# 03MESMET - 3 CHAPTER Latent Variables and Simultaneous...

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3 : CHAPTER Latent Variables and Simultaneous Equations In the previous chapter, we introduced, through a brief example, the idea of a signal buried in noise; and we indicated a method of extracting the signal which required some prior knowledge of the statistical parameters of the error process. The example was expressed in the terminology of communications engineering; and, in this field, it is usually possible to obtain information on the error process by experimental means. The problem of signal extraction has an analogy in the so-called errors-in-variables problem of econometrics. There are good reasons for expecting economic variables to be measured with error. There are bound to be numerous recording errors in the process of compiling economic indices from individual measurements of prices and quan- tities. Moreover, econometricians are often constrained to use alternative or proxy measurements in place of those which they would ideally employ. The difference between a proxy measurement and an ideal measurement is akin to an error of observation. A major problem in dealing with measurement errors in econometrics is the difficulty in determining the parameters of the error processes. There is usually no way of telling the extent to which recording errors and errors of compilation afflict the official economic statistics. Reflection on the question of the accuracy of economic statistics raises philosophical problems. An economic index, such as the level of investment, is often the empirical counterpart of an abstract construct which would have little meaning apart from the context of an economic model. Only rarely do the facts which are gathered in a statistical enquiry match perfectly with the categories of economic analysis. In order to bring theory and reality together, the data gatherers must arbitrate on matters which cannot be expressed in terms of the theory; and therefore economic indices are unavoidably imprecise. The idea, borrowed from the physical sciences, that there is an exact quantity underlying each erroneous measurement is difficult to sustain. Thus there are both practical and philosophical reasons for why the no- tion of errors in variables has few direct applications in econometrics; and, in 1

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D.S.G. POLLOCK : ECONOMETRIC THEORY many textbooks, the topic is accorded only minor importance. Nevertheless the so-called errors-in-variables model does have a important role to play in classi- cal econometric theory. The mathematical structure of the errors-in-variables model is identical to that of a structural equation within a simultaneous- equation econometric model. The simultaneous-equation model has occupied a position of prime importance within econometric theory. In this chapter, we shall begin by developing the error-in-variables model in its own right, as if it were directly applicable to a practical problem. Then we shall apply our results to the problem of estimating a simultaneous-equation system which we shall exemplify with a model of supply and demand. Finally,
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