Merge pull request #1042 from JohannesPfeifer/discretionary
Various fixes for discretionary_policytime-shift
MichelJuillard
commit
866ab33575
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@ -35,6 +35,6 @@ if options_.noprint == 0
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end
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%oo_ = evaluate_planner_objective(oo_.dr,M_,oo_,options_);
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oo_.planner_objective_value = evaluate_planner_objective(M_,options_,oo_);
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options_ = oldoptions;
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@ -1,6 +1,6 @@
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function [dr,ys,info]=discretionary_policy_1(oo_,Instruments)
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% Copyright (C) 2007-2012 Dynare Team
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% Copyright (C) 2007-2015 Dynare Team
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%
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% This file is part of Dynare.
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%
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@ -78,7 +78,8 @@ it_ = MaxLag + 1 ;
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if exo_nbr == 0
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oo_.exo_steady_state = [] ;
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end
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[junk,jacobia_] = feval([M_.fname '_dynamic'],z, [oo_.exo_simul ...
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[junk,jacobia_] = feval([M_.fname '_dynamic'],z, [zeros(size(oo_.exo_simul)) ...
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oo_.exo_det_simul], M_.params, zeros(endo_nbr,1), it_);
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if any(junk~=0)
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error(['discretionary_policy: the model must be written in deviation ' ...
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@ -88,6 +89,13 @@ end
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eq_nbr= size(jacobia_,1);
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instr_nbr=endo_nbr-eq_nbr;
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if instr_nbr==0
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error('discretionary_policy:: There are no available instruments, because the model has as many equations as variables.')
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end
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if size(Instruments,1)~= instr_nbr
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error('discretionary_policy:: There are more declared instruments than omitted equations.')
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end
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instr_id=nan(instr_nbr,1);
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for j=1:instr_nbr
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vj=deblank(Instruments(j,:));
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@ -126,24 +134,49 @@ if info
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dr=[];
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return
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else
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Hold=H;
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Hold=H; %save previous solution
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% Hold=[]; use this line if persistent command is not used.
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end
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% update the following elements
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LLI=lead_lag_incidence;
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LLI(MaxLag,:)=any(H);
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LLI(MaxLag,:)=any(H); %check if variable drops out in solution
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LLI=LLI';
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tmp=find(LLI);
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LLI(tmp)=1:numel(tmp);
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LLI(tmp)=1:numel(tmp); %renumber
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M_.lead_lag_incidence = LLI';
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M_.lead_lag_incidence = LLI'; %update lead_lag_incidence
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%update info in M_
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max_lag = M_.maximum_endo_lag;
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endo_nbr = M_.endo_nbr;
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lead_lag_incidence = M_.lead_lag_incidence;
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fwrd_var = find(lead_lag_incidence(max_lag+2:end,:))';
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if max_lag > 0
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pred_var = find(lead_lag_incidence(1,:))';
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both_var = intersect(pred_var,fwrd_var);
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pred_var = setdiff(pred_var,both_var);
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fwrd_var = setdiff(fwrd_var,both_var);
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stat_var = setdiff([1:endo_nbr]',union(union(pred_var,both_var),fwrd_var)); % static variables
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else
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pred_var = [];
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both_var = [];
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stat_var = setdiff([1:endo_nbr]',fwrd_var);
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end
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M_.nstatic=length(stat_var);
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M_.nfwrd=length(fwrd_var);
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M_.npred=length(pred_var);
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M_.nboth=length(both_var);
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M_.nspred=M_.npred+M_.nboth;
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M_.nsfwrd=M_.nfwrd+M_.nboth;
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M_.ndynamic=M_.endo_nbr-M_.nstatic;
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% set the state
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dr=oo_.dr;
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dr.ys =zeros(endo_nbr,1);
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dr=set_state_space(dr,M_,options_);
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dr=set_state_space(dr,M_,options_); %relies on M_.lead_lag_incidence being updated
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order_var=dr.order_var;
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T=H(order_var,order_var);
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@ -1,16 +1,54 @@
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function [H,G,retcode]=discretionary_policy_engine(AAlag,AA0,AAlead,BB,bigw,instr_id,beta,solve_maxit,discretion_tol,qz_criterium,H00,verbose)
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% Solves the discretionary problem for a model of the form:
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% AAlag*yy_{t-1}+AA0*yy_t+AAlead*yy_{t+1}+BB*e=0, with W the weight on the
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% variables in vector y_t and instr_id is the location of the instruments
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% in the yy_t vector.
