Economic theory defines and constrains admissible functional form
and functional structure throughout the economy. Constraints on
behavioral functions of individual economic agents and on the
recursive nesting of those behavioral functions often are derived
directly from economic theory. Theoretically implied constraints on
the properties of equilibrium stochastic solution paths also are
common, although are less directly derived. In both cases, the
restrictions on relevant function spaces have implications for
econometric modeling and for the choice of hypotheses to be tested
and potentially imposed. This book contains state-of-the-art
cumulative research and results on functional structure,
approximation, and estimation: for (1) individual economic agents,
(2) aggregation over those agents, and (3) equilibrium solution
stochastic processes.
A: Functional Structure Modeling, Aggregation, and Estimation.
Over the past 25 years, William Barnett, who is a coeditor of this
volume, has advanced the state of the art of this subject in many
directions. He has contributed many new modeling and inference
approaches, such as the Laurent series flexible functional form
approach, the Mntz-Szatz series seminonparametric approach, the
generalized hypocycloidal utility tree approach, and an aggregated
convergence approach within the space of stochastic differential
equations. Many of Barnett's innovations contain the earlier Taylor
series and CES approaches as nested special cases. He also has
contributed extensively to the literature on aggregation over
approximating specifications in econometrics, as well as to
aggregation over economic agents and goods in economic theory.
Inaddition, his work in those areas has motivated new approaches by
others, such as the generalized symmetric Barnett approach
originated by Diewert and Wales (1987).
Part 1 of this book contains Barnett's contributions to functional
structure modeling and estimation for consumers, while Part 2
contains his contributions on those subjects for firms.
B: Statistical Theory.
Barnett's contributions to statistical theory provide much of the
asymptotic statistical theory needed to apply econometric inference
procedures to the literature on economic functional structure and
approximation. His contributions to the relevant statistical theory
include discovery of the measure theoretic foundations for
confidence regions in sampling theoretic statistics and the
derivation of the asymptotic theory for joint maximum likelihood
inference with closed-form systemwide models. He originated a
multivariate extension of the Kolmogorov-Smirnov test to permit
testing the disturbances of an equation system for multivariate
normality.
Part 3 contains relevant results in statistical theory.
C: Nonlinear Time Series.
Analogous approximation and function space problems arise in time
series approaches. A Volterra expansion in the time domain with a
finite number of terms cannot span the space of possible
time-series solution processes from the state space structures of
economic theory. Hence when sample size is finite, all structural
and time-series approximating specifications, whether dynamic or
static, drive an unavoidable wedge between econometrics and
economic theory. No easy solution exists to this inherently deep
problem in econometric modeling and testing.
Inthe time series literature, Barnett has designed and run a
competition among tests for nonlinear and chaotic structure. The
purpose was to investigate paradoxes that arose in that literature
following his publication of findings of nonlinearity and chaos in
some economic time series. The literature on modeling and filtering
out linear structure from time series is now highly advanced. But
many unsolved problems remain in the literature on modeling or
filtering out various forms of nonlinear structure from time
series. The results of Barnett's competition have cast much needed
light on those problems and the relative properties of the various
available competing approaches.
Contributions to time series modeling and inference in the time
domain and the frequency domain are provided in Part 4.
General
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