298 lines
13 KiB
Matlab
298 lines
13 KiB
Matlab
function [Pi,H0multi,Hpmulti,H0invmulti,Hpinvmulti] ...
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= fn_rnrprior_covres_dobs(nvar,q_m,lags,xdgel,mu,indxDummy,hpmsmd,indxmsmdeqn,nexo,asym0,asymp)
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% Differs from fn_rnrprior_covres_dobs_tv(): no linear restrictions (Ui and Vi) have applied yet to this function, but
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% linear restrictions are incorported in fn_rnrprior_covres_dobs_tv().
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%
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% Only works for the nexo=1 (constant term) case. To extend this to other exogenous variables, see fn_dataxy.m. 01/14/03.
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% Differs from fn_rnrprior_covres.m in that dummy observations are included as part of the explicit prior. See Forcast II, pp.68-69b.
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% More general than fn_rnrprior.m because when hpmsmd=0, fn_rnrprior_covres() is the same as fn_rnrprior().
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% Allows for prior covariances for the MS and MD equations to achieve liquidity effects.
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% Exports random Bayesian prior of Sims and Zha with asymmetric rior (but no linear restrictions yet)
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% See Waggoner and Zha's Gibbs sampling paper and TVBVAR NOTES p. 71k.0.
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%
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% nvar: number of endogenous variables
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% q_m: quarter or month
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% lags: the maximum length of lag
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% xdgel: T*nvar endogenous-variable matrix of raw or original data (no manipulation involved) with sample size including lags.
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% Order of columns: (1) nvar endogenous variables; (2) constants will be automatically put in the last column.
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% Used only to get variances of residuals for mu(1)-mu(5) and for dummy observations mu(5) and mu(6).
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% mu: 6-by-1 vector of hyperparameters (the following numbers for Atlanta Fed's forecast), where
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% mu(5) and mu(6) are NOT used here. See fn_dataxy.m for using mu(5) and mu(6).
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% mu(1): overall tightness and also for A0; (0.57)
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% mu(2): relative tightness for A+; (0.13)
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% mu(3): relative tightness for the constant term; (0.1). NOTE: for other
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% exogenous terms, the variance of each exogenous term must be taken into
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% acount to eliminate the scaling factor.
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% mu(4): tightness on lag decay; (1)
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% mu(5): weight on nvar sums of coeffs dummy observations (unit roots); (5)
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% mu(6): weight on single dummy initial observation including constant
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% (cointegration, unit roots, and stationarity); (5)
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% indxDummy: 1: uses dummy observations to form part of an explicit prior; 0: no dummy observations as part of the prior.
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% 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.
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% 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.
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% 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.
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% This will give us a liquidity effect.
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% indxmsmdeqn: 4-by-1 index for the locations of the MS and MD equation and for the locations of M and R.
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% indxmsmdeqn(1) for MS and indxmsmdeqn(2) for MD.
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% indxmsmdeqn(3) for M and indxmsmdeqn(4) for R.
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% nexo: number of exogenous variables (if not specified, nexo=1 (constant) by default).
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% The constant term is always put to the last of all endogenous and exogenous variables.
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% asym0: nvar-by-nvar asymmetric prior on A0. Column -- equation.
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% If ones(nvar,nvar), symmetric prior; if not, relative (asymmetric) tightness on A0.
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% asymp: ncoef-1-by-nvar asymmetric prior on A+ bar constant. Column -- equation.
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% If ones(ncoef-1,nvar), symmetric prior; if not, relative (asymmetric) tightness on A+.
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% --------------------
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% Pi: ncoef-by-nvar matrix for the ith equation under random walk. Same for all equations
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% H0multi: nvar-by-nvar-by-nvar; H0 for different equations under asymmetric prior
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% Hpmulti: ncoef-by-ncoef-by-nvar; H+ for different equations under asymmetric prior
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% H0invmulti: nvar-by-nvar-by-nvar; inv(H0) for different equations under asymmetric prior
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% Hpinvmulti: ncoef-by-ncoef-by-nvar; inv(H+) for different equations under asymmetric prior
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%
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% Tao Zha, February 2000. Revised, September 2000, February, May 2003.
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% Copyright (C) 2000-2011 Tao Zha
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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 <http://www.gnu.org/licenses/>.
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if (nargin<=8), nexo=1; end
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ncoef = nvar*lags+nexo; % number of coefficients in *each* equation, RHS coefficients only.
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H0multi=zeros(nvar,nvar,nvar); % H0 for different equations under asymmetric prior
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Hpmulti=zeros(ncoef,ncoef,nvar); % H+ for different equations under asymmetric prior
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H0invmulti=zeros(nvar,nvar,nvar); % inv(H0) for different equations under asymmetric prior
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Hpinvmulti=zeros(ncoef,ncoef,nvar); % inv(H+) for different equations under asymmetric prior
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%*** Constructing Pi for the ith equation under the random walk assumption
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Pi = zeros(ncoef,nvar); % same for all equations
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Pi(1:nvar,1:nvar) = eye(nvar); % random walk
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%
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%@@@ Prepared for Bayesian prior
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%
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%
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% ** monthly lag decay in order to match quarterly decay: a*exp(bl) where
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% ** l is the monthly lag. Suppose quarterly decay is 1/x where x=1,2,3,4.
