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function [LIK,lik] = gaussian_filter(ReducedForm,Y,start,DynareOptions)
% Evaluates the likelihood of a non-linear model approximating the
% predictive (prior) and filtered (posterior) densities for state variables
% by gaussian distributions.
% Gaussian approximation is done by:
% - a Kronrod-Paterson gaussian quadrature with a limited number of nodes.
% Multidimensional quadrature is obtained by the Smolyak operator (ref: Winschel & Kratzig, 2010).
% - a spherical-radial cubature (ref: Arasaratnam & Haykin, 2008,2009).
% - a scaled unscented transform cubature (ref: )
% - Monte-Carlo draws from a multivariate gaussian distribution.
% First and second moments of prior and posterior state densities are computed
% from the resulting nodes/particles and allows to generate new distributions at the
% following observation.
% => The use of nodes is much faster than Monte-Carlo Gaussian particle and standard particles
% filters since it treats a lesser number of particles and there is no need
% of resampling.
% However, estimations may reveal biaised if the model is truly non-gaussian
% since predictive and filtered densities are unimodal.
%
% INPUTS
% reduced_form_model [structure] Matlab's structure describing the reduced form model.
% reduced_form_model.measurement.H [double] (pp x pp) variance matrix of measurement errors.
% reduced_form_model.state.Q [double] (qq x qq) variance matrix of state errors.
% reduced_form_model.state.dr [structure] output of resol.m.
% Y [double] pp*smpl matrix of (detrended) data, where pp is the maximum number of observed variables.
% start [integer] scalar, likelihood evaluation starts at 'start'.
% smolyak_accuracy [integer] scalar.
%
% OUTPUTS
% LIK [double] scalar, likelihood
% lik [double] vector, density of observations in each period.
%
% REFERENCES
%
% NOTES
% The vector "lik" is used to evaluate the jacobian of the likelihood.
% Copyright (C) 2009-2013 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/>.
persistent init_flag mf0 mf1
persistent nodes2 weights2 weights_c2 number_of_particles
persistent sample_size number_of_state_variables number_of_observed_variables
verbose = 1;
% Set default
if isempty(start)
start = 1;
end
% Set persistent variables.
if isempty(init_flag)
mf0 = ReducedForm.mf0;
mf1 = ReducedForm.mf1;
sample_size = size(Y,2);
number_of_state_variables = length(mf0);
number_of_observed_variables = length(mf1);
number_of_particles = DynareOptions.particle.number_of_particles;
init_flag = 1;
end
% compute gaussian quadrature nodes and weights on states and shocks
if isempty(nodes2) && strcmpi(DynareOptions.particle.approximation_method,'quadrature')
[nodes2,weights2] = nwspgr('GQN',number_of_state_variables,DynareOptions.particle.smolyak_accuracy) ;
weights_c2 = weights2 ;
end
if isempty(nodes2) && strcmpi(DynareOptions.particle.approximation_method,'cubature')
[nodes2,weights2] = spherical_radial_sigma_points(number_of_state_variables) ;
weights_c2 = weights2 ;
end
if isempty(nodes2) && strcmpi(DynareOptions.particle.approximation_method,'unscented')
[nodes2,weights2,weights_c2] = unscented_sigma_points(number_of_state_variables,DynareOptions) ;
end
if isempty(nodes2) && strcmpi(DynareOptions.particle.approximation_method,'monte-carlo')
set_dynare_seed('default');
end
% Get covariance matrices
Q = ReducedForm.Q;
H = ReducedForm.H;
if isempty(H)
H = 0;
H_lower_triangular_cholesky = 0;
else
H_lower_triangular_cholesky = reduced_rank_cholesky(H)';
end
% Get initial condition for the state vector.
