1function [Pi_bar,H0tldcell_inv,Hptldcell_inv] ... 2 = fn_rnrprior_covres_dobs_tv2(nvar,nStates,indxScaleStates,q_m,lags,xdgel,mu,indxDummy,Ui,Vi,hpmsmd,indxmsmdeqn,nexo,asym0,asymp) 3% Differs from fn_rnrprior_covres_dobs(): linear restrictions (Ui and Vi) have been incorported in fn_rnrprior_covres_dobs_tv?(). 4% Differs from fn_rnrprior_covres_dobs_tv(): allows an option to scale up the prior variance by nStates or not scale at all, 5% so that the prior value is the same as the constant VAR when the parameters in all states are the same. 6% 7% Only works for the nexo=1 (constant term) case. To extend this to other exogenous variables, see fn_dataxy.m. 01/14/03. 8% Differs from fn_rnrprior_covres_tv.m in that dummy observations are included as part of the explicit prior. See Forcast II, pp.68-69b. 9% Exports random Bayesian prior of Sims and Zha with asymmetric rior with linear restrictions already applied 10% and with dummy observations (i.e., mu(5) and mu(6)) used as part of an explicit prior. 11% This function allows for prior covariances for the MS and MD equations to achieve liquidity effects. 12% See Waggoner and Zha's Gibbs sampling paper and TVBVAR NOTES pp. 71k.0 and 50-61. 13% 14% nvar: number of endogenous variables 15% nStates: Number of states. 16% indxScaleStates: if 0, no scale adjustment in the prior variance for the number of states in the function fn_rnrprior_covres_dobs_tv2(); 17% if 1: allows a scale adjustment, marking the prior variance bigger by the number of states. 18% q_m: quarter or month 19% lags: the maximum length of lag 20% xdgel: T*nvar endogenous-variable matrix of raw or original data (no manipulation involved) with sample size including lags. 21% Order of columns: (1) nvar endogenous variables; (2) constants will be automatically put in the last column. 22% Used only to get variances of residuals for mu(1)-mu(5) and for dummy observations mu(5) and mu(6). 23% mu: 6-by-1 vector of hyperparameters (the following numbers for Atlanta Fed's forecast), where 24% mu(5) and mu(6) are NOT used here. See fn_dataxy.m for using mu(5) and mu(6). 25% mu(1): overall tightness and also for A0; (0.57) 26% mu(2): relative tightness for A+; (0.13) 27% mu(3): relative tightness for the constant term; (0.1). NOTE: for other 28% exogenous terms, the variance of each exogenous term must be taken into 29% acount to eliminate the scaling factor. 30% mu(4): tightness on lag decay; (1) 31% mu(5): weight on nvar sums of coeffs dummy observations (unit roots); (5) 32% mu(6): weight on single dummy initial observation including constant 33% (cointegration, unit roots, and stationarity); (5) 34% NOTE: for this function, mu(5) and mu(6) are not used. See fn_dataxy.m for using mu(5) and mu(6). 35% indxDummy: 1: uses dummy observations to form part of an explicit prior; 0: no dummy observations as part of the prior. 36% Ui: nvar-by-1 cell. In each cell, nvar-by-qi*si orthonormal basis for the null of the ith 37% equation contemporaneous restriction matrix where qi is the number of free parameters 38% within the state and si is the number of free states. 39% With this transformation, we have ai = Ui*bi or Ui'*ai = bi where ai is a vector 40% of total original parameters and bi is a vector of free parameters. When no 41% restrictions are imposed, we have Ui = I. There must be at least one free 42% parameter left for the ith equation in the order of [a_i for 1st state, ..., a_i for last state]. 43% Vi: nvar-by-1 cell. In each cell, k-by-ri*ti orthonormal basis for the null of the ith 44% equation lagged restriction matrix where k is a total of exogenous variables and 45% ri is the number of free parameters within the state and ti is the number of free states. 