12-FinEcon-Model Sel Page 316 Wednesday, February 4, 2004 12:59 PM
316 The Mathematics of Financial Modeling and Investment ManagementSince the pioneering work of Simon, many different search strate-
gies have been proposed by statisticians and researchers in artificial
intelligence. Most approaches to searching strategies are based on mini-
mizing a “distance” from an objective. In the case of econometrics, the
objective of searching is to find the best model that describes data.
Searches are implemented by optimization of some functional.
The problem with the search approach is that the search space is infi-
nite. Even if the search space can be made finite by applying some sort of
discretization, its size for real-life problems is enormous. Any practical
application of the idea of automatic searches requires that the search
space is constrained. Econometrics, as well as statistics and data mining,
constrains the search space by searching within given families of models.
In econometrics, the selection of the model family is typically per-
formed on the basis of theoretical considerations as in the physical sci-
ences. There is no way that an unconstrained search for models might
yield positive results. Various tools might help to decide what family of
models to adopt but, ultimately, model selection is a creative decision
based on theoretical grounds. Once a family of models is selected, there
are still choices to be made as regards the constraints to apply.
A typical top-down approach to constraining searches consists of
starting with a broad family of unrestricted models, for instance, as
explained later in this chapter, Vector Autoregressive Models (VAR),
and then proceeding by constraining them, for instance by applying
error correction constraints as discussed later. A typical bottom-up
approach starts with a family of highly constrained models suggested by
theory and then progressively relaxes constraints.
As there is a large amount of uncertainty in econometrics, model
selection is never definitive and many different models may coexist as
competing or synergic explanations of the same empirical facts, leading
to model uncertainty. One can deal with this by giving weights to vari-
ous models, e.g., predict with the weighted average of the prediction
from several models. This process can be performed under a classical
statistical framework or under a Bayesian statistical framework if prior
probabilities can be assigned to models.^2 In this sense, econometrics is
quite different from the physical sciences where the coexistence of com-
peting theories is a rare event.
Econometric models generally entail the selection of parameters or
even the selection of a specific model within a family. This is the realm of
algorithmic searches, generally in the form of optimization procedures.(^2) A classical reference to Bayesian statistics with emphasis on statistical inference as
decision theory is: Josè M. Bernardo and Adrian F.M. Smith, Bayesian Theory
(Chichester, U.K.: John Wiley & Sons., 2000).