382 lines
18 KiB
HTML
382 lines
18 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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"http://www.w3.org/TR/REC-html40/loose.dtd">
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<html>
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<head>
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<title>Description of olr2</title>
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<meta name="keywords" content="olr2">
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<meta name="description" content="Copyright (C) 2003 Michel Juillard">
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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
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<meta name="generator" content="m2html © 2003 Guillaume Flandin">
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<meta name="robots" content="index, follow">
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<link type="text/css" rel="stylesheet" href="../m2html.css">
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</head>
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<body>
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<a name="_top"></a>
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<div><a href="../index.html">Home</a> > <a href="index.html">.</a> > olr2.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>olr2
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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>Copyright (C) 2003 Michel Juillard</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 dr=olr2(dr,olr_inst,bet,obj_var,W) </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"> Copyright (C) 2003 Michel Juillard
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computes an optimal policy as the optimal linear regulator</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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<li><a href="union.html" class="code" title="function x = union(a,b)">union</a> </li></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="olr1.html" class="code" title="function dr = olr1(ys,algo,olr_inst,bet,obj_var,W)">olr1</a> Copyright (C) 2001 Michel Juillard</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 <span class="comment">% Copyright (C) 2003 Michel Juillard</span>
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0002 <span class="comment">%</span>
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0003 <span class="comment">% computes an optimal policy as the optimal linear regulator</span>
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0004 <span class="comment">%</span>
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0005 <a name="_sub0" href="#_subfunctions" class="code">function dr=olr2(dr,olr_inst,bet,obj_var,W)</a>
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0006
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0007 <span class="keyword">global</span> M_ options_ oo_
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0008 <span class="keyword">global</span> it_ stdexo_
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0009
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0010 xlen = M_.maximum_lead + M_.maximum_lag + 1;
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0011 klen = M_.maximum_lag + M_.maximum_lead + 1;
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0012 iyv = M_.lead_lag_incidence';
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0013 iyv = iyv(:);
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0014 iyr0 = find(iyv) ;
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0015 it_ = M_.maximum_lag + 1 ;
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0016
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0017 inst_nbr = size(olr_inst,1);
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0018 inst_i = zeros(inst_nbr,1);
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0019 <span class="keyword">for</span> i=1:inst_nbr
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0020 k = strmatch(olr_inst(i,:),M_.endo_names,<span class="string">'exact'</span>);
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0021 <span class="keyword">if</span> isempty(k)
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0022 error(sprintf(<span class="string">'OLR_INST %s isn''t a declared variable'</span>));
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0023 <span class="keyword">else</span>
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0024 inst_i(i) = k;
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0025 <span class="keyword">end</span>
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0026 <span class="keyword">end</span>
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0027
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0028 <span class="keyword">if</span> M_.maximum_lead == 0
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0029 error (<span class="string">'OLR : No forward variable: no point in using OLR'</span>) ;
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0030 <span class="keyword">end</span>
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0031
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0032 <span class="keyword">if</span> find(any(M_.lead_lag_incidence([1:M_.maximum_lag M_.maximum_lag+2:M_.maximum_lead],inst_i),2))
