Further expectation rules¶

Beyond the baseline pair of RE agents and AR(1) learners, the toolbox has seven further expectation rules from the bounded-rationality literature. Each one is a scenario in scenarios/sw07/br_portfolio/ that starts from a1_learn and adds a few lines, so it inherits the project baseline and runs Application 1 (the forecast from the 2022Q2 peak).

No.	Rule	Scenario	What it changes	Value and source
5b	Cognitive discounting on the learning set	`br05b_cd_learn`	inflation expectations of both types multiplied by \(m^h\)	\(m = 0.94\), Meggiorini and Milani (2021)
7	Myopia and anchoring	`br07_myopia_anchoring`	the inflation expectation of the price Phillips curve	\((\omega_f, b) = (0.52, 0.434)\), Angeletos and Huo (2021)
10	Diagnostic expectations	`br10_diagnostic`	the RE agents' expectations become diagnostic	\(\theta = 0.44\), L'Huillier, Singh and Yoo (2024)
19	Target believers	`br19_target_believers`	a third type that always forecasts the steady state	shares 1/3 each at the start; Hommes and Lustenhouwer (2019)
23	Intercept-only learners	`br23_intercept_only`	learners forecast the RE forecast plus a learned mean of its error	baseline gain; Gáti (2023), Carvalho, Eusepi, Moench and Preston (2023)
24	Mean-only learners	`br24_mean_only`	the learners' PLM keeps only the constant	baseline gain
26	Experience-based gain	`br26_experience`	the AR(1) learners use a constant gain of 0.018	Malmendier and Nagel (2016), Nagel and Xu (2022)

Rules 24 and 26 use the PLM and gain options of the earlier pages. The other five use three groups of options, all off by default:

Options	What they do
`expectations.types`	the population of forecasting types (rules 19 and 23)
`shares.init`, `shares.fixed`	initial or fixed shares by type name
`expectations.transforms`	transforms of the expectations as they enter the model (rules 5b, 7, 10)

A scenario that sets none of them 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 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.

Shares over N types¶

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

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

with \(\chi\) = 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. The scenarios exclude the SW07 flexible-price block, FLEX = {'invef', 'pkf', 'rkf', 'cf', 'labf'}, so the flexible-price economy behind potential output keeps its RE expectations.

The seven rules¶

5b. Cognitive discounting on the learning set¶

Agents shrink their forecasts toward the steady state (Gabaix 2020). Here both types' inflation expectations are multiplied by \(m^h\); with t-1 information \(h = 2\), so the factor is \(0.94^2\). \(m = 0.94\) is Meggiorini and Milani's (2021) estimate for the US, 1960-2007, from their own model rather than SW07.

function cfg = br05b_cd_learn()
cfg = a1_learn();
cfg.id = 'br05b_cd_learn';
cfg.title = 'BR 5b: cognitive discounting on the learning set, m = 0.94';
FLEX = {'invef', 'pkf', 'rkf', 'cf', 'labf'};
cfg.expectations.transforms = struct('kind', 'multiplier', 'types', 'all', ...
    'value', 0.94, 'scope', 'learning', 'exclude', {FLEX});
cfg.compare_with = {'a1_learn', 'a1_re'};
end

7. Myopia and anchoring¶

Angeletos and Huo (2021) show that sticky, noisy information makes the Phillips curve behave as if expectations were myopic and anchored to the past: the expectation becomes \(\omega_f E\pi_{t+1} + b\,\pi_{t-1}\). The transform acts only in the price Phillips curve, for all types, with \((\omega_f, \omega_b) = (0.52, 0.43)\) from their Table 1 and \(b = 0.43/0.99 = 0.434\). SW07 keeps its own price indexation, so this backward weight comes on top of it.

function cfg = br07_myopia_anchoring()
cfg = a1_learn();
cfg.id = 'br07_myopia_anchoring';
cfg.title = 'BR 7: myopia and anchoring in the price Phillips curve (0.52, 0.434)';
cfg.expectations.transforms = struct('kind', 'myopia_anchoring', 'types', 'all', 'value', [0.52 0.434], ...
    'scope', {{'pinf@Phillips curve, price setting'}});
cfg.compare_with = {'a1_learn', 'a1_re'};
end

10. Diagnostic expectations¶

Diagnostic agents overreact to news: their forecast is \((1+\theta) E_t x_{t+1} - \theta E_{t-1} x_{t+1}\) (one-period memory). Here they replace the RE agents for every forward-looking expectation outside the flexible-price block; the learners are unchanged. Because the baseline RE type uses t-1 information, the reference forecast of inflation is the anchored \(E_{t-2}\pi_{t+1}\). \(\theta = 0.44\) is L'Huillier, Singh and Yoo's (2024) estimate in SW07 with Rotemberg pricing.

function cfg = br10_diagnostic()
cfg = a1_learn();
cfg.id = 'br10_diagnostic';
cfg.title = 'BR 10: diagnostic expectations replace RE, theta = 0.44';
FLEX = {'invef', 'pkf', 'rkf', 'cf', 'labf'};
cfg.expectations.transforms = struct('kind', 'diagnostic', 'types', {{'RE'}}, 'value', 0.44, ...
    'scope', 'all', 'exclude', {FLEX});
cfg.compare_with = {'a1_learn', 'a1_re'};
end

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'.

