Abstract
A new strategy for deriving the three-stage least squares (3SLS) estimator of the simultaneous equations model (SEM) is proposed. The main numerical tool employed is the generalized singular value decomposition. This provides a numerical estimation procedure which can tackle efficiently the particular case when the variance-covariance matrix is singular. The proposed algorithm is further adapted to deal with the special case of the block-recursive SEM. The block diagonal structure of the variance-covariance matrix is exploited in order to reduce significantly the computational burden. Experimental results are presented to illustrate the computational efficiency of the new estimation strategy when compared with the equivalent method that ignores the block-recursive structure of the SEM.
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References
Abdi, H. (2007). Singular value decomposition (SVD) and generalized singular value decomposition (GSVD). In N. J. Salkind (Ed.), Encyclopedia of measurement and statistics (pp. 907–912). Thousand Oaks: SAGE.
Ando, T., & Zellner, A. (2010). Hierarchical bayesian analysis of the seemingly unrelated regression and simultaneous equations models using a combination of direct monte carlo and importance sampling techniques. Bayesian Analysis, 5(1), 65–96.
Andrews, H. C., & Kane, J. (1970). Kronecker matrices, computer implementation, and generalized spectra. Journal of the ACM, 17(2), 260–268.
Angrist, J. D., Graddy, K., & Imbens, G. W. (2000). The interpretation of instrumental variables estimators in simultaneous equations models with an application to the demand for fish. Review of Economic Studies, 67(3), 499–527.
Bai, Z., & Demmel, J. W. (1991). Computing the generalized singular value decomposition. SIAM Journal on Scientific Computing, 14, 1464–1486.
Belsley, D. A. (1988). Two- or three-stage least squares? Computer Science in Economics and Management, 1(1), 21–30.
Belsley, D. A. (1992). Paring 3SLS calculation down to manageable proportions. Computer Science Economics Management, 5, 157–169.
Dent, W. (1976). Information and computation in simultaneous equations estimation. Journal of Econometrics, 4(1), 89–95.
Dhrymes, P. J. (1978). Introductory econometrics. New York, LLC: Springer.
Golub, G., Solna, K., & Dooren, P. V. (2000). Computing the SVD of a general matrix product/quotient. SIAM Journal on Matrix Analysis Applications, 22, 1–19.
Golub, G. H., & Loan, C. F. V. (1996). Matrix computations (3rd ed.). Baltimore: Johns Hopkins University Press.
Greene, W. (2008). Econometric analysis (7th ed.). Upper Saddle River: Prentice Hall.
Hausman, J. A. (1983). Specification and estimation of simultaneous equation models. Handbook of econometrics, Volume 1, Chapter 7 (pp. 391–448). Amsterdam: Elsevier.
Imbens, G. W., & Newey, W. K. (2009). Identification and estimation of triangular simultaneous equations models without additivity. Econometrica, 77(5), 1481–1512.
Jennings, L. S. (1980). Simultaneous equations estimation : Computational aspects. Journal of Econometrics, 12(1), 23–39.
Kontoghiorghes, E. J. (1997). Computing 3SLS solutions of simultaneous equation models with a possible singular variance-convariance matrix. Computational Economics, 10, 231–250.
Kontoghiorghes, E. J. (2000). Parallel algorithms for linear models: Numerical methods and estimation problems. Advances in computational economics. Dordrecht: Kluwer Academic.
Lloyd, W. P., & Lee, C. F. (1976). Block recursive systems in asset pricing models. American Finance Association, 31(4), 1101–1113.
Lopez-Espin, J. J., Vidal, A. M., & Gimenez, D. (2012). Two-stage least squares and indirect least squares algorithms for simultaneous equations models. Journal of Computational and Applied Mathematics, 236(15), 3676–3684.
Matzkin, R. L. (2008). Identification in nonparametric simultaneous equations models. Econometrica, 76(5), 945–978.
Paige, C. C. (1985). The general linear model and the generalized singular value decomposition. Linear Algebra and its Applications, 70, 269–284.
Paige, C. C., & Saunders, M. A. (1981). Toward a generalized singular value decomposition. SIAM Journal on Numerical Analysis, 18, 398–405.
Srivastava, V. K., & Tiwari, R. (1978). Efficiency of two-stage and three-stage least squares estimators. Econometrica, 46(6), 1495–98.
Stewart, G. W. (1982). Computing the CS decomposition of a partitioned orthonormal matrix. Numerische Mathematik, 40, 297–306.
Van Loan, C. F. (1976). Generalizing the singular value decomposition. SIAM Journal Numerical Analysis, 13, 76–83.
Zellner, A., & Theil, H. (1962). Three-stage least squares: Simultaneous estimation of simultaneous equations. Econometrica, 30, 54–78.
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This work is in part supported by the Romanian National Authority for Scientific Research Project PN-II-RU-TE-2011-3-0242, the COST Action IC1408 and the Project GRUPIN14-005.
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Cosbuc, M.I., Gatu, C., Colubi, A. et al. A Generalized Singular Value Decomposition Strategy for Estimating the Block Recursive Simultaneous Equations Model. Comput Econ 50, 503–515 (2017). https://doi.org/10.1007/s10614-016-9595-y
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DOI: https://doi.org/10.1007/s10614-016-9595-y