The Estimation of the Linear Expenditure System

From the point of view of consumer demand theory the linearity of the linear expenditure system is one of its most attractive and interesting features. But when estimation problems are discussed, the descriptive adjective is more notable for its irony than its accuracy. Indeed the non-linearity of the model in terms of its parameters presents formidable estimation problems especially when a large number of goods is being considered. Although it is true that many econometricians are now prepared to deal with non-linearity as an everyday occurrence, the successful and convincing estimation of large non-linear models is still a comparative rarity. And since the time, now twenty years ago, when Stone (1954) first calculated parameter estimates for the linear expenditure system, a considerable amount of both human and machine intelligence has been devoted to the development and improvement of viable estimation techniques for the model. In this chapter I shall review the techniques which have been suggested for the estimation of the parameters of the linear expenditure system, from Stone’s method to the application of modern algorithms of non-linear estimation. I shall also put forward a particular modification to the latter which results in a considerable increase in efficiency when dealing with the linear expenditure system and which thus permits the estimation of much larger systems than has been possible to date. These matters are dealt with in section 4.3. Before this, in section 4.2, the theory behind the estimators is discussed; in particular, various types of maximum likelihood estimation are discussed, as well as that specification which leads to ordinary least squares. First, however, some notation must be introduced.