Why we need non-linear time series models and why we are not using them so often

The Wold Decomposition theorem says that under fairly general conditions, a stationary time series $X_t$ has a unique linear causal representation in terms of uncorrelated random variables. However, The Wold Decomposition theorem gives us a representation, not a model for $X_t$, in the sense that we can only recover uniquely the moments of $X_t$ up to second order from this representation, unless the input series is a Gaussian sequence. If we look for models for $X_t$, then we should look for such model within the class of convergent Volterra series expansions. If we have to go beyond second order properties, and many real data sets from financial and environmental sciences indicate that we should, then linear models with iid Gaussian input are a very tiny, insignificant fraction of possible models for a stationary time series, corresponding to the first term of the infinite order Volterra expansion. On the other hand, Volterra series expansions are not particularly useful as a possible class of models, as conditions of stationarity and invertibility are hard to check, if not impossible, therefore they have very limited use as models for time series, unless the input series is observable. From a prediction point of view, the Projection Theorem for Hilbert spaces tells us how to obtain the best linear predictor for $X_{t+k}$ within the linear span of $\{X_t, X_{t-1}, \dots,\}$ , but when linear predictors are not sufficiently good, it is not straightforward to find, if possible at all, the best predictor within richer subspaces constructed over $\{X_t, X_{t-1}, \dots,\}$. It is therefore important to look for classes of nonlinear models to improve upon the linear predictor, which are sufficiently general, but at the same time are sufficiently flexible to work with. There are many ways a time series can be nonlinear. As a consequence, there are many classes of nonlinear models to explain such nonlinearities, but whose probabilistic characteristics are difficult to study, not to mention the difficulties associated with modeling issues. Likelihood based inference is particularly a difficult issue as for most nonlinear processes, we can not even write the likelihood. However, recently there has been very exciting advances in simulation based inferential methods such as sequential Markov Chain Monte Carlo, Particle filters and Approximate Bayesian Computation methods for generalized state space models which we will mention briefly.