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]

## Contents

* 1 Usage
* 2 See also
* 3 References
* 4 Further reading

## 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

[math]\displaystyle{ \beta_{k+1}=\beta_{k}-\lambda_{k}A_{k}\frac{\partial Q}{\partial \beta}(\beta_{k}), }[/math],

where [math]\displaystyle{ \beta_{k} }[/math] is the parameter estimate at step k, and [math]\displaystyle{ \lambda_{k} }[/math] 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

[math]\displaystyle{ Q = \sum_{i=1}^{N} Q_i }[/math]

and A is calculated using

[math]\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} . }[/math]

In other cases, e.g. Newton–Raphson, [math]\displaystyle{ A_{k} }[/math] 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

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

## References

1. ↑ Henningsen, A.; Toomet, O. (2011). "maxLik: A package for maximum likelihood estimation in R". Computational Statistics 26 (3): 443–458 [p. 450]. doi:10.1007/s00180-010-0217-1.
2. ↑ Berndt, E.; Hall, B.; Hall, R.; Hausman, J. (1974). "Estimation and Inference in Nonlinear Structural Models". Annals of Economic and Social Measurement 3 (4): 653–665. https://www.nber.org/chapters/c10206.pdf.

## Further reading

* 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. https://archive.org/details/advancedeconomet00amem/page/137.
* 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. https://books.google.com/books?id=gqI-pAP2JZ8C&pg=PA452.
* Harvey, A. C. (1990). The Econometric Analysis of Time Series (Second ed.). Cambridge: MIT Press. pp. 137–138. ISBN 0-262-08189-X.

* v
* t
* e

Optimization: Algorithms, methods, and heuristics

Unconstrained nonlinear

| … functions|

* Golden-section search
* Interpolation methods
* Line search
* Nelder–Mead method
* Successive parabolic interpolation

|

… and gradients|

| Convergence|

* Trust region
* Wolfe conditions

|

Quasi–Newton|

* Berndt–Hall–Hall–Hausman
* Broyden–Fletcher–Goldfarb–Shanno and L-BFGS
* Davidon–Fletcher–Powell
* Symmetric rank-one (SR1)

Other methods|

* Gauss–Newton
* Gradient
* Levenberg–Marquardt
* Conjugate gradient
* Truncated Newton

… and Hessians|

* Newton's method

Constrained nonlinear

| General|

* Barrier methods
* Penalty methods

|

Differentiable|

* Augmented Lagrangian methods
* Sequential quadratic programming
* Successive linear programming

Convex optimization

| Convex
minimization|

* Cutting-plane method
* Reduced gradient (Frank–Wolfe)
* Subgradient method

|

Linear and
quadratic|

| Interior point|

* Affine scaling
* Ellipsoid algorithm of Khachiyan
* Projective algorithm of Karmarkar

|

Basis-exchange|

* Simplex algorithm of Dantzig
* Revised simplex algorithm
* Criss-cross algorithm
* Principal pivoting algorithm of Lemke

Combinatorial

| Paradigms|

* Approximation algorithm
* Dynamic programming
* Greedy algorithm
* Integer programming
* Branch and bound/cut

|

Graph
algorithms|

| Minimum
spanning tree|

* Borůvka
* Prim
* Kruskal

|

* Bellman–Ford
* Dijkstra
* Floyd–Warshall

Network flows|

* Dinic
* Edmonds–Karp
* Ford–Fulkerson
* Push–relabel maximum flow

Metaheuristics

* Evolutionary algorithm
* Hill climbing
* Local search
* Simulated annealing
* Tabu search

* Software

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