 2011 the last decades due to their great importance for the domestic economies of developing countries as they consist a large proportion of their export value and due to their notable volatility14.The empirical findings regarding the long range dependence of the commodity futures prices contradict one another preventing a confirmation of the existence of long memory in those prices.
　　The value of commodities is driven by factors such as weather and geopolitical conditions, supply constraints in the physical production, and event risks different from those that determine the value of stocks and bonds 15, 16.  In fact, a number of empirical studies have confirmed this type of correlation over certain periods of time 17-20. Consequently, diversification benefits may emerge (for instance a risk reduction for any given level of expected return). However, there is evidence that the growing presence of index funds in commodities markets integrates the commodity markets with the stock and bond ones 21, 22.
　　The abundance of the factors affecting the futures’ prices formation may account for the complexity in their behaviour necessitating the use of newly developed financial models for their description. Chaotic behaviour may be a rational approach for them. However, the discovery of chaotic processes is difficult; one can find randomness or noisy chaos that is a dynamic chaotic system disturbed by random noise. Modelling such processes is a difficult issue. In a bulk of studies it has been argued that daily futures returns are not normal as they have in most cases fatter tails than a normal distribution and skewed. In addition returns are not independent and exhibit non-linear dynamics 23. Thus, nonlinear models may see price fluctuations triggered by an interaction between a stabilizing force driving prices back towards fundamental values when the market is dominated by fundamentalists, while destabilizing forces drive prices away from their fundamental values when the market is dominated by speculative traders, i.e. when the long speculative to hedging ratio is several times unity. Another study 24, that examined futures price series, provided insights as to how such problems can be dealt with; even if the underlying dynamics of a system is totally (chaotic) deterministic, the resulting movements of price variability can be described statistically as stochastic. This confirms the fact that chaotic series can mimic real financial series properties 25.     The stock prices behaviour as well as the futures commodity price behaviour has been extensively described by GARCH  models 1, 23, 26-31. Low dimension chaos in the futures prices of agricultural products has been a subject of a recent study 31. The findings of the study confirm that there is a non-linear dependence, although no long lasting chaotic structure can be argued. The results of non-linearity may be explained through ARCH-type processes along with seasonality control and contract maturity effects. The futures’ volatility1 as confirmed by another study, displays the self-similarity property consistent with long memory and  futures volatilities exhibit persistent long memory with finite unconditional variance 1.
　　The present study makes an effort to trace the existence of nonlinearities in the futures price formation through a number of univariate tests and to model their behaviour with the assistance of a noisy GARCH chaotic model.
　　                                 Data
　　For the empirical analysis, daily observations of the CT Cotton No. 2 ICE FUT US  (Cotton), has been used. These data have been obtained from the Reuters DataLink database of the Thomson Reuters Company. The period studied extends from 1.1.2002 to 1.31.2009. The stationarity of the returns of the cotton futures  is evident from  Figure 1. This result is confirmed with the application of the ADF test and presented in the following section. The data are characterized by a leptokyrtic distribution a fact that is revealed with the low value of the kyrtosis coefficient (6.209) while the value of the Jarque-Bera coefficient is equal to 880.47, implying a non-normal distribution.
　　In order to describe fully our data, an autocorrelation test was also employed. Based on the aforementioned results we concluded, that the time series studied suffers from autocorrelation. This result necessitates modelling the futures’ returns behaviour through a model that will eliminate this problem.
　　                       Methodology
　　The test applied for the survey of nonlinearities in the cotton returns is the BDS test 32, an evolution of correlation dimension test33. The test mentioned above is very demanding regarding the size of the sample, since enormous amount of data is needed in case of high dimensional chaos.
　　BDS test tests the null hypothesis of whiteness (independently and identically distributed observations) against an unspecified alternative through a nonparametric technique. To be more specific the BDS statistic is given by the following equation: W(T, m, ε ) = √ T) , , () , 1, ( ) , , (H VH HmT C m T C7š                                                 (1) where  m = embedding dimension, C(T, m, ε ) = the correlation integral,Vš (T, m,  ε) = an estimate of the asymptotic standard deviation.
