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Most modern statistical models contain restricted parameters. These include stationarity constraints in time dependent processes, and constraints on the positiveness of variances such as the conditional variances in GARCH models.

Transformations of parameters and penalty methods have been customarily used to enforce constraints in statistical models. Convergence to a solution with these methods, however, has not always been reliable.

Han (1977) proposed the Sequential Quadratic Programming (SQP) method for the optimization of functions with general equality and inequality constraints. This method was applied to a statistical problem by Jamshidian, et al., (1993). Software implementations followed: Matlab's optimization toolbox, SAS's Proc NLP, and Aptech System's CML. CML is the first implementation of the SQP method explicitly for the maximum likelihood estimation of constrained statistical models.

R. Schoenberg
Fri Sep 12 09:47:41 PDT 1997