Instruments, generalised transformations, and ring theory

I proposed last week that the procedure used in a recent paper for testing convergence could be split into several stages. The first stage was to transform the equations under consideration into a form where only a small set of quantities are present and those quantities can be analysed exactly with available data. I suggested that mathematical group theory may be applicable to examine which quantities could be derived.

With slightly more reflection, I think that mathematical ring theory may be a better tool than group theory, in view of the transformations used being both additive and multiplicative. In maths terms, the quantity derived represents an ideal, and the largest number of quantities which can be simultaneously estimated is the size of the maximal ideal of the ring consisting of the original equations under multiplication and addition. The size of the quotient ring may help to determine this maximum number; I am not familiar enough with ring theory to say for sure. The ring less the maximal ideal (the minimal ideal is the name I presume) is the set of inestimable elements, at least while the maximal ideal elements are estimated. The maximal ideal may not be unique.

The ring analysis can apply to conventional examination of identification in two stage least squares, where each endogenous variable should have at least one instrumental variable allocated to it, and no subset of n endogenous variables should have less than n instruments allocated to it. The required row non-degeneracy of the instrument-endogenous variable cross-product is the same as the requirement that the rows of the matrix span a space of dimension equal to the number of endogenous variables. Equivalently, under addition, the instrumental variables should cover a space equal to the space covered by the endogenous variables. If they cover a smaller subspace, its dimension indicates the maximum number of endogenous variables whose coefficients can be estimated.

The required operations (addition) for instrumental variable identification are fewer than for the convergence transformation (addition and multiplication) so in this ring theory viewpoint, the latter may be considered one example of a theory generalising the transformations of instrumental variable estimation.