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0033 error(<span class="string">'OLR: instruments can only appear at the current period'</span>);
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0034 <span class="keyword">end</span>
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0035
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0036 non_inst_i = setdiff([1:M_.endo_nbr],inst_i);
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0037 iy1_ = M_.lead_lag_incidence(:,non_inst_i);
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0038 endo_nbr_1 = M_.endo_nbr - inst_nbr;
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0039
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0040 <span class="keyword">if</span> M_.exo_nbr == 0
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0041 oo_.exo_steady_state = [] ;
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0042 <span class="keyword">end</span>
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0043
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0044 <span class="keyword">if</span> ~ M_.lead_lag_incidence(M_.maximum_lag+1,:) > 0
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0045 error (<span class="string">'Error in model specification: some variables don''t appear as current'</span>) ;
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0046 <span class="keyword">end</span>
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0047
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0048 <span class="keyword">if</span> xlen > 1
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0049 error ([<span class="string">'SS: stochastic exogenous variables must appear only at the'</span> <span class="keyword">...</span>
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0050 <span class="string">' current period. Use additional endogenous variables'</span>]) ;
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0051 <span class="keyword">end</span>
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0052
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0053 tempex = oo_.exo_simul;
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0054
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0055 it_ = M_.maximum_lag + 1;
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0056 dr.ys = oo_.steady_state;
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0057 z = repmat(dr.ys,1,klen);
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0058 z = z(iyr0) ;
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0059
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0060 <span class="comment">%M_.jacobia=real(jacob_a('ff1_',[z; oo_.exo_steady_state])) ;</span>
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0061 [junk,M_.jacobia] = feval([M_.fname <span class="string">'_dynamic'</span>],z,oo_.exo_simul);
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0062
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0063 oo_.exo_simul = tempex ;
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0064 tempex = [];
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0065
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0066 nz = size(z,1);
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0067 fwrd_var = find(any(M_.lead_lag_incidence(M_.maximum_lag+2:<span class="keyword">end</span>,:),1))';
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0068 <span class="keyword">if</span> M_.maximum_lag > 0
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0069 pred_var = find(any(M_.lead_lag_incidence(1:M_.maximum_lag,:),1))';
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0070 both_var = intersect(pred_var,fwrd_var);
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0071 pred_var = setdiff(pred_var,both_var);
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0072 fwrd_var = setdiff(fwrd_var,both_var);
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0073 stat_var = setdiff([1:M_.endo_nbr]',<a href="union.html" class="code" title="function x = union(a,b)">union</a>(<a href="union.html" class="code" title="function x = union(a,b)">union</a>(pred_var,both_var),fwrd_var)); <span class="comment">% static variables</span>
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0074 <span class="keyword">else</span>
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0075 pred_var = [];
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0076 both_var = [];
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0077 stat_var = setdiff([1:M_.endo_nbr]',fwrd_var);
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0078 <span class="keyword">end</span>
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0079
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0080 stat_var = setdiff(stat_var,inst_i);
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0081 <span class="comment">% static variables in objective function</span>
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0082 [stat_obj_var] = intersect(obj_var,stat_var);
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0083 n_stat_obj_var = length(stat_obj_var);
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0084 pred_var = [stat_obj_var; pred_var];
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0085 nboth = length(both_var);
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0086 npred = length(pred_var);