46% With this transformation, we have fi = Vi*gi 47% or Vi'*fi = gi where fi is a vector of total original parameters and gi is a 48% vector of free parameters. The ith equation is in the order of [nvar variables 49% for 1st lag and 1st state, ..., nvar variables for last lag and 1st state, const for 1st state, nvar 50% variables for 1st lag and 2nd state, nvar variables for last lag and 2nd state, const for 2nd state, and so on]. 51% hpmsmd: 2-by-1 hyperparameters with -1<h1=hpmsmd(1)<=0 for the MS equation and 0<=h2=hpmsmd(2)<1 the MD equation. Consider a1*R + a2*M. 52% The term h1*var(a1)*var(a2) is the prior covariance of a1 and a2 for MS, equivalent to penalizing the same sign of a1 and a2. 53% The term h2*var(a1)*var(a2) is the prior covariance of a1 and a2 for MD, equivalent to penalizing opposite signs of a1 and a2. 54% This will give us a liquidity effect. If hpmsmd=0, no such restrictions will be imposed. 55% indxmsmdeqn: 4-by-1 index for the locations of the MS and MD equation and for the locations of M and R. 56% indxmsmdeqn(1) for MS and indxmsmdeqn(2) for MD. 57% indxmsmdeqn(3) for M and indxmsmdeqn(4) for R. 58% nexo: number of exogenous variables (if not specified, nexo=1 (constant) by default). 59% The constant term is always put to the last of all endogenous and exogenous variables. 60% asym0: nvar-by-nvar asymmetric prior on A0. Column -- equation. 61% If ones(nvar,nvar), symmetric prior; if not, relative (asymmetric) tightness on A0. 62% asymp: ncoef-1-by-nvar asymmetric prior on A+ bar constant. Column -- equation. 63% If ones(ncoef-1,nvar), symmetric prior; if not, relative (asymmetric) tightness on A+. 64% -------------------- 65% Pi_bar: ncoef-by-nvar matrix for the ith equation under random walk. Same for all equations 66% H0tldcell_inv: cell(nvar,1). The ith cell represents the ith equation, where the dim is 67% qi*si-by-qi*si. The inverse of H0tld on p.60. 68% Hptldcell_inv: cell(nvar,1). The ith cell represents the ith equation, where the dim is 69% ri*ti-by-ri*ti.The inverse of Hptld on p.60. 70% 71% Differs from fn_rnrprior_covres_dobs(): linear restrictions (Ui and Vi) have been incorported in fn_rnrprior_covres_dobs_tv?(). 72% Differs from fn_rnrprior_covres_dobs_tv(): allows an option to scale up the prior variance by nStates or not scale at all. 73% so that the prior value is the same as the constant VAR when the parameters in all states are the same. 74% Tao Zha, February 2000. Revised, September 2000, 2001, February, May 2003, May 2004. 75% 76% Copyright (C) 1997-2012 Tao Zha 77% 78% This free software: you can redistribute it and/or modify 79% it under the terms of the GNU General Public License as published by 80% the Free Software Foundation, either version 3 of the License, or 81% (at your option) any later version. 82% 83% It is distributed in the hope that it will be useful, 84% but WITHOUT ANY WARRANTY; without even the implied warranty of 85% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 86% GNU General Public License for more details. 87% 88% If you did not received a copy of the GNU General Public License 89% with this software, see <http://www.gnu.org/licenses/>. 90% 91 92 93if nargin<=12, nexo=1; end % <<>>1 94ncoef = nvar*lags+nexo; % Number of coefficients in *each* equation for each state, RHS coefficients only. 95ncoefsts = nStates*ncoef; % Number of coefficients in *each* equation in all states, RHS coefficients only. 96 97H0tldcell_inv=cell(nvar,1); % inv(H0tilde) for different equations under asymmetric prior. 98Hptldcell_inv=cell(nvar,1); % inv(H+tilde) for different equations under asymmetric prior. 99 100%*** Constructing Pi_bar for the ith equation under the random walk assumption 101Pi_bar = zeros(ncoef,nvar); % same for all equations 102Pi_bar(1:nvar,1:nvar) = eye(nvar); % random walk 103 104% 105%@@@ Prepared for Bayesian prior 106% 107% 108% ** monthly lag decay in order to match quarterly decay: a*exp(bl) where 109% ** l is the monthly lag. Suppose quarterly decay is 1/x where x=1,2,3,4. 