Implement Andreasen et al. 2013 pruning at order
Implements and documents the Andreasen et al. 2013 pruning at order=3 and sets it as the default at this order. Michel's pruning based on the approximation of the forecast function has been assigned the option pruning_forecast_approximation. The preprocessor-interface still needs to be added for this option. Moreover, more documentation/a reference what this option does is needed. At a later point, we might change the default to Michel's approach.time-shift
Johannes Pfeifer
parent
efaa3c71c5
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cdb7b01879
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@ -3234,11 +3234,14 @@ period(s). The periods must be strictly positive. Conditional variances are give
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decomposition provides the decomposition of the effects of shocks upon
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impact. The results are stored in
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@code{oo_.conditional_variance_decomposition}
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(@pxref{oo_.conditional_variance_decomposition}). The variance decomposition is only conducted, if theoretical moments are requested, i.e. using the @code{periods=0}-option. In case of @code{order=2}, Dynare provides a second-order accurate approximation to the true second moments based on the linear terms of the second-order solution (see @cite{Kim, Kim, Schaumburg and Sims (2008)}).
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(@pxref{oo_.conditional_variance_decomposition}). The variance decomposition is only conducted, if theoretical moments are requested, i.e. using the @code{periods=0}-option. In case of @code{order=2}, Dynare provides a second-order accurate approximation to the true second moments based on the linear terms of the second-order solution (see @cite{Kim, Kim, Schaumburg and Sims (2008)}).
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@item pruning
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Discard higher order terms when iteratively computing simulations of
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the solution, as in @cite{Kim, Kim, Schaumburg and Sims (2008)}.
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Discard higher order terms when iteratively computing simulations of
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the solution. At second order, Dynare uses the algorithm of @cite{Kim, Kim, Schaumburg and Sims (2008)}, while at third order its generalization by @cite{Andreasen, Fernández-Villaverde and Rubio-Ramírez (2013)} is used.
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@item pruning_forecast_approximation
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This type of pruning generalizes the @cite{Kim, Kim, Schaumburg and Sims (2008)} approach to higher orders based on the k-order approximation of the forecasting function.
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@item partial_information
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@anchor{partial_information}
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@ -7725,6 +7728,9 @@ internals --test particle/local_state_iteration
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Aguiar, Mark and Gopinath, Gita (2004): ``Emerging Market Business
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Cycles: The Cycle is the Trend,'' @i{NBER Working Paper}, 10734
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@item
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Andreasen, Martin M., Fernández-Villaverde, Jesús and Juan Rubio-Ramírez (2013): ``The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications,'' @i{NBER Working Paper}, 18983
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@item
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Backus, David K., Patrick J. Kehoe, and Finn E. Kydland (1992):
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``International Real Business Cycles,'' @i{Journal of Political
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@ -259,6 +259,8 @@ options_.minimal_solving_periods = 1;
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% Solution
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options_.order = 2;
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options_.pruning = 0;
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options_.pruning = 0;
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options_.pruning_forecast_approximation = 0;
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options_.solve_algo = 2;
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options_.linear = 0;
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options_.replic = 50;
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@ -39,7 +39,7 @@ switch(order)
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dr.g_1 = g_1;
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dr.g_2 = g_2;
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case 3
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if options.pruning
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if options.pruning || options_.pruning_forecast_approximation
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[err, g_0, g_1, g_2, g_3, derivs] = k_order_perturbation(dr, ...
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M,options);
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dr.ghx = derivs.gy;
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@ -67,7 +67,7 @@ switch(order)
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error('order > 3 isn''t implemented')
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end
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if options.pruning
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if options.pruning || options_.pruning_forecast_approximation
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return
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end
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117
matlab/simult_.m
117
matlab/simult_.m
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@ -53,7 +53,7 @@ if ~options_.k_order_solver
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end
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end
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if options_.k_order_solver && ~options_.pruning % Call dynare++ routines.
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if options_.k_order_solver && ~options_.pruning && ~options_.pruning_forecast_approximation% Call dynare++ routines.
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ex_ = [zeros(1,exo_nbr); ex_];
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switch options_.order
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case 1
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@ -107,7 +107,7 @@ else
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y_ = bsxfun(@plus,y_,dr.ys);
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case 2
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constant = dr.ys(order_var)+.5*dr.ghs2;
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if options_.pruning
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if options_.pruning || options_.pruning_forecast_approximation
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y__ = y0;
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for i = 2:iter+M_.maximum_lag
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yhat1 = y__(order_var(k2))-dr.ys(order_var(k2));
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@ -157,38 +157,87 @@ else
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threads = options_.threads.kronecker.A_times_B_kronecker_C;
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nspred = M_.nspred;
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ipred = M_.nstatic+(1:nspred);
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yhat1 = y0(order_var(k2))-dr.ys(order_var(k2));
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yhat2 = zeros(nspred,1);
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yhat3 = zeros(nspred,1);
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for i=2:iter+M_.maximum_lag
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u = ex_(i-1,:)';
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[gyy, err] = A_times_B_kronecker_C(ghxx,yhat1,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[guu, err] = A_times_B_kronecker_C(ghuu,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyu, err] = A_times_B_kronecker_C(ghxu,yhat1,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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y2a = kron(yhat1,yhat1);
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[gyyy, err] = A_times_B_kronecker_C(ghxxx,y2a,yhat1,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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u2a = kron(u,u);
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[guuu, err] = A_times_B_kronecker_C(ghuuu,u2a,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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yu = kron(yhat1,u);
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[gyyu, err] = A_times_B_kronecker_C(ghxxu,yhat1,yu,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyuu, err] = A_times_B_kronecker_C(ghxuu,yu,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyy12, err] = A_times_B_kronecker_C(ghxx,yhat1,yhat2,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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yhat3 = ghx*yhat3 + gyyy + guuu + 3*(gyyu + gyuu ...
