# BHHH algorithm

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BHHH is an optimization algorithm in numerical optimization 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. 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, βk 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 λk is determined by calculations within a given iterative step, involving a line-search until a point βk+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.[citation needed]

## 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.