Wold H. (ed.) The Fix-point approach to interdependent systems

Amsterdam: North Holland Publishing Company, 1981. — 337 p.In modern econometric research, an important part is played by the discussion of systems of linear relations assumed to hold between various quantities characteristic of the economic life of a community. Some of these quantities have predetermined values, while others are regarded as observed values of stochastic variables with unknown mean values, sometimes assumed to have a multinormal distribution. At least some of the coefficients occurring in the linear relations are unknown.
Given a set of statistical observations on the variables entering into such a system, it is required to estimate the unknown coefficients and mean values. This may be performed by different methods. In order that a proposed method of estimation should be regarded as satisfactory, the requisite numerical work should not be too cumbersome, and the resulting estimates should satisfy some convenient tests of consistency and accuracy.
In the present volume, the family of estimation methods known as the Fix-Point (FP) methods introduced by Herman Wold in 1965 is closely studied. The FP methods work with a system of linear relations of the form known as structural, where each of the stochastic (endogenous) variables under investigation is expressed as a linear combination of the other (explanatory) stochastic variables, the predetermined variables, and a residual. The residuals are assumed to be uncorrelated with all the predetermined variables. In general such a system is called a system of interdependent (ID) relations. The direct application of ordinary least squares (OLS) regression to such a system has been criticized, as it does not give a consistent forecast of the variables occurring in the time series under investigation.