Disequilibrium models constitute an extensive area of research, about which numerous papers have been written. Some of the early econometric models have been surveyed by Maddala and Nelson (1974). A more extensive and
up-to-date survey has been given by Quandt (1982). See, also, the article by Hartley (1976a) for a connection between a disequilibrium model and the standard Tobit model. Here we shall mention two basic models first discussed in the pioneering work of Fair and JafFee (1972).
The simplest disequilibrium model of Fair and JafFee is a special case of the Type 5 model (10.10.1), in which y*,- is the quantity demanded in the ith period, y*i is the quantity supplied in the ith period, and y* = y* — y*. Thus the actual quantity sold, which a researcher observes, is the minimum of supply and demand. The fact that the variance-covariance matrix of (у*, у*, у *) is only of rank 2 because of the linear relationship above does not essentially change the nature of the model because the likelihood function
(10.10.2) involves only bivariate densities.
In another model Fair and JafFee added the price equation to the model of the preceding paragraphs as
Лі “ У(У*і – Уз)), (10.10.12)
where yM denotes a change in the price at the ith period. The likelihood function of this model can be written as18
£ = П І /з(К, УзіУ4і)ЯУлі) dyb (10.10.13)
The form of the likelihood function does not change if we add a normal error term to the right-hand side of (10.10.12). In either case the model may be schematically characterized by
P(yt < 0, Уз, y4) • Р(Уі > о, y2, y4), (10.10.14)
which is a simple generalization of the Type 5 model.