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git-svn-id: https://www.dynare.org/svn/dynare/dynare_v4@1567 ac1d8469-bf42-47a9-8791-bf33cf982152time-shift
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function [LIK, lik] = DiffuseLikelihoodH1(T,R,Q,H,Pinf,Pstar,Y,trend,start)
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% M. Ratto added lik in output
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% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
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% function [LIK, lik] = DiffuseLikelihoodH1(T,R,Q,H,Pinf,Pstar,Y,trend,start)
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% Computes the diffuse likelihood (H=measurement error) in the case of a non-singular var-cov matrix
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%
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% INPUTS
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% T: mm*mm matrix
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% R: mm*rr matrix
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% Q: rr*rr matrix
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% H: pp*pp matrix
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% Pinf: mm*mm diagonal matrix with with q ones and m-q zeros
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% Pstar: mm*mm variance-covariance matrix with stationary variables
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% Y: pp*1 vector
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% trend
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% start: likelihood evaluation at 'start'
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%
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% OUTPUTS
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% LIK: likelihood
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% lik: density vector in each period
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%
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% SPECIAL REQUIREMENTS
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% See "Filtering and Smoothing of State Vector for Diffuse State Space
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% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
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% Analysis, vol. 24(1), pp. 85-98).
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%
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% THE PROBLEM:
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%
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% y_t = Z_t * \alpha_t + \varepsilon_t
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% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
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%
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% with:
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%
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% \alpha_1 = a + A*\delta + R_0*\eta_0
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%
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% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
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% columns constitue all the columns of the m*m identity matrix) so that
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%
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% A'*R_0 = 0 and A'*\alpha_1 = \delta
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%
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% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
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% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
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% its variance is P, with
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%
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% P = \kappa*P_{\infty} + P_{\star}
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%
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% P_{\infty} = A*A'
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% P_{\star} = R_0*Q_0*R_0'
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%
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% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
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%
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%
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% and where:
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%
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% y_t is a pp*1 vector
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% \alpha_t is a mm*1 vector
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% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
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% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
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% a_1 is a mm*1 vector
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%
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% Z_t is a pp*mm matrix
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% T_t is a mm*mm matrix
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% H_t is a pp*pp matrix
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% R_t is a mm*rr matrix
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% Q_t is a rr*rr matrix
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% P_1 is a mm*mm matrix
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%
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%
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% FILTERING EQUATIONS:
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%
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% v_t = y_t - Z_t* a_t
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% F_t = Z_t * P_t * Z_t' + H_t
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% K_t = T_t * P_t * Z_t' * F_t^{-1}
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% L_t = T_t - K_t * Z_t
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% a_{t+1} = T_t * a_t + K_t * v_t
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% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
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%
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%
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% DIFFUSE FILTERING EQUATIONS:
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%
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% a_{t+1} = T_t*a_t + K_{\infty,t}v_t
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% P_{\infty,t+1} = T_t*P_{\infty,t}*L_{\infty,t}'
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% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\infty,t}' - K_{\infty,t}*F_{\infty,t}*K_{\ast,t}' + R_t*Q_t*R_t'
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% K_{\infty,t} = T_t*P_{\infty,t}*Z_t'*F_{\infty,t}^{-1}
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% v_t = y_t - Z_t*a_t
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% L_{\infty,t} = T_t - K_{\infty,t}*Z_t
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% F_{\infty,t} = Z_t*P_{\infty,t}*Z_t'
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% K_{\ast,t} = (T_t*P_{\ast,t}*Z_t' + K_{\infty,t}*F_{\ast,t})*F_{\infty,t}^{-1}
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% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
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%
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% Matrix Finf is assumed to be non singular. If this is not the case we have
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% to switch to another algorithm (NewAlg=3).
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%
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% start = options_.presample
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% Analysis, vol. 24(1), pp. 85-98).
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%
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% part of DYNARE, copyright Dynare Team (2005-2008)
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% Gnu Public License.
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% M. Ratto added lik in output
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global bayestopt_ options_
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mf = bayestopt_.mf;
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