Berndt–Hall–Hall–Hausman algorithm

Berndt–Hall–Hall–Hausman (BHHH) algorithm is an iterative method used in econometrics and statistics for estimating the parameters of a model. It is particularly useful in the context of maximum likelihood estimation when the Hessian matrix of second derivatives of the likelihood function is difficult to compute. The BHHH algorithm, named after its developers Econometricians Ernst R. Berndt, Bronwyn H. Hall, Robert E. Hall, and Jerry A. Hausman, offers an efficient alternative to the more computationally intensive Newton-Raphson algorithm by approximating the Hessian matrix in a simpler way.

Overview

The BHHH algorithm is based on the insight that the outer product of the gradient of the likelihood function can provide a good approximation to the Hessian matrix under certain conditions. This approach simplifies the computation involved in parameter estimation, making the algorithm particularly attractive for large datasets or complex models where the exact Hessian is difficult to calculate.

Algorithm

The steps of the BHHH algorithm are as follows:

1. Start with an initial estimate of the parameter vector, \(\theta^{(0)}\). 2. At iteration \(k\), compute the gradient of the log-likelihood function, \(\nabla l(\theta^{(k)})\), for each observation in the dataset. 3. Calculate the average outer product of the gradient vectors, which serves as an approximation to the Hessian matrix. 4. Update the parameter estimate using the formula: \(\theta^{(k+1)} = \theta^{(k)} + \lambda^{(k)} [(\sum \nabla l(\theta^{(k)}) \nabla l(\theta^{(k)})')^{-1} \sum \nabla l(\theta^{(k)})]\), where \(\lambda^{(k)}\) is a step size parameter chosen to ensure convergence. 5. Repeat steps 2-4 until convergence is achieved, typically when the change in the log-likelihood function between iterations is below a predetermined threshold.

Applications

The BHHH algorithm is widely used in econometrics for estimating the parameters of models where the likelihood function is complex. Its applications range from simple regression models to more sophisticated econometric models used in microeconomics, macroeconomics, and finance. The algorithm's efficiency and simplicity make it a valuable tool for researchers and practitioners in these fields.

Advantages

- **Simplicity**: The BHHH algorithm is simpler to implement than the Newton-Raphson algorithm, especially when the Hessian matrix is difficult to compute. - **Efficiency**: It is computationally efficient, making it suitable for large datasets and complex models. - **Robustness**: The algorithm is robust to the choice of initial parameter estimates, often converging to the maximum likelihood estimate from a wide range of starting points.

Limitations

- **Convergence**: While the BHHH algorithm generally converges to the maximum likelihood estimate, convergence is not guaranteed in all cases, especially if the log-likelihood function has multiple local maxima. - **Step Size**: The choice of step size \(\lambda^{(k)}\) can affect the speed of convergence. Improper selection can lead to slow convergence or divergence.

See Also

• Maximum likelihood estimation
• Newton-Raphson algorithm
• Econometrics
• Statistics