Predicting Wheat Futures Prices in India

“Derivatives are an extremely efficient tool for risk management”

“Derivatives are financial weapons of mass destruction”

–Warren Buffett

Abstract

Futures markets perform their economic roles of price discovery and hedging only when they are efficient. One of the important features of efficient market is that one cannot make abnormal profits from the futures markets by trading in it. This paper addresses the question of whether Indian wheat futures prices can be forecast. This would add to our knowledge whether wheat futures market is efficient, and would enable brokers, traders and speculators to develop profitable trading strategy. We employ the economic variable model to predict the wheat futures prices, and employ out of sample point forecasts. We also evaluate the robustness of our results by employing several alternative specifications, viz. ARMA process and artificial neural network technique. We then test the statistical significance of point forecast using the Diebold and Mariano test. We consider random walk orecast as the bench mark. In order to predict the evolution of wheat futures prices, we use traders’ expectations about the futures prices, a number of economic variables and futures prices (lagged) of wheat. The study finds that the futures price of wheat cannot be forecast, and the wheat futures market is efficient.

Appendices

Appendix

Architecture of Artificial Neural Network (ANN)

The neural network comprises three layers: the input layer, the (hidden) intermediate layer, and the output layer. While the input layer is analogous to the regressor variables in the traditional regression framework, and the output layer corresponds to the regressand, it is the (hidden) intermediate layer that is the distinguishing component of a neural network framework. This intermediate layer tends to get denser and, indeed, multi-layered as the phenomenon of interest (or the data-generating process as it were) gets more complicated.

An artificial neural network consists of processing elements, we call it neurons.¹⁰ Each neuron is like a variable. The arrows linking the neurons signify the parameters of neurons (Fig. 1). The parameters of the neurons are the weights associated with them. In this technique, we connect the input to the output data via a system of neurons and these neurons learn from other neurons. The following are the steps used in constructing an artificial neural network.

Fig. 1

Artificial neural network

First, we decide the structure of the network which is explained in Fig. 1. It includes number of input nodes, number of hidden layers and nodes, the transfer function and the output node. The number of input nodes are our lagged explanatory variables such as the gram futures prices, the real rate of interest, the basis defined as the difference between spot and futures prices, and the futures price of wheat in the USA, ratio of high to low price. The first lag of the wheat futures prices is also an explanatory variable. Therefore, the number of input nodes are six. We have one hidden layer (three neurons), and one output node, i.e., the wheat futures prices. The output node is our dependent variable.

Second, we split the data on explanatory variables and dependent variables into two groups. The first group is used to train the network, and the second group is used to validate the network using out-of-sample forecast. Our sample size is 891. We have used the initial 800 sample points for fitting the network, and the remaining 91 sample points to validate the model.

Third, we scale all explanatory as well as output variables in the range of 0 to 1. It is required because of the mathematical structure of the network. The rescaling of the data has no impact on the pattern. We rescale the data using the following formula:

$$Scaled\, value = \frac{Actual - Minimum}{{Maximum - Minimum}}$$

where Minimum $=$ Expected minimum value in data. Maximum $=$ Expected maximum value in data.

Fourth, we set the initial training weight and begin a training epoch. An epoch implies the computation of errors and change of weights by processing the entire sample in the training data set. In addition, we give initial values to all the weights in the network. The initial weight affects the final solution. The good starting point of the initial weight is unknown. Most programs recommend the randomization of the training weights between − 1 and 1 (DeLurgio 1998). We have used RATS software, and it determines the initial weight randomly between − 1 and 1.

Fifth, we provide the scaled inputs at neurons 0, 1, 2, 3, 4 and 5 (Fig. 1), and we designate these neurons as $I_{0}$, $I_{1}$, $I_{2}$, $I_{3}$, $I_{4}$ and $I_{5}$, respectively. Generally, we name input node as $I_{j}$ and output node as $O_{j}$. At input layer, there is no transformation of explanatory variables, so output of an input neuron is the same as its input value.

Sixth, every neuron in the input layer receives its scaled value and transmits it to every neuron in the hidden layer. These scaled inputs are weighted and provide input to hidden neurons in step (7). There is parallel processing of inputs at other two hidden neurons. In our network, there is one hidden layer with three neurons.

Seven, we give weights and sum inputs to receiving nodes.¹¹ This implies that at every hidden neuron, we give weights to the output of input neurons and sum it. Therefore, the input to neuron 6 is written as:

$$I_{6} = W_{06} O_{0} + W_{16} O_{1} + W_{26} O_{2} + W_{36} O_{3} + W_{46} O_{4} + W_{56} O_{5}$$

We determine the weights and values after several iterations of the training data. The weights are memory and intelligencef the network. They are similar to the coefficients of the regression model. But, weights in the neural network are hard to interpret.

Eight, we transform the weighted inputs at each hidden node to outputs ranged from 0 to 1. We employ the logistic function to express the relationship between input and output of neuron. This is a nonlinear transformation.¹² Output at neuron 6 is

$$O_{6} = \frac{1}{{\left( {1 + e^{{ - I_{6} }} } \right)}}$$

where $e$ is the natural number whose value is 2.718…

The output value of $O_{6}$ is one input for neuron $I_{9}$ (Fig. 1).

Nine, we give weights and add hidden neuron outputs as input to output node. The output neuron is the final neuron. From Fig. 1

$$I_{9} = W_{69 } O_{6} + W_{79 } O_{7} + W_{89} O_{8} + \theta_{3}$$

where $\theta_{3}$ is the bias of the forecast. It is like a constant term in regression equation.

Ten, we transform the weighted inputs $I_{9}$ to output of $O_{9}$ ranged from 0 to 1. $O_{9}$ is the final output of the neural network. Again we employ the logistic function. Output at neuron 9 is

$$O_{9} = \frac{1}{{\left( {1 + e^{{ - I_{9} }} } \right)}}$$

Eleven, we then calculate errors in the output. We compare the scaled ANN output value $O_{9}$ to the true output (scaled) $D_{9}$, and compute the error.

$$e_{i} = D_{9} - O_{9}$$

where $i$ denotes the observation number in the training data.

Twelve, in this step, there is back propagation of errors, i.e., there is adjustment of weights in the training so as to achieve the minimum residual mean standard error (RMS) value (DeLurgio 1998). The method of spreading information about errors from the output layer back to hidden layer is termed as back propagation.

Thirteen, we continue the epoch by repeating the steps from fifth to twelfth for all input data.

Fourteen, there is calculation of epoch RMS value. If we get the low value of the RMS,¹³ we move to the next step, otherwise we repeat the steps (5) to (14).

Fifteen, we validate the ANN results with out-of-sample data. The model is valid if the training RMS is consistent with the out-of-sample RMS.

Sixteen, we then use the model for forecasting.