Abstract
Suppose that demand is specified as
and that supply is 
.
Consider the situation where a market does not clear quickly and
that the quantity sold is 
.
It if cannot be certain
that a particular quantity sold
is on the supply or demand curve, the best
that can be determined is the probability 
and the
resulting parameter estimates can be obtained by maximum likelihood
estimates. A test case constructed by Maddala and Nelson [Maddala and Nelson1974]
is examined in this research.
The derivation of analytic expressions for the gradient vectors and Hessian matrix for many likelihood problems can be difficult, time consuming, and prone to mathematical and coding errors. The difficulties with finding analytical methods can be avoided by using approximation techniques, but being approximations they have their own source of error which can be excessive. In any case, values for the derivatives are what is needed for the purposes of estimation, not the expressions themselves. Automatic differentiation has proved to be a more reliable method of obtaining these results, both from the standpoint of eliminating user error and from the standpoint of eliminating approximation error when using approximation methods rather than analytical expressions. Once a library of automatic derivatives has been built, then all that the user need do is to code the function, the automatic differentiation method will provide values for the derivatives. When the library has been implemented in a language with operator overloading, such as C++ or Fortran 90, the effort required by the user is often trivial.
Interval arithmetic is a method of global optimization which has proven to be effective optimizing many mathematical functions [Hansen1992]. The combination of automatic differentiation and interval arithmetic offer users a easy method of obtaining a global solution to estimation problems. The cost for this solution can be an increase in computer execution time which can be considerable.
Previous research has shown that the disequilibrium model is characterized by multiple local optima. This research indicates that it is characterized by multiple local optima, maxima and minima, as well as possible saddle points. Further, that these points are clustered closely together with values near to each other. Given this, it is not surprising that the disequilibrium model has been difficult to solve. The interval method was able to reject all of these false optima and find a global optima within the starting region. It does require extensive computer time however.