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0087 nfwrd = length(fwrd_var);
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0088 nstatic1 = length(stat_var);
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0089 nstatic = nstatic1-n_stat_obj_var;
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0090
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0091 order_var = [ setdiff(stat_var,obj_var); pred_var; both_var; fwrd_var];
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0092 k1 = M_.lead_lag_incidence(find([1:klen] ~= M_.maximum_lag+1),:);
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0093 b = M_.jacobia(1:endo_nbr_1,M_.lead_lag_incidence(M_.maximum_lag+1,order_var));
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0094 a = b\M_.jacobia(1:endo_nbr_1,nonzeros(k1'));
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0095 <span class="keyword">if</span> M_.exo_nbr
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0096 fu = -b\M_.jacobia(1:endo_nbr_1,nz+1:end);
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0097 <span class="keyword">end</span>
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0098 <span class="comment">% instruments' effects</span>
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0099 b = -b\M_.jacobia(1:endo_nbr_1,M_.lead_lag_incidence(M_.maximum_lag+1,inst_i));
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0100 <span class="comment">% building kmask for z state vector in t+1</span>
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0101 <span class="keyword">if</span> M_.maximum_lag > 0
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0102 <span class="keyword">if</span> M_.maximum_lead > 0
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0103 kmask = [flipud(cumsum(M_.lead_lag_incidence(1:M_.maximum_lag,order_var),1))] ;
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0104 kmask = [kmask; flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_lag+2:<span class="keyword">end</span>,order_var)),1))] ;
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0105 <span class="keyword">end</span>
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0106 <span class="keyword">else</span>
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0107 kmask = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_lag+2:klen,order_var)),1));
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0108 <span class="keyword">end</span>
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0109 <span class="comment">% static variables in objective function</span>
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0110 kmask(1,nstatic+1:nstatic1) = 1;
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0111 kmask = kmask';
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0112
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0113 <span class="comment">% lags aren't ordered as in dr1 !</span>
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0114 <span class="comment">% this is necessary to have the zeros on R's diagonal aligned with jump variables</span>
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0115 kmask = kmask(:);
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0116 i_kmask = find(kmask); <span class="comment">% index of nonzero entries in kmask</span>
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0117 nd = size(i_kmask,1); <span class="comment">% size of the state vector</span>
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0118 kmask(i_kmask) = [1:nd];
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0119
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0120 <span class="comment">% auxiliary equations</span>
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0121
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0122 <span class="comment">% elements that are both in z(t+1) and z(t)</span>
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0123 kad = [];
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0124 kae = [];
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0125 k1 = find([kmask(1:endo_nbr_1) & kmask(M_.maximum_lag*endo_nbr_1+1:(M_.maximum_lag+1)*endo_nbr_1)] );
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0126 <span class="keyword">if</span> ~isempty(k1)
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0127 kad = kmask(k1);
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0128 kae = kmask(k1+M_.maximum_lag*endo_nbr_1);
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0129 <span class="keyword">end</span>
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0130
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0131 <span class="keyword">if</span> M_.maximum_lag > 1
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0132 k1 = find([kmask(endo_nbr_1+1:M_.maximum_lag*endo_nbr_1) & kmask(1:(M_.maximum_lag-1)*endo_nbr_1)] );
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0133 <span class="keyword">if</span> ~isempty(k1)
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0134 kad = [kad; kmask(k1+endo_nbr_1)];
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0135 kae = [kae; kmask(k1)];
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0136 <span class="keyword">end</span>
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0137 <span class="keyword">end</span>
