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function [condX, rankX, ind0, indno, ixno, Mco, Pco, jweak, jweak_pair] = identification_checks(X, test_flag, tol_rank, tol_sv, param_nbr)
% function [condX, rankX, ind0, indno, ixno, Mco, Pco, jweak, jweak_pair] = identification_checks(X, test_flag, tol_rank, tol_sv, param_nbr)
% -------------------------------------------------------------------------
% Checks rank criteria of identification tests and finds out parameter sets
% that are not identifiable via the nullspace, pairwise correlation
% coefficients and multicorrelation coefficients
% =========================================================================
% INPUTS
% * X [matrix] dependent on test_flag:
% test_flag = 0: Sample information matrix (Ahess)
% test_flag = 1: Jacobian of Moments (J), reduced-form (dTAU) or dynamic model (dLRE)
% test_flag = 2: Jacobian of minimal system (D)
% test_flag = 3: Gram matrix (hessian or correlation type matrix) of spectrum (G)
% -------------------------------------------------------------------------
% OUTPUTS
% * cond [double] condition number of X
% * rank [double] rank of X with tolerance tol_rank
% * ind0 [vector] binary indicator for non-zero columns of H
% * indno [matrix] index of non-identified params
% * ixno [integer] number of rows in indno
% * Mco [matrix] multicollinearity coefficients
% * Pco [matrix] pairwise correlations
% * jweak [matrix] gives 1 if the parameter has Mco=1 (with tolerance tol_rank)
% * jweak_pair [(vech) matrix] gives 1 if a couple parameters has Pco=1 (with tolerance tol_rank)
% -------------------------------------------------------------------------
% This function is called by
% * identification_analysis.m
% -------------------------------------------------------------------------
% This function calls
% * cosn
% * dyn_vech
% * vnorm
% =========================================================================
% Copyright (C) 2010-2019 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/>.
% =========================================================================
if issparse(X)
X = full(X);
end
if nargin < 3 || isempty(tol_rank) || strcmp(tol_rank,'robust')
tol_rank = max(size(X)) * eps(norm(X)); %tolerance level used for rank computations
end
if nargin < 4 || isempty(tol_sv)
tol_sv = 1.e-3; %tolerance level for zero singular value
end
if nargin < 5 || isempty(param_nbr)
param_nbr = size(X,2);
end
indno = zeros(1,param_nbr);
rankrequired = size(X,2);
if test_flag == 2
% Komunjer and Ng's D
Xpar = X(:,1:param_nbr);
Xrest = X(:,(param_nbr+1):end);
else
Xpar = X;
Xrest = [];
end
% find non-zero columns at machine precision
if size(Xpar,1) > 1
ind1 = find(vnorm(Xpar) >= eps);
else
ind1 = find(abs(Xpar) >= eps); % if only one parameter
end
if test_flag == 3
Xparnonzero = Xpar(ind1,ind1); % focus on non-zero rows and columns for Qu and Tkachenko's G
else
Xparnonzero = Xpar(:,ind1); % focus on non-zero columns
end
[eu, ee2, ee1] = svd( [Xparnonzero Xrest], 0 );
condX = cond([Xparnonzero Xrest]);
rankX = rank(X, tol_rank);
icheck = 0; %initialize flag which is equal to 0 if we already found all single parameters that are not identified
if param_nbr > 0 && (rankX<rankrequired)
% search for singular values associated to ONE individual parameter
% Compute an orthonormal basis for the null space using the columns of ee1 that correspond
% to singular values equal to zero and associated to an individual parameter
