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From |
"Scott Merryman" <smerryman@kc.rr.com> |

To |
<statalist@hsphsun2.harvard.edu> |

Subject |
st: RE: sample code in -heckprob- |

Date |
Mon, 5 Sep 2005 12:22:02 -0500 |

> -----Original Message----- > From: owner-statalist@hsphsun2.harvard.edu [mailto:owner- > statalist@hsphsun2.harvard.edu] On Behalf Of Katsuhide Isa > Sent: Tuesday, August 30, 2005 7:57 AM > To: Statalist > Subject: st: sample code in -heckprob- <snip> > It looks like a symmetric positive-definite matrix is first > geneated, then is decomposed into the product of lower triangular > matrix and its transpose, whose [2,1] and [2,2] elements are > used to construct correlated two (bivariate normal) error > terms. > > But I don't see why Cholesky decomposition is required here > to create these error terms. > I'm afraid no persuasive explanations can be found in the > manual, though it might be just a standard routine for > programmers. > Generating Correlated Random Normal Variables Suppose we have 3 independent normal variables x_i with zero mean, unit variance, and zero covariance; X ~ MN(0, I). We want to transform so that they will have correlation structure P. : P [symmetric] 1 2 3 +-------------------+ 1 | 1 | 2 | .25 1 | 3 | .5 .75 1 | +-------------------+ Since P is a symmetric nonnegative definite matrix it can be decomposed into P = AA' where A is a lower triangular matrix. If we multiply the original data X times A' R = X*A' The resulting variables in R will have the desired correlation. I believe the argument is as follows: cov(R) = E(R'*R) = E(A*X'*X*A') = A*E(X'*X)A = A*I*A' = P Below is an example of generating 3 random variables with a given correlation matrix, P (though this uses -mata-, it could translated to in Stata 8.2). -------------------------------------------------- clear matrix C = (1, 0, 0\0, 1, 0\0, 0, 1) corr2data x1 x2 x3, n(20000) corr(C) clear corr x* mata P =(1, .25, .5\.25,1 , .75 \ .5, .75 , 1) A = cholesky(P) X= st_data( ., .) R = X*A' corr = correlation(R,1) st_matrix ("corr", corr) st_matrix ("R", R) end matrix list corr svmat R corr R* ------------------------------------------------- . clear . matrix C = (1, 0, 0\0, 1, 0\0, 0, 1) . corr2data x1 x2 x3, n(20000) corr(C) clear (obs 20000) . corr x* (obs=20000) | x1 x2 x3 -------------+--------------------------- x1 | 1.0000 x2 | 0.0000 1.0000 x3 | -0.0000 -0.0000 1.0000 . mata ------------------------------------------------- mata (type end to exit) -- : P =(1, .25, .5\.25,1 , .75 \ .5, .75 , 1) : A = cholesky(P) : X= st_data( ., .) : R = X*A' : corr = correlation(R,1) : st_matrix ("corr", corr) : st_matrix ("R", R) : end ---------------------------------------------------------------------------- . matrix list corr symmetric corr[3,3] c1 c2 c3 r1 1 r2 .25 1 r3 .5 .75 1 . svmat R . corr R* (obs=20000) | R1 R2 R3 -------------+--------------------------- R1 | 1.0000 R2 | 0.2500 1.0000 R3 | 0.5000 0.7500 1.0000 I hope this helps, Scott * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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