Demand and Price Analysis, Research Methods/Statistical Methods

T HE MODERN theory of consumer demand has been in the core of economic theory from its very beginning around the 1870's. Somewhat earlier (1857) Engel [6] published his study for the Kingdom of Saxony, which marks the start of systematic measurement of consumer behaviour. Both theory and measurement have developed enormously since their beginnings. Remarkably enough, the links between these two branches of study of consumer demand have remained rather weak. Nobody will deny the importance of those classics on theory and measurement of consumer demand as the monumental monographs by Schultz [15], Wold [22] and Stone [17], but even there the relation between the exposition of the theory and the derivation of the empirical results is frequently superficial. In this connection we can quote Cramer [4]:

1. IN THIS NOTE we consider the estimation of linear models when the coefficients of the structural form are not the same for all observations for which the model is postulated to be valid. An example of such a model is given in [3], where some structural relations have a piecewise linear form. Another example is the water melon market model of Suits [2] where there are two alternative harvest supply schedules. Also discussed here is the case where for one part of the sample period one or more variables are endogenously determined while for another part they are exogenous, for instance, the wage rate or the rate of exchange. Such a change in the nature of the model can also be interpreted as a change in the coefficients of the structural form. It is assumed throughout that it is known a priori for what observations each specification holds. 2. A shift in the value of the coefficients of a predetermined variable does not cause special problems. If, say, only one shift occurs, one defines two predetermined variables to replace the original one. The vector of observations for the first of these consists of the observations on the original variable with the exception of those observations for which the second value is supposed to hold. These latter observations are replaced by zero's. The vector of observations on the second variable is simply the difference between the vector of observations for the original variable and the one for the first variable. 3. Next, consider the case where there is a shift in the value of one or more structural coefficients associated with an endogenous variable. Let the model in structural form be (1) Yt = y'B + x'C + ur where yt is an M-element vector of endogenous or jointly dependent variables, xt an L-element vector of predetermined variables, while ut is the M-element vector of structural disturbances. The matrix B is the M x M matrix of coefficients associated with the endogenous variables and C is the L x M matrix of coefficients associated with the predetermined variables. It is assumed that one or more elements of B take for some observations a different value than for others. The superscripts a and b are used to distinguish between the two situations. The following partitioning of the sets of T observations in two subsets of T7 and Tb observations, respectively, are introduced:

A. P. Barten, T. Kloek, F. B. Lempers; A Note on a Class of Utility and Production Functions Yielding Everywhere Differentiable Demand Functions, The Revie