A Comparison of Linear and Nonlinear Models in Forecasting Market Risk: The Evidence from Turkish Derivative Exchange

: This paper aims to compare the volatility forecasting performance of linear and nonlinear models for ISE-30 future index which is traded in Turkish Derivatives Exchangefor the period between 04.02.2005-17.06.2011. As a result of analyses, we conclude that ANN model has better forecasting performance than traditional ARCH-GARCH models. This result is important in many fields of finance such as investment decisions, asset pricing, portfolio allocation and risk management


Introduction
Volatility is defined as the fluctations in security prices. As a barometer of the market risk, volatility is important for investment decisions, asset pricing, portfolio allocation and risk management in finance. In this respect, it is crucial to forecast volatility accurately in finance literature. Associated with the increasing importance of volatility, different volatility models come into use in the finance literature. Conditional heteroscedasticity models are the most commonly used volatility models in forecasting financial assets' volatility. In volatility forecasting, in addition to Engle's (1982) Autoregressive Conditional Heteroscedasticity Model (henceforth ARCH) and Bollerslev's (1986) Generalized Autoregressive Heteroscedasticity Model (henceforth GARCH), Artifical Neural Network (henceforth ANN)model is being used in the literature.ANN model which simply mimics the human brain function is successful in the estimation of the stock price behaviour due to its feature of learning from tha data. Model is working with multiple variables, easy adaptation to the noisy data and handling complex and nonlinear problems (Karaatlı etal. (2005); Cinko and Avcı (2007). The paper aims to compare the performance of linear and nonlinear models in forecasting the volatility of ISE-30 future contracts which is traded in Turkish Derivatives Exchange. Future markets are important in terms of reducing the uncertainty about the future, forecasting the future values of prices and interest rates, providing efficienct risk management and supporting the spot markets. In the literatüre, the papers mostly focus on comparing forecasting performance of volatility models in spot markets. Different from the existing literature, this paper focus on forecasting volatility in future markets. Therefore, the findings of paper will contribute to the existing literature. The paper organized as follows section 2 summarise the literature, section 3 describes the methodology used, section 4 explains the dataset used, section 5 explain and interpret the emprical findings and section 6 concludes.
In Turkey, (Egeli et al., 2003;Diler, 2003;Cinko & Avcı, 2007;Yıldız et al., 2008) forecast different indices in Istanbul Stock Exchange by using ANN model. Dutta andShekhar (1988); Moody and Utans (1991); Surkan andSingleton(1991); Kimetal.(1993); Maher andSen (1997) make risk assessment of fixed income securities by employing ANN model. Recently, it is possible to find different papers that compare the performance of ANN model and traditional models in forecasting financial data. Boyd etal. (1996) find that ARIMA modelis superior to ANN model in forecasting commodity prices. Chiang et al. (1996) conclude that traditional statistical methods are better than ANN model in forecasting the end of year value of mutual funds. Dutta and Shekhar (1988) find that ANN model is more successful in rating and pricing bonds. Yoon and Swales (1990) conlude that ANN model has better performance than discriminant analysis in forecasting stock prices. Altay and Satman (2005) state that linear regression model has better forecasting performance than ANN model in ISE-30 and ISE-100 indices. Karaatlı etal. (2005) find that ANN model is superior to the linear regression model in forecasting of ISE-100 index. Cinko and Avcı (2007) indicate that ANN model is more successful than linear regression model in daily and seasonal forecasting of ISE-100 index. Dunis et al. (2012) (2011) compare the performance of traditional ARCH-GARCH models and ANN model and conclude that ANN model has better performance than traditional ARCH-GARCH models. Different from these papers, Mantri etal.(2010), Anwar and Mikami (2011) state that ARCH-GARCH models are superior to ANN model.

Methodology
Linear Models: In linear models, we firstlyuse AR, MA, ARMA model to identify the best fitted model in modelling conditional mean. AR model assumes that a time series is explained by its lagged values or an error term. A simple AR(p) model is given in Equation (1).
In equation (1), yt denotes a time series, denotes error term. MA model is a function of the lag values of error term and unpredictable error term. MA model is given in Equation (2).
(2) ARMA models are a combination of AR and MA models. An ARMA model predicts a value in a response time series as a linear combination of its own past values and past errors and defined as in Equation (3) (Enders, 2004) : After identification of appropriate ARMA model, it is necessary to examine whether time series include Autoregressive Conditional Heteroscedasticity. In the literature, the most suggested models to test the existence of ARCH effect is Engle's (1982) (4), square them ( t  2 ) and model them as a dependent variable in equation (5). yt = δ + Ø 1yt-1 + Ø 2yt-2 +…….+ Ø pyp-1 + t  (4) t  2 = δ + α1ε 2 t-1+ α2 ε 2 t-2+.......+ αq ε 2 t-q+ vt (5) As the ARCH effect is detectedin stock exchange indices, the volatilityis modelled by using ARCH-GARCH models. ARCH model which is suggested by Engle (1982) assumes that the variance of "u" at time t, σt 2 depends on the square of error term, u 2 (t-1) at time t-1.
In this context, ARCH(q) and GARCH(q,p)models are given in Equation (6), >0, = +vt (6) GARCH models which is the generalized version of ARCH models are introduced by Engle (1982) and Bollerslev (1986). GARCH models include conditional variance equation in addition to conditional mean equation. GARCH model can be defined as in Equation (7).
The variance restrictions of the model are as follows: Artificial Neural Networks: ANN is a flexible non-linear modeling tool. The human being's learning ability is transferred to a computer environment with ANN. ANN is composed of a number of processing elements, which come together within the frame of particular rules which are called neurons or nodes (Haykin, 1994;Zhang et al., 1998). An ANN generally consists of three layers of interconnected neurons. A three layer ANN is shown in Fig. 1. The first layer is called the input layer where external information is received. Each neuron in the input layer sends signals to the hidden layer. Information received from the input layer is processed in the hidden layer. The output layer transmits the information outside of the network.

