Mishra, SK (2004): On generating correlated random variables with a given valid or invalid Correlation matrix.

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Abstract
In simulation we often have to generate correlated random variables by giving a reference intercorrelation matrix, R or Q. The matrix R is positive definite and a valid correlation matrix. The matrix Q may appear to be a correlation matrix but it may be invalid (negative definite). With R(m,m) it is easy to generate X(n,m), but Q(m,m) cannot give real X(n,m). So, Q has to be converted into the nearmost R matrix by some procedure.
NJ Higham (2002) provides a method to generate R from Q that satisfies the minimum Frobenius norm condition for (QR). Ali AlSubaihi (2004) gives another method, but his method does not produce an optimal R from Q.
In this paper we propose an algorithm to produce an optimal R from Q by minimizing the maximum norm of (QR). A Computer program (in FORTRAN) also has been provided.
Having obtained R from Q, the paper gives an algorithm to obtain X(n,m) from R(m,m). The proposed algorithm is based on factorization of R, yet it is different from the Kaiser Dichman (1962) procedure. A computer program also has been given.
Item Type:  MPRA Paper 

Institution:  NorthEastern Hill University, Shillong (India) 
Original Title:  On generating correlated random variables with a given valid or invalid Correlation matrix 
Language:  English 
Keywords:  Positive semidefinite; negative definite; maximum norm; frobenius norm; correlated random variables; intercorrelation matrix; correlation matrix; Monte Carlo experiment; multicollinearity; cointegration; computer program; multivariate analysis; simulation; generation of collinear sample data 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63  Computational Techniques ; Simulation Modeling C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology ; Computer Programs > C87  Econometric Software C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C65  Miscellaneous Mathematical Tools C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology ; Computer Programs > C88  Other Computer Software C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology ; Computer Programs > C82  Methodology for Collecting, Estimating, and Organizing Macroeconomic Data ; Data Access 
Item ID:  1782 
Depositing User:  Sudhanshu Kumar Mishra 
Date Deposited:  13 Feb 2007 
Last Modified:  26 Sep 2019 12:42 
References:  · AlSubaihi, AA (2004). “Simulating Correlated Multivariate Pseudorandom Numbers”, At www.jstatsoft.org/counter.php?id=85& url=v09/i04/ paper.pdf&ct =1 Searched by http://www.google.com on 28th July, 2004. · Ferré, M (2004). “The Johansen Test and the Transitivity Property”, Economics Bulletin, Vol. 3 (27), pp. 17. · Filzmoser, P and C Croux (2002). “A Projection Algorithm for Regression with Collinearity”, in K Jajuga, A Sokolowski, and HH Bock (eds), Classification, Clustering, and Data Analysis, SpringerVerlag, Berlin, pp. 227234. · Fleishman, A (1978). “A Method for Simulating NonNormal Distributions”, Psychometrica, 43(4), pp. 521532. · Gillett, BE (1979). Introduction to Operations Research. Tata McGraw Hill, New Delhi. · Goldberg, DE (1989). Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley, Reading, Mass. 11 · Headrick TC and SS Sawilowski (1999). “Simulating Correlated Multivariate Nonnormal Distributions extending the Fleishman Power Method”, Psychometrica, 64(1), pp. 2535. · Higham, NJ (2002). “Computing the Nearest Correlation Matrix – A Problem from Finance”, IMA Journal of Numerical Analysis, 22, pp. 329343. · Hoffman, PJ (1924). “Generating Variables with Arbitrary Parameters”, Psychometrica, 24, pp. 265267. · Holland, J (1975). Adaptation in Natural and Artificial Systems, Univ. of Michigan Press, Ann Arbor, USA. · Kaiser, HF and K Dichman (1962). “Sample and Population Score Matrices and Sample Correlation Matrices from an Arbitrary Population Correlation Matrix”, Psychometrica, 27(2), pp. 179182. · Kapur, J N and H C Saxena (1982). Mathematical Statistics. S Chand & Co. New Delhi. · Kendall, MG and A Stuart (1968). The Advanced Theory of Statistics, Vol. 3. Charles Griffin & Co. London. · Knuth, DE (1969). The Art of Computer Programming. Addison Wesley, London. · Krishnamurthy, EV and SK Sen (1976). ComputerBased Numerical Algorithms, Affiliated EastWest Press, New Delhi. · Mishra, SK (2004). “Multicollinearity and Modular Maximum Entropy Leuven Estimator”, Social Science Research Network at http://ssrn.com/author=353253 · Paris, Q (2001). “Multicollinearity and Maximum Entropy Estimators”, Economics Bulletin, Vol. 3 (11), pp. 19. · Takayama, A (1974). Mathematical Economics, The Dryden Press, Illinois. · Texas Instruments Inc (1979). TI Programmable 58C/59 Master Library, Texas Instruments Inc. Texas. · Tadikamalla, PR (1980). “On Simulating Non normal Distributions”, Psychometrica, 45(2), pp. 273279. · Theil, H (1971). Principles of Econometrics, Wiley, New York. · Vale, CD and VA Maurelli (1983). “Simulating Multivariate Nonnormal Distributions ”, Psychometrica, 48(3), pp. 465471. · Wright, AH (1991). “Genetic Algorithms for Real Parameter Optimization”, in GJE Rawlings (ed) Foundations of Genetic Algorithms, Morgam Kauffmann Publishers, San Mateo, CA, pp. 205218. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/1782 