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%
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% Loss=E_0 sum_{t=0}^{\infty} beta^t [y_t'*W*y+x_t'*Q*x_t]
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% subject to
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% AAlag*yy_{t-1}+AA0*yy_t+AAlead*yy_{t+1}+BB*e=0
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%
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% with W the weight on the variables in vector y_t.
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%
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% The solution takes the form
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% y_t=H*y_{t-1}+G*e_t
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% where H=[H1;F1] and G=[H2;F2].
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%
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% We use the Dennis (2007, Macroeconomic Dynamics) algorithm and so we need
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% to re-write the model in the form
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% A0*y_t=A1*y_{t-1}+A2*y_{t+1}+A3*x_t+A4*x_{t+1}+A5*e_t, with W the
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% weight on the y_t vector and Q the weight on the x_t vector of
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% instruments.
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%
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% Inputs:
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% AAlag [double] matrix of coefficients on lagged
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% variables
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% AA0 [double] matrix of coefficients on
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% contemporaneous variables
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% AAlead [double] matrix of coefficients on
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% leaded variables
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% BB [double] matrix of coefficients on
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% shocks
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% bigw [double] matrix of coefficients on variables in
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% loss/objective function; stacks [W and Q]
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% instr_id [double] location vector of the instruments in the yy_t vector.
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% beta [scalar] planner discount factor
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% solve_maxit [scalar] maximum number of iterations
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% discretion_tol [scalar] convergence criterion for solution
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% qz_criterium [scalar] tolerance for QZ decomposition
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% H00
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% verbose [scalar] dummy to control verbosity
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%
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% Outputs:
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% H [double] (endo_nbr*endo_nbr) solution matrix for endogenous
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% variables, stacks [H1 and H1]
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% G [double] (endo_nbr*exo_nbr) solution matrix for shocks, stacks [H2 and F2]
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%
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% retcode [scalar] return code
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%
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% Algorithm:
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% Dennis, Richard (2007): Optimal policy in rational expectations models: new solution algorithms,
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% Macroeconomic Dynamics, 11, 31–55.
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% Copyright (C) 2007-2012 Dynare Team
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% Copyright (C) 2007-2015 Dynare Team
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%
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% This file is part of Dynare.
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%
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@ -53,14 +91,15 @@ end
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[A0,A1,A2,A3,A4,A5,W,Q,endo_nbr,exo_nbr,aux,endo_augm_id]=GetDennisMatrices(AAlag,AA0,AAlead,BB,bigw,instr_id);
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% aux is a logical index of the instruments which appear with lags in the
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% model. Their location in the state vector is instr_id(aux)
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% model. Their location in the state vector is instr_id(aux);
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% endo_augm_id is index (not logical) of locations of the augmented vector
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% of non-instrumental variables
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AuxiliaryVariables_nbr=sum(aux);
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H0=zeros(endo_nbr+AuxiliaryVariables_nbr);
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if ~isempty(H00)
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H0(1:endo_nbr,1:endo_nbr)=H00;clear H00
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H0(1:endo_nbr,1:endo_nbr)=H00;
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clear H00
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end
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H10=H0(endo_augm_id,endo_augm_id);
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@ -69,6 +108,7 @@ F10=H0(instr_id,endo_augm_id);
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iter=0;
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H1=H10;
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F1=F10;
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%solve equations (20) and (22) via fixed point iteration
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while 1
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iter=iter+1;
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P=SylvesterDoubling(W+beta*F1'*Q*F1,beta*H1',H1,discretion_tol,solve_maxit);
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@ -79,11 +119,11 @@ while 1
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return
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end
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end
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D=A0-A2*H1-A4*F1;
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D=A0-A2*H1-A4*F1; %equation (20)
