12-FinEcon-Model Sel Page 345 Wednesday, February 4, 2004 12:59 PM
Financial Econometrics: Model Selection, Estimation, and Testing 345processes is a critical issue. Usual tests for cointegration cannot be
applied to large portfolios such as the S&P 500 given the computational
cost: The space of possible cointegrating relationships is simply too
large to be searched effectively.
Effective methods to reduce the search space are needed. The dis-
covery of cointegrating relationships is a tremendous advantage from a
trading point of view. As discussed by Alexander, it allows, for instance,
to engineer parsimonious portfolios for index tracking and to create
profitable trading strategies for hedge funds. Possible solutions to this
problem remain proprietary. The consideration of the equivalence of
cointegration and state-space modeling might be a step in this direction.
Effective algorithms for determining state space models are described in
the engineering and, more recently, in the econometric literature.^29NONSTATIONARY MODELS OF FINANCIAL TIME SERIES
Let’s now proceed to explore a number of nonlinear models. The exist-
ence of nonlinearities in financial time series has been documented in
many works.^30 However identifying and estimating a reasonable non-
linear model remains a highly challenging task. The key problem is the
explosion of the search space, the so called “curse of dimensionality”
entailed by nonlinear models.
Models based on neural networks and many other families of uni-
versal function approximators have been explored both in the literature
and in the practice of financial trading. These models try to estimate a
nonlinear DGP. We will not deal with these models which are highly
specialized and often used as proprietary trading models.
However, a number of relatively simple nonlinear models have dem-
onstrated their ability to capture important nonlinear phenomena. The
first (and perhaps the best known) of such models, is the ARCH/
GARCH family of models. Another class of nonlinear models are the
Markov switching models, where a Markov chain drives discrete
changes in the model parameters. Perhaps the best known of these mod-
els is the Hamilton model, though a variety of Markov switching VAR
models have been proposed. These models are appealing because they
implement, in a coherent statistical framework, the idea of structural
change which is reasonable from an economic standpoint.(^29) D. Bauer and M. Wagner, “Estimating Cointegrated Systems Using Subspace Al -
gorithms,” Journal of Econometrics 111 (2002), pp. 47–84.
(^30) Campbell, Lo, and MacKinley, The Econometrics of Financial Markets.