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% ** Let the decay of l1 (a*exp(b*l1)) match that of x1 (say, beginning: 1/1)
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% ** and the decay of l2 (a*exp(b*l2)) match that of x2 (say, end: 1/5),
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% ** we can solve for a and b which are
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% ** b = (log_x1-log_x2)/(l1-l2), and a = x1*exp(-b*l1).
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if q_m==12
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l1 = 1; % 1st month == 1st quarter
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xx1 = 1; % 1st quarter
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l2 = lags; % last month
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xx2 = 1/((ceil(lags/3))^mu(4)); % last quarter
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%xx2 = 1/6; % last quarter
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% 3rd quarter: i.e., we intend to let decay of the 6th month match
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% that of the 3rd quarter, so that the 6th month decays a little
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% faster than the second quarter which is 1/2.
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if lags==1
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b = 0;
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else
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b = (log(xx1)-log(xx2))/(l1-l2);
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end
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a = xx1*exp(-b*l1);
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end
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%
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% *** specify the prior for each equation separately, SZ method,
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% ** get the residuals from univariate regressions.
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%
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sgh = zeros(nvar,1); % square root
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sgsh = sgh; % square
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nSample=size(xdgel,1); % sample size-lags
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yu = xdgel;
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C = ones(nSample,1);
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for k=1:nvar
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[Bk,ek,junk1,junk2,junk3,junk4] = sye([yu(:,k) C],lags);
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clear Bk junk1 junk2 junk3 junk4;
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sgsh(k) = ek'*ek/(nSample-lags);
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sgh(k) = sqrt(sgsh(k));
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end
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% ** prior variance for A0(:,1), same for all equations!!!
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sg0bid = zeros(nvar,1); % Sigma0_bar diagonal only for the ith equation
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for j=1:nvar
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sg0bid(j) = 1/sgsh(j); % sgsh = sigmai^2
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end
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% ** prior variance for lagged and exogeous variables, same for all equations
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sgpbid = zeros(ncoef,1); % Sigma_plus_bar, diagonal, for the ith equation
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for i = 1:lags
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if (q_m==12)
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lagdecay = a*exp(b*i*mu(4));
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end
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%
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for j = 1:nvar
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if (q_m==12)
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% exponential decay to match quarterly decay
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sgpbid((i-1)*nvar+j) = lagdecay^2/sgsh(j); % ith equation
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elseif (q_m==4)
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sgpbid((i-1)*nvar+j) = (1/i^mu(4))^2/sgsh(j); % ith equation
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elseif (q_m==1)
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sgpbid((i-1)*nvar+j) = (1/(i*4)^mu(4))^2/sgsh(j); % ith equation
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else
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error('Incompatibility with lags, check the possible errors!!!')
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%warning('Incompatibility with lags, check the possible errors!!!')
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%return
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end
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end
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end
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%
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if indxDummy % Dummy observations as part of the explicit prior.
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ndobs=nvar+1; % Number of dummy observations: nvar unit roots and 1 cointegration prior.
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phibar = zeros(ndobs,ncoef);
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%* constant term
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const = ones(nvar+1,1);
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const(1:nvar) = 0.0;
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phibar(:,ncoef) = const; % the first nvar periods: no or zero constant!
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xdgelint = mean(xdgel(1:lags,:),1); % mean of the first lags initial conditions
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%* Dummies
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for k=1:nvar
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for m=1:lags
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phibar(ndobs,nvar*(m-1)+k) = xdgelint(k);
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phibar(k,nvar*(m-1)+k) = xdgelint(k);
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% <<>> multiply hyperparameter later
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end
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end
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phibar(1:nvar,:) = 1*mu(5)*phibar(1:nvar,:); % standard Sims and Zha prior
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phibar(ndobs,:) = mu(6)*phibar(ndobs,:);
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[phiq,phir]=qr(phibar,0);
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xtxbar=phir'*phir; % phibar'*phibar. ncoef-by-ncoef. Reduced (not full) rank. See Forcast II, pp.69-69b.
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end
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%=================================================
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% Computing the (prior) covariance matrix for A0, no data yet.
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% As proved in pp.69a-69b, Forecast II, the following prior covariance of A0
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% will remain the same after the dummy observations prior is incorporated.
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% The dummy observation prior only affects the prior covariance of A+|A0.
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% See pp.69a-69b for the proof.
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%=================================================
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%
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%
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% ** set up the conditional prior variance sg0bi and sgpbi.
