Berndt–Hall–Hall–Hausman algorithm

The Berndt–Hall–Hall–Hausman (BHHH) algorithm is a numerical optimization algorithm similar to the Gauss–Newton algorithm. It is named after 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. A number of optimisation algorithms have the following general structure. Suppose that the function to be optimized is Q(β). Then the algorithms are iterative, defining a sequence of approximations, β_k given by

$\beta _{k+1}=\beta _{k}-\lambda _{k}A_{k}{\frac {\partial Q}{\partial \beta }}(\beta _{k}),$,

where $\beta _{k}$ is the parameter estimate at step k, and $\lambda _{k}$ is a parameter (called step size) which partly determines the particular algorithm. For the BHHH algorithm λ_k is determined by calculations within a given iterative step, involving a line-search until a point β_k+1 is found satisfying certain criteria. In addition, for the BHHH algorithm, Q has the form

$Q=\sum _{i=1}^{N}Q_{i}$

and A is calculated using

$A_{k}=\left[\sum _{i=1}^{N}{\frac {\partial \ln Q_{i}}{\partial \beta }}(\beta _{k}){\frac {\partial \ln Q_{i}}{\partial \beta }}(\beta _{k})'\right]^{-1}.$

In other cases, e.g. Newton-Raphson, $A_{k}$ can have other forms. The BHHH algorithm has the advantage that, if certain conditions apply, convergence of the iterative procedure is guaranteed.

Literature

• Berndt, E., B. Hall, R. Hall, and J. Hausman, (1974), “Estimation and Inference in Nonlinear Structural Models”, Annals of Economic and Social Measurement, 3, 653–665.
• Luenberger, D. (1972), Introduction to Linear and Nonlinear Programming, Addison Wesley, Reading Massachusetts.
• 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.