456 lines
20 KiB
Matlab
456 lines
20 KiB
Matlab
function [Ui,Vi,n0,np] = ftd_2s_caseall_upperchol6v(lags,nvar,nStates,indxEqnTv_m,nexo)
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% Case 2&3: Policy a0j and a+j have only time-varying structural variances -- Case 2.
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% All policy and nonpolicy a0j's and the corresponding constant terms are completely time-varying and only the scale
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% of each variable in d+j,1** (excluding the constant term) is time-varying -- Case 3.
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%
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% Variables: Pcom, M2, FFR, y, P, U. Equations: information, policy, money demand, y, P, U.
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% Exporting orthonormal matrices for the deterministic linear restrictions
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% (equation by equation) with time-varying A0 and D+** equations.
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% See Model II.3 on pp.71k-71r in TVBVAR NOTES (and Waggoner and Zha's Gibbs sampling paper and TVBVAR NOTES p.58).
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%
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% lags: Maximum length of lag.
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% nvar: Number of endogeous variables.
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% nStates: Number of states.
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% indxEqnTv_m: nvar-by-2. Stores equation characteristics.
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% 1st column: labels of equations [1:nvar]'.
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% 2nd column: labels of time-varying features with
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% 1: indxConst -- all coefficients are constant,
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% 2: indxStv -- only shocks are time-varying,
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% 3: indxTva0pv -- a0 are freely time-varying and each variable i for d+ is time-varying only by the scale lambda_i(s_t).
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% 4: indxTva0ps -- a0 are freely time-varying and only the scale for the whole of d+ is time-varying.
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% 5: indxTv -- time-varying for all coeffficients (a0 and a+) where the lag length for a+ may be shorter.
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% nexo: number of exogenous variables. If nexo is not supplied, nexo=1 as default for a constant.
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% So far this function is written to handle one exogenous variable, which is a constant term.
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%-----------------
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% Ui: nvar-by-1 cell. In each cell, nvar*nStates-by-qi*si orthonormal basis for the null of the ith
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% equation contemporaneous restriction matrix where qi is the number of free parameters
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% within the state and si is the number of free states.
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% With this transformation, we have ai = Ui*bi or Ui'*ai = bi where ai is a vector
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% of total original parameters and bi is a vector of free parameters. When no
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% restrictions are imposed, we have Ui = I. There must be at least one free
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% parameter left for the ith equation in the order of [a_i for 1st state, ..., a_i for last state].
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% Vi: nvar-by-1 cell. In each cell, k*nStates-by-ri*si orthonormal basis for the null of the ith
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% equation lagged restriction matrix where k is a total of exogenous variables and
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% ri is the number of free parameters within the state and si is the number of free states.
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% With this transformation, we have fi = Vi*gi or Vi'*fi = gi where fi is a vector of total original
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% parameters and gi is a vector of free parameters. The ith equation is in the order of [nvar variables
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% for 1st lag and 1st state, ..., nvar variables for last lag and 1st state, const for 1st state, nvar
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% variables for 1st lag and 2nd state, nvar variables for last lag and 2nd state, const for 2nd state, and so on].
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% n0: nvar-by-1, whose ith element represents the number of free A0 parameters in ith equation in *all states*.
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% np: nvar-by-1, whose ith element represents the number of free D+ parameters in ith equation in *all states*.
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%
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% Tao Zha, February 2003
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Ui = cell(nvar,1); % initializing for contemporaneous endogenous variables
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Vi = cell(nvar,1); % initializing for lagged and exogenous variables
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n0 = zeros(nvar,1); % ith element represents the number of free A0 parameters in ith equation in all states.
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np = zeros(nvar,1); % ith element represents the number of free D+ parameters in ith equation in all states.
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if (nargin==3)
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nexo = 1; % 1: constant as default where nexo must be a nonnegative integer
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end
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n = nvar*nStates;
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kvar=lags*nvar+nexo; % Maximum number of lagged and exogenous variables in each equation under each state.
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k = kvar*nStates; % Maximum number of lagged and exogenous variables in each equation in all states.
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Qi = zeros(n,n,nvar); % 3rd dim: nvar contemporaneous equations.
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Ri = zeros(k,k,nvar); % 1st and 2nd dims: lagged and exogenous equations.
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% Row corresponds to equation with nvar variables for state 1, ..., nvar variables for state nState.
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% 0 means no restriction.
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% 1 and -1 or any other number means the linear combination of the corresponding parameters is restricted to 0.
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% 1 (only 1) means that the corresponding parameter is restricted to 0.
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%nfvar = 6; % number of foreign (Granger causing) variables
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%nhvar = nvar-nfvar; % number of home (affected) variables.