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Dinv=inv(D);
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A3DPD=A3'*Dinv'*P*Dinv;
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F1=-(Q+A3DPD*A3)\(A3DPD*A1);
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H1=Dinv*(A1+A3*F1);
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A3DPD=A3'*Dinv'*P*Dinv;
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F1=-(Q+A3DPD*A3)\(A3DPD*A1); %component of (26)
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H1=Dinv*(A1+A3*F1); %component of (27)
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[rcode,NQ]=CheckConvergence([H1;F1]-[H10;F10],iter,solve_maxit,discretion_tol);
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if rcode
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@ -97,16 +137,17 @@ while 1
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F10=F1;
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end
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%check if successful
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retcode = 0;
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switch rcode
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case 3 % nan
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retcode=63;
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retcode(2)=10000;
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if verbose
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disp([mfilename,':: NAN elements in the solution'])
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disp([mfilename,':: NaN elements in the solution'])
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end
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case 2% maxiter
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retcode = 61
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retcode = 61;
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if verbose
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disp([mfilename,':: Maximum Number of Iterations reached'])
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end
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@ -125,8 +166,8 @@ if retcode(1)
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H=[];
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G=[];
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else
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F2=-(Q+A3DPD*A3)\(A3DPD*A5);
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H2=Dinv*(A5+A3*F2);
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F2=-(Q+A3DPD*A3)\(A3DPD*A5); %equation (29)
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H2=Dinv*(A5+A3*F2); %equation (31)
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H=zeros(endo_nbr+AuxiliaryVariables_nbr);
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G=zeros(endo_nbr+AuxiliaryVariables_nbr,exo_nbr);
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H(endo_augm_id,endo_augm_id)=H1;
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@ -159,6 +200,7 @@ end
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end
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function [A00,A11,A22,A33,A44,A55,WW,Q,endo_nbr,exo_nbr,aux,endo_augm_id]=GetDennisMatrices(AAlag,AA0,AAlead,BB,bigw,instr_id)
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%get the matrices to use the Dennis (2007) algorithm
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[eq_nbr,endo_nbr]=size(AAlag);
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exo_nbr=size(BB,2);
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y=setdiff(1:endo_nbr,instr_id);
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@ -211,7 +253,7 @@ end
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function v = SylvesterHessenbergSchur(d,g,h)
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%
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% DSYLHS Solves a discrete time sylvester equation using the
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% DSYLHS Solves a discrete time sylvester equation using the
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% Hessenberg-Schur algorithm
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%
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% v = DSYLHS(g,d,h) computes the matrix v that satisfies the
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@ -11,7 +11,7 @@ function planner_objective_value = evaluate_planner_objective(M,options,oo)
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% SPECIAL REQUIREMENTS
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% none
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% Copyright (C) 2007-2012 Dynare Team
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% Copyright (C) 2007-2015 Dynare Team
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%
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% This file is part of Dynare.
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%
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@ -29,19 +29,10 @@ function planner_objective_value = evaluate_planner_objective(M,options,oo)
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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dr = oo.dr;
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endo_nbr = M.endo_nbr;
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exo_nbr = M.exo_nbr;
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nstatic = M.nstatic;
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nspred = M.nspred;
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lead_lag_incidence = M.lead_lag_incidence;
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beta = get_optimal_policy_discount_factor(M.params,M.param_names);
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if options.ramsey_policy
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i_org = (1:M.orig_endo_nbr)';
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else
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i_org = (1:M.endo_nbr)';
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end
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ipred = find(lead_lag_incidence(M.maximum_lag,:))';
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order_var = dr.order_var;
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Gy = dr.ghx(nstatic+(1:nspred),:);
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Gu = dr.ghu(nstatic+(1:nspred),:);
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@ -51,11 +42,9 @@ gu(dr.order_var,:) = dr.ghu;
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ys = oo.dr.ys;
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u = oo.exo_simul(1,:)';
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[U,Uy,Uyy] = feval([M.fname '_objective_static'],ys,zeros(1,exo_nbr), ...
