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. This approximation is based on the information matrix equality and therefore only valid while maximizing a likelihood function.^[1] The BHHH algorithm is named after the four originators: Ernst R. Berndt, Bronwyn Hall, Robert Hall, and Jerry Hausman.^[2]

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

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

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 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

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

and A is calculated using

${\displaystyle A_{k}=\left[\sum _{i=1}^{N}{\frac {\partial \ln Q_{i}}{\partial \beta }}(\beta _{k}){\frac {\partial \ln Q_{i}}{\partial \beta }}(\beta _{k})'\right]^{-1}.}$

In other cases, e.g. Newton–Raphson, ${\displaystyle A_{k}}$ 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.