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%-------------------------------------------------------------
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% Beginning the manual input of the restrictions one quation at a time for A0_s.
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%-------------------------------------------------------------
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%
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%======== The first equation ===========
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eqninx = 1;
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nreseqn = 5; % Number of linear restrictions for A0(:,eqninx) for each state.
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if (indxEqnTv_m(eqninx, 2)<=2)
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%**** For constant A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:(nStates-1)*nvar+nreseqn,:,eqninx) = [
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1 0 0 0 0 0 -1 0 0 0 0 0
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0 1 0 0 0 0 0 -1 0 0 0 0
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0 0 1 0 0 0 0 0 -1 0 0 0
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0 0 0 1 0 0 0 0 0 -1 0 0
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0 0 0 0 1 0 0 0 0 0 -1 0
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0 0 0 0 0 1 0 0 0 0 0 -1
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0 0 0 0 0 0 0 1 0 0 0 0
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0 0 0 0 0 0 0 0 1 0 0 0
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0 0 0 0 0 0 0 0 0 1 0 0
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For constant D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else % Time-varying equations at least for A0_s. For D+_s, constant-parameter equations in general.
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%**** For time-varying A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:nreseqn*nStates,:,eqninx) = [
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0 1 0 0 0 0 0 0 0 0 0 0
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0 0 1 0 0 0 0 0 0 0 0 0
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0 0 0 1 0 0 0 0 0 0 0 0
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0 0 0 0 1 0 0 0 0 0 0 0
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0 0 0 0 0 1 0 0 0 0 0 0
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0 0 0 0 0 0 0 1 0 0 0 0
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0 0 0 0 0 0 0 0 1 0 0 0
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0 0 0 0 0 0 0 0 0 1 0 0
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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if (indxEqnTv_m(eqninx, 2)==3) % For constant D+** except the constant term. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar-1 % -1: no restrictions on the constant term, which is freely time-varying.
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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elseif (indxEqnTv_m(eqninx, 2)==4) % For constant D+**. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else
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error('.../ftd_2s_caseall_simszha5v.m: Have not got time to deal with the simple case indxEqnTv_m(eqninx, 2)=5.')
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end
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end
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%======== The second equation ===========
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eqninx = 2;
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nreseqn = 4; % Number of linear restrictions for A0(:,eqninx) for each state.
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if (indxEqnTv_m(eqninx, 2)<=2)
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%**** For constant A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:(nStates-1)*nvar+nreseqn,:,eqninx) = [
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1 0 0 0 0 0 -1 0 0 0 0 0
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0 1 0 0 0 0 0 -1 0 0 0 0
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0 0 1 0 0 0 0 0 -1 0 0 0
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0 0 0 1 0 0 0 0 0 -1 0 0
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0 0 0 0 1 0 0 0 0 0 -1 0
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0 0 0 0 0 1 0 0 0 0 0 -1
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0 0 0 0 0 0 0 0 1 0 0 0
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0 0 0 0 0 0 0 0 0 1 0 0
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For constant D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else % Time-varying equations at least for A0_s. For D+_s, constant-parameter equations in general.
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%**** For time-varying A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:nreseqn*nStates,:,eqninx) = [
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0 0 1 0 0 0 0 0 0 0 0 0
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0 0 0 1 0 0 0 0 0 0 0 0
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0 0 0 0 1 0 0 0 0 0 0 0
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0 0 0 0 0 1 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 1 0 0 0
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0 0 0 0 0 0 0 0 0 1 0 0
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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if (indxEqnTv_m(eqninx, 2)==3) % For constant D+** except the constant term. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar-1 % -1: no restrictions on the constant term, which is freely time-varying.
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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elseif (indxEqnTv_m(eqninx, 2)==4) % For constant D+**. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else
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error('.../ftd_2s_caseall_simszha5v.m: Have not got time to deal with the simple case indxEqnTv_m(eqninx, 2)=5.')
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end
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end
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%======== The third equation ===========
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eqninx = 3;
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nreseqn = 3; % Number of linear restrictions for A0(:,eqninx) for each state.
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if (indxEqnTv_m(eqninx, 2)<=2)
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%**** For constant A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:(nStates-1)*nvar+nreseqn,:,eqninx) = [
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1 0 0 0 0 0 -1 0 0 0 0 0
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0 1 0 0 0 0 0 -1 0 0 0 0
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0 0 1 0 0 0 0 0 -1 0 0 0
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0 0 0 1 0 0 0 0 0 -1 0 0
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0 0 0 0 1 0 0 0 0 0 -1 0
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0 0 0 0 0 1 0 0 0 0 0 -1
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0 0 0 0 0 0 0 0 0 1 0 0
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For constant D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else % Time-varying equations at least for A0_s. For D+_s, constant-parameter equations in general.
