183 lines
8.3 KiB
Matlab
183 lines
8.3 KiB
Matlab
function [LIK,lik] = nonlinear_kalman_filter(ReducedForm, Y, start, ParticleOptions, ThreadsOptions, DynareOptions, Model)
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% Evaluates the likelihood of a non-linear model approximating the
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% predictive (prior) and filtered (posterior) densities for state variables
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% by a Kalman filter.
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% Gaussian distribution approximation is done by:
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% - a spherical-radial cubature (ref: Arasaratnam & Haykin, 2009).
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% - a scaled unscented transform cubature (ref: Julier & Uhlmann 1995)
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% - Monte-Carlo draws from a multivariate gaussian distribution.
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% First and second moments of prior and posterior state densities are computed
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% from the resulting nodes/particles and allows to generate new distributions at the
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% following observation.
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% Pros: The use of nodes is much faster than Monte-Carlo Gaussian particle and standard particles
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% filters since it treats a lesser number of particles.
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% Cons: 1. Application a linear projection formulae in a nonlinear context.
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% 2. Parameter estimations may be biaised if the model is truly non-gaussian since predictive and
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% filtered densities are unimodal.
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%
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% INPUTS
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% Reduced_Form [structure] Matlab's structure describing the reduced form model.
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% Y [double] matrix of original observed variables.
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% start [double] structural parameters.
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% ParticleOptions [structure] Matlab's structure describing options concerning particle filtering.
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% ThreadsOptions [structure] Matlab's structure.
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%
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% OUTPUTS
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% LIK [double] scalar, likelihood
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% lik [double] vector, density of observations in each period.
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%
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% REFERENCES
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%
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% NOTES
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% The vector "lik" is used to evaluate the jacobian of the likelihood.
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% Copyright © 2009-2022 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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% Set default
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if isempty(start)
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start = 1;
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end
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order = DynareOptions.order;
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if ReducedForm.use_k_order_solver
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dr = ReducedForm.dr;
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udr = ReducedForm.udr;
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else
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% Set local state space model (first-order approximation).
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ghx = ReducedForm.ghx;
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ghu = ReducedForm.ghu;
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% Set local state space model (second-order approximation).
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ghxx = ReducedForm.ghxx;
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ghuu = ReducedForm.ghuu;
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ghxu = ReducedForm.ghxu;
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ghs2 = ReducedForm.ghs2;
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if (order == 3)
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% Set local state space model (third order approximation).
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ghxxx = ReducedForm.ghxxx;
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ghuuu = ReducedForm.ghuuu;
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ghxxu = ReducedForm.ghxxu;
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ghxuu = ReducedForm.ghxuu;
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ghxss = ReducedForm.ghxss;
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ghuss = ReducedForm.ghuss;
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end
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end
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constant = ReducedForm.constant;
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steadystate = ReducedForm.steadystate;
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state_variables_steady_state = ReducedForm.state_variables_steady_state;
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mf0 = ReducedForm.mf0;
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mf1 = ReducedForm.mf1;
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sample_size = size(Y,2);
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number_of_state_variables = length(mf0);
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number_of_observed_variables = length(mf1);
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number_of_structural_innovations = length(ReducedForm.Q);
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% compute gaussian quadrature nodes and weights on states and shocks
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if ParticleOptions.proposal_approximation.montecarlo
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nodes = randn(ParticleOptions.number_of_particles,number_of_state_variables+number_of_structural_innovations);
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weights = 1/ParticleOptions.number_of_particles;
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weights_c = weights;
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elseif ParticleOptions.proposal_approximation.cubature
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[nodes,weights] = spherical_radial_sigma_points(number_of_state_variables+number_of_structural_innovations);
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weights_c = weights;
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elseif ParticleOptions.proposal_approximation.unscented
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[nodes,weights,weights_c] = unscented_sigma_points(number_of_state_variables+number_of_structural_innovations,ParticleOptions);
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else
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error('Estimation: This approximation for the proposal is not implemented or unknown!')
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end
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if ParticleOptions.distribution_approximation.montecarlo
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set_dynare_seed('default');
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end
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% Get covariance matrices
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H = ReducedForm.H;
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H_lower_triangular_cholesky = chol(H)' ;
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Q_lower_triangular_cholesky = chol(ReducedForm.Q)';
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% Get initial condition for the state vector.
