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: Ernst R. Berndt, B. Hall, Robert Hall, and Jerry Hausman.
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
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 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
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.
- Berndt, E.; Hall, B.; Hall, R.; Hausman, J. (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. Reading, Massachusetts: Addison Wesley.
- Gill, P.; Murray, W.; Wright, M. (1981). Practical Optimization. London: Harcourt Brace.
- Sokolov, S. N.; Silin, I. N. (1962). "Determination of the coordinates of the minima of functionals by the linearization method". Joint Institute for Nuclear Research preprint D-810, Dubna.