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<div><a href="../index.html">Home</a> > <a href="index.html">.</a> > DiffuseLikelihoodH1.m</div>
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<!--<table width="100%"><tr><td align="left"><a href="../index.html"><img alt="<" border="0" src="../left.png"> Master index</a></td>
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<td align="right"><a href="index.html">Index for . <img alt=">" border="0" src="../right.png"></a></td></tr></table>-->
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<h1>DiffuseLikelihoodH1
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</h1>
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<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>M. Ratto added lik in output</strong></div>
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<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>function [LIK, lik] = DiffuseLikelihoodH1(T,R,Q,H,Pinf,Pstar,Y,trend,start) </strong></div>
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<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre class="comment"> M. Ratto added lik in output
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stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
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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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THE PROBLEM:
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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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with:
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\alpha_1 = a + A*\delta + R_0*\eta_0
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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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A'*R_0 = 0 and A'*\alpha_1 = \delta
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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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P = \kappa*P_{\infty} + P_{\star}
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P_{\infty} = A*A'
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P_{\star} = R_0*Q_0*R_0'
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P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
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and where:
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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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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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FILTERING EQUATIONS:
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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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DIFFUSE FILTERING EQUATIONS:
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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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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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start = options_.presample</pre></div>
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<!-- crossreference -->
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<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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This function calls:
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<ul style="list-style-image:url(../matlabicon.gif)">
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</ul>
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This function is called by:
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<ul style="list-style-image:url(../matlabicon.gif)">
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<li><a href="DsgeLikelihood.html" class="code" title="function [fval,cost_flag,ys,trend_coeff,info] = DsgeLikelihood(xparam1,gend,data)">DsgeLikelihood</a> stephane.adjemian@cepremap.cnrs.fr [09-07-2004]</li></ul>
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<!-- crossreference -->
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<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre>0001 <a name="_sub0" href="#_subfunctions" class="code">function [LIK, lik] = DiffuseLikelihoodH1(T,R,Q,H,Pinf,Pstar,Y,trend,start)</a>
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0002 <span class="comment">% M. Ratto added lik in output</span>
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0003 <span class="comment">% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]</span>
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0004 <span class="comment">%</span>
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0005 <span class="comment">% See "Filtering and Smoothing of State Vector for Diffuse State Space</span>
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0006 <span class="comment">% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series</span>
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0007 <span class="comment">% Analysis, vol. 24(1), pp. 85-98).</span>
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0008 <span class="comment">%</span>
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0009 <span class="comment">% THE PROBLEM:</span>
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0010 <span class="comment">%</span>
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0011 <span class="comment">% y_t = Z_t * \alpha_t + \varepsilon_t</span>
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0012 <span class="comment">% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t</span>
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0013 <span class="comment">%</span>
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0014 <span class="comment">% with:</span>
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0015 <span class="comment">%</span>
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0016 <span class="comment">% \alpha_1 = a + A*\delta + R_0*\eta_0</span>
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0017 <span class="comment">%</span>
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0018 <span class="comment">% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their</span>
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0019 <span class="comment">% columns constitue all the columns of the m*m identity matrix) so that</span>
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0020 <span class="comment">%</span>
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0021 <span class="comment">% A'*R_0 = 0 and A'*\alpha_1 = \delta</span>
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0022 <span class="comment">%</span>
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0023 <span class="comment">% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)</span>
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0024 <span class="comment">% for a given \kappa > 0. So that the expectation of \alpha_1 is a and</span>
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0025 <span class="comment">% its variance is P, with</span>