Figure 1: A three layer ANN
Hidden Layer

Output Layer
Although ANN can be applied successfully in many fields, it has some disadvantages. ANN requires a long training process in developing the optimal model. ANN has also been criticized for lack of theory. There is no opportunity to explain the result produced by ANN, in other words, the model acts as a black box (Chen and Huang 2003;Piramuthu, 1999;Trippi and Turban 1996;West, 2000).

Data:
The dataset includes the closing prices of ISE-30 future index, which is traded in Turkish Derivatives Exchange, for the period between 04.02.2005-17.06.2011 providing 1606 observations. Return series of ISE-30 index futures are calculated as in Equation (11). Pt denotes the closing price of ISE-30 index futures at time, t.

ARCH-GARCH Model:
The summary statistics of the return series are given in Table.1. The mean return and standard deviation of ISE-30 index futures are 0.000469 and 0.020123, respectively. According to the skewness and kurtosis values, it is clear that ISE-30 future index return do not have normal distribution.  We investigate the stationary of the return series by using Augmented Dickey-Fuller unit root test. According to the findings in Table.2, t-values are greater than 1%, 5% and 10% critical values for trended and untrended models. Therefore, we reject the null hypothesis of "series have unit root" and conclude that return series are stationary. For this reason, we use return series in the subsequent analysis. After testing the stationary of return series, we use Autocorrelation Function (henceforth ACF) and Partial Autocorrelation Function (henceforth PACF) to identify the best fitted ARMA model. In addition to ACF and PACF functions, we also use Akaike and Schwarz criterion to determine the appropriate ARIMA model. The appropriate ARIMA models are presented in Table 3.

Table 4: ARCH-LM Test
The next stage is to test the existence of ARCH effects in residuals by using ARCH-LM test. According to the findings in Table.4, prob values of F-statistic is less than 0.05 and so we reject the null hypothesis. This finding supports the presence of ARCH effect in residuals and consequently we use ARCH model to model volatility in ISE-30 future index.The best ARCH model for ISE-30 index futures is summarized in Table 5.   Vellido et al., (1999) point out that more than 75% of business applications using ANN will adopt the BPN training algorithm; this study also uses the feed forward multilayer perceptrons with the BPN training algorithm. As recommended by Zhang et al., (1998) the single hidden layer network is sufficient to model any complex system, therefore, the designed network will have only one hidden layer. Egeli et al., (2003) also report that 1 hidden layer is better than 2, 3 and 4 hidden layer in modeling of the Istanbul Stock Exchange. Generally, the learning rate is set between 0.01 and 0.4, the momentum is set between 0.8 and 0.99 and the training lengths ranging from 1000 to 10000 epochs (Chuang and Lin, 2009). Determining the number of hidden nodes is generally associated with input nodes. The most commonly used way in determining the number of hidden nodes is via experiments or trial and error process. The number of hidden nodes to be tested 2n, 2n±1 and 2n±2, n denotes input nodes (Hecht-Nielsen, 1990). To determine the optimal number of hidden nodes, for 5 time lags network structure, 8, 9, 10, 11, 12was tested, when the learning rate, momentum and training epochs are set to 0.1, 0.9 and 2,000, respectively. In other words, we develop 5 different models for this purpose.  Table 6 are multiplied by 10 4 .

169
The MAE of the 5 time lags ANN model is shown in Table 6. The network architecture with the lowest MAE (5-10-1) is considered as the optimal network architecture. After determining the optimal network architecture, various learning parameters are applied on this architecture. Nine different learning rates (from 0.001 to 0.5) and four different momentum rates (from 0.7 to 0.99) have been tested, so here we have also developed 36 different models. The MAE of various learning parameters for the 5-10-1 architecture is shown in Table 7. The minimum MAE of the 5 time lags ANN model is 0.000350652 when the learning rate and momentum are set to 0.5 and 0.7 respectively. 1

Conclusion
Volatility is sudden movements in stocks price levels and high volatility in stock markets affects investment decisions directly.Market participants consider especially high volatile markets to increase gain from investment. Therefore different volatility models and that's performance have become important for investors. In this paper, we forecast the volatility of ISE-30 future index return by applying ARCH-GARCH models and ANN model and compare the relative performance of models according to their MAE values. The findings of the paper support that ANN model, which represents the nonlinear models, has better forecasting performance than traditional ARCH-GARCH models. This result is consistent with those of Lim and McNelis (1998); Schittenkopf et al., (1998) In conclusion, volatility forecasting is important for many financial applications such asinvestment, asset pricing, portfolio allocation and risk management. From this perspective, the findings of the paper is important for market participants since they can increase the success of their financial decisions by using ANN model in forecasting market risk.