110% ** Let the decay of l1 (a*exp(b*l1)) match that of x1 (say, beginning: 1/1) 111% ** and the decay of l2 (a*exp(b*l2)) match that of x2 (say, end: 1/5), 112% ** we can solve for a and b which are 113% ** b = (log_x1-log_x2)/(l1-l2), and a = x1*exp(-b*l1). 114if q_m==12 115 l1 = 1; % 1st month == 1st quarter 116 xx1 = 1; % 1st quarter 117 l2 = lags; % last month 118 xx2 = 1/((ceil(lags/3))^mu(4)); % last quarter 119 %xx2 = 1/6; % last quarter 120 % 3rd quarter: i.e., we intend to let decay of the 6th month match 121 % that of the 3rd quarter, so that the 6th month decays a little 122 % faster than the second quarter which is 1/2. 123 if lags==1 124 b = 0; 125 else 126 b = (log(xx1)-log(xx2))/(l1-l2); 127 end 128 a = xx1*exp(-b*l1); 129end 130 131 132 133% 134% *** specify the prior for each equation separately, SZ method, 135% ** get the residuals from univariate regressions. 136% 137sgh = zeros(nvar,1); % square root 138sgsh = sgh; % square 139nSample=size(xdgel,1); % sample size-lags 140yu = xdgel; 141C = ones(nSample,1); 142for k=1:nvar 143 [Bk,ek,junk1,junk2,junk3,junk4] = sye([yu(:,k) C],lags); 144 clear Bk junk1 junk2 junk3 junk4; 145 sgsh(k) = ek'*ek/(nSample-lags); 146 sgh(k) = sqrt(sgsh(k)); 147end 148% ** prior variance for A0(:,1), same for all equations!!! 149sg0bid = zeros(nvar,1); % Sigma0_bar diagonal only for the ith equation 150for j=1:nvar 151 sg0bid(j) = 1/sgsh(j); % sgsh = sigmai^2 152end 153% ** prior variance for lagged and exogeous variables, same for all equations 154sgpbid = zeros(ncoef,1); % Sigma_plus_bar, diagonal, for the ith equation 155for i = 1:lags 156 if (q_m==12) 157 lagdecay = a*exp(b*i*mu(4)); 158 end 159 % 160 for j = 1:nvar 161 if (q_m==12) 162 % exponential decay to match quarterly decay 163 sgpbid((i-1)*nvar+j) = lagdecay^2/sgsh(j); % ith equation 164 elseif (q_m==4) 165 sgpbid((i-1)*nvar+j) = (1/i^mu(4))^2/sgsh(j); % ith equation 166 else 167 error('Incompatibility with lags, check the possible errors!!!') 168 %warning('Incompatibility with lags, check the possible errors!!!') 169 %return 170 end 171 end 172end 173% 174if indxDummy % Dummy observations as part of the explicit prior. 175 ndobs=nvar+1; % Number of dummy observations: nvar unit roots and 1 cointegration prior. 176 phibar = zeros(ndobs,ncoef); 177 %* constant term 178 const = ones(nvar+1,1); 179 const(1:nvar) = 0.0; 180 phibar(:,ncoef) = const; % the first nvar periods: no or zero constant! 181 182 xdgelint = mean(xdgel(1:lags,:),1); % mean of the first lags initial conditions 183 %* Dummies 184 for k=1:nvar 185 for m=1:lags 186 phibar(ndobs,nvar*(m-1)+k) = xdgelint(k); 187 phibar(k,nvar*(m-1)+k) = xdgelint(k); 188 % <<>> multiply hyperparameter later 189 end 190 end 191 phibar(1:nvar,:) = 1*mu(5)*phibar(1:nvar,:); % standard Sims and Zha prior 192 phibar(ndobs,:) = mu(6)*phibar(ndobs,:); 193 [phiq,phir]=qr(phibar,0); 194 xtxbar=phir'*phir; % phibar'*phibar. ncoef-by-ncoef. Reduced (not full) rank. See Forcast II, pp.69-69b. 195end 196 197 198 199 200 201%================================================= 202% Computing the (prior) covariance matrix for the posterior of A0, no data yet 203%================================================= 204% 205% 206% ** set up the conditional prior variance sg0bi and sgpbi. 207sg0bida = mu(1)^2*sg0bid; % ith equation 208sgpbida = mu(1)^2*mu(2)^2*sgpbid; 209sgpbida(ncoef-nexo+1:ncoef) = mu(1)^2*mu(3)^2; 210 %<<>> No scaling adjustment has been made for exogenous terms other than constant 211sgppbd = sgpbida(nvar+1:ncoef); % corresponding to A++, in a Sims-Zha paper 212 213Hptd = zeros(ncoef); 214Hptdi=Hptd; 215Hptd(ncoef,ncoef)=sgppbd(ncoef-nvar); 216Hptdinv(ncoef,ncoef)=1./sgppbd(ncoef-nvar); 217 % condtional on A0i, H_plus_tilde 218 219 220if nargin<14 % <<>>1 Default is no asymmetric information 221 asym0 = ones(nvar,nvar); % if not ones, then we have relative (asymmetric) tightness 222 asymp = ones(ncoef-1,nvar); % for A+. Column -- equation 223end 224 225%**** Asymmetric Information 226%asym0 = ones(nvar,nvar); % if not