84 lines
2.9 KiB
Matlab
84 lines
2.9 KiB
Matlab
function [y,dy] = bivmom(p,rho)
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% Computes the product moment (and its derivative with respect to standard
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% errors and correlation parameters) of X_1^{p_1}X_2^{p_2}, where X_1 and X_2
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% are standard bivariate normally distributed.
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% n : dimension of X
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% rho: correlation coefficient between X_1 and X_2
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% =========================================================================
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% INPUTS
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% p [2 by 1] powers of X_{1} and X_{2}
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% rho [1 by 1] correlation coefficient between X_1 and X_2
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% -------------------------------------------------------------------------
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% OUTPUTS
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% y [1 by 1] product moment E[X_1^{p_1}X_2^{p_2}]
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% dy [1 by 1] derivative of y wrt to rho
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% -------------------------------------------------------------------------
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% This function is based upon bivmom.m which is part of replication codes
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% of the following paper:
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% Kan, R.: "From moments of sum to moments of product." Journal of
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% Multivariate Analysis, 2008, vol. 99, issue 3, pages 542-554.
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% bivmom.m can be retrieved from http://www-2.rotman.utoronto.ca/~kan/papers/prodmom.zip
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% Further references:
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% Kotz, Balakrishnan, and Johnson (2000), Continuous Multivariate Distributions, Vol. 1, p.261
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% Note that there is a typo in Eq.(46.25), there should be an extra rho in front
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% of the equation.
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% =========================================================================
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% Copyright © 2008-2015 Raymond Kan <kan@chass.utoronto.ca>
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% Copyright © 2019-2020 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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% =========================================================================
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s1 = p(1);
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s2 = p(2);
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rho2 = rho^2;
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if nargout > 1
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drho2 = 2*rho;
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end
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if rem(s1+s2,2)==1
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y = 0;
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return
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end
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r = fix(s1/2);
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s = fix(s2/2);
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y = 1;
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c = 1;
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if nargout > 1
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dy = 0;
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dc = 0;
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end
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odd = 2*rem(s1,2);
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for j=1:min(r,s)
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if nargout > 1
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dc = 2*dc*(r+1-j)*(s+1-j)*rho2/(j*(2*j-1+odd)) + 2*c*(r+1-j)*(s+1-j)*drho2/(j*(2*j-1+odd));
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end
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c = 2*c*(r+1-j)*(s+1-j)*rho2/(j*(2*j-1+odd));
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y = y+c;
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if nargout > 1
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dy = dy + dc;
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end
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end
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if odd
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if nargout > 1
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dy = y + dy*rho;
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end
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y = y*rho;
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end
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y = prod(1:2:s1)*prod(1:2:s2)*y;
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if nargout > 1
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dy = prod(1:2:s1)*prod(1:2:s2)*dy;
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end
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