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%**** For time-varying A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:nreseqn*nStates,:,eqninx) = [
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0 0 0 1 0 0 0 0 0 0 0 0
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0 0 0 0 1 0 0 0 0 0 0 0
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0 0 0 0 0 1 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 1 0 0
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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if (indxEqnTv_m(eqninx, 2)==3) % For constant D+** except the constant term. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar-1 % -1: no restrictions on the constant term, which is freely time-varying.
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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elseif (indxEqnTv_m(eqninx, 2)==4) % For constant D+**. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else
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error('.../ftd_3s_case3a.m: Have not got time to deal with the simple case indxEqnTv_m(eqninx, 2)=5.')
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end
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%==== For freely time-varying A+ for only the first 6 lags.
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%==== Lagged restrictions: zeros on all lags except the first 6 lags in the MS equation.
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% nlagsno0 = 6; % Number of lags to be nonzero.
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% for si=1:nStates
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% for ki = 1:lags-nlagsno0
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% for kj=1:nvar
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% Ri(kvar*(si-1)+nlagsno0*nvar+nvar*(ki-1)+kj,kvar*(si-1)+nlagsno0*nvar+nvar*(ki-1)+kj,2) = 1;
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% end
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% end
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% end
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%**** For constant D+_s except the first two lags and the constant term. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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% for si=1:nStates-1
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% for ki=[2*nvar+1:kvar-1]
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% Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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% end
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% end
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end
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%======== The fourth equation ===========
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eqninx = 4;
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nreseqn = 2; % Number of linear restrictions for the equation for each state.
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if (indxEqnTv_m(eqninx, 2)<=2)
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%**** For constant A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:(nStates-1)*nvar+nreseqn,:,eqninx) = [
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1 0 0 0 0 0 -1 0 0 0 0 0
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0 1 0 0 0 0 0 -1 0 0 0 0
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0 0 1 0 0 0 0 0 -1 0 0 0
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0 0 0 1 0 0 0 0 0 -1 0 0
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0 0 0 0 1 0 0 0 0 0 -1 0
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0 0 0 0 0 1 0 0 0 0 0 -1
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For constant D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else % Time-varying equations at least for A0_s. For D+_s, constant-parameter equations in general.
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%**** For time-varying A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:nreseqn*nStates,:,eqninx) = [
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0 0 0 0 1 0 0 0 0 0 0 0
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0 0 0 0 0 1 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 1 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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if (indxEqnTv_m(eqninx, 2)==3) % For constant D+** except the constant term. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar-1 % -1: no restrictions on the constant term, which is freely time-varying.
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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elseif (indxEqnTv_m(eqninx, 2)==4) % For constant D+**. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else
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error('.../ftd_2s_caseall_simszha5v.m: Have not got time to deal with the simple case indxEqnTv_m(eqninx, 2)=5.')
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end
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end
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%======== The fifth equation ===========
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eqninx = 5;
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nreseqn = 1; % Number of linear restrictions for the equation for each state.
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if (indxEqnTv_m(eqninx, 2)<=2)
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%**** For constant A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:(nStates-1)*nvar+nreseqn,:,eqninx) = [
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1 0 0 0 0 0 -1 0 0 0 0 0
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0 1 0 0 0 0 0 -1 0 0 0 0
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0 0 1 0 0 0 0 0 -1 0 0 0
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0 0 0 1 0 0 0 0 0 -1 0 0
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0 0 0 0 1 0 0 0 0 0 -1 0
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0 0 0 0 0 1 0 0 0 0 0 -1
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For constant D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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else % Time-varying equations at least for A0_s. For D+_s, constant-parameter equations in general.
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%**** For time-varying A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
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Qi(1:nreseqn*nStates,:,eqninx) = [
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0 0 0 0 0 1 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 0 1
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];
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%**** For D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
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if (indxEqnTv_m(eqninx, 2)==3) % For constant D+** except the constant term. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar-1 % -1: no restrictions on the constant term, which is freely time-varying.
