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Looking at a very simple example of an error-in-variables model, I was surprised at the effect that standard dynamic features (in the form of autocorre 11 lation. in the variables) could have on the state of identification of the model. It became apparent that identification of error-in-variables models was less of a problem when some dynamic features were present, and that the cathegory of "pre determined variables" was meaningless, since lagged endogenous and truly exogenous variables had very different identification properties. Also, for'the models I was considering, both necessary and sufficient conditions for identification could be expressed as simple counting rules, trivial to compute. These results seemed somewhat striking in the context of traditional econometrics literature, and p- vided the original motivation for this monograph. The monograph, therefore, atempts to analyze econometric identification of models when the variables are measured with error and when dynamic features are present. In trying to generalize the examples I was considering, although the final results had very simple expressions, the process of formally proving them became cumbersome and lengthy (in particular for the "sufficiency" part of the proofs). Possibly this was also due to a lack of more high-powered analytical tools and/or more elegant derivations, for which I feel an apology coul be appropiate. With some minor modifications, this monograph is a Ph. D. dissertation presented to the Department of Economics of the University of Wisconsin, Madison. Thanks are due to. Dennis J. Aigner and Arthur S.
This book presents an overview of the different errors-in-variables (EIV) methods that can be used for system identification. Readers will explore the properties of an EIV problem. Such problems play an important role when the purpose is the determination of the physical laws that describe the process, rather than the prediction or control of its future behaviour. EIV problems typically occur when the purpose of the modelling is to get physical insight into a process. Identifiability of the model parameters for EIV problems is a non-trivial issue, and sufficient conditions for identifiability are given. The author covers various modelling aspects which, taken together, can find a solution, including the characterization of noise properties, extension to multivariable systems, and continuous-time models. The book finds solutions that are constituted of methods that are compatible with a set of noisy data, which traditional approaches to solutions, such as (total) least squares, do not find. A number of identification methods for the EIV problem are presented. Each method is accompanied with a detailed analysis based on statistical theory, and the relationship between the different methods is explained. A multitude of methods are covered, including: instrumental variables methods; methods based on bias-compensation; covariance matching methods; and prediction error and maximum-likelihood methods. The book shows how many of the methods can be applied in either the time or the frequency domain and provides special methods adapted to the case of periodic excitation. It concludes with a chapter specifically devoted to practical aspects and user perspectives that will facilitate the transfer of the theoretical material to application in real systems. Errors-in-Variables Methods in System Identification gives readers the possibility of recovering true system dynamics from noisy measurements, while solving over-determined systems of equations, making it suitable for statisticians and mathematicians alike. The book also acts as a reference for researchers and computer engineers because of its detailed exploration of EIV problems.
1.1. Introduction Solving systems of nonlinear equations has since long been of great interest to researchers in the field of economics, mathematics, en gineering, and many other professions. Many problems such as finding an equilibrium, a zero point, or a fixed point, can be formulated as the problem of finding a solution to a system of nonlinear equations. There are many methods to solve the nonlinear system such as Newton's method, the homotopy method, and the simplicial method. In this monograph we mainly consider the simplicial method. Traditionally, the zero point and fixed point problem have been solved by iterative methods such as Newton's method and modifications thereof. Among the difficulties which may cause an iterative method to perform inefficiently or even fail are: the lack of good starting points, slow convergence, and the lack of smoothness of the underlying function. These difficulties have been partly overcome by the introduction of homo topy methods.
This book treats the subject of global optimization with minimal restrictions on the behavior on the objective functions. In particular, optimal conditions were developed for a class of noncontinuous functions characterized by their having level sets that are robust. The integration-based approach contrasts with existing approaches which require some degree of convexity or differentiability of the objective function. Some computational results on a personal computer are presented.