Non-Gaussian time series variables are prevalent in the economic and finance spheres, with state space models often employed to analyze such variables and, ultimately, to produce forecasts. A review of the relevant literature reveals that existing methods are characterized by a reliance on (potentially incorrect) parametric assumptions and are often computationally expensive. The primary aim of this thesis is to develop a non-parametric approach to forecasting - within the state space framework

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... - with computational ease an important focus. With a view to capturing all relevant information about the likely future values of the variable of interest, the approach is used to produce non-parametric estimates of the full forecast distribution over any time horizon. Simulation experiments are used to document the accuracy of the non-parametric method relative to both correctly and incorrectly specified parametric alternatives, in a variety of relevant settings. Applying a range of methods for evaluating and comparing distributional forecasts, the non-parametric method is shown to perform significantly better, overall, than misspecified parametric alternatives while remaining competitive with correctly specified parametric estimators. Focus is then given to the development of a new non-Gaussian state space model for observed realized volatility from which estimates of forecast distributions of future volatility are produced using the non-parametric method. In an empirical illustration, the non-parametric method is used to produce sequential estimates of the out-of-sample one-step-ahead forecast distribution of realized volatility on the S&P500 index during the recent financial crisis. A resampling technique for measuring sampling variation in an estimated forecast distribution is also demonstrated. The proposed filtering algorithm is further extended to cater, in particular, for multi-step-ahead forecasting and multivariate systems. A simulation-based version of the algorithm is also illustrated, with the algorithm i [...]