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0138 <span class="keyword">if</span> M_.maximum_lead > 1
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0139 k1 = find([kmask((M_.maximum_lag+1)*endo_nbr_1+1:end) & kmask(M_.maximum_lag*endo_nbr_1+1:end-endo_nbr_1)] );
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0140 <span class="keyword">if</span> ~isempty(k1)
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0141 kad = [kad; kmask(M_.maximum_lag*endo_nbr_1+k1)];
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0142 kae = [kae; kmask((M_.maximum_lag+1)*endo_nbr_1+k1)];
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0143 <span class="keyword">end</span>
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0144 <span class="keyword">end</span>
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0145
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0146 <span class="comment">% composition of state vector</span>
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0147 <span class="comment">% col 1: variable; col 2: lead/lag in z(t+1);</span>
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0148 <span class="comment">% col 3: A cols for t+1 (D); col 4: A cols for t (E)</span>
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0149 kstate = [ repmat([1:endo_nbr_1]',klen-1,1) <span class="keyword">...</span>
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0150 [kron([M_.maximum_lag+1:-1:2]',ones(endo_nbr_1,1)); <span class="keyword">...</span>
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0151 kron([M_.maximum_lag+2:klen]',ones(endo_nbr_1,1))] <span class="keyword">...</span>
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0152 zeros((klen-1)*endo_nbr_1,2)];
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0153 kiy = [flipud(M_.lead_lag_incidence(1:M_.maximum_lag+1,order_var)); M_.lead_lag_incidence(M_.maximum_lag+2:<span class="keyword">end</span>,order_var)]';
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0154 kiy = kiy(:);
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0155 i1 = find(kiy((M_.maximum_lag+1)*endo_nbr_1+1:end));
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0156 kstate(i1+M_.maximum_lag*endo_nbr_1,3) = kiy(i1+(M_.maximum_lag+1)*endo_nbr_1)-M_.endo_nbr;
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0157 kstate(1:M_.maximum_lag*endo_nbr_1,4) = kiy(endo_nbr_1+1:(M_.maximum_lag+1)*endo_nbr_1);
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0158 <span class="comment">% put in E only the current variables that are not already in D</span>
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0159 kstate = kstate(i_kmask,:);
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0160
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0161 sdyn = M_.endo_nbr-nstatic-inst_nbr;
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0162
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0163 <span class="comment">% buildind D and E</span>
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0164 d = zeros(nd,nd) ;
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0165 e = d ;
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0166
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0167 <span class="comment">% dynamical system</span>
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0168 k = find(kstate(:,2) >= M_.maximum_lag+2 & kstate(:,3));
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0169 d(1:sdyn,k) = a(nstatic+1:<span class="keyword">end</span>,kstate(k,3)) ;
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0170 k1 = find(kstate(:,2) == M_.maximum_lag+2);
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0171 a1 = eye(sdyn);
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0172 e(1:sdyn,k1) = -a1(:,kstate(k1,1)-nstatic);
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0173 k = find(kstate(:,2) <= M_.maximum_lag+1 & kstate(:,4));
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0174 e(1:sdyn,k) = -a(nstatic+1:<span class="keyword">end</span>,kstate(k,4)) ;
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0175 k2 = find(kstate(:,2) == M_.maximum_lag+1);
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0176 k2 = k2(~ismember(kstate(k2,1),kstate(k1,1)));
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0177 d(1:sdyn,k2) = a1(:,kstate(k2,1)-nstatic);
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0178
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0179 <span class="keyword">if</span> ~isempty(kad)
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0180 <span class="keyword">for</span> j = 1:size(kad,1)
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0181 d(sdyn+j,kad(j)) = 1 ;
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0182 e(sdyn+j,kae(j)) = 1 ;
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0183 <span class="keyword">end</span>
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0184 <span class="keyword">end</span>
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0185 [Q,R] = qr(d);
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0186 C = Q'*[e [[b(nstatic+1:<span class="keyword">end</span>,:) fu(nstatic+1:<span class="keyword">end</span>,:)];<span class="keyword">...</span>
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0187 zeros(nd-sdyn,inst_nbr+M_.exo_nbr)]];
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0188
