TheBerndt–Hall–Hall–Hausman (BHHH)algorithm is anumerical optimizationalgorithm similar to theNewton–Raphson algorithm, but it replaces the observed negativeHessian matrix with theouter product of thegradient. This approximation is based on theinformation matrix equality and therefore only valid while maximizing alikelihood function.^[1] The BHHH algorithm is named after the four originators:Ernst R. Berndt,Bronwyn Hall,Robert Hall, andJerry Hausman.^[2]

If anonlinear model is fitted to thedata one often needs to estimatecoefficients throughoptimization. A number of optimization algorithms have the following general structure. Suppose that the function to be optimized isQ(β). Then the algorithms are iterative, defining a sequence of approximations,β_k given by

${\displaystyle {\beta }_{k+1}={\beta }_k-{\lambda }_k{A}_k\frac{\mathrm{\partial }Q}{\mathrm{\partial }\beta }({\beta }_k),}$,

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

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

andA is calculated using

${\displaystyle A_k={\left[\sum _{i=1}^N\frac{\mathrm{\partial }\ln Q_i}{\mathrm{\partial }\beta }({\beta }_k)\frac{\mathrm{\partial }\ln Q_i}{\mathrm{\partial }\beta }({\beta }_k{)}^{\prime }\right]}^{-1}.}$

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

• Davidon–Fletcher–Powell (DFP) algorithm
• Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm

• V. Martin, S. Hurn, and D. Harris,Econometric Modelling with Time Series, Chapter 3 'Numerical Estimation Methods'. Cambridge University Press, 2015.
• 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.

• Estimation methods
• Optimization algorithms and methods

• All articles with unsourced statements
• Articles with unsourced statements from March 2013