Further expectation rules

Beyond the baseline pair of rational expectations (RE) agents and learners, three groups of options build further expectation rules: more than two types of agents, shares over those types, and transforms of the expectations as they enter the model. All are off by default. Application 7 uses them for seven rules from the bounded-rationality literature, with the code and results of each.

Options	What they do
expectations.types	the population of forecasting types
shares.init, shares.fixed	initial or fixed shares by type name
expectations.transforms	transforms of the expectations as they enter the model

Two of the seven rules (mean-only learners and the experience-based gain) need none of these options: they use the perceived law of motion and gain options. A scenario that sets none of the three groups runs the two-type code unchanged; the test tests/test_br_regression.m checks that its results are bit-identical.

Agent types: expectations.types

A cell array of structs, one per type:

Field	Values	Meaning
name	letters and digits, starting with a letter; required	Suffix of the type's forecast <x>_exp1_<name> and of its share n_<name>_<x>
kind	're', 'plm', 'rule', 're_mean'; required	The RE forecast (only the type named RE); a learned perceived law of motion (PLM); a fixed linear rule; the RE forecast plus a learned mean of its own error
rule	text, or struct('expr', <text>, 'params', struct(...))	'rule' types only: a linear expression in x{-k}, other transition variables with lags, numbers and named constants from params

Each kind takes its settings from the global options: 're' from expectations.re_variant and .re_info; 'plm' from beliefs.plm, beliefs.info, beliefs.update and gain; 'rule' from beliefs.info; 're_mean' from the RE type and, for its learned mean, from beliefs.update and gain. Every type forecasts every learning variable. The baseline learners are the type named BR (bounded rationality) of kind 'plm'.

% Target believers beside RE agents and learners (Application 7)
cfg = a1_learn(); cfg.id = 'my_target_believers';
cfg.expectations.types = {struct('name', 'RE', 'kind', 're'), struct('name', 'BR', 'kind', 'plm'), ...
    struct('name', 'TB', 'kind', 'rule', 'rule', '0')};
cfg.shares.init = struct('RE', 1/3, 'BR', 1/3, 'TB', 1/3);
res = hetlearn.run(cfg, 'save', false);

Shares over N types

The baseline discrete choice extends to N types in each share group:

\[ \zeta_{k,t} = \omega\, \zeta_{k,t-1} - (1-\omega) FE_{k,t}, \qquad n_{k,t} = \frac{\exp(\chi\, \zeta_{k,t})}{\sum_j \exp(\chi\, \zeta_{j,t})}, \]

with \(\zeta_{k,t}\) the fitness of type k, \(\chi\) = switching.intensity, \(\omega\) = switching.memory and \(FE_{k,t}\) the squared error of type k's forecast as it enters the model (after any transform).

Option	Values (default first)	Meaning
shares.init	[], struct by type name	Initial shares, summing to 1; [] = shares.init_n2 for the RE/BR pair, equal shares otherwise
shares.fixed	[], struct by type name	Mode 'fixed_shares'; not together with shares.fixed_n2

Forecast transforms: expectations.transforms

One struct per transform, applied to the one-step expectations as they enter the model:

kind	Expectation of a base forecast \(f\)	value (identity)
'multiplier'	\(m^h f\), with \(h = 2\) for a forecast made on t-1 information, else 1	\(m\) (1)
'diagnostic'	\((1+\theta) f - \theta f^{ref}\), \(f^{ref}\) = the same forecast made one quarter earlier	\(\theta\) (0)
'myopia_anchoring'	\(\omega_f f + b\, x_{t-1}\)	[omega_f b] ([1 0])

Field	Values	Meaning
types	'all' or type names	Which types' forecasts are transformed; 'all' also covers the model-consistent expectations outside the learning set
scope	'learning' (default), 'all', or a cellstr of variables and 'x@<label>' entries	Which expectations; 'x@<label>' acts in one equation only, named by its exact label in the model file
exclude	cellstr	Variables taken out of the scope

The values are model parameters, so one generated file serves every value.

% Cognitive discounting of both types' inflation expectations, m = 0.94,
% outside the flexible-price block of the Smets-Wouters model
FLEX = {'invef', 'pkf', 'rkf', 'cf', 'labf'};
cfg = a1_learn(); cfg.id = 'my_cd';
cfg.expectations.transforms = struct('kind', 'multiplier', 'types', 'all', ...
    'value', 0.94, 'scope', 'learning', 'exclude', {FLEX});
res = hetlearn.run(cfg, 'save', false);

A diagnostic transform must be the first transform on its expectation and acts on RE types and model-consistent expectations only. With period-t RE information the anchored reference is not built: keep expectations.re_info = 'lagged' (the baseline) or use re_variant = 'model_consistent'.

Where the rules can be used

Stage	With types or transforms
History, history.mode = 're_smoother' or 'kalman'	supported
Forecast and scenario stage, conditions, shocks	supported
Density forecasts	supported (each draw re-runs the forecast)
hetlearn.evaluate	supported
hetlearn.decompose	supported, except for results of history.mode = 'kalman'
History, history.mode = 'inversion'	not supported (hetlearn:options:unsupported)
Ramsey policy	not supported (hetlearn:options:unsupported)

Outputs

Output	Content
res.learning.<x>	for each type k: n_<k> (share at the end of the quarter), n_model_<k> (share in the model), U_<k> (fitness \(\zeta_k\)), FE_<k> (squared error), f_<k> (forecast as it enters the model), fb_<k> (before transforms, with transforms only)
res.learning.<target>__<k>	beliefs, gain and moments of each learning type, e.g. res.learning.pinf__BR.alpha_pinf__BR, res.learning.pinf_feRE__IO.alpha_pinf_feRE__IO
res.paths.<x>_exp1_<k>	each type's forecasts
res.meta.types, res.meta.model_values	the resolved types and the transform values

The generated model of a run with types or transforms is models/<name>/generated/<name>_ext_n<hash>.model.

Further reading

• docs/USER_GUIDE.md, section 9: every option of this page with its checks.
• Technical appendix: the formulas and timing of each rule.