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M.params);
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%second order terms
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Uyy = full(Uyy);
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[Uyygygy, err] = A_times_B_kronecker_C(Uyy,gy,gy,options.threads.kronecker.A_times_B_kronecker_C);
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@ -65,7 +54,7 @@ mexErrCheck('A_times_B_kronecker_C', err);
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[Uyygygu, err] = A_times_B_kronecker_C(Uyy,gy,gu,options.threads.kronecker.A_times_B_kronecker_C);
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mexErrCheck('A_times_B_kronecker_C', err);
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Wbar =U/(1-beta);
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Wbar =U/(1-beta); %steady state welfare
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Wy = Uy*gy/(eye(nspred)-beta*Gy);
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Wu = Uy*gu+beta*Wy*Gu;
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Wyy = Uyygygy/(eye(nspred*nspred)-beta*kron(Gy,Gy));
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@ -75,7 +64,7 @@ mexErrCheck('A_times_B_kronecker_C', err);
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mexErrCheck('A_times_B_kronecker_C', err);
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Wuu = Uyygugu+beta*Wyygugu;
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Wyu = Uyygygu+beta*Wyygygu;
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Wss = beta*Wuu*M.Sigma_e(:)/(1-beta);
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Wss = beta*Wuu*M.Sigma_e(:)/(1-beta); % at period 0, we are in steady state, so the deviation term only starts in period 1, thus the beta in front
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% initialize yhat1 at the steady state
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yhat1 = oo.steady_state;
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@ -92,7 +81,6 @@ if ~isempty(M.endo_histval)
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end
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end
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yhat1 = yhat1(dr.order_var(nstatic+(1:nspred)),1)-dr.ys(dr.order_var(nstatic+(1:nspred)));
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yhat2 = yhat2(dr.order_var(nstatic+(1:nspred)),1)-dr.ys(dr.order_var(nstatic+(1:nspred)));
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u = oo.exo_simul(1,:)';
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[Wyyyhatyhat1, err] = A_times_B_kronecker_C(Wyy,yhat1,yhat1,options.threads.kronecker.A_times_B_kronecker_C);
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@ -104,6 +92,7 @@ mexErrCheck('A_times_B_kronecker_C', err);
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planner_objective_value(1) = Wbar+Wy*yhat1+Wu*u+Wyuyhatu1 ...
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+ 0.5*(Wyyyhatyhat1 + Wuuuu+Wss);
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if options.ramsey_policy
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yhat2 = yhat2(dr.order_var(nstatic+(1:nspred)),1)-dr.ys(dr.order_var(nstatic+(1:nspred)));
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[Wyyyhatyhat2, err] = A_times_B_kronecker_C(Wyy,yhat2,yhat2,options.threads.kronecker.A_times_B_kronecker_C);
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mexErrCheck('A_times_B_kronecker_C', err);
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[Wyuyhatu2, err] = A_times_B_kronecker_C(Wyu,yhat2,u,options.threads.kronecker.A_times_B_kronecker_C);
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@ -115,9 +104,13 @@ end
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if ~options.noprint
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skipline()
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disp('Approximated value of planner objective function')
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disp([' - with initial Lagrange multipliers set to 0: ' ...
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if options.ramsey_policy
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disp([' - with initial Lagrange multipliers set to 0: ' ...
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num2str(planner_objective_value(2)) ])
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disp([' - with initial Lagrange multipliers set to steady state: ' ...
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num2str(planner_objective_value(1)) ])
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disp([' - with initial Lagrange multipliers set to steady state: ' ...