ones, then we have relative (asymmetric) tightness 227%asymp = ones(ncoef-1,nvar); % pp: plus without constant. Column -- equation 228%>>>>>> B: asymmetric prior variance for asymp <<<<<<<< 229% 230%for i = 1:lags 231% rowif = (i-1)*nvar+1; 232% rowil = i*nvar; 233% idmatw0 = 0.5; % weight assigned to idmat0 in the formation of asymp 234% if (i==1) 235% asymp(rowif:rowil,:)=(1-idmatw0)*ones(nvar)+idmatw0*idmat0; % first lag 236% % note: idmat1 is already transposed. Column -- equation 237% else 238% %asymp(rowif:rowil,1:nvar) = (1-idmatw0)*ones(nvar)+idmatw0*idmat0; 239% % <<<<<<< toggle + 240% % Note: already transposed, since idmat0 is transposed. 241% % Meaning: column implies equation 242% asymp(rowif:rowil,1:nvar) = ones(nvar); 243% % >>>>>>> toggle - 244% end 245%end 246% 247%>>>>>> E: asymmetric prior variance for asymp <<<<<<<< 248 249 250%================================================= 251% Computing the final covariance matrix (S1,...,Sm) for the prior of A0, 252% and final Gb=(G1,...,Gm) for A+ if asymmetric prior or for 253% B if symmetric prior for A+ 254%================================================= 255% 256for i = 1:nvar 257 %------------------------------ 258 % Introduce prior information on which variables "belong" in various equations. 259 % In this first trial, we just introduce this information here, in a model-specific way. 260 % Eventually this info has to be passed parametricly. In our first shot, we just damp down 261 % all coefficients except those on the diagonal. 262 263 %*** For A0 264 factor0=asym0(:,i); 265 266 sg0bd = sg0bida.*factor0; % Note, this only works for the prior variance Sg(i) 267 % of a0(i) being diagonal. If the prior variance Sg(i) is not 268 % diagonal, we have to the inverse to get inv(Sg(i)). 269 %sg0bdinv = 1./sg0bd; 270 % * unconditional variance on A0+ 271 H0td = diag(sg0bd); % unconditional 272 %=== Correlation in the MS equation to get a liquidity effect. 273 if (i==indxmsmdeqn(1)) 274 H0td(indxmsmdeqn(3),indxmsmdeqn(4)) = hpmsmd(1)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4))); 275 H0td(indxmsmdeqn(4),indxmsmdeqn(3)) = hpmsmd(1)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4))); 276 elseif (i==indxmsmdeqn(2)) 277 H0td(indxmsmdeqn(3),indxmsmdeqn(4)) = hpmsmd(2)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4))); 278 H0td(indxmsmdeqn(4),indxmsmdeqn(3)) = hpmsmd(2)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4))); 279 end 280 H0tdinv = inv(H0td); 281 %H0tdinv = diag(sg0bdinv); 282 % 283 if indxScaleStates 284 H0tldcell_inv{i}=(Ui{i}'*kron(eye(nStates),H0tdinv/nStates))*Ui{i}; 285 else 286 H0tldcell_inv{i}=(Ui{i}'*kron(eye(nStates),H0tdinv))*Ui{i}; 287 end 288 289 290 291 %*** For A+ 292 if ~(lags==0) % For A1 to remain random walk properties 293 factor1=asymp(1:nvar,i); 294 sg1bd = sgpbida(1:nvar).*factor1; 295 sg1bdinv = 1./sg1bd; 296 % 297 Hptd(1:nvar,1:nvar)=diag(sg1bd); 298 Hptdinv(1:nvar,1:nvar)=diag(sg1bdinv); 299 if lags>1 300 factorpp=asymp(nvar+1:ncoef-1,i); 301 sgpp_cbd = sgppbd(1:ncoef-nvar-1) .* factorpp; 302 sgpp_cbdinv = 1./sgpp_cbd; 303 Hptd(nvar+1:ncoef-1,nvar+1:ncoef-1)=diag(sgpp_cbd); 304 Hptdinv(nvar+1:ncoef-1,nvar+1:ncoef-1)=diag(sgpp_cbdinv); 305 % condtional on A0i, H_plus_tilde 306 end 307 end 308 %--------------- 309 % The dummy observation prior affects only the prior covariance of A+|A0, 310 % but not the covariance of A0. See pp.69a-69b for the proof. 311 %--------------- 312 if indxDummy % Dummy observations as part of the explicit prior. 313 Hptdinv2 = Hptdinv + xtxbar; % Rename Hptdinv to Hptdinv2 because we want to keep Hptdinv diagonal in the next loop of i. 314 else 315 Hptdinv2 = Hptdinv; 316 end 317 if (indxScaleStates) 318 Hptldcell_inv{i}=(Vi{i}'*kron(eye(nStates),Hptdinv2/nStates))*Vi{i}; 319 else 320 Hptldcell_inv{i}=(Vi{i}'*kron(eye(nStates),Hptdinv2))*Vi{i}; 321 end 322 %Hptdinv_3 = kron(eye(nStates),Hptdinv); % ????? 323end 324 325 326