Density forecasts

A density forecast runs the scenario many times with random shocks drawn on top of it and reports quantiles of every observable and of the learning states.

Option	Values (default first)	Meaning
density.enabled	false	Switch on
density.shocks	cellstr	Shocks to draw (required when enabled)
density.sigma	'calibrated', 'smoothed'	The model's std_<shock>, or the sample standard deviation of the RE-smoothed shocks over sigma_range
density.sigma_range	[]	[] = the filter range
density.method	'normal', 'bootstrap'	Normal draws, or resampled whole quarters of smoothed shocks
density.demean	true	Bootstrap: demean the pool first
density.ndraws	200	Number of draws
density.seed	0	Random seed
density.quantiles	[0.05 0.16 0.5 0.84 0.95]	Reported quantiles
density.range	[]	Quarters with draws; [] = the horizon
density.keep_draws	false	Store every draw in density.mat
density.fast_path	true	Mode 'rational' without conditions: one linear simulation per draw

How it works

The draws come on top of the scenario's own shocks and conditions. With learning, each draw re-runs the quarter-by-quarter forecast, so beliefs, gains and shares respond to the drawn shocks; the projection facility applies in each draw. An RE scenario without conditions takes the fast path, one linear simulation per draw. On SW07, 500 draws take about 30 seconds under RE and 4-6 minutes with learning.

Examples

% Application 5: price and wage mark-up shocks, normal, calibrated sigma, 500 draws
cfg = a1_learn(); cfg.id = 'my_markup_learn';
cfg.density.enabled = true;
cfg.density.shocks = {'epinf', 'ew'};
cfg.density.sigma = 'calibrated';
cfg.density.method = 'normal';
cfg.density.ndraws = 500;
cfg.density.seed = 0;
res = hetlearn.run(cfg);

% Use the same draws for the RE run: copy the block
r = a1_re(); r.id = 'my_markup_re';
r.density = cfg.density;
hetlearn.run(r);

% Bootstrap whole quarters of smoothed shocks from 1984-2019, draws only in the first two years
cfg = a1_learn(); cfg.id = 'my_bootstrap';
cfg.density.enabled = true;
cfg.density.shocks = {'ea', 'eb', 'eg', 'eqs', 'em', 'epinf', 'ew'};
cfg.density.method = 'bootstrap';
cfg.density.sigma_range = [qq(1984, 1), qq(2019, 4)];
cfg.density.demean = true;
cfg.density.range = qq(2022, 3):qq(2024, 2);
cfg.density.quantiles = [0.1 0.5 0.9];
cfg.density.ndraws = 200;

Outputs

Output	Content
res.density	Quantiles, means, standard deviations and point forecasts of every summarised series, draw information, events
results/<id>/density.mat	The same, saved apart from results.mat; every draw too with keep_draws = true
results/<id>/density_quantiles.csv	variable, date, quantile, value_model, value_report
results/<id>/density_events.csv	One row per draw: draw, status, projection_plm, projection_alm, projection_alm_shares, explosive_events, rls_missing, shares_nan, gain_bound, max_abs, message

A later run of the same id without density deletes the old density files.

q = readtable(fullfile('results', 'my_markup_learn', 'density_quantiles.csv'));
q(strcmp(q.variable, 'pinfobs') & q.quantile == 0.95, :)

In the project

Application 5 runs 500 draws for mark-up shocks and for demand shocks. With mark-up shocks the average 90% inflation band is 14.59 pp with learning against 4.76 pp under RE; with demand shocks the two are about the same (1.16 and 1.18 pp). See Application 5.