The Bayesian evidence framework is applied in this paper to least squares support vector machine (LS-SVM) regression in order to infer nonlinear models for predicting a financial time series and the related volatility. On the first level of inference, a statistical framework is related to the LS-SVM formulation which allows one to include the time-varying volatility of the market by an appropriate choice of several hyper-parameters. The hyper-parameters of the model are inferred on the second level of inference. The inferred hyper-parameters, related to the volatility, are used to construct a volatility model within the evidence framework. Model comparison is performed on the third level of inference in order to automatically tune the parameters of the kernel function and to select the relevant inputs. The LS-SVM formulation allows one to derive analytic expressions in the feature space and practical expressions are obtained in the dual space replacing the inner product by the related kernel function using Mercer's theorem. The one step ahead prediction performances obtained on the prediction of the weekly 90-day T-bill rate and the daily DAX30 closing prices show that significant out of sample sign predictions can be made with respect to the Pesaran-Timmerman test statistic.
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