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.

In recent years, there has been renewed interest in index number and aggregation theory, since the two previously divergent fields have been successfully unified. The underlying aggregator functions which are weakly separable subfunctions of utility and production functions, are the building blocks of economic theory, and the derivation of index numbers based upon their ability to track those building blocks is now called the "economic theory of index numbers."

William Barnett, the coeditor of this volume, introduced modern economic index number theory into monetary economics. His merger of economic index number theory, with monetary theory was based upon the use of Diewert's approach to producing "superlative" nonparametric approximations to the theoretically exact aggregator functions. This book comprises a focussed and unified collection of Barnett's most important publications in this area.

The papers in the book have been organized into logical sections, with unifying introductions and overviews. The result is a systematic development of the state of the art in monetary and financial aggregation theory. The sections cover the origin of the user cost price of monetary services. Exact aggregation of monetary assets on the demand side for consumers and firms, and on the supply side for financial intermediaries, general equilibrium of all economic agents' demands and supplies, dynamic solution of the exact system, and extension to monetary aggregation under risk. The extension of index number theory to the case of risk is completely general, and can be applied to tracking any exact economic aggregator under risk. In all cases, the criterion used for evaluation isthe tracking ability of the approximation to the exact aggregator function of economic theory.

Many of the empirical and policy puzzles in monetary economics disappear when simple sum monetary aggregates are replaced by index numbers that are coherent with theory. Simple sum monetary aggregates became incoherent with theory, when monetary assets began paying interest and therefore could no longer be viewed as perfect substitutes.

This is a useful tool to those associated with economics departments within universities, business schools, central banks and federal governments, financial institutions including underwriters, bankers and stockbrokers.