BHHH algorithm

BHHH is an optimization algorithm in econometrics similar to Gauss-Newton algorithm. It is an acronym of the four originators: Berndt, B. Hall, R. Hall, and Jerry Hausman.

Usage
If a nonlinear model is fitted to the data one often needs to estimate coefficients through optimization. In general $$\beta_{k+1}=\beta_{k}-\lambda_{k}A_{k}\frac{\partial Q}{\partial \beta}(\beta_{k})$$, where $$\beta_{k}$$ is the coefficient at step k, $$\lambda_{k}$$ is a parameter, $$Q = \sum_{i=1}^{N} Q_i$$ is the objective function (the negative of the likelihood function) and $$A_{k}=\left[1/N\sum_{i=1}^{N}\frac{\partial \ln Q_i}{\partial \beta}(\beta_{k})\frac{\partial \ln Q_i}{\partial \beta}(\beta_{k})'\right]^{-1}$$ in the case of BHHH. In other cases, e.g. Newton-Raphson, $$A_{k}$$ can have other forms.

Literature

 * Berndt, E., B. Hall, R. Hall, and J. Hausman, (1974), “Estimation and Inference in Nonlinear Structural Models”, Annals of Social Measurement, Vol. 3, 653-665.
 * Luenberger, D. (1972), Introduction to Linear and Nonlinear Programming, Addison Wesley, Reading Massacusetts.
 * Gill, P., W. Murray, and M. Wright, (1981), Practical Optimization, Harcourt Brace and Company, London
 * Sokolov, S.N., and I.N. Silin (1962), “Determination of the coordinates of the minima of functionals by the linearization method”, Joint Institute for Nuclear Research preprint D-810, Dubna.