108 lines
3.1 KiB
Matlab
108 lines
3.1 KiB
Matlab
%
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% prodmom.m Date: 4/29/2006
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% This Matlab program computes the product moment of X_{i_1}^{nu_1}X_{i_2}^{nu_2}...X_{i_m}^{nu_m},
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% where X_{i_j} are elements from X ~ N(0_n,V).
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% V only needs to be positive semidefinite.
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% V: variance-covariance matrix of X
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% ii: vector of i_j
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% nu: power of X_{i_j}
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% Reference: Triantafyllopoulos (2003) On the Central Moments of the Multidimensional
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% Gaussian Distribution, Mathematical Scientist
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% Kotz, Balakrishnan, and Johnson (2000), Continuous Multivariate
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% 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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% Usage: prodmom(V,[i1 i2 ... ir],[nu1 nu2 ... nur])
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% Example: To get E[X_2X_4^3X_7^2], use prodmom(V,[2 4 7],[1 3 2])
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%
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% Retrieved from http://www-2.rotman.utoronto.ca/~kan/papers/prodmom.zip
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% This function is part of replication codes 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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% =========================================================================
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% Copyright © 2008-2015 Raymond Kan <kan@chass.utoronto.ca>
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% Copyright © 2019-2023 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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function y = prodmom(V,ii,nu)
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if nargin<3
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nu = ones(size(ii));
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end
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s = sum(nu);
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if s==0
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y = 1;
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return
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end
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if rem(s,2)==1
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y = 0;
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return
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end
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nuz = nu==0;
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nu(nuz) = [];
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ii(nuz) = [];
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m = length(ii);
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V = V(ii,ii);
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s2 = s/2;
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%
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% Use univariate normal results
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%
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if m==1
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y = V^s2*prod(1:2:s-1);
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return
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end
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%
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% Use bivariate normal results when there are only two distinct indices
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%
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if m==2
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if V(1,1)==0 || V(2,2)==0
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y=0;
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return
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end
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rho = V(1,2)/sqrt(V(1,1)*V(2,2));
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y = V(1,1)^(nu(1)/2)*V(2,2)^(nu(2)/2)*pruned_SS.bivmom(nu,rho);
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return
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end
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%
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% Regular case
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%
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[nu,inu] = sort(nu,2,'descend');
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V = V(inu,inu); % Extract only the relevant part of V
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x = zeros(1,m);
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V = V./2;
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nu2 = nu./2;
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p = 2;
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q = nu2*V*nu2';
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y = 0;
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for i=1:fix(prod(nu+1)/2)
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y = y+p*q^s2;
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for j=1:m
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if x(j)<nu(j)
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x(j) = x(j)+1;
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p = -round(p*(nu(j)+1-x(j))/x(j));
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q = q-2*(nu2-x)*V(:,j)-V(j,j);
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break
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else
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x(j) = 0;
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if rem(nu(j),2)==1
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p = -p;
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end
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q = q+2*nu(j)*(nu2-x)*V(:,j)-nu(j)^2*V(j,j);
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end
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end
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end
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y = y/prod(1:s2);
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