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num2str(planner_objective_value(1)) ])
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elseif options.discretionary_policy
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fprintf('with discretionary policy: %10.8f',planner_objective_value(1))
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end
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skipline()
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end
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@ -46,6 +46,7 @@ MODFILES = \
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optimal_policy/Ramsey/Gali_commitment.mod \
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optimal_policy/RamseyConstraints/test1.mod \
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discretionary_policy/dennis_1.mod \
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discretionary_policy/Gali_discretion.mod \
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initval_file/ramst_initval_file.mod \
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ramst_normcdf_and_friends.mod \
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ramst_vec.mod \
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@ -0,0 +1,149 @@
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/*
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* This file implements the baseline New Keynesian model of Jordi Galí (2008): Monetary Policy, Inflation,
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* and the Business Cycle, Princeton University Press, Chapter 5
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*
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* This implementation was written by Johannes Pfeifer.
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*
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* Please note that the following copyright notice only applies to this Dynare
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* implementation of the model.
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*/
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/*
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* Copyright (C) 2013-15 Johannes Pfeifer
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*
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* This 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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* It 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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* For a copy of the GNU General Public License,
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* see <http://www.gnu.org/licenses/>.
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*/
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var pi
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y_gap
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r_e
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y_e
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r_nat
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i
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u
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a
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p
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;
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varexo eps_a
|
||||
eps_u;
|
||||
|
||||
parameters alppha
|
||||
betta
|
||||
rho_a
|
||||
rho_u
|
||||
siggma
|
||||
phi
|
||||
phi_y
|
||||
eta
|
||||
epsilon
|
||||
theta
|
||||
;
|
||||
%----------------------------------------------------------------
|
||||
% Parametrization, p. 52
|
||||
%----------------------------------------------------------------
|
||||
siggma = 1;
|
||||
phi=1;
|
||||
phi_y = .5/4;
|
||||
theta=2/3;
|
||||
rho_u = 0;
|
||||
rho_a = 0.9;
|
||||
betta = 0.99;
|
||||
eta =4;
|
||||
alppha=1/3;
|
||||
epsilon=6;
|
||||
|
||||
|
||||
|
||||
%----------------------------------------------------------------
|
||||
