We introduce new methods of filtering and forecasting for the causal-noncausal convolution model. This model represents the dynamics of stationary processes with local explosions, such as spikes and bubbles, which characterize the time series of commodity prices, cryptocurrency exchange rates, and other financial and macroeconomic variables. The convolution model is a structural mixture of independent latent causal and noncausal component series. We propose an algorithm that recovers the latent components by evaluating the filtering density of one component, conditional on the observed past, present, and future values of the time series. Forecasts of the observed time series are obtained as a combination of filtered causal and noncausal component forecasts. The new filtering and forecasting methods are illustrated in a simulation study and compared with the results obtained from the mixed causal-noncausal autoregressive MAR model in application to WTI crude oil prices.