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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elseif (indxEqnTv_m(eqninx, 2)==4) % For constant D+**. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
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for si=1:nStates-1
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for ki=1:kvar
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Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
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end
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end
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|
else
|
|
error('.../ftd_2s_caseall_simszha5v.m: Have not got time to deal with the simple case indxEqnTv_m(eqninx, 2)=5.')
|
|
end
|
|
end
|
|
|
|
|
|
%======== The sixth equation ===========
|
|
eqninx = 6;
|
|
nreseqn = 0; % Number of linear restrictions for the equation for each state.
|
|
if (indxEqnTv_m(eqninx, 2)<=2)
|
|
%**** For constant A0_s. In the order of [a0j(1),...,a0j(nStates)] for the 2nd dim of Qi.
|
|
Qi(1:(nStates-1)*nvar+nreseqn,:,eqninx) = [
|
|
1 0 0 0 0 0 -1 0 0 0 0 0
|
|
0 1 0 0 0 0 0 -1 0 0 0 0
|
|
0 0 1 0 0 0 0 0 -1 0 0 0
|
|
0 0 0 1 0 0 0 0 0 -1 0 0
|
|
0 0 0 0 1 0 0 0 0 0 -1 0
|
|
0 0 0 0 0 1 0 0 0 0 0 -1
|
|
];
|
|
%**** For constant D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
|
|
for si=1:nStates-1
|
|
for ki=1:kvar
|
|
Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
|
|
end
|
|
end
|
|
else % Time-varying equations at least for A0_s. For D+_s, constant-parameter equations in general.
|
|
%**** For D+_s. In the order of [aj+(1),...,aj+(nStates)] for the 2nd dim of Ri.
|
|
if (indxEqnTv_m(eqninx, 2)==3) % For constant D+** except the constant term. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
|
|
for si=1:nStates-1
|
|
for ki=1:kvar-1 % -1: no restrictions on the constant term, which is freely time-varying.
|
|
Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
|
|
end
|
|
end
|
|
elseif (indxEqnTv_m(eqninx, 2)==4) % For constant D+**. In the order of [dj**(1),...,dj**(nStates)] for the 2nd dim of Ri.
|
|
for si=1:nStates-1
|
|
for ki=1:kvar
|
|
Ri(kvar*(si-1)+ki,[kvar*(si-1)+ki si*kvar+ki],eqninx) = [1 -1];
|
|
end
|
|
end
|
|
else
|
|
error('.../ftd_2s_caseall_simszha5v.m: Have not got time to deal with the simple case indxEqnTv_m(eqninx, 2)=5.')
|
|
end
|
|
end
|
|
|
|
|
|
|
|
|
|
|
|
%===== Lagged restrictions in foreign (Granger causing) block
|
|
%nfbres = lags*(nvar-nfvar); % number of block restrictions in each foreign equation
|
|
%bfor = zeros(nfbres,k); % each foreign equation
|
|
%cnt=0;
|
|
%for ki = 1:lags
|
|
% for kj=1:nvar-nfvar
|
|
% cnt=cnt+1;
|
|
% bfor(cnt,nvar*(ki-1)+nfvar+kj) = 1;
|
|
% end
|
|
%end
|
|
%%
|
|
%if cnt~=nfbres
|
|
% error('Check lagged restrictions in foreign equations!')
|
|
%end
|
|
%%
|
|
%for kj=1:nfvar
|
|
% Ri(1:nfbres,:,kj) = bfor;
|
|
%end
|
|
|
|
|
|
%===== Lagged restrictions in home (affected) block
|
|
%
|
|
%~~~~~ selected domestic equations
|
|
%dlrindx = nfvar+1; %[nfvar+1 nfvar+2]; % index for relevant home equations
|
|
%rfvindx = []; %[6]; %[1 2 3 5]; % index for restricted foreign variables (e.g., Poil, M2, FFR, P).
|
|
%%nf2hvar = nfvar-length(rfvindx); % number of free parameters -- foreign variables entering the home sector
|
|
%nhbres = lags*length(rfvindx); % number of block restrictions in each home equation
|
|
%bhom = zeros(nhbres,k); % each home equation
|
|
%cnt=0;
|
|
%for ki = 1:lags
|
|
% for kj=1:length(rfvindx)
|
|
% cnt=cnt+1;
|
|
% bhom(cnt,nvar*(ki-1)+rfvindx(kj)) = 1;
|
|
% end
|
|
%end
|
|
%%
|
|
%if cnt~=nhbres
|
|
% error('Check lagged restrictions in domestic equations!')
|
|
%end
|
|
%%
|
|
%for kj=dlrindx
|
|
% Ri(1:nhbres,:,kj) = bhom;
|
|
%end
|
|
|
|
|
|
|
|
for ki=1:nvar % initializing loop for each equation
|
|
Ui{ki} = null(Qi(:,:,ki));
|
|
Vi{ki} = null(Ri(:,:,ki));
|
|
n0(ki) = size(Ui{ki},2);
|
|
np(ki) = size(Vi{ki},2);
|
|
end
|