Inequality Constraints, Multicollinearity and Models of Police Expenditure

Abstract

Inequality constraints have been used very infrequently in the analysis of linear statistical models. This is not due to an inability to construct estimators. Rather, it is a result of an inability to say very much about the properties of the inequality constrained estimator. Hanson [15] proved the existence of inequality constrained maximum likelihood estimators. A year later Judge and Takayama [17] demonstrated the value of quadratic programming for obtaining estimates of the parameters in a model constrained by inequalities. More recently, Liew [20; 21] and Klemm and Sposito [19] have suggested closed form estimates. The work by Liew makes some reference to the properties of the inequality constrained least squares (ICLS) estimator. A more formal effort to determine the properties of the ICLS estimator was made by Lovell and Prescott [23]. Their paper considered only the case of one location parameter being restricted to either the positive or negative half of the real line. In a more recent paper Wardle [32] has considered a Bayesian approach to the problem of determining the small sample properties of the ICLS estimator. Our more classical results are a confirmation of his work.