Halbert White - US grants

Cognitive scientists have recently developed a rich and interesting class of nonlinear models inspired by the neural architecture of the brain (neural network models). These networks are capable of learning through interaction with their environment, in a process which can be viewed as a recursive statistical estimation procedure. The promise of these models and associated estimation procedures and the excitement evident across a spectrum of disciplines including psychology, computer science, genetics, linguistics and engineering is founded on the demonstrated success of neural network modeling in solving a diverse range of difficult problems. Especially impressive have been solutions to problems which had previously resisted conventional attempts at solution, as well as relatively quick and reliable solutions to problems which had previously yielded comparable effective solutions grudgingly, and after several man- years of more conventional effort. The objectives of this project are (1) to investigate the applicability of neural network models to the study of economic phenomena and to refine and extend these models in directions suitable to the study of economic phenomena, (2) to refine and extend the learning methods (estimation procedures) used to train the networks so as to obtain parameter estimates which converge quickly and reliably when faced with economic data, and (3) to apply model specification and selection techniques developed by the investigator in previous funded research to neural network models in order to develop techniques for choosing between competing neural network architectures for particular problems. This is an exciting project because no one has ever applied neural network models to economics. These new methods will dramatically reduce the computational time needed to solve complex economic problems. The neural network models will provide a new methodology for studying the way economic agents learn from their environment. Neural networks appear to be particularly well suited to nonlinear economic forecasting, so these new methods could provide us with better predictions of the economic future.

Using methods recently developed in econometrics and mathematical statistics, this project will investigate the theory of "learning" in a class of "artificial neural network" models. Models are to be developed that will yield potentially more efficient and powerful estimation and testing procedures. The research falls into four interrelated ares: large-sample properties of nonparametric estimation procedures based on neural network models, relative computational properties of different methods for optimizing the network learning (estimation) objective function, application of nonparametric network-based procedures to construct semi-parametric estimators and perform specification tests, and comparison of network-based procedures to standard nonparametric and semi-parametric procedures based on kernel estimators and series estimators, among others. The techniques developed in this field have been successfully used to forecast in engineering and technology. Some specific applications have included various industrial settings and in defense to accomplish tasks that had been insoluble. Some examples include bomb detection in airports, distinguishing between sonar signals generated from rocks and mines, vibration damping in machinery, handwriting character recognition and robot coordination and control. Further work in this area should continue to expand the fields of applications beyond economics.

This award will support joint research between US and French researchers on the topic of multinational econometric policy analysis. The US investigators are: Dr. Robert F. Engle, Dr. Clive W. Granger, and Dr. Halbert L. White, University of California at San Diego. The collaborators in France are: Dr. Michel Lubrano, Dr. Luc Bauwens and Dr. Russell Davidson, University of Aix-Marseille II and III. The objective of the project is to develop and apply statistical techniques to evaluate policy models. Potentially one of the most important activities by econometricians and applied economists is the production of models to be used by policy makers to help them determine an appropriate policy. However, these models have to pass not only standard specification tests and evaluation procedures, but they also have to show that they are relevant for policy purposes. Testing procedures for these models are very preliminary and are in need of both further theoretical and empirical development. In this project, the investigators will examine and apply econometric methods to study existing models. Theoretical developments will be coupled with empirical studies, which examine the co-movements of output and asset prices across countries. Both the US and French groups have extensive experience in both the theory and practice of econometrics, as well as in the development and application of tests and testing procedures. In particular, the French investigators bring to the project experience of testing procedures based on methods associated with the use of artificial regressions, as well as Bayesian specification testing.

This project is concerned with techniques for active selection of training examples for neural network learning, while simultaneously growing the network to fit the data. The approach uses a statistical sampling criterion, Integrated Mean Squared Error, to derive a "greedy" selection criterion which picks the next training example that maximizes the decrement in this measure. This selection criterion is usable for a wide class estimators. A practical realization of this schemes for multi- layer neural networks is demonstrated. //

Artificial neural networks (ANN's) are a rich and interesting class of nonlinear models developed recently by cognitive scientists. Although inspired by certain features of the neural architecture of the brain, these models have considerable potential for use in econometrics and economics because of their simplicity and great flexibility. Specifically, ANN models show promise as the basis for nonparmetric estimation and for specification testing because of their ability to approximate arbitrary elements of general function spaces to arbitrary accuracy. Under previous NSF grants, this project focused on establishing the mathematical and statistical foundations necessary for using ANN models in nonparametric estimation and specification testing. The research proposed in this accomplishment based renewal is to continue development of these foundations, to establish properties of ANN- based nonparametric estimators, and to apply the resulting theory to construct model specification tests. The areas in which mathematical and statistical theory will be pursued are two. The first area is the establishment of degree of approximation results for ANN models in Sobolev spaces. The second area is to obtain central limit theorems and invariance principles for random elements constructed as partial sums of a dependent process (specifically, a mixingale process) in a Hilbert space. The application areas make direct use of the proposed theoretical results to model learning by economic agents.

Halbert White SBR-9511253 The P.I. proposes to use the bootstrap to determine the sampling distribution of model specification error tests. The bootstrap has been increasing rapidly in statistics and specification tests, where critical values based on asymptotic distributions appear to often lead to over-rejection of correct specifications, are a natural place to apply this powerful method. Examples of tests that will be studied are: information matrix tests, tests based on nonparametric estimators and neural net estimators, and test of whether a particular subset of variables belong in a regression. The P.I. also proposes to develop high breakdown estimators that are insensitive to significant amounts of data contamination. The P.I. also intends to apply the methods developed to a number of empirical applications.

Economic time series are often difficult to predict and as a result different forecasters, faced with predicting the same economic variable, often come up with very different answers, reflecting their use of separate forecasting models, information sets and estimation methods. It is very rare for any individual forecast to systematically dominate the others. Many studies have found that a simple equal-weighted combination of forecasts produces better predictions than those generated by individual models. This project develops both theoretical tools and empirical techniques for explaining why simple equal-weighted forecasts do so well in practice and compares these with wider classes of combinations.
The research project develops estimation and forecast combination methods that can compete with the equal-weighted forecast combination. We establish conditions under which our proposed methods can be expected to produce improved forecast. The research project also considers the performance of different forecast combination methods under a host of economic circumstances, including prediction of economic variables in the near, medium and distant future, prediction of the entire probability distribution of an economic variable and prediction with loss functions that are tailored to individual policy makers or economic decision makers. The proposal also investigates the possibility of letting the forecast combination weights vary over time since some models may work better in some situations (e.g. when the economy is in a recession) and others may prove to be better in different circumstances (e.g. an expansion state).
Forecast combination techniques have already been found to be useful in practical situations. This proposal provides further understanding of why this is so and explores various alternative approaches both theoretically and in practice, particularly when many forecasts are involved, and expands the approach in new ways. The interest in developing these methods is in part driven by practical concerns of the Principal Investigators that has come through interaction with the various Federal Reserve banks and other international organizations such as the IMF and through considering their needs and requests. We expect to continue strong relations with these and other organizations through the grant period.