I have a purely deterministic system-theoretic background, so please bear with me if this is elementary.
The question is related to: How to test whether a series data follow Ornstein-Uhlenbeck process (OU process)?, but my case is more generic.
I have fitted a parametric (1-5 parameters, case-specific) dynamic model to financial time series data; in the simplest case, the model reduces to an Ornstein-Uhlenbeck specification. I do not (yet) aim to predict anything. I want to confirm - or refute - the hypothesis that this model describes the data. How can I do it? The model is based on sound economic theory (under certain circumstances), and I am now investigating its validity in practice.
My initial thought was to study the statistical properties of the residual process, i.e. $y_{actual}(t) - y_{model}(t)$. But should I test for stationarity (e.g. Augmented Dickey-Fuller)? Or normality (e.g. Kolmogorov-Smirnov)? Basically, the residual process should be unpredictable for the model to hold true, right?
On the one hand, I am haunted by the Weierstrass approximation theorem, which implies that by increasing model complexity, I can always find a model to describe any sufficiently well-behaved data series. On the other hand, I keep reading journal papers (e.g. the widely cited ones by D. Sornette and coworkers on bubbles and crashes) where specific deterministic models are fit to explain dynamical phenomena of financial markets. As far as I understand, a major argument in favour of such models is that a certain fixed model (with case-specific parameter values) appears to provide a good description of the target series in several different circumstances and time periods. This reasoning would be sufficient for my case, too, as indicated above, but I fear I am missing something important.