% First Order Conditions
|
||||
%----------------------------------------------------------------
|
||||
|
||||
model(linear);
|
||||
//Composite parameters
|
||||
#Omega=(1-alppha)/(1-alppha+alppha*epsilon); //defined on page 47
|
||||
#psi_n_ya=(1+phi)/(siggma*(1-alppha)+phi+alppha); //defined on page 48
|
||||
#lambda=(1-theta)*(1-betta*theta)/theta*Omega; //defined on page 47
|
||||
#kappa=lambda*(siggma+(phi+alppha)/(1-alppha)); //defined on page 49
|
||||
#alpha_x=kappa/epsilon; //defined on page 96
|
||||
#phi_pi=(1-rho_u)*kappa*siggma/(alpha_x)+rho_u; //defined on page 101
|
||||
|
||||
r_e=siggma*(y_e(+1)-y_e);
|
||||
y_e=psi_n_ya*a;
|
||||
pi=betta*pi(+1)+kappa*y_gap + u;
|
||||
y_gap=-1/siggma*(i-pi(+1)-r_e)+y_gap(+1);
|
||||
//3. Interest Rate Rule eq. (25)
|
||||
% i=r_e+phi_pi*pi;
|
||||
|
||||
r_nat=siggma*psi_n_ya*(a(+1)-a);
|
||||
u=rho_u*u(-1)+eps_u;
|
||||
a=rho_a*a(-1)+eps_a;
|
||||
|
||||
pi=p-p(-1);
|
||||
end;
|
||||
|
||||
%----------------------------------------------------------------
|
||||
% define shock variances
|
||||
%---------------------------------------------------------------
|
||||
|
||||
|
||||
shocks;
|
||||
var eps_u = 1;
|
||||
end;
|
||||
|
||||
planner_objective pi^2 +(((1-theta)*(1-betta*theta)/theta*((1-alppha)/(1-alppha+alppha*epsilon)))*(siggma+(phi+alppha)/(1-alppha)))/epsilon*y_gap^2;
|
||||
|
||||
discretionary_policy(instruments=(i),irf=20,planner_discount=betta,discretionary_tol=1e-12) y_gap pi p u;
|
||||
|
||||
verbatim;
|
||||
%% Check correctness
|
||||
Omega=(1-alppha)/(1-alppha+alppha*epsilon); %defined on page 47
|
||||
lambda=(1-theta)*(1-betta*theta)/theta*Omega; %defined on page 47
|
||||
kappa=lambda*(siggma+(phi+alppha)/(1-alppha)); %defined on page 49
|
||||
alpha_x=kappa/epsilon; %defined on page 96
|
||||
Psi=1/(kappa^2+alpha_x*(1-betta*rho_u)); %defined on page 99
|
||||
Psi_i=Psi*(kappa*siggma*(1-rho_u)+alpha_x*rho_u); %defined on page 101
|
||||
phi_pi=(1-rho_u)*kappa*siggma/(alpha_x)+rho_u; %defined on page 101
|
||||
|
||||
%Compute theoretical solution
|
||||
var_pi_theoretical=(alpha_x*Psi)^2; %equation (6), p.99
|
||||
var_y_gap_theoretical=(-kappa*Psi)^2; %equation (7), p.99
|
||||
|
||||
pi_pos=strmatch('pi',var_list_,'exact');
|
||||
y_gap_pos=strmatch('y_gap',var_list_,'exact');
|
||||
if abs(oo_.var(pi_pos,pi_pos)-var_pi_theoretical)>1e-10 || abs(oo_.var(y_gap_pos,y_gap_pos)-var_y_gap_theoretical)>1e-10
|
||||
error('Variances under optimal policy are wrong')
|
||||
end
|
||||
|
||||
%Compute theoretical objective function
|
||||
V=betta/(1-betta)*(var_pi_theoretical+alpha_x*var_y_gap_theoretical); %evaluate at steady state in first period
|
||||
|
||||
if abs(V-oo_.planner_objective_value)>1e-10
|
||||
error('Computed welfare deviates from theoretical welfare')
|
||||
end
|
||||
end;
|
||||
|
||||
%% repeat exercise with initial shock of 1 to check whether planner objective is correctly specified
|
||||
initval;
|
||||
eps_u = 1;
|
||||
end;
|
||||
|
||||
%Compute theoretical objective function
|
||||
V=var_pi_theoretical+alpha_x*var_y_gap_theoretical+ betta/(1-betta)*(var_pi_theoretical+alpha_x*var_y_gap_theoretical); %evaluate at steady state in first period
|
||||
|
||||
discretionary_policy(instruments=(i),irf=20,discretionary_tol=1e-12,planner_discount=betta) y_gap pi p u;
|
||||
if abs(V-oo_.planner_objective_value)>1e-10
|
||||
error('Computed welfare deviates from theoretical welfare')
|
||||
end
|
||||
|
|
@ -19,7 +19,7 @@ y = y(+1) -(omega/sigma)*(i-pi(+1))+g;
|
|||
pi = beta*pi(+1)+kappa*y+u;
|
||||
pi_c = pi+(alpha/(1-alpha))*(q-q(-1));
|
||||
q = q(+1)-(1-alpha)*(i-pi(+1))+(1-alpha)*e;
|
||||
i = 1.5*pi;
|
||||
% i = 1.5*pi;
|
||||
end;
|
||||
|
||||
shocks;
|
||||
|
|
@ -29,6 +29,5 @@ var e; stderr 1;
|
|||
end;
|
||||
|
||||
planner_objective pi_c^2 + y^2;
|
||||
|
||||
discretionary_policy(instruments=(i),irf=0,qz_criterium=0.999999);
|
||||
discretionary_policy(instruments=(i),irf=0,planner_discount=beta);
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue