Tests for a Global Maximum

Robert E. Dorsey and Walter J. Mayer
University of Mississippi
Dorsey@bus.olemiss.edu
http://www.bus.olemiss.edu/dorsey/dorsey.htm

Abstract

In almost all applied econometric work, the promise of meaningful inferences requires that the parameters have been at least consistently estimated. Consistent estimators such as maximum likelihood and generalized method of moments, in turn, require finding the global maximum of a specified objective function. While the global maximizer of a likelihood function, for example, is consistent and asymptotically efficient under general conditions, local maxima of the same function might have no useful properties.

Many applications such as the standard probit and tobit models have globally concave likelihood functions and, consequently, convergence to a stationary point guarantees that the global maximizer has been found. In some other important applications, however, the likelihood functions have an unknown number of local maxima and, thus, the researcher can rarely be certain that a found local maximum is global. Examples include econometric models of rational expectations and disequilibrium (see Dorsey and Mayer (1995)).

Given that the quality of inferences is at stake, how should one decide if a candidate maximum is global? The conventional approach is compare the candidate maximum to values generated from a large number of random draws from the parameter space: If values greater than the candidate or some ad hoc threshold are found, the candidate is rejected. While this approach is intuitively appealing, it lacks a formal framework to determine test size and other properties.

The purpose of the present paper is to propose such a framework and use it to devise and assess a variety of tests for a global maximum that are applicable to a wide range of problems. In particular, we propose using a generalized beta distribution as a basis for constructing Lagrange Multiplier, Wald, Likelihood Ratio and other tests for a global maximum. This framework contains the test of Veall (1990) and de Haan (1981) as a special case. We report monte carlo evidence on the relative performance of several tests, and address a number of theoretical issues that arise. The theoretical issues include that of testing on the boundary of the parameter space.

References

1

Dorsey, R. E. and W.J. Mayer, 1995. `Genetic Algorithms for estimation problems with multiple optima, nondifferentiability and other irregular features', Journal of Business and Economic Statistics 13, 53-66.

2

De Haan, L., 1981. `Estimation of the minimum of a function using order statistics', Journal of the American Statistical Association 76, 467-469.

3

Veall, M.R., 1990. `Testing for a global maximum in an econometric context', Econometrica 58, 1459-1465.