　　The statistic BDS under the null hypothesis is asymptotically normal34. This test was applied on our data, while the time series studied was investigated with the ADF test in order to examine the stationarity of the series and was found stationary in first differences and not in levels.
　　The next step involves the survey of autocorrelations in the time series studied. This survey is based on the Q test for 36 lags of the time series studied. The next univariate tests employed for the survey of nonlinearities are the MC Leod-Li 35 and Engle 36tests.
　　The first test may be conducted with the application of the Box- Ljung Q statistic of the squared residuals. An ARMA process is preceded aiming at the filtering of our data. The initial (raw) data 0  100  200  300  400  500  600  -0.05  0.00   0.05 0.10 0.15Series: CottonSample 1 1937Observations 1937Mean    0.000267 Median    0.000000 Maximum   0.147800 Minimum -0.073900 Std. Dev.    0.019633 Skewness    0.390531 Kurtosis    6.209238 Jarque-Bera  880.4690 Probability  0.000000 Figure 1.  Returns of the cotton futures. Journal of Food, Agriculture & Environment, Vol.9 (2), April 2011  553 may be examined with the use of the k autocorrelation coefficients for {xt} and {x2t}. The Q statistic is used aiming at the examination of the existence of serial correlation.
　　According to this method, a time series follows an i.i.d. process (under the null hypothesis) if for a fixed L the following equation: T1/2ρ2(k) =  [ρ2(1), ...,  ρ2(L)]                                                                 (2)is asymptotically a multivariate unit normal. As a consequence, for a high value of L, the Box– Ljung statistic Q ∼ x2(L). Q is given by the following equation: Q  = T(T+2) ¦  7Ljj1]} {[ 22)NU                                                                                             (3) The null hypothesis is that of a linear generating mechanism for the data. The other test 36 examines the non-linearity through a Lagrange multiplier test in the second moments. This test preconditions the regression of the squared residuals: tjj t jt u a  ¦ šUH D H1202   (4)
　　The non-existence of ARCH-type effects means that the coefficients are non-significant and thus the regression will have a limited explanatory power while the coefficient of determination is very low. According to the null hypothesis given that the sample size is T, there are no ARCH-type effects and the statistic used for this test is the TxR2∼  x2p. If TxR2 is large enough then we reject the null hypothesis under which there is no ARCH-type errors.
　　The final step employed in the methodology applied involves the estimation of Mackey GARCH (p,q) model. In order to model the observed dynamics in financial markets we use an equation that includes two parts. The first part is deterministic (intrinsic deterministic dynamics) and the other one is the stochastic part (the random noise). The model might have the following form: Xt = f (Xt-1,...) +  εt                                                                                   (6) where Xt = observable non-linear function, f = a deterministic non-linear function and εt∼ iid.
　　In order to detect the complexity of the financial time series we are going to use a version of the Mackey-Glass equation 37. This model is a noisy Mackey-Glass equation whose errors follow a GARCH (p, q) process. That is why it is called an MG-GARCH (p, q) model. The particular model provides us with an econometric tool to study volatility-clustering phenomena. Its main characteristic is that volatility-clustering is interpreted as an endogenous phenomenon. The existence of volatility clustering stems from the interactions between fundamentalists and noisy traders38. The behaviour of the noisy traders cannot be explained and thus it is given exogenously. This is the stochastic part of the Mackey-Glass equation. The deterministic part of the MG-GARCH (p, q) model represents the determinant stock prices given that the market is dominated by fundamentalists. As fundamentalists, are considered the traders who have rational expectations (are based on the information from the macroeconomic environment), regarding the future price and dividends of an asset. This fact is not valid in reality though and that is why this equation cannot represent all the traders. The model is given by the following equations25:Rt = α  -cttRRWW1-δRt-1 + bRt-j(1-Rt-j)+ εt                                                         (7) ht = α0 + α1ε2t-1 + β1ht-1                                                                                                 (8) where t, j denote the delays, c = constant, Rt= the returns, ht = the GARCH variance. The part  a -Rt- W 1+ Rct- W - δ Rt-1 filters the dynamics that positive feedback trading causes, consisting the discretized version of the model under preview, whereas the part bRt-j(1-Rt-j) models structures25 that may be attributed to negative feedback in the market. The positive feedback describes the investors that buy when the price rises and sell when the price falls. The negative feedback describes the opposite investors’ behaviour.
