Abstract
Computationally efficient and numerically stable methods for solving Seemingly Unrelated Regression (SUR) models are proposed. The iterative feasible generalized least squares estimator of SUR módels where the regression equations have common exogenous variables is derived. At each iteration an estimator of the SUR model is obtained from the solution of a generalized linear least squares problem. The proposed methods, which have as a basic tool the generalized QR decomposition (GQRD), are also found to be efficient in the general case where the number of linear independent regressors is smaller than the number of observations.
Parallel strategies based on compound disjoint Givens rotations are designed for computing the main two factorizations that are used in the GQRD. The first factorization requires the triangularization of a set of upper-triangular after deleting columns. The second factorization is equivalent in updating a lower-triangular matrix with a matrix having a block lower-triangular structure. Theoretical measures of complexity and examples are used for comparing and investigating the various parallel strategies.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anderson, E., Bai, Z., and Dongarra, J. J. (1992). Generalized QR factorization and its applications. Linear Algebra and its Applications, 162: 243–271.
Andrews, H. C. and Kane, J. (1970). Kronecker matrices, computer implementation, and generalized spectra. Journal of the ACM, 17 (2): 260–268.
Belsely, D. A., Foschi, P., and Kontoghiorghes, E. J. (2002). Numerical estimation of seemingly unrelated regression models. Computational Economics. (To be submitted for publication).
Blackford, L. S., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., and Whaley, R. (1997). ScaLAPACK Users’ Guide. SIAM, Philadelphia.
Cosnard, M. and Daoudi, M. (1994). Optimal algorithms for parallel Givens factorization on a coarse—grained PRAM. Journal of the ACM, 41 (2): 399–421.
Cosnard, M., Muller, J.-M., and Robert, Y (1986). Parallel QR decomposition of a rectangular matrix. Numerische Mathematik, 48: 239–249.
Dhrymes, P. J. (1994). Topics in Advanced Econometrics, volume Vol. 2: Linear and Nonlinear Simultaneous Equations. Springer—Verlag, New York.
Foschi, P. and Kontoghiorghes, E. J. (2001). Estimation of VAR(p) models: computational aspects. Computational Economics. (In press).
Foschi, P. and Kontoghiorghes, E. J. (2002). Estimation of seemingly unrelated regression models with unequal size of observations: computational aspects. Computational Statistics and Data Analysis. (forthcoming).
Garin, L. (1999). The QR decomposition of trapezoidal matrices after deleting columns. Diploma Thesis, Institut d’Informatique, Université de Neuchâtel, Switzerland.
Gatu, C. and Kontoghiorghes, E. J. (2002). Parallel algorithms for computing all possible subset regression models using the QR decomposition. Parallel Computing. (forthcoming).
Graham, A. (1986). Kronecker products and matrix calculus: with applications. Ellis Horwood Series in Mathematics and its Applications. Chichester: Ellis Horweod Limited, Publishers; New York: Halsted Press, a division of John Wiley & Sons.
Kontoghiorghes, E. J. (1995). New parallel strategies for block updating the QR decomposition. Parallel Algorithms and Applications, 5 (42): 229–239.
Kontoghiorghes, E. J. (1999). Parallel strategies for computing the orthogonal factorizations used in the estimation of econometric models. Algorithmica, 25: 58–74.
Kontoghiorghes, E. J. (2000a). Parallel Algorithms for Linear Models: Numerical Methods and Estimation Problems, volume 15 of Advances in Computational Economics. Kluwer Academic Publishers, Boston, MA.
Kontoghiorghes, E. J. (2000b). Parallel Givens sequences for solving the general linear model on a EREW PRAM. Parallel Algorithms and Applications, 15 (1–2): 57–75.
Kontoghiorghes, E. J. (2000c). Parallel strategies for solving SURE models with variance inequalities and positivity of correlations constraints. Computational Economics, 15 (42): 89–106.
