Statistical aspects of W. D. Fisher's method of optimal aggregation
Tóm tắt
W. D. Fisher's approach to approximating a known linear model y=Ax by an aggregation-disaggregation sequence involves a moment matrix M which is supposed to be known. This paper investigates the statistical problems arising when M is not known. As an example we consider the open static Leontief model with A being the Leontief inverse. Under the assumption that the exogenous variables are multinormal with zero mean, tests and confidence intervals for the loss of information are given. The use of maximum entropy estimation of M for insufficient data is discussed. Statistical procedures to evaluate the goodness of aggregation-disaggregation of a time series of matrices A are also described.
Tài liệu tham khảo
ANDERSON, T.W. (1963). Asymptotic Theory for Principal Component Analysis. Annals of Mathematical Statistics 34: 122–148.
ANDREWS, D. F., R. GNANADESIKAN and J. L. WARNER (1973). Methods for Assessing Multivariate Normality; in P. R. Krishnaiah (ed.), Multivariate Analysis III: 95–116.
CHIPMAN, J. S. (1976). Estimation and aggregation in econometrics: an application of the theory of generalized inverses, in: Nashed (ed.): Generalized inverses and applications, Academic Press, New York.
CHIPMAN, J. S. (1977). Statistical Problems Arising in the Theory of Aggregation; in P. R. Krishnaiah (ed.), Applications of Statistics, North-Holland, Amsterdam.
DEUTSCHES INSTITUT FÜR WIRTSCHAFTSFORSCHUNG (DIW) (1979). Baumgart, E., Staeglin, R., Weiß, J. P., Wessels, H., et al., Jährliche nominale Input-Output-Tabellen, Import-Matrizen und Investitionsmatrizen für die Bundesrepublik Deutschland 1960 bis 1974, Berlin.
FISHER, W. D. (1962). Optimal Aggregation in Multi-Equation Prediction Models. Econometrica 30: 744–769.
FISHER, W. D. (1969). Clustering and Aggregation in Economics, John Hopkins, Baltimore.
IJIRI, Y. (1968). The Linear Aggregation Coefficient as the Dual of the Linear Correlation Coefficient. Econometrica 36: 252–259.
JAMES, A. T. (1969). Tests of Equality of Latent Roots of the Covariance Matrix; in P. R. Krishnaiah (ed.), Multivariate Analysis II, Academic Press, New York: 205–218.
JUDGE, G. G., W. E. GRIFFITHS, R. C. HILL, R. C. LEE (1980). The Theory and Practice of Econometrics, John Wiley, New York.
LAISNEY, F. (1982 a). Théorie et pratique de l'agrégation linéaire dans les systémes linéaires, thèse de doctorat de 3-ême cycle, Universités d'Aix-Marseille II et III.
LAISNEY, F. (1982 b). Input-Output Projections and Optimal Aggregation. Paper presented at the IXth International Conference of Applied Econometrics, Budapest, March 29–April 1, 1982.
LAISNEY, F. (1984). Theory and practice in optimal aggregation of linear models: application of W. D. Fisher's method to input-output models, Economic Letters 15, 315–24.
LAWLEY, D. N. (1956). Tests of Significance for the Latent Roots of Covariance and Correlation Matrices. Biometrika 43: 128–136.
MARDIA, K. V. (1974). Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhya B, 36: 115–128.
MARDIA, K. V., J. R. KENT and J. M. BIBBY (1979). Multivariate Analysis, Academic Press, London.
MUIRHEAD, R. J. (1978). Latent Roots and Matrix Variates: A Review of Some Asymptotic Results. The Annals of Statistics 6: 5–33.
SONDERMANN, D. (1973). Optimale Aggregation von großen linearen Gleichungssystemen; Zeitschrift für National-ökonomie 33: 235–250.
SONDERMANN, D. (1980). Best Approximate Solutions to Matrix Equations under Rank Restrictions. Report 23/80, Hebrew University, Jerusalem.
SRIVASTAVA, M. S., E. M. CARTER (1980). Asymptotic Distribution of Latent Roots and Applications; in R. P. Gupta (ed.), Multivariate Statistical Analysis (North-Holland, New-York): 219–236.
THEIL, H. and K. LAITINEN (1980). Singular Moment Matrices in Applied Econometrics; in P. R. Krishnaiah (ed.), Multivariate Analysis V, North-Holland, Amsterdam: 629–649.