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0026 <span class="comment">%</span>
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0027 <span class="comment">% P = \kappa*P_{\infty} + P_{\star}</span>
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0028 <span class="comment">%</span>
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0029 <span class="comment">% P_{\infty} = A*A'</span>
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0030 <span class="comment">% P_{\star} = R_0*Q_0*R_0'</span>
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0031 <span class="comment">%</span>
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0032 <span class="comment">% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.</span>
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0033 <span class="comment">%</span>
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0034 <span class="comment">%</span>
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0035 <span class="comment">% and where:</span>
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0036 <span class="comment">%</span>
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0037 <span class="comment">% y_t is a pp*1 vector</span>
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0038 <span class="comment">% \alpha_t is a mm*1 vector</span>
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0039 <span class="comment">% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))</span>
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0040 <span class="comment">% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))</span>
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0041 <span class="comment">% a_1 is a mm*1 vector</span>
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0042 <span class="comment">%</span>
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0043 <span class="comment">% Z_t is a pp*mm matrix</span>
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0044 <span class="comment">% T_t is a mm*mm matrix</span>
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0045 <span class="comment">% H_t is a pp*pp matrix</span>
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0046 <span class="comment">% R_t is a mm*rr matrix</span>
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0047 <span class="comment">% Q_t is a rr*rr matrix</span>
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0048 <span class="comment">% P_1 is a mm*mm matrix</span>
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0049 <span class="comment">%</span>
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0050 <span class="comment">%</span>
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0051 <span class="comment">% FILTERING EQUATIONS:</span>
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0052 <span class="comment">%</span>
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0053 <span class="comment">% v_t = y_t - Z_t* a_t</span>
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0054 <span class="comment">% F_t = Z_t * P_t * Z_t' + H_t</span>
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0055 <span class="comment">% K_t = T_t * P_t * Z_t' * F_t^{-1}</span>
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0056 <span class="comment">% L_t = T_t - K_t * Z_t</span>
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0057 <span class="comment">% a_{t+1} = T_t * a_t + K_t * v_t</span>
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0058 <span class="comment">% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'</span>
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0059 <span class="comment">%</span>
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0060 <span class="comment">%</span>
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0061 <span class="comment">% DIFFUSE FILTERING EQUATIONS:</span>
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0062 <span class="comment">%</span>
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0063 <span class="comment">% a_{t+1} = T_t*a_t + K_{\infty,t}v_t</span>
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0064 <span class="comment">% P_{\infty,t+1} = T_t*P_{\infty,t}*L_{\infty,t}'</span>
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0065 <span class="comment">% 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'</span>
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0066 <span class="comment">% K_{\infty,t} = T_t*P_{\infty,t}*Z_t'*F_{\infty,t}^{-1}</span>
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0067 <span class="comment">% v_t = y_t - Z_t*a_t</span>
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0068 <span class="comment">% L_{\infty,t} = T_t - K_{\infty,t}*Z_t</span>
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0069 <span class="comment">% F_{\infty,t} = Z_t*P_{\infty,t}*Z_t'</span>
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0070 <span class="comment">% K_{\ast,t} = (T_t*P_{\ast,t}*Z_t' + K_{\infty,t}*F_{\ast,t})*F_{\infty,t}^{-1}</span>
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0071 <span class="comment">% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t</span>
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0072 <span class="comment">%</span>
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0073 <span class="comment">% Matrix Finf is assumed to be non singular. If this is not the case we have</span>
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0074 <span class="comment">% to switch to another algorithm (NewAlg=3).</span>
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0075 <span class="comment">%</span>
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0076 <span class="comment">% start = options_.presample</span>
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0077 <span class="keyword">global</span> bayestopt_ options_
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0078
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0079 mf = bayestopt_.mf;
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0080 smpl = size(Y,2);
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0081 mm = size(T,2);
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0082 pp = size(Y,1);
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0083 a = zeros(mm,1);
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0084 dF = 1;
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0085 QQ = R*Q*transpose(R);
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0086 t = 0;
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0087 lik = zeros(smpl+1,1);
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0088 LIK = Inf;
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0089 lik(smpl+1) = smpl*pp*log(2*pi);
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0090 notsteady = 1;
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0091 crit = options_.kalman_tol;
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0092 reste = 0;
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0093 <span class="keyword">while</span> rank(Pinf,crit) & t < smpl
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0094 t = t+1;