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0189 nyf = sum(kstate(:,2) > M_.maximum_lag+1);
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0190
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0191
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0192 np = nd - nyf;
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0193
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0194
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0195 dd = zeros(2*nd+inst_nbr,2*nd+inst_nbr);
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0196 ee = dd;
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0197
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0198 dd(1:nd,1:np) = R(:,1:np);
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0199 ee(1:nd,1:np) = C(:,1:np);
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0200 dd(1:nd,nd+1:nd+nyf) = R(:,np+1:end);
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0201 ee(1:nd,nd+1:nd+nyf+inst_nbr) = C(:,np+1:nd+inst_nbr);
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0202
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0203
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0204 <span class="comment">% derivatives with respect to x</span>
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0205 <span class="comment">% building QQ and UU</span>
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0206 m = 1;
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0207 v0 = kstate(find(kstate(:,2)==M_.maximum_lag+2),1);
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0208 <span class="keyword">for</span> i=1:nd;
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0209 <span class="keyword">if</span> (kstate(i,2) == M_.maximum_lag+1 & isempty(find(kstate(i,1)==v0)))
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0210 k = find(order_var(kstate(i,1))==obj_var);
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0211 <span class="keyword">if</span> ~isempty(k)
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0212 iq(m) = i;
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0213 m = m+1;
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0214 <span class="keyword">end</span>
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0215 <span class="keyword">elseif</span> kstate(i,2) == M_.maximum_lag+2
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0216 k = find(order_var(kstate(i,1))==obj_var);
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0217 <span class="keyword">if</span> ~isempty(k)
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0218 iq(m) = i;
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0219 m = m+1;
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0220 <span class="keyword">end</span>
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0221 <span class="keyword">end</span>
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0222 <span class="keyword">end</span>
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0223 QQ1 = zeros(nd,nd);
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0224 QQ1(iq,iq) = W(obj_var,obj_var);
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0225 UU1 = zeros(nd,inst_nbr);
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0226 UU1(iq,:) = W(obj_var,inst_i);
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0227 RR = W(inst_i,inst_i);
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0228 offset = nd;
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0229 ee(offset+1:2*offset,1:np) = bet*QQ1(1:np,:)';
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0230 dd(offset+1:2*offset,np+1:nd) = bet*C(np+1:nd,1:nd)';
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0231 ee(offset+1:2*offset,nd+1:nd+nyf) = bet*QQ1(np+1:<span class="keyword">end</span>,:)';
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0232 dd(offset+1:2*offset,nd+nyf+inst_nbr+1:end)=bet*C(1:np,1:nd)';
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0233 ee(offset+1:2*offset,np+1:nd) = R(np+1:<span class="keyword">end</span>,:)';
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0234 ee(offset+1:2*offset,nd+nyf+1:nd+nyf+inst_nbr) = -bet*UU1;
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0235 ee(offset+1:2*offset,nd+nyf+inst_nbr+1:end) = R(1:np,:)';
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0236
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0237 <span class="comment">%derivatives with respect to u</span>
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0238 offset = 2*nd;
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0239 dd(offset+1:<span class="keyword">end</span>,np+1:nd) = -C(np+1:<span class="keyword">end</span>,nd+1:nd+inst_nbr)';
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0240 dd(offset+1:<span class="keyword">end</span>,nd+nyf+inst_nbr+1:end) = -C(1:np,nd+1:nd+inst_nbr)';
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0241 ee(offset+1:<span class="keyword">end</span>,1:np) = UU1(1:np,:)';
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0242 ee(offset+1:<span class="keyword">end</span>,nd+1:nd+nyf) = UU1(np+1:<span class="keyword">end</span>,:)';
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0243 ee(offset+1:<span class="keyword">end</span>,nd+nyf+1:nd+nyf+inst_nbr) = RR;
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0244
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0245 [ss,tt,w,sdim,oo_.eigenvalues,info] = mjdgges(ee,dd);