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sg0bida = mu(1)^2*sg0bid; % ith equation
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sgpbida = mu(1)^2*mu(2)^2*sgpbid;
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sgpbida(ncoef-nexo+1:ncoef) = mu(1)^2*mu(3)^2;
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%<<>> No scaling adjustment has been made for exogenous terms other than constant
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sgppbd = sgpbida(nvar+1:ncoef); % corresponding to A++, in a Sims-Zha paper
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Hptd = zeros(ncoef);
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Hptdi=Hptd;
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Hptd(ncoef,ncoef)=sgppbd(ncoef-nvar);
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Hptdinv(ncoef,ncoef)=1./sgppbd(ncoef-nvar);
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% condtional on A0i, H_plus_tilde
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if nargin<10 % the default is no asymmetric information
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asym0 = ones(nvar,nvar); % if not ones, then we have relative (asymmetric) tightness
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asymp = ones(ncoef-1,nvar); % for A+. Column -- equation
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end
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%**** Asymmetric Information
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%asym0 = ones(nvar,nvar); % if not ones, then we have relative (asymmetric) tightness
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%asymp = ones(ncoef-1,nvar); % pp: plus without constant. Column -- equation
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%>>>>>> B: asymmetric prior variance for asymp <<<<<<<<
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%
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%for i = 1:lags
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% rowif = (i-1)*nvar+1;
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% rowil = i*nvar;
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% idmatw0 = 0.5; % weight assigned to idmat0 in the formation of asymp
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% if (i==1)
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% asymp(rowif:rowil,:)=(1-idmatw0)*ones(nvar)+idmatw0*idmat0; % first lag
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% % note: idmat1 is already transposed. Column -- equation
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% else
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% %asymp(rowif:rowil,1:nvar) = (1-idmatw0)*ones(nvar)+idmatw0*idmat0;
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% % <<<<<<< toggle +
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% % Note: already transposed, since idmat0 is transposed.
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% % Meaning: column implies equation
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% asymp(rowif:rowil,1:nvar) = ones(nvar);
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% % >>>>>>> toggle -
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% end
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%end
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%
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%>>>>>> E: asymmetric prior variance for asymp <<<<<<<<
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%=================================================
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% Computing the final covariance matrix (S1,...,Sm) for the prior of A0,
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% and final Gb=(G1,...,Gm) for A+ if asymmetric prior or for
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% B if symmetric prior for A+
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%=================================================
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%
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for i = 1:nvar
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%------------------------------
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% Introduce prior information on which variables "belong" in various equations.
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% In this first trial, we just introduce this information here, in a model-specific way.
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% Eventually this info has to be passed parametricly. In our first shot, we just damp down
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% all coefficients except those on the diagonal.
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%*** For A0
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factor0=asym0(:,i);
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sg0bd = sg0bida.*factor0; % Note, this only works for the prior variance Sg(i)
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% of a0(i) being diagonal. If the prior variance Sg(i) is not
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% diagonal, we have to the inverse to get inv(Sg(i)).
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%sg0bdinv = 1./sg0bd;
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% * unconditional variance on A0+
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H0td = diag(sg0bd); % unconditional
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%=== Correlation in the MS equation to get a liquidity effect.
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if (i==indxmsmdeqn(1))
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H0td(indxmsmdeqn(3),indxmsmdeqn(4)) = hpmsmd(1)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4)));
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H0td(indxmsmdeqn(4),indxmsmdeqn(3)) = hpmsmd(1)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4)));
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elseif (i==indxmsmdeqn(2))
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H0td(indxmsmdeqn(3),indxmsmdeqn(4)) = hpmsmd(2)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4)));
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H0td(indxmsmdeqn(4),indxmsmdeqn(3)) = hpmsmd(2)*sqrt(sg0bida(indxmsmdeqn(3))*sg0bida(indxmsmdeqn(4)));
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end
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H0tdinv = inv(H0td);
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%H0tdinv = diag(sg0bdinv);
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%
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H0multi(:,:,i)=H0td;
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H0invmulti(:,:,i)=H0tdinv;
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%*** For A+
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if ~(lags==0) % For A1 to remain random walk properties
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factor1=asymp(1:nvar,i);
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sg1bd = sgpbida(1:nvar).*factor1;
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sg1bdinv = 1./sg1bd;
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%
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Hptd(1:nvar,1:nvar)=diag(sg1bd);
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Hptdinv(1:nvar,1:nvar)=diag(sg1bdinv);
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if lags>1
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factorpp=asymp(nvar+1:ncoef-1,i);
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sgpp_cbd = sgppbd(1:ncoef-nvar-1) .* factorpp;
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sgpp_cbdinv = 1./sgpp_cbd;
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Hptd(nvar+1:ncoef-1,nvar+1:ncoef-1)=diag(sgpp_cbd);
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Hptdinv(nvar+1:ncoef-1,nvar+1:ncoef-1)=diag(sgpp_cbdinv);
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% condtional on A0i, H_plus_tilde
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end
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end
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%---------------
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% The dummy observation prior affects only the prior covariance of A+|A0,
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% but not the covariance of A0. See pp.69a-69b for the proof.
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%---------------
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if indxDummy % Dummy observations as part of the explicit prior.
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Hpinvmulti(:,:,i)=Hptdinv + xtxbar;
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Hpmulti(:,:,i) = inv(Hpinvmulti(:,:,i));
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else
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Hpmulti(:,:,i)=Hptd;
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Hpinvmulti(:,:,i)=Hptdinv;
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end
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end
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