Estimating conditional volatility with neural networks

Ian T. Nabney, H. W. Cheng

Research output: Unpublished contribution to conferenceUnpublished Conference Paperpeer-review


It is well known that one of the obstacles to effective forecasting of exchange rates is heteroscedasticity (non-stationary conditional variance). The autoregressive conditional heteroscedastic (ARCH) model and its variants have been used to estimate a time dependent variance for many financial time series. However, such models are essentially linear in form and we can ask whether a non-linear model for variance can improve results just as non-linear models (such as neural networks) for the mean have done. In this paper we consider two neural network models for variance estimation. Mixture Density Networks (Bishop 1994, Nix and Weigend 1994) combine a Multi-Layer Perceptron (MLP) and a mixture model to estimate the conditional data density. They are trained using a maximum likelihood approach. However, it is known that maximum likelihood estimates are biased and lead to a systematic under-estimate of variance. More recently, a Bayesian approach to parameter estimation has been developed (Bishop and Qazaz 1996) that shows promise in removing the maximum likelihood bias. However, up to now, this model has not been used for time series prediction. Here we compare these algorithms with two other models to provide benchmark results: a linear model (from the ARIMA family), and a conventional neural network trained with a sum-of-squares error function (which estimates the conditional mean of the time series with a constant variance noise model). This comparison is carried out on daily exchange rate data for five currencies.
Original languageEnglish
Number of pages15
Publication statusPublished - 1997
EventFourth Internation Conference: Forecasting Financial Markets -
Duration: 1 Jan 19971 Jan 1997


ConferenceFourth Internation Conference: Forecasting Financial Markets

Bibliographical note



  • heteroscedasticity
  • autoregressive
  • time series
  • neural networks
  • Density Networks
  • Bayesian approach
  • conditional mean
  • exchange rate


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