The “traditional” tools of time series analysis (acf’s, spectral analysis) may find no evidence that we could use a linear model, but the data may still not be independent.
Portmanteau tests for non-linear dependence have been developed.
The simplest is Ramsey’s RESET test, which took the form:
Here the dependent variable is the residual series and the independent variables are the squares, cubes, …, of the fitted values.
Many other non-linearity tests are available - e.g., the BDS and bispectrum test
BDS is a pure hypothesis test. That is, it has as its null hypothesis that the data are pure noise (completely random)
It has been argued to have power to detect a variety of departures from randomness – linear or non-linear stochastic processes, deterministic chaos, etc)
The BDS test follows a standard normal distribution under the null
The test can also be used as a model diagnostic on the residuals to ‘see what is left’
If the proposed model is adequate, the standardised residuals should be white noise.
Chaos theory is a notion taken from the physical sciences
It suggests that there could be a deterministic, non-linear set of equations underlying the behaviour of financial series or markets
Such behaviour will appear completely random to the standard statistical tests
A positive sighting of chaos implies that while, by definition, long-term forecasting would be futile, short-term forecastability and controllability are possible, at least in theory, since there is some deterministic structure underlying the data
Varying definitions of what actually constitutes chaos can be found in the literature.
