  Using a time series model to mimic an observed time series has a long
history. However, with regard to this objective, conventional estimation
methods for discrete-time dynamical models are frequently found to be wanting.
In fact, they are characteristically misguided in at least two respects: (i)
assuming that there is a true model; (ii) evaluating the efficacy of the
estimation as if the postulated model is true. There are numerous examples of
models, when fitted by conventional methods, that fail to capture some of the
most basic global features of the data, such as cycles with good matching
periods, singularities of spectral density functions (especially at the origin)
and others. We argue that the shortcomings need not always be due to the model
formulation but the inadequacy of the conventional fitting methods. After all,
all models are wrong, but some are useful if they are fitted properly. The
practical issue becomes one of how to best fit the model to data. Thus, in the
absence of a true model, we prefer an alternative approach to conventional
model fitting that typically involves one-step-ahead prediction errors. Our
primary aim is to match the joint probability distribution of the observable
time series, including long-term features of the dynamics that underpin the
data, such as cycles, long memory and others, rather than short-term
prediction. For want of a better name, we call this specific aim feature
matching.