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0095 v = Y(:,t)-a(mf)-trend(:,t);
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0096 Finf = Pinf(mf,mf);
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0097 <span class="keyword">if</span> rcond(Finf) < crit
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0098 <span class="keyword">if</span> ~all(abs(Finf(:))<crit)
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0099 <span class="keyword">return</span>
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0100 <span class="keyword">else</span>
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0101 iFstar = inv(Pstar(mf,mf)+H);
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0102 dFstar = det(Pstar(mf,mf)+H);
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0103 Kstar = Pstar(:,mf)*iFstar;
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0104 lik(t) = log(dFstar) + transpose(v)*iFstar*v;
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0105 Pinf = T*Pinf*transpose(T);
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0106 Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kstar))*transpose(T)+QQ;
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0107 a = T*(a+Kstar*v);
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0108 <span class="keyword">end</span>
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0109 <span class="keyword">else</span>
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0110 lik(t) = log(det(Finf));
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0111 iFinf = inv(Finf);
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0112 Kinf = Pinf(:,mf)*iFinf; <span class="comment">%% premultiplication by the transition matrix T is removed (stephane)</span>
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0113 Fstar = Pstar(mf,mf)+H;
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0114 Kstar = (Pstar(:,mf)-Kinf*Fstar)*iFinf; <span class="comment">%% premultiplication by the transition matrix T is removed (stephane)</span>
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0115 Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kinf)-Pinf(:,mf)*transpose(Kstar))*transpose(T)+QQ;
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0116 Pinf = T*(Pinf-Pinf(:,mf)*transpose(Kinf))*transpose(T);
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0117 a = T*(a+Kinf*v);
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0118 <span class="keyword">end</span>
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0119 <span class="keyword">end</span>
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0120 <span class="keyword">if</span> t == smpl
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0121 error([<span class="string">'There isn''t enough information to estimate the initial'</span> <span class="keyword">...</span><span class="comment"> </span>
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0122 <span class="string">' conditions of the nonstationary variables'</span>]);
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0123 <span class="keyword">end</span>
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0124 F_singular = 1;
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0125 <span class="keyword">while</span> notsteady & t < smpl
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0126 t = t+1;
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0127 v = Y(:,t)-a(mf)-trend(:,t);
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0128 F = Pstar(mf,mf)+H;
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0129 oldPstar = Pstar;
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0130 dF = det(F);
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0131 <span class="keyword">if</span> rcond(F) < crit
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0132 <span class="keyword">if</span> ~all(abs(F(:))<crit)
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0133 <span class="keyword">return</span>
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0134 <span class="keyword">else</span>
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0135 a = T*a;
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0136 Pstar = T*Pstar*transpose(T)+QQ;
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0137 <span class="keyword">end</span>
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0138 <span class="keyword">else</span>
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0139 F_singular = 0;
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0140 iF = inv(F);
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0141 lik(t) = log(dF)+transpose(v)*iF*v;
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0142 K = Pstar(:,mf)*iF; <span class="comment">%% premultiplication by the transition matrix T is removed (stephane)</span>
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0143 a = T*(a+K*v); <span class="comment">%% --> factorization of the transition matrix...</span>
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0144 Pstar = T*(Pstar-K*Pstar(mf,:))*transpose(T)+QQ; <span class="comment">%% ... idem (stephane)</span>
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0145 <span class="keyword">end</span>
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0146 notsteady = ~(max(max(abs(Pstar-oldPstar)))<crit);
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0147 <span class="keyword">end</span>
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0148 <span class="keyword">if</span> F_singular == 1
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0149 error([<span class="string">'The variance of the forecast error remains singular until the'</span> <span class="keyword">...</span>
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0150 <span class="string">'end of the sample'</span>])
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0151 <span class="keyword">end</span>
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0152 reste = smpl-t;
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0153 <span class="keyword">while</span> t < smpl
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0154 t = t+1;
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0155 v = Y(:,t)-a(mf)-trend(:,t);
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0156 a = T*(a+K*v);
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0157 lik(t) = transpose(v)*iF*v;
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0158 <span class="keyword">end</span>
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0159 lik(t) = lik(t) + reste*log(dF);
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0160 LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);<span class="comment">% Minus the</span>
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0161 <span class="comment">% log-likelihood.</span>
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0162</pre></div>
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<hr><address>Generated on Fri 16-Jun-2006 09:09:06 by <strong><a href="http://www.artefact.tk/software/matlab/m2html/">m2html</a></strong> © 2003</address>
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