19. Target believers¶

A third type of agent always forecasts the inflation target, the steady state (0 in deviations); its share measures the credibility of the target (Hommes and Lustenhouwer 2019). The three types switch as in the baseline (\(\chi = 2\), memory 0.8), starting from a third each.

function cfg = br19_target_believers()
cfg = a1_learn();
cfg.id = 'br19_target_believers';
cfg.title = 'BR 19: target believers added beside RE agents and learners';
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);
cfg.compare_with = {'a1_learn', 'a1_re'};
end

The rule can be any fixed linear expression, for example a naive forecaster struct('name', 'NV', 'kind', 'rule', 'rule', 'x{-1}'), or one with named constants, 'rule', struct('expr', 'a*x{-1}', 'params', struct('a', 0.5)).

23. Intercept-only learners¶

The learners know the structure of the economy (they use the RE forecast) but not its long-run level: they forecast the RE forecast plus a learned mean \(\bar a\) of that forecast's own error, \(x_t - E^{RE}_{t-1} x_t\) (Gáti 2023; Carvalho, Eusepi, Moench and Preston 2023). The mean is learned with the baseline's KOZ gain. They replace the AR(1) learners and switch against the RE agents as in the baseline. Gáti's smooth anchoring function is not part of this rule, so the paths stay close to RE.

function cfg = br23_intercept_only()
cfg = a1_learn();
cfg.id = 'br23_intercept_only';
cfg.title = 'BR 23: intercept-only learners with RE slopes replace the AR(1) learners';
cfg.expectations.types = {struct('name', 'RE', 'kind', 're'), struct('name', 'IO', 'kind', 're_mean')};
cfg.compare_with = {'a1_learn', 'a1_re'};
end

With \(\bar a\) held at 0 this type forecasts exactly as the RE type. Its learned mean is res.learning.pinf_feRE__IO.alpha_pinf_feRE__IO.

24. Mean-only learners¶

The learners' inflation PLM keeps only the constant, so their forecast is the learned mean: adaptive expectations by exponential smoothing, \(\alpha_t = \alpha_{t-1} + g_t(\pi_t - \alpha_{t-1})\), with the baseline's KOZ gain. No new option is needed.

function cfg = br24_mean_only()
cfg = a1_learn();
cfg.id = 'br24_mean_only';
cfg.title = 'BR 24: mean-only inflation learners (exponential smoothing)';
cfg.beliefs.plm.by_var.pinf = struct('regressors', {{}}, 'constant', true);
cfg.compare_with = {'a1_learn', 'a1_re'};
end

26. Experience-based gain¶

In Malmendier and Nagel (2016) people weight the inflation they have lived through; in the aggregate this is close to constant-gain learning with a gain of about 0.018 per quarter (Nagel and Xu 2022; Malmendier and Nagel's best-matching constant gain is 0.0175). The AR(1) learners use this gain instead of the KOZ gain. Cohorts are not modelled.

function cfg = br26_experience()
cfg = a1_learn();
cfg.id = 'br26_experience';
cfg.title = 'BR 26: experience-based learning in aggregate, constant gain 0.018';
cfg.gain.rule = 'constant';
cfg.gain.value = 0.018;
cfg.compare_with = {'a1_learn', 'a1_re'};
end

Results¶

Run all seven and the two references:

run_br_app1('run')          % a1_re, a1_learn, then the seven rules; about two minutes
run_br_app1('csv')          % results/br_portfolio/app1_*.csv
run_br_portfolio('irf')     % impulse responses from each rule's own 2019Q4 state
run_br_portfolio('rmse')    % Kalman histories: one-step RMSE and log likelihood, 1966Q1-2026Q2
run_br_portfolio('csv')     % results/br_portfolio/irf_*.csv, rmse_*.csv

Each rule on its own: res = hetlearn.run(br19_target_believers());.

Application 1 and the in-sample fit, from results/br_portfolio/app1_summary.csv and rmse_long.csv (runs of 2026-10-22):

Rule	Avg. inflation 2022Q3-2026Q2 (%)	Peak gap to RE (pp)	Shares at the end of 2022Q2	Inflation RMSE, one-step (pp)	RMSE / RE(t-1)	Log likelihood
RE(t-1), `a1_re`	2.59			1.483	1.000	−2182.1
Baseline, `a1_learn`	3.45	3.54 (2023Q1)	RE 0.409, BR 0.591	1.237	0.834	−1839.8
5b cognitive discounting	4.09	3.23 (2023Q1)	RE 0.416, BR 0.584	1.120	0.755	−1709.4
7 myopia and anchoring	3.42	3.68 (2023Q1)	RE 0.483, BR 0.517	1.209	0.816	−1825.3
10 diagnostic	4.75	4.74 (2023Q1)	RE 0.456, BR 0.544	1.325	0.894	−2025.7
19 target believers	3.38	2.36 (2022Q4)	RE 0.340, BR 0.491, TB 0.169	1.121	0.756	−1723.2
23 intercept-only	2.94	0.69 (2022Q4)	RE 0.388, IO 0.612	1.488	1.004	−2192.8
24 mean-only	3.08	0.89 (2023Q4)	RE 0.710, BR 0.290	1.125	0.759	−1702.6
26 experience-based gain	3.06	2.11 (2022Q4)	RE 0.425, BR 0.575	1.247	0.841	−1857.6

The RMSE is of annualised inflation (percentage points), from the Kalman history (history.mode = 'kalman') over 1966Q1-2026Q2. The projection facility acts in none of these runs.

The two portfolio reports draw these results: reports/sw07_br_app1_report.html (Application 1: paths, shares, settings of each rule) and reports/sw07_br_portfolio_report.html (impulse responses to each SW07 shock from 2019Q4, and the RMSE by subsample). They are built locally by python reports/build_br_app1_report.py and python reports/build_br_portfolio_report.py from results/br_portfolio/; generated HTML is not kept in the repository.

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), `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.