# Berndt–Hall–Hall–Hausman algorithm

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.^{[1]}

## Contents

## Usage[edit]

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

- ,

where is the parameter estimate at step k, and 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

and *A* is calculated using

In other cases, e.g. Newton–Raphson, can have other forms. The BHHH algorithm has the advantage that, if certain conditions apply, convergence of the iterative procedure is guaranteed.^{[citation needed]}

## See also[edit]

## References[edit]

**^**Berndt, E.; Hall, B.; Hall, R.; Hausman, J. (1974). "Estimation and Inference in Nonlinear Structural Models" (PDF).*Annals of Economic and Social Measurement*.**3**(4): 653–665.

## Further reading[edit]

- Amemiya, Takeshi (1985).
*Advanced Econometrics*. Cambridge: Harvard University Press. pp. 137–138. ISBN 0-674-00560-0. - Gill, P.; Murray, W.; Wright, M. (1981).
*Practical Optimization*. London: Harcourt Brace. - Gourieroux, Christian; Monfort, Alain (1995). "Gradient Methods and ML Estimation".
*Statistics and Econometric Models*. New York: Cambridge University Press. pp. 452–458. ISBN 0-521-40551-3. - Harvey, A. C. (1990).
*The Econometric Analysis of Time Series*(Second ed.). Cambridge: MIT Press. pp. 137–138. ISBN 0-262-08189-X.