　　The main advantage of this model is that two nonlinear trading strategies may capture more complicated dynamics. To be more specific this model consists a more realistic approach given the assumption of the existence of more than one type of investor as the driving force of endogenous perturbations is more realistic.
　　                           Results and Discussion
　　Initially, within the effort to survey the time series studied we employed the ADF stationarity test on the closing prices as well as on the time series of returns generated by the closing prices. For the first time series we confirm that the time series is I(1), while for the time series of returns stationarity is confirmed. The aforementioned results are presented in Table 1. The application of the BDS independence test is the second stage within our methodology and has given the results presented in Table 2.
　　 Based on the aforementioned results the null hypothesis of independent and identically distributed observations is rejected. This result can confirm neither nonlinearity nor chaotic behavior39. This test provides indirect evidence regarding the nonlinear dependence either chaotic or stochastic a necessary but not a sufficient condition for chaos 40.
　　Employing Mc Leod-Li and Engle tests has given us the results presented in Table 3. The null hypothesis in Mc Leod-Li test suggests that there is a linear generating mechanism while in Engle test the null hypothesis suggests that there are no ARCH-type effects. Rejection of null hypotheses in both tests confirms the existence of nonlinearities and ARCH heteroscedasticity for the time series studied, respectively. Variable ADF statistic cp -1.855875 ǻcp -34.07714 r -43.14551 ǻr - Table 1. Results of ADF test. m  İ 2 0.009083 3 0.015522 4 0.019424 5 0.023342 6 0.025219 Table 2.  BDS statistics at dimensions (M) 2-6.      554  Journal of Food, Agriculture & Environment, Vol.9 (2), April 2011
　　Based on the aforementioned results we may model the returns behaviour with a nonlinear GARCH model. The best fit GARCH model is the GARCH (1,1) model while the mean equation is non- linear. The results of the estimated model are presented in Table 4. All the estimated coefficients are statistically significant confirming the nonlinearity of the mean equation in the Mackey GARCH (1,1) model. Furthermore, noisy chaotic structures are responsible for nonlinearity in the mean of the time series under preview. Based on the aforementioned model we conducted a forecast, while the two coefficients RMSE and MAPE have given us the following results: RMSE=0.000631MAPE=859.46
　　Based on the aforementioned results we may conclude that the particular model provides us with a satisfactory forecasting tool at least in the short term. The residuals of the estimated equation do not have a problem of low kyrtosis coefficient or a problem of normality nor a problem of autocorrelation, as confirmed by the use of different tests. The attractor of this equation has heteroscedastic errors given that the particular equation follows a GARCH (1, 1) process.
　　                           Conclusions
　　Chaotic behaviour of macroeconomic data has been a subject of extensive study in the past but the evidence for its existence is extremely weak 31. On the other hand, the studies of commodity prices have generally found evidence consistent with low dimension chaos. The chaos on the other hand is confirmed in many cases of the commodity prices. One could argue that seasonality or short data spans may account for this diversity in the results found.  Nonlinearities were detected with a number of univariate tests. For the tests we employed daily data of cotton futures’ prices for a time period of seven years, derived by Reuters. Independence of the observations has been rejected according to the results of the BDS test. This result, though, provides no evidence for non-linearities in the price behaviour. Thus, other tests were implemented in order to confirm the existence of non- linearities such as Engle and McLeod-Li tests. The results of those tests confirmed that the behaviour of the time series is nonlinear, a result that is consistent with other studies that examined the nonlinearity of other financial time series 42, 40, 26. In addition 32 in another study where the futures prices of agricultural products Series McLeod – Li (L=24) Engle (p=5) ǻc 0.000 0.000 Table 3. Significance level for Mcleod-Li and Engle tests. Coefficient  Estimate  z - statistic Į  0.179967 5.194686  į  -0.203272 6.054522  b 0.197301 8.998634 Į0 3.05E-07 2.949286 Į1 0.026419 1.828737 ȕ1 0.609027 4.637427 Table 4 . CMG – Garch (1,1) estimation results.
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