StateVectorMean = ReducedForm.StateVectorMean;
StateVectorVarianceSquareRoot = reduced_rank_cholesky(ReducedForm.StateVectorVariance)';
state_variance_rank = size(StateVectorVarianceSquareRoot,2);
Q_lower_triangular_cholesky = reduced_rank_cholesky(Q)';
% Initialization of the likelihood.
const_lik = (2*pi)^(number_of_observed_variables/2) ;
lik = NaN(sample_size,1);
LIK = NaN;
SampleWeights = 1/number_of_particles ;
ks = 0 ;
%Estimate = zeros(number_of_state_variables,sample_size) ;
%V_Estimate = zeros(number_of_state_variables,number_of_state_variables,sample_size) ;
for t=1:sample_size
% build the proposal
[PredictedStateMean,PredictedStateVarianceSquareRoot,StateVectorMean,StateVectorVarianceSquareRoot] = ...
gaussian_filter_bank(ReducedForm,Y(:,t),StateVectorMean,StateVectorVarianceSquareRoot,Q_lower_triangular_cholesky,H_lower_triangular_cholesky,H,DynareOptions) ;
%Estimate(:,t) = PredictedStateMean ;
%V_Estimate(:,:,t) = PredictedStateVarianceSquareRoot ;
if strcmpi(DynareOptions.particle.approximation_method,'quadrature') || ... % sparse grids approximations
strcmpi(DynareOptions.particle.approximation_method,'cubature') || ...
strcmpi(DynareOptions.particle.approximation_method,'unscented')
StateParticles = bsxfun(@plus,StateVectorMean,StateVectorVarianceSquareRoot*nodes2') ;
IncrementalWeights = ...
gaussian_densities(Y(:,t),StateVectorMean,...
StateVectorVarianceSquareRoot,PredictedStateMean,...
PredictedStateVarianceSquareRoot,StateParticles,H,const_lik,...
weights2,weights_c2,ReducedForm,DynareOptions) ;
SampleWeights = weights2.*IncrementalWeights ;
SumSampleWeights = sum(SampleWeights) ;
lik(t) = log(SumSampleWeights) ;
SampleWeights = SampleWeights./SumSampleWeights ;
else % Monte-Carlo draws
StateParticles = bsxfun(@plus,StateVectorVarianceSquareRoot*randn(state_variance_rank,number_of_particles),StateVectorMean) ;
IncrementalWeights = ...
gaussian_densities(Y(:,t),StateVectorMean,...
StateVectorVarianceSquareRoot,PredictedStateMean,...
PredictedStateVarianceSquareRoot,StateParticles,H,const_lik,...
1/number_of_particles,1/number_of_particles,ReducedForm,DynareOptions) ;
SampleWeights = SampleWeights.*IncrementalWeights ;
SumSampleWeights = sum(SampleWeights) ;
%VarSampleWeights = IncrementalWeights-SumSampleWeights ;
%VarSampleWeights = VarSampleWeights*VarSampleWeights'/(number_of_particles-1) ;
lik(t) = log(SumSampleWeights) ; %+ .5*VarSampleWeights/(number_of_particles*(SumSampleWeights*SumSampleWeights)) ;
SampleWeights = SampleWeights./SumSampleWeights ;
Neff = 1/sum(bsxfun(@power,SampleWeights,2)) ;
if (Neff<.5*sample_size && strcmpi(DynareOptions.particle.resampling.status,'generic')) || ...
strcmpi(DynareOptions.particle.resampling.status,'systematic')
ks = ks + 1 ;
StateParticles = resample(StateParticles',SampleWeights,DynareOptions)' ;
StateVectorMean = mean(StateParticles,2) ;
StateVectorVarianceSquareRoot = reduced_rank_cholesky( (StateParticles*StateParticles')/(number_of_particles-1) - StateVectorMean*(StateVectorMean') )';
SampleWeights = 1/number_of_particles ;
elseif strcmpi(DynareOptions.particle.resampling.status,'none')
StateVectorMean = (sampleWeights*StateParticles)' ;
temp = sqrt(SampleWeights').*StateParticles ;
StateVectorVarianceSquareRoot = reduced_rank_cholesky( temp'*temp - StateVectorMean*(StateVectorMean') )';
end
end
end
LIK = -sum(lik(start:end));
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