bvar_forecast.m

function bvar_forecast(nlags)

    global options_
    
    options_ = set_default_option(options_, 'bvar_replic', 2000);
    if options_.forecast == 0
        error('bvar_forecast: you must specify "forecast" option')
    end

    [ny, nx, posterior, prior, forecast_data] = bvar_toolbox(nlags);
        
    sims_no_shock = NaN(options_.forecast, ny, options_.bvar_replic);
    sims_with_shocks = NaN(options_.forecast, ny, options_.bvar_replic);
    
    S_inv_upper_chol = chol(inv(posterior.S));

    % Option 'lower' of chol() not available in old versions of
    % Matlab, so using transpose
    XXi_lower_chol = chol(posterior.XXi)';
    
    k = ny*nlags+nx;
    
    for d = 1:options_.bvar_replic
        Sigma = rand_inverse_wishart(ny, posterior.df, S_inv_upper_chol);

        % Option 'lower' of chol() not available in old versions of
        % Matlab, so using transpose
        Sigma_lower_chol = chol(Sigma)';

        Phi = rand_matrix_normal(k, ny, posterior.PhiHat, XXi_lower_chol, Sigma_lower_chol);
        
        % Without shocks
        lags_data = forecast_data.initval;
        for t = 1:options_.forecast
            X = [ reshape(flipdim(lags_data, 1)', 1, ny*nlags) forecast_data.xdata(t, :) ];
            y = X * Phi;
            lags_data = [ lags_data(2:end, :); y ];
            sims_no_shock(t, :, d) = y;
        end
    
        % With shocks
        lags_data = forecast_data.initval;
        for t = 1:options_.forecast
            X = [ reshape(flipdim(lags_data, 1)', 1, ny*nlags) forecast_data.xdata(t, :) ];
            shock = (Sigma_lower_chol * randn(ny, 1))';
            y = X * Phi + shock;
            lags_data = [ lags_data(2:end, :); y ];
            sims_with_shocks(t, :, d) = y;
        end
    end

    % Plot graphs
    sims_no_shock_mean = mean(sims_no_shock, 3);

    sort_idx = round((0.5 + [-options_.conf_sig, options_.conf_sig]/2) * options_.bvar_replic);
    
    sims_no_shock_sort = sort(sims_no_shock, 3);
    sims_no_shock_down_conf = sims_no_shock_sort(:, :, sort_idx(1));
    sims_no_shock_up_conf = sims_no_shock_sort(:, :, sort_idx(2));

    sims_with_shocks_sort = sort(sims_with_shocks, 3);
    sims_with_shocks_down_conf = sims_with_shocks_sort(:, :, sort_idx(1));
    sims_with_shocks_up_conf = sims_with_shocks_sort(:, :, sort_idx(2));

    dynare_graph_init(sprintf('BVAR forecasts (nlags = %d)', nlags), ny, {'b-' 'g-' 'g-' 'r-' 'r-'});
    
    for i = 1:ny
        dynare_graph([ sims_no_shock_mean(:, i) ...
                       sims_no_shock_up_conf(:, i) sims_no_shock_down_conf(:, i) ...
                       sims_with_shocks_up_conf(:, i) sims_with_shocks_down_conf(:, i) ], ...
                     options_.varobs(i, :));
    end
    
    dynare_graph_close;

    
    % Compute RMSE
    
    if ~isempty(forecast_data.realized_val)
        
        sq_err_cumul = zeros(1, ny);
        
        lags_data = forecast_data.initval;
        for t = 1:size(forecast_data.realized_val, 1)
            X = [ reshape(flipdim(lags_data, 1)', 1, ny*nlags) forecast_data.realized_xdata(t, :) ];
            y = X * posterior.PhiHat;
            lags_data = [ lags_data(2:end, :); y ];
            sq_err_cumul = sq_err_cumul + (y - forecast_data.realized_val(t, :)) .^ 2;
        end
        
        rmse = sqrt(sq_err_cumul / size(sq_err_cumul, 1));
        
        fprintf('RMSE of BVAR(%d):\n', nlags);
        
        for i = 1:size(options_.varobs, 1)
            fprintf('%s: %10.4f\n', options_.varobs(i, :), rmse(i));
        end
    end
    
    
function G = rand_inverse_wishart(m, v, H_inv_upper_chol)
% rand_inverse_wishart  Pseudo random matrices drawn from an
%                       inverse Wishart distribution
%
% G = rand_inverse_wishart(m, v, H_inv_upper_chol)
%
% Returns an m-by-m matrix drawn from an inverse-Wishart distribution.
%
% m:          dimension of G and H_inv_upper_chol.
% v:          degrees of freedom, greater or equal than m.
% H_inv_chol: upper cholesky decomposition of the inverse of the
%             matrix parameter.
%             The upper cholesky of the inverse is requested here
%             in order to avoid to recompute it at every random draw.
%             H_inv_upper_chol = chol(inv(H))
%
% In other words:
%  G ~ IW(m, v, H) where H = inv(H_inv_upper_chol'*H_inv_upper_chol)
% or, equivalently, using the correspondence between Wishart and
% inverse-Wishart:
%  inv(G) ~ W(m, v, S) where S = H_inv_upper_chol'*H_inv_upper_chol = inv(H)

    X = NaN(v, m);
    for i = 1:v
        X(i, :) = randn(1, m) * H_inv_upper_chol;
    end
    
    % At this point, X'*X is Wishart distributed
    % G = inv(X'*X);

    % Rather compute inv(X'*X) using the SVD
    [U,S,V] = svd(X, 0);
    SSi = 1 ./ (diag(S) .^ 2);
    G = (V .* repmat(SSi', m, 1)) * V';


function B = rand_matrix_normal(n, p, M, Omega_lower_chol, Sigma_lower_chol)
% rand_matrix_normal  Pseudo random matrices drawn from a
%                     matrix-normal distribution
%
% B = rand_matrix_normal(n, p, M, Omega_lower_chol, Sigma_lower_chol)
%
% Returns an n-by-p matrix drawn from a Matrix-normal distribution
% Same notations than: http://en.wikipedia.org/wiki/Matrix_normal_distribution
% M is the mean, n-by-p matrix
% Omega_lower_chol is n-by-n, lower Cholesky decomposition of Omega
% (Omega_lower_chol = chol(Omega, 'lower'))
% Sigma_lower_chol is p-by-p, lower Cholesky decomposition of Sigma
% (Sigma_lower_chol = chol(Sigma, 'lower'))
%
% Equivalent to vec(B) ~ N(vec(Mu), kron(Sigma, Omega))
%
    
    B1 = randn(n * p, 1);
    B2 = kron(Sigma_lower_chol, Omega_lower_chol) * B1;
    B3 = reshape(B2, n, p);
    B = B3 + M;