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Shares and switching

In mode 'switching' the shares of RE agents and learners move each quarter with their recent forecast performance (Özden 2025, eqs. 14-15).

Option Values (default first) Meaning
switching.rule 'discrete_choice' The switching rule; a function in toolbox/+hetlearn/+switching/
switching.intensity 0.51 Intensity of choice \(\chi\), >= 0. Project baseline: 2
switching.memory 0.6 Memory \(\omega\) of the fitness measure, in [0, 1). Project baseline: 0.8
shares.structure 'per_variable', 'universal' One learners' share per learning variable, or one share for all
shares.master name or cellstr 'universal' only (required there): the variables whose squared errors drive the share
shares.init_n2 0.5 Learners' share at beliefs.init.date
shares.fixed_n2 number, or struct by variable Mode 'fixed_shares' (required there): the learners' share

The rule

For each share group \(G\) (one learning variable, or the master list), with \(F^i_{v,t-1}\) type \(i\)'s forecast of \(v_t\) made in t-1:

\[ FE^i_t = \sum_{v \in G}\left(v_t - F^i_{v,t-1}\right)^2, \qquad U^i_t = \omega U^i_{t-1} - (1-\omega) FE^i_t , \]
\[ n_{1,t} = \frac{\exp(\chi U^{RE}_t)}{\exp(\chi U^{RE}_t) + \exp(\chi U^{BR}_t)}, \qquad n_{2,t} = 1 - n_{1,t}. \]

The shares computed at the end of t are in force in t+1. Initial fitness: \(U^{RE}_0 = 0\) and \(U^{BR}_0\) such that \(n_{2,0}\) = shares.init_n2.

  • \(\chi = 0\) gives equal shares whatever the errors; a large \(\chi\) sends almost everyone to the type with the better recent record.
  • \(\omega\) close to 1 means a long memory of past errors.

Units of the intensity of choice

\(\chi\) multiplies squared forecast errors, so its meaning depends on the units of the model variables. SW07 and the data of Özden (2025) are in percent per quarter, so the estimate 0.51 applies as published. With annualised variables, divide \(\chi\) by 16 for the same switching.

Examples

% Intensity of choice and memory (Ozden 2025, Table 1: chi 0.51, memory 0.62)
cfg = a1_learn(); cfg.id = 'a1_ijcb_switching';
cfg.switching.intensity = 0.51;
cfg.switching.memory = 0.62;
res = hetlearn.run(cfg, 'save', false);

% Start the share recursion at 0.3 learners
cfg = a1_learn(); cfg.id = 'a1_init03';
cfg.shares.init_n2 = 0.3;

% Fixed shares: 30% learners in every quarter
cfg = a1_learn(); cfg.id = 'a1_fixed30';
cfg.expectations.mode = 'fixed_shares';
cfg.shares.fixed_n2 = 0.3;

% Fixed shares that differ by variable
cfg = a1_learn(); cfg.id = 'a1_fixed_byvar';
cfg.expectations.variables = {'pinf', 'w'};
cfg.expectations.mode = 'fixed_shares';
cfg.shares.fixed_n2 = struct('pinf', 0.3, 'w', 0.6);

One share for several variables

By default each learning variable has its own share, driven by its own errors. As in Özden (2025), where one share is driven by the summed squared errors of inflation and the output gap, you can use one share for all learning variables:

cfg = a1_learn(); cfg.id = 'a1_universal';
cfg.expectations.variables = {'pinf', 'w'};
cfg.shares.structure = 'universal';
cfg.shares.master = {'pinf', 'w'};      % squared errors of both drive the one share
res = hetlearn.run(cfg, 'save', false);

Every master variable must be a learning variable (error hetlearn:engine:master otherwise).

Share names in the outputs

Type 1 is the RE agents, type 2 the learners; the shares sum to 1.

Name Where Meaning
n1_<x>, n2_<x> model parameters RE and learners' shares of x's expectation in the model
n1, n2 res.learning.<x> Discrete-choice shares computed at the end of each quarter
n1_model res.learning.<x> RE share actually in the model that quarter (= n1 of the quarter before, unless the projection facility kept the shares)
zetaRE, zetaBR res.learning.<x> Fitness \(U^{RE}\), \(U^{BR}\)
FE_RE, FE_BR res.learning.<x> Squared one-step errors
fRE, fBR res.learning.<x> The one-step forecasts 
res = hetlearn.run(a1_learn(), 'save', false);
L = res.learning.pinf;
L.n2(qq(2022, 2))              % 0.591, computed at the end of 2022Q2
1 - L.n1_model(qq(2022, 3))    % 0.591, in force in 2022Q3

In modes other than 'switching', n1 and n2 are still computed as a diagnostic; the model uses the fixed shares.

With zero future shocks

With no shocks over the forecast, both types' errors shrink and both fitness measures decay toward zero, so the shares drift back toward 0.5 by construction. There is no cost term that would favour one type. In Application 1 the learners' share falls from its peak of 0.685 toward 0.516 by 2026Q2 for this reason.

More than two types

N types of agents with their own rules and shares, shares.init and shares.fixed by type name, as in the target-believers rule: Further expectation rules.