We now begin our ascent toward the main peak of our effort, although there are still interesting and useful peaks to climb in the remaining chapters. The theories of stochastic processes and probability are put to work below to address the central question: can we reliably deduce a model or class of dynamic models from a single time series, where “systematically rerunning the experiment” is impossible? If so, then how, and what are the main pitfalls to be avoided along the way? With detrended data in mind, the two classes of dynamics of interest are those with and without increment autocorrelations: Martingales vs everything else. We will also see that Wigner's analysis applies (Chapter 1): unless we can find an inherent statistical repetitiveness to exploit, then the effort is doomed in advance. We will exhibit the required statistical repetition for FX data, and also show how a class of diffusive models is implied. Because we work with detrended time series (this would be impossible were the increments correlated), attention must first be paid to restrictions on detrending Ito processes.

Detrending economic variables

Prices are recorded directly but we'll study log returns of prices. The use of logarithms of prices is common both in finance and macroeconomics. Before the transformation can be made from price to log returns, a price scale must be defined so that the argument of the logarithm is dimensionless.