###
07/03/2012, 14:30 — 15:30 — Room P3.10, Mathematics Building

K F Turkman, *CEAUL - DEIO - FCUL - University of Lisbon*

```
```###
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.