A nonlinear recurrent neural network estimation of conditional mean and variance

In the context of nonlinear financial time series, both conditional mean and variance (volatility) tend to evolve over time and depend on previous values. Commonly, the objective function used in Artificial Neural Networks (ANNs) is the sum of squared errors. This requires the target and forecasted output vector to have the same dimension. It is therefore of interest to consider recurrent neural networks with two-dimensional output even though the target data are one-dimensional. The idea of the optimization algorithm can be extended to this situation. In additional, the negative log-likelihood based on a parametric statistical model is a possible alternative to the traditional least squares objective. It has been found that the Root Mean Square Prediction Error (RMSPE) for the mean and variance prediction is smaller for of the developed recurrent neural network (S-GRNN) than others in the majority cases. S-GRNN also provides better performance to other for the real data set (S&P500 Index)