1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
|
%
% prodmom.m Date: 4/29/2006
% This Matlab program computes the product moment of X_{i_1}^{nu_1}X_{i_2}^{nu_2}...X_{i_m}^{nu_m},
% where X_{i_j} are elements from X ~ N(0_n,V).
% V only needs to be positive semidefinite.
% V: variance-covariance matrix of X
% ii: vector of i_j
% nu: power of X_{i_j}
% Reference: Triantafyllopoulos (2003) On the Central Moments of the Multidimensional
% Gaussian Distribution, Mathematical Scientist
% Kotz, Balakrishnan, and Johnson (2000), Continuous Multivariate
% Distributions, Vol. 1, p.261
% Note that there is a typo in Eq.(46.25), there should be an extra rho in front
% of the equation.
% Usage: prodmom(V,[i1 i2 ... ir],[nu1 nu2 ... nur])
% Example: To get E[X_2X_4^3X_7^2], use prodmom(V,[2 4 7],[1 3 2])
%
% Retrieved from http://www-2.rotman.utoronto.ca/~kan/papers/prodmom.zip
% This function is part of replication codes of the following paper:
% Kan, R.: "From moments of sum to moments of product." Journal of
% Multivariate Analysis, 2008, vol. 99, issue 3, pages 542-554.
% =========================================================================
% Copyright (C) 2008-2015 Raymond Kan <kan@chass.utoronto.ca>
% Copyright (C) 2019-2020 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
% =========================================================================
function y = prodmom(V,ii,nu);
if nargin<3
nu = ones(size(ii));
end
s = sum(nu);
if s==0
y = 1;
return
end
if rem(s,2)==1
y = 0;
return
end
nuz = nu==0;
nu(nuz) = [];
ii(nuz) = [];
m = length(ii);
V = V(ii,ii);
s2 = s/2;
%
% Use univariate normal results
%
if m==1
y = V^s2*prod([1:2:s-1]);
return
end
%
% Use bivariate normal results when there are only two distinct indices
%
if m==2
rho = V(1,2)/sqrt(V(1,1)*V(2,2));
y = V(1,1)^(nu(1)/2)*V(2,2)^(nu(2)/2)*bivmom(nu,rho);
return
end
%
% Regular case
%
[nu,inu] = sort(nu,2,'descend');
V = V(inu,inu); % Extract only the relevant part of V
x = zeros(1,m);
V = V./2;
nu2 = nu./2;
p = 2;
q = nu2*V*nu2';
y = 0;
for i=1:fix(prod(nu+1)/2)
y = y+p*q^s2;
for j=1:m
if x(j)<nu(j)
x(j) = x(j)+1;
p = -round(p*(nu(j)+1-x(j))/x(j));
q = q-2*(nu2-x)*V(:,j)-V(j,j);
break
else
x(j) = 0;
if rem(nu(j),2)==1
p = -p;
end
q = q+2*nu(j)*(nu2-x)*V(:,j)-nu(j)^2*V(j,j);
end
end
end
y = y/prod([1:s2]);
|
