# Berndt–Hall–Hall–Hausman algorithm

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The Berndt–Hall–Hall–Hausman (BHHH) algorithm is a numerical optimization algorithm similar to the Newton–Raphson algorithm, but it replaces the observed negative Hessian matrix with the outer product of the gradient. It is named after the four originators: Ernst R. Berndt, Bronwyn Hall, Robert 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]