Generalized Simultaneous Equation Models

On the optimal use of forecasts in economic policy decisions

Least squares estimation and quadratic openloop optimal control share both algebra and intuition; this paper explores the duality to develop the parallels in each. Estimation results suggest control analogs to correlation measures, significance tests, distributed lags and problems of multicollinearity, tests of compatability between restrictions and data, and a version of the Kalman filter that is sometimes computationally expedient. Conversely, the control literature provides methods of incorporating certain types of inequality restrictions and a development of recursive estimation based on the Kalman filter. A by-product is a method of solving control problems using least squares computer programs.

Estimation Analogies in Control

Generalized Inverses and the Timing of Policy Instruments

Optimal Macro-Economic Policies

The selection of macro policy instruments over time

In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application which has relevance to the statistical problem of finding ‘best’ approximate solutions of inconsistent systems of equations by the method of least squares. Some suggestions for computing this generalized inverse are also given.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100030929

On best approximate solutions of linear matrix equations

IT DOES NOT SEEM to be generally known that E. H. Moore generalized the notion of the inverse of a matrix to include all matrices, rectangular as well as square. This was done in Part 1 of his work "General Analysis," which was published in 1935 (with the editorial collaboration of R. W. Barnard) as Volume 1 of the Memoirs of the American Philosophical Society.8 This generalized inverse was recently rediscovered independently by Penrose [1]. It was called by Moore the "general reciprocal," while Max Woodbury has suggested the more expressive term pseudoinverse which is used here. There seems to be some interest in an elementary presentation of the subject, and this is the purpose of the present article, which makes no claim to originality. This discussion is limited to matrices with real elements, both for ease of exposition and because it is in such cases that the possibility of practical application is most apparent. The extension of the theory to matrices with complex elements is easy. The writer wishes to express his indebtedness to Max Woodbury for drawing his attention to Moore's work and to G. E. Forsythe for a number of valuable suggestions.

Generalized Simultaneous Equation Models

Citations

On the optimal use of forecasts in economic policy decisions

Estimation Analogies in Control

Generalized Inverses and the Timing of Policy Instruments

Optimal Macro-Economic Policies

The selection of macro policy instruments over time

References

On best approximate solutions of linear matrix equations

The Pseudoinverse of a Rectangular or Singular Matrix and Its Application to the Solution of Systems of Linear Equations

Quantitative Planning of Economic Policy.

Related Papers (5)

On the generalized linear ordinary differential equation

Generalized Estimating Equations

Generalized differential equations

Generalized systems of linear differential equations

On generalized random differential equations