Application 7: further expectation rules¶

Question. Beyond the baseline pair of rational expectations (RE) agents and AR(1) learners, how do seven expectation rules from the bounded-rationality literature change the forecast from the 2022Q2 inflation peak, and how well does each fit the history?

Each rule 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 options they use are on the Further expectation rules options page.

Rule	Scenario	What it changes	Value and source
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)
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)
Diagnostic expectations	`br10_diagnostic`	the RE agents' expectations become diagnostic	\(\theta = 0.44\), L'Huillier, Singh and Yoo (2024)
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)
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)
Mean-only learners	`br24_mean_only`	the learners' perceived law of motion keeps only the constant	baseline gain
Experience-based gain	`br26_experience`	the AR(1) learners use a constant gain of 0.018	Malmendier and Nagel (2016), Nagel and Xu (2022)

The transforms leave the expectations of the flexible-price block of the Smets and Wouters (2007) model (SW07) out of their scope, FLEX = {'invef', 'pkf', 'rkf', 'cf', 'labf'}, so the flexible-price economy behind potential output keeps its RE expectations.

The seven rules¶

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';
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

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.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

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';
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

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 (intensity of choice 2, memory 0.8), starting from a third each.

function cfg = br19_target_believers()
cfg = a1_learn();
cfg.id = 'br19_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);
cfg.compare_with = {'a1_learn', 'a1_re'};
end

BR is the code's name for the baseline learners (bounded rationality). 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)).

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 endogenous gain of Kostyshyna, Özden and Zhang (2024). 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.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.

Mean-only learners¶

The learners' inflation perceived law of motion 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 endogenous gain. No new option is needed.

function cfg = br24_mean_only()
cfg = a1_learn();
cfg.id = 'br24_mean_only';
cfg.beliefs.plm.by_var.pinf = struct('regressors', {{}}, 'constant', true);
cfg.compare_with = {'a1_learn', 'a1_re'};
end

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 baseline's endogenous gain. Cohorts are not modeled.

function cfg = br26_experience()
cfg = a1_learn();
cfg.id = 'br26_experience';
cfg.gain.rule = 'constant';
cfg.gain.value = 0.018;
cfg.compare_with = {'a1_learn', 'a1_re'};
end

Running the rules¶

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());.

Results¶

Application 1 and the in-sample fit, from results/br_portfolio/app1_summary.csv and rmse_long.csv. The gap is in percentage points (pp); the root-mean-square error (RMSE) is of annualized inflation, in pp, from the Kalman history (history.mode = 'kalman') over 1966Q1-2026Q2; RE(t-1) is the RE model with t-1 information for inflation.

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
Cognitive discounting	4.09	3.23 (2023Q1)	RE 0.416, BR 0.584	1.120	0.755	−1709.4
Myopia and anchoring	3.42	3.68 (2023Q1)	RE 0.483, BR 0.517	1.209	0.816	−1825.3
Diagnostic expectations	4.75	4.74 (2023Q1)	RE 0.456, BR 0.544	1.325	0.894	−2025.7
Target believers	3.38	2.36 (2022Q4)	RE 0.340, BR 0.491, TB 0.169	1.121	0.756	−1723.2
Intercept-only learners	2.94	0.69 (2022Q4)	RE 0.388, IO 0.612	1.488	1.004	−2192.8
Mean-only learners	3.08	0.89 (2023Q4)	RE 0.710, BR 0.290	1.125	0.759	−1702.6
Experience-based gain	3.06	2.11 (2022Q4)	RE 0.425, BR 0.575	1.247	0.841	−1857.6

The projection facility acts in none of these runs.