ee0 = [rankX+1:size([Xparnonzero Xrest],2)]; %look into last columns with singular values of problematic parameter sets (except single parameters)
ind11 = ones(length(ind1),1); %initialize
for j=1:length(ee0)
% check if nullspace is spanned by only one parameter
if length(find(abs(ee1(:,ee0(j))) > tol_sv))==1 %note that tol_sv is not the same tolerance used for rank computations
icheck=1; %indicate that additional single parameters are found
if test_flag == 2
temp = (abs(ee1(:,ee0(j))) <= tol_sv);
ind11 = ind11.*temp(1:(end-size(Xrest,2))); % find non-zero columns
else
ind11 = ind11.*(abs(ee1(:,ee0(j))) <= tol_sv); % find non-zero columns
end
end
end
ind1 = ind1(find(ind11)); % find non-zero columns
end
if icheck
%if we found additional single parameters we need to recheck if we found all parameters
if test_flag == 3
Xparnonzero = Xpar(ind1,ind1); % focus on non-zero rows and columns for Qu and Tkachenko's G
else
Xparnonzero = Xpar(:,ind1); % focus on non-zero columns
end
[eu, ee2, ee1] = svd( [Xparnonzero Xrest], 0 );
condX = cond([Xparnonzero Xrest]);
rankX = rank(X,tol_rank);
end
ind0 = zeros(1,param_nbr); %initialize
ind0(ind1) = 1;
% find near linear dependence problems via multicorrelation coefficients
if test_flag == 0 || test_flag == 3 % G is a Gram matrix and hence should be a correlation-like matrix
if test_flag == 3 % For Qu and Tkachenko's G matrix we need to keep track of all parameters
Mco = NaN(param_nbr,1);
end
Pco=NaN(param_nbr,param_nbr); % pairwise correlation coefficient
deltaX = sqrt(diag(X(ind1,ind1)));
tildaX = X(ind1,ind1)./((deltaX)*(deltaX'));
Mco(ind1,1)=(1-1./diag(inv(tildaX))); % multicorrelation coefficent
Pco(ind1,ind1)=inv(X(ind1,ind1));
sd=sqrt(diag(Pco));
Pco = abs(Pco./((sd)*(sd')));
else
Mco = NaN(param_nbr,1);
for ii = 1:size(Xparnonzero,2)
Mco(ind1(ii),:) = cosn([Xparnonzero(:,ii) , Xparnonzero(:,find([1:1:size(Xparnonzero,2)]~=ii)), Xrest]);
end
end
%% find out which parameters are involved
ixno = 0; %initialize number of rows
if param_nbr>0 && (rankX<rankrequired || min(1-Mco)<tol_rank)
if length(ind1)<param_nbr
% single parameters with zero columns
ixno = ixno + 1;
indno(ixno,:) = (~ismember([1:param_nbr],ind1));
end
ee0 = [rankX+1:size([Xparnonzero Xrest],2)]; %look into last columns with singular values of problematic parameter sets (except single parameters)
for j=1:length(ee0)
% linearely dependent parameters
ixno = ixno + 1;
if test_flag == 2
temp = (abs(ee1(:,ee0(j))) > tol_sv)';
indno(ixno,ind1) = temp(1:(end-size(Xrest,2)));
else
indno(ixno,ind1) = (abs(ee1(:,ee0(j))) > tol_sv)';
end
end
end
%% here there is no exact linear dependence, but there are several near-dependencies, mostly due to strong pairwise colliniearities
jweak = zeros(1,param_nbr);
jweak_pair = zeros(param_nbr,param_nbr);
if test_flag ~= 0 || test_flag ~= 0
% these tests only apply to Jacobians, not to Gram matrices, i.e. Hessian-type or 'covariance' matrices
Pco = NaN(param_nbr,param_nbr);
for ii = 1:size(Xparnonzero,2)
Pco(ind1(ii),ind1(ii)) = 1;
for jj = ii+1:size(Xparnonzero,2)
Pco(ind1(ii),ind1(jj)) = cosn([Xparnonzero(:,ii),Xparnonzero(:,jj),Xrest]);
Pco(ind1(jj),ind1(ii)) = Pco(ind1(ii),ind1(jj));
end
end
for j=1:param_nbr
if Mco(j)>(1-tol_rank)
jweak(j)=1;
[~, jpair] = find(Pco(j,j+1:end)>(1-tol_rank));
for jx=1:length(jpair)
jweak_pair(j, jpair(jx)+j)=1;
jweak_pair(jpair(jx)+j, j)=1;
end
end
end
end
jweak_pair=dyn_vech(jweak_pair)'; % focus only on unique combinations
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