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StateVectorMean = ReducedForm.StateVectorMean;
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StateVectorVarianceSquareRoot = chol(ReducedForm.StateVectorVariance)';
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% Initialization of the likelihood.
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lik = NaN(sample_size,1);
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LIK = NaN;
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for t=1:sample_size
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xbar = [StateVectorMean ; zeros(number_of_structural_innovations,1) ] ;
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sqr_Px = [StateVectorVarianceSquareRoot zeros(number_of_state_variables,number_of_structural_innovations);
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zeros(number_of_structural_innovations,number_of_state_variables) Q_lower_triangular_cholesky];
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sigma_points = bsxfun(@plus,xbar,sqr_Px*(nodes'));
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StateVectors = sigma_points(1:number_of_state_variables,:);
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epsilon = sigma_points(number_of_state_variables+1:number_of_state_variables+number_of_structural_innovations,:);
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yhat = bsxfun(@minus,StateVectors,state_variables_steady_state);
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if ReducedForm.use_k_order_solver
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tmp = local_state_space_iteration_k(yhat, epsilon, dr, Model, DynareOptions, udr);
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else
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if order == 2
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tmp = local_state_space_iteration_2(yhat, epsilon, ghx, ghu, constant, ghxx, ghuu, ghxu, ThreadsOptions.local_state_space_iteration_2);
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elseif order == 3
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tmp = local_state_space_iteration_3(yhat, epsilon, ghx, ghu, ghxx, ghuu, ghxu, ghs2, ghxxx, ghuuu, ghxxu, ghxuu, ghxss, ghuss, steadystate, ThreadsOptions.local_state_space_iteration_3, false);
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end
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end
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PredictedStateMean = tmp(mf0,:)*weights ;
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PredictedObservedMean = tmp(mf1,:)*weights;
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if ParticleOptions.proposal_approximation.cubature || ParticleOptions.proposal_approximation.montecarlo
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PredictedStateMean = sum(PredictedStateMean,2);
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PredictedObservedMean = sum(PredictedObservedMean,2);
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dState = bsxfun(@minus,tmp(mf0,:),PredictedStateMean)'.*sqrt(weights);
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dObserved = bsxfun(@minus,tmp(mf1,:),PredictedObservedMean)'.*sqrt(weights);
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big_mat = [dObserved dState ; [H_lower_triangular_cholesky zeros(number_of_observed_variables,number_of_state_variables)] ];
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[~, mat] = qr2(big_mat,0);
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mat = mat';
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PredictedObservedVarianceSquareRoot = mat(1:number_of_observed_variables,1:number_of_observed_variables);
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CovarianceObservedStateSquareRoot = mat(number_of_observed_variables+(1:number_of_state_variables),1:number_of_observed_variables);
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StateVectorVarianceSquareRoot = mat(number_of_observed_variables+(1:number_of_state_variables),number_of_observed_variables+(1:number_of_state_variables));
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PredictionError = Y(:,t) - PredictedObservedMean;
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StateVectorMean = PredictedStateMean + (CovarianceObservedStateSquareRoot/PredictedObservedVarianceSquareRoot)*PredictionError;
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else
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dState = bsxfun(@minus,tmp(mf0,:),PredictedStateMean);
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dObserved = bsxfun(@minus,tmp(mf1,:),PredictedObservedMean);
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PredictedStateVariance = dState*diag(weights_c)*dState';
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PredictedObservedVariance = dObserved*diag(weights_c)*dObserved' + H;
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PredictedStateAndObservedCovariance = dState*diag(weights_c)*dObserved';
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PredictionError = Y(:,t) - PredictedObservedMean;
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KalmanFilterGain = PredictedStateAndObservedCovariance/PredictedObservedVariance;
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StateVectorMean = PredictedStateMean + KalmanFilterGain*PredictionError;
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StateVectorVariance = PredictedStateVariance - KalmanFilterGain*PredictedObservedVariance*KalmanFilterGain';
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[StateVectorVarianceSquareRoot, p]= chol(StateVectorVariance,'lower');
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if p
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LIK=-Inf;
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lik(t)=-Inf;
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return
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end
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[~, p]= chol(PredictedObservedVariance,'lower');
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if p
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LIK=-Inf;
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lik(t)=-Inf;
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return
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end
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end
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lik(t) = log( sum(probability2(Y(:,t),H_lower_triangular_cholesky,tmp(mf1,:)).*weights,1) ) ;
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end
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LIK = -sum(lik(start:end));
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