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+ gyy12 + ghxss*yhat1 + ghuss*u);
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yhat2 = ghx*yhat2 + gyy + guu + 2*gyu + ghs2;
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yhat1 = ghx*yhat1 + ghu*u;
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y_(order_var,i) = constant + yhat1 + (1/2)*yhat2 + (1/6)*yhat3;
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yhat1 = yhat1(ipred);
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yhat2 = yhat2(ipred);
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yhat3 = yhat3(ipred);
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if options_.pruning_forecast_approximation
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yhat1 = y0(order_var(k2))-dr.ys(order_var(k2));
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yhat2 = zeros(nspred,1);
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yhat3 = zeros(nspred,1);
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for i=2:iter+M_.maximum_lag
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u = ex_(i-1,:)';
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[gyy, err] = A_times_B_kronecker_C(ghxx,yhat1,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[guu, err] = A_times_B_kronecker_C(ghuu,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyu, err] = A_times_B_kronecker_C(ghxu,yhat1,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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y2a = kron(yhat1,yhat1);
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[gyyy, err] = A_times_B_kronecker_C(ghxxx,y2a,yhat1,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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u2a = kron(u,u);
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[guuu, err] = A_times_B_kronecker_C(ghuuu,u2a,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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yu = kron(yhat1,u);
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[gyyu, err] = A_times_B_kronecker_C(ghxxu,yhat1,yu,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyuu, err] = A_times_B_kronecker_C(ghxuu,yu,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyy12, err] = A_times_B_kronecker_C(ghxx,yhat1,yhat2,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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yhat3 = ghx*yhat3 + gyyy + guuu + 3*(gyyu + gyuu ...
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+ gyy12 + ghxss*yhat1 + ghuss*u);
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yhat2 = ghx*yhat2 + gyy + guu + 2*gyu + ghs2;
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yhat1 = ghx*yhat1 + ghu*u;
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y_(order_var,i) = constant + yhat1 + (1/2)*yhat2 + (1/6)*yhat3;
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yhat1 = yhat1(ipred);
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yhat2 = yhat2(ipred);
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yhat3 = yhat3(ipred);
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end
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else
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%construction follows Andreasen et al (2013), Technical
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%Appendix, Formulas (65) and (66)
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%split into first, second, and third order terms
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yhat1 = y0(order_var(k2))-dr.ys(order_var(k2));
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yhat2 = zeros(nspred,1);
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yhat3 = zeros(nspred,1);
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for i=2:iter+M_.maximum_lag
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u = ex_(i-1,:)';
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%construct terms of order 2 from second order part, based
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%on linear component yhat1
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[gyy, err] = A_times_B_kronecker_C(ghxx,yhat1,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[guu, err] = A_times_B_kronecker_C(ghuu,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyu, err] = A_times_B_kronecker_C(ghxu,yhat1,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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%construct terms of order 3 from second order part, based
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%on order 2 component yhat2
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[gyy12, err] = A_times_B_kronecker_C(ghxx,yhat1,yhat2,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gy2u, err] = A_times_B_kronecker_C(ghxu,yhat2,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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%construct terms of order 3, all based on first order component yhat1
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y2a = kron(yhat1,yhat1);
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[gyyy, err] = A_times_B_kronecker_C(ghxxx,y2a,yhat1,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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u2a = kron(u,u);
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[guuu, err] = A_times_B_kronecker_C(ghuuu,u2a,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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yu = kron(yhat1,u);
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[gyyu, err] = A_times_B_kronecker_C(ghxxu,yhat1,yu,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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[gyuu, err] = A_times_B_kronecker_C(ghxuu,yu,u,threads);
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mexErrCheck('A_times_B_kronecker_C', err);
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%add all terms of order 3, linear component based on third
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%order yhat3
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yhat3 = ghx*yhat3 +gyy12 ... prefactor is 1/2*2=1, see (65) Appendix Andreasen et al.
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+ gy2u ... prefactor is 1/2*2=1, see (65) Appendix Andreasen et al.
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+ 1/6*(gyyy + guuu + 3*(gyyu + gyuu + ghxss*yhat1 + ghuss*u)); %note: s is treated as variable, thus xss and uss are third order
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yhat2 = ghx*yhat2 + 1/2*(gyy + guu + 2*gyu + ghs2);
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yhat1 = ghx*yhat1 + ghu*u;
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y_(order_var,i) = dr.ys(order_var)+yhat1 + yhat2 + yhat3; %combine terms again
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yhat1 = yhat1(ipred);
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yhat2 = yhat2(ipred);
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yhat3 = yhat3(ipred);
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end
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end
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end
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end
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