Model Selection 293
of any financial system and economy change too much over time to confirm
that laws in financial economics are time-invariant laws of nature. One is,
therefore, inclined to believe that only approximate laws can be discovered.
The attention of the modeler has therefore to switch from discovering deter-
ministic paths to determining the time evolution of probability distributions.^2
The adoption of probability as a descriptive framework is not with-
out a cost: discovering probabilistic laws with confidence requires working
with very large populations (or samples). In physics, this is not a problem
as physicists have very large populations of particles.^3 In finance, however,
populations are typically too small to allow for a safe estimate of prob-
ability laws; small changes in the sample induce changes in the laws. We
can, therefore, make the following general statement: Financial data are too
scarce to allow one to make probability estimates with complete certainty.
(The exception is the ultra high-frequency intraday data, five seconds or
faster trading.)
As a result of the scarcity of financial data, many statistical models, even
simple ones, can be compatible with the same data with roughly the same
level of statistical confidence. For example, if we consider stock price pro-
cesses, many statistical models—including the random walk—compete to
describe each process with the same level of significance. Before discussing
the many issues surrounding model selection and estimation, we will briefly
discuss the subject of machine learning and the machine learning approach
to modeling.
Model Complexity and Sample Size
Let’s now discuss three basic approaches to financial modeling, namely the
- Machine learning approach
- Theoretical approach
- Machine learning theoretical approach
The machine learning theoretical approach is a hybrid of the two former
approaches.
(^2) In physics, this switch was made at the end of the nineteenth century, with the intro-
duction of statistical physics. It later became an article of scientific faith that one can
arrive at no better than a probabilistic description of nature.
(^3) Although this statement needs some qualification because physics has now reached
the stage where it is possible to experiment with small numbers of elementary par-
ticles, it is sufficient for our discussion here.