For the period 1826--2000, we analyze the spectral characteristics of the winter North Atlantic Oscillation (NAO) index and its predictability based on Singular Spectral Analysis (SSA) and Autoregressive Moving Average (ARMA) models. In the first part, SSA is applied to the winter NAO index to isolate its main spectral characteristics. Based on the SSA, a reconstruction (filtering) of the winter NAO index series was carried out. Results of the SSA indicate that the winter NAO index can be broken down into some modulated amplitude oscillations with periods around 7.7 and 4.8 years, some oscillations associated with a broadband peak of period around 2.4 years along with nonlinear trends. The sum of these components, the SSA-filtered series, explains 56% of the variance of the raw winter NAO index. The SSA-filtered series is particularly reliable, reproducing the NAO phase during extreme events (winter NAO index ≥ 1 or ≤ -1); for this subset of events, the phase of the actual and SSA-filtered series shows to be the same in 91% of the cases. The high positive values observed in the winter NAO index in the last eighties and nineties appear to be associated with the simultaneous presence of a positive trend, starting in the early eighties and of unprecedented steepness, and an oscillation with period around 7.7 years, having very high amplitude. In the second part, an ARMA model has been fitted to the filtered winter NAO index and a forecasting experiment was conducted; results are tested against the raw winter NAO index. Results show that the ARMA modeling has useful 1-year-ahead forecasting abilities. Particularly, over the period 1986--2000, not used to fit the model, the model skill is 27.8% better than climatology and 43.3% better than persistence (38.5% and 47.6%, respectively, when taking into account only extreme NAO events). Additionally, percentage of cases in which the NAO phase was accurately predicted proved to be 80% (88% for extreme NAO events). For 2001/2002 and 2002/2003 winters, persistence in the negative phase of the NAO is predicted, having an index value close to -1.