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0246
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0247 nba = sum(abs(oo_.eigenvalues) > 1+1e-5);
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0248 nyf1 = nd+inst_nbr;
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0249 nd = 2*nd+inst_nbr;
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0250
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0251 <span class="keyword">if</span> nba ~= nyf1;
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0252 warning(<span class="string">'OLR: Blanchard-Kahn conditions are not satisfied.'</span>);
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0253 <span class="keyword">end</span>
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0254
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0255 np1 = nd - nyf1;
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0256 n2 = np1 + 1;
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0257 n3 = np1;
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0258 n4 = n3 + 1;
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0259
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0260 <span class="comment">% derivatives with respect to dynamic state variables</span>
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0261 <span class="comment">% forward variables</span>
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0262 gx = -w(n4:nd,n2:nd)'\w(1:n3,n2:nd)';
|
|
0263 <span class="comment">% predetermined variables</span>
|
|
0264 hx = w(n4:nd,1:np1)'*gx+w(1:n3,1:np1)';
|
|
0265 hx = (tt(1:np1,1:np1)*hx)\(ss(1:np1,1:np1)*hx);
|
|
0266
|
|
0267 <span class="comment">% including Lagrange multipliers in M_.endo_names, order_var and kstate</span>
|
|
0268 <span class="keyword">for</span> i=1:M_.maximum_lead
|
|
0269 k = find(kstate(:,2)==M_.maximum_lag+1+i);
|
|
0270 temp = strcat(<span class="string">'mult_'</span>,M_.endo_names(order_var(kstate(k,1)),:));
|
|
0271 temp = strcat(temp,[<span class="string">'_'</span> int2str(i)]);
|
|
0272 M_.endo_names = strvcat(M_.endo_names,temp);
|
|
0273 <span class="keyword">end</span>
|
|
0274
|
|
0275 <span class="keyword">if</span> nyf > nboth
|
|
0276 order_var = [order_var(1:nstatic+npred+nboth);[M_.endo_nbr+1:M_.endo_nbr+nyf]'; <span class="keyword">...</span>
|
|
0277 order_var(nstatic+npred+nboth+1:end); inst_i];
|
|
0278 <span class="keyword">else</span>
|
|
0279 order_var = [order_var(1:nstatic+npred+nboth);[M_.endo_nbr+1:M_.endo_nbr+nyf]'; <span class="keyword">...</span>
|
|
0280 inst_i];
|
|
0281 <span class="keyword">end</span>
|
|
0282 kstate = [kstate(1:np,:);zeros(nyf,4);kstate(np+1:<span class="keyword">end</span>,:);zeros(inst_nbr+np,4)];
|
|
0283 k = find(kstate(:,2) <= M_.maximum_lag+1);
|
|
0284 kstate(np+1:np+nyf,1:2) = [[nstatic+npred+nboth+1:nstatic+npred+nboth+nyf]' <span class="keyword">...</span>
|
|
0285 (M_.maximum_lag+1)*ones(nyf,1)];
|
|
0286 kstate(np+2*nyf+1:np+2*nyf+inst_nbr,1:2) = [[endo_nbr_1+1:endo_nbr_1+inst_nbr]' <span class="keyword">...</span>
|
|
0287 (M_.maximum_lag+2)*ones(inst_nbr,1)];
|
|
0288 kstate(np+2*nyf+inst_nbr+1:<span class="keyword">end</span>,1:2) = [[M_.endo_nbr+1:M_.endo_nbr+np]' (M_.maximum_lag+2)*ones(np,1)];
|
|
0289
|
|
0290 k1 = find(kstate(:,2) == M_.maximum_lag+1);
|
|
0291 k2 = find(kstate(:,2) == M_.maximum_lag+2)-np-nyf;
|
|
0292 dr.ghx = [hx(k1,:); gx(k2(nboth+1:end),:)];
|
|
0293
|
|
0294 <span class="comment">%lead variables actually present in the model</span>
|
|
0295 <span class="comment">% derivatives with respect to exogenous variables</span>
|
|
0296 <span class="keyword">if</span> M_.exo_nbr
|
|
0297 n1 = find(kstate(:,2) > M_.maximum_lag+1);
|
|
0298 ghu = [dd(:,n1(1):end)*gx+dd(:,1:n1(1)-1) -ee(:,np+nyf+1:end)]\[C(:,end-M_.exo_nbr+1:end); zeros(size(dd,1)-size(C,1),M_.exo_nbr)];
|
|
0299 dr.ghu = [ghu(k1,:);ghu(k2(nboth+1:end)+np+nyf,:)];
|
|
0300 <span class="keyword">end</span>
|
|
0301
|
|
0302 <span class="comment">% static variables</span>
|
|
0303 <span class="keyword">if</span> nstatic > 0
|
|
0304 j3 = nonzeros(kstate(:,3));
|
|
0305 j4 = find(kstate(:,3))-np-nyf;
|
|
0306 temp = -a(1:nstatic,j3)*gx(j4,:)*hx;
|
|
0307 temp = temp + b(1:nstatic,:)*gx(nyf+1:nyf+inst_nbr,:);
|
|
0308 j5 = find(kstate(:,4));
|
|
0309 temp(:,j5) = temp(:,j5)-a(1:nstatic,nonzeros(kstate(:,4)));
|
|
0310 dr.ghx = [temp; dr.ghx];
|
|
0311 temp = -a(1:nstatic,j3)*gx(j4,:)*ghu(1:np+nyf,:);
|
|
0312 temp = temp + b(1:nstatic,:)*ghu(np+2*nyf+1:np+2*nyf+inst_nbr,:);
|
|
0313 temp = temp + fu(1:nstatic,:);
|
|
0314 dr.ghu = [temp; dr.ghu];
|
|
0315 temp = [];
|
|
0316 <span class="keyword">end</span>
|
|
0317
|
|
0318 dr.ghx = dr.ghx(1:M_.endo_nbr+nyf,:);
|
|
0319 dr.ghu = dr.ghu(1:M_.endo_nbr+nyf,:);
|
|
0320
|
|
0321 dr.ys = [dr.ys; zeros(nyf,1)];
|
|
0322 dr.nstatic1 = nstatic1;
|
|
0323 dr.nstatic = nstatic;
|
|
0324 dr.npred = npred+nyf+nboth;
|
|
0325 dr.kstate = kstate;
|
|
0326 dr.order_var = order_var;
|
|
0327 M_.endo_nbr = M_.endo_nbr+nyf;
|
|
0328
|
|
0329 <span class="comment">% 05/29/03 MJ replaced diffext by jacobia (much faster)</span>
|
|
0330 <span class="comment">% corrected kmask for static variables in objective function</span>
|
|
0331
|
|
0332
|
|
0333
|
|
0334</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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