Kontoghiorghes, E. J. and Clarke, M. R. B. (1993a). Parallel reorthogonalization of the QR decomposition after deleting columns. Parallel Computing, 19 (6): 703–707.
Kontoghiorghes, E. J. and Clarke, M. R. B. (1993b). Solving the updated and downdated ordinary linear model on massively parallel SIMD systems. Parallel Algorithms and Applications, 1 (2): 243–252.
Kontoghiorghes, E. J. and Clarke, M. R. B. (1993c). Stable parallel algorithms for computing and updating the QR decomposition. In Proceedings of the IEEE TENCON’93, pages 656–659, Beijing. International Academic Publishers.
Kontoghiorghes, E. J. and Clarke, M. R. B. (1995a). An alternative approach for the numerical solution of seemingly unrelated regression equations models. Computational Statistics & Data Analysis, 19 (4): 369–377.
Kontoghiorghes, E. J. and Clarke, M. R. B. (1995b). Solving the general linear model on a SIMD array processor. Computers and Artificial Intelligence, 14 (4): 353–370.
Kontoghiorghes, E. J., Clint, M., and Dinenis, E. (1996). Parallel strategies for estimating the parameters of a modified regression model on a SIMD array processor. In Prat, A., editor, COMPSTAT,Proceedings in Computational Statistics, pages 319–324. Physical Verlag.
Kontoghiorghes, E. J. and Dinenis, E. (1996). Solving triangular seemingly unrelated regression equations models on massively parallel systems. In Gilli, M., editor, Computational Economic Systems: Models, Methods & Econometrics,volume 5 of Advances in Computational Economics,pages 191–201. Kluwer Academic Publishers.
Kontoghiorghes, E. J. and Dinenis, E. (1997). Computing 3SLS solutions of simultaneous equation models with a possible singular variance—covariance matrix. Computational Economics, 10: 231–250.
Kourouklis, S. and Paige, C. C. (1981). A constrained least squares approach to the general Gauss—Markov linear model. Journal of the American Statistical Association, 76 (375): 620–625.
Modi, J. J. and Clarke, M. R. B. (1984). An alternative Givens ordering. Numerische Mathematik, 43: 83–90.
Paige, C. C. (1978). Numerically stable computations for general univariate linear models. Communications on Statistical and Simulation Computation, 7 (5): 437–453.
Paige, C. C. (1979). Fast numerically stable computations for generalized linear least squares problems. SLIM Journal on Numerical Analysis, 16 (1): 165–171.
Paige, C. C. (1990). Some aspects of generalized QR factorizations. In Cox, M. G. and Hammarling, S. J., editors, Reliable Numerical Computation, pages 71–91. Clarendon Press, Oxford, UK.
Regalia, P. A. and Mitra, S. K. (1989). Kronecker products, unitary matrices and signal processing applications. SIAM Review, 31 (4): 586–613.
Sameh, A. H. and Kuck, D. J. (1978). On stable parallel linear system solvers. Journal of the ACM, 25 (1): 81–91.
Srivastava, V. K. and Dwivedi, T. D. (1979). Estimation of seemingly unrelated regression equations Models: a brief survey. Journal of Econometrics, 10: 15–32.
Srivastava, V. K. and Giles, D. E. A. (1987). Seemingly Unrelated Regression Equations Models: Estimation and Inference (Statistics: Textbooks and Monographs), volume 80. Marcel Dekker, Inc.
Zellner, A. (1962). An efficient method of estimating seemingly unrelated regression equations and tests for aggregation bias. Journal of the American Statistical Association, 57: 348–368.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Foschi, P., Garin, L., Kontoghiorghes, E.J. (2002). Numerical and Computational Strategies for Solving Seemingly Unrelated Regression Models. In: Kontoghiorghes, E.J., Rustem, B., Siokos, S. (eds) Computational Methods in Decision-Making, Economics and Finance. Applied Optimization, vol 74. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3613-7_21
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3613-7_21
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5230-1
Online ISBN: 978-1-4757-3613-7
eBook Packages: Springer Book Archive