Gains¶

The gain sets how much weight the learners put on the newest observation when they update their beliefs. With a gain \(g_t\) and regressors \(z_{t-1}\), recursive least squares is

\[ R_t = R_{t-1} + g_t\left(z_{t-1} z_{t-1}' - R_{t-1}\right), \qquad \phi_t = \phi_{t-1} + g_t R_t^{-1} z_{t-1}\left(v_t - \phi_{t-1}' z_{t-1}\right), \]

equation by equation, each with its own \(R\). A constant gain \(g\) means the learners weight past data with a geometric decay; \(1/g\) is roughly their memory in quarters (0.05: about five years).

Option	Values (default first)	Meaning
`gain.rule`	`'constant'`, `'koz'`	Constant gain, or the endogenous gain of Kostyshyna, Özden and Zhang (2024). The name is a function in `toolbox/+hetlearn/+gain/`. Project baseline: `'koz'`
`gain.value`	`0.0585`	The constant gain (Özden 2025, posterior mean), and the start of `'koz'`. In (0, 1]. Project baseline: 0.05
`gain.by_var.<x>`	number	Gain of one variable: with `'constant'` its gain, with `'koz'` and `'per_variable'` its start value (ignored with a warning under `'common'`)
`gain.koz.mu`	`0.001`	Step size of the gain process; 0 gives the constant gain exactly
`gain.koz.gamma0`	`[]`	Start gain; `[]` = `gain.value`
`gain.koz.Rg0`	`0`	Start value of \(R^g\)
`gain.koz.bounds`	`[0.001 0.25]`	The gain stays inside; a `gain_bound` event each quarter a bound binds
`gain.koz.error`	`'rls'`, `'forecast'`	Signed error that drives the gain: the PLM's one-step residual, or the learners' expectation error \(x_t - F^{BR}_{t-1}\)
`gain.koz.error_scale`	`1`	Factor on that error; 1 = model units (percent per quarter for SW07)
`gain.koz.structure`	`'common'`, `'per_variable'`	One gain process for every PLM equation, or one per equation driven by its own error
`gain.koz.master`	`''`	`'common'` only: the variable whose error drives the gain; `''` = the first learning variable; a list = the mean of their errors, so a quarter in which any of them is missing skips the gain update

The endogenous gain¶

Kostyshyna, Özden and Zhang (2024, eq. 9), the adaptive step-size algorithm of Kostyshyna (2012). After \(x_t\) is observed, with \(e_t\) the signed error (times error_scale) and \(P\) the projection on bounds:

\[ \gamma_t = P\left[\gamma_{t-1} + \mu R^g_{t-1} e_t\right], \qquad R^g_t = (1 - \gamma_{t-1}) R^g_{t-1} + e_t . \]

The RLS update at the end of t uses \(g_t = \gamma_{t-1}\). The gain rises when errors keep the same sign and falls when they alternate. It updates from the observed error whatever the projection facility does with the beliefs.

Things to know:

mu = 0 reproduces the constant gain bit for bit.
mu is scale dependent: the error enters both equations, so mu acts roughly with the square of error_scale. An error_scale of 4 (annualised percent) works like a 16 times larger mu.
With 'koz', res.learning.<x> also holds gain_R (\(R^g_t\)).
KOZ's Table 1 "R0 = 0.001" is the initial RLS moment matrix, not \(R^g_0\). To use it, set beliefs.init.method = 'user' and only beliefs.init.values.<x>.R (example below).

Examples¶

% Constant gain at the toolbox default (the previous project baseline)
cfg = a1_learn(); cfg.id = 'a1_const0585';
cfg.gain.rule = 'constant';
cfg.gain.value = 0.0585;

% Constant gain 0.05, the baseline's start gain (model variant v4_const)
cfg = a1_learn(); cfg.id = 'v4_const';
cfg.gain.rule = 'constant';
cfg.gain.value = 0.05;

% A different constant gain for one variable
cfg = a1_learn(); cfg.id = 'a1_pinf_gain';
cfg.gain.rule = 'constant';
cfg.gain.by_var.pinf = 0.02;

% Endogenous gain with a larger step and the learners' forecast error as the driver
cfg = a1_learn(); cfg.id = 'a1_koz_fast';
cfg.gain.rule = 'koz';
cfg.gain.value = 0.05;
cfg.gain.koz.mu = 0.01;
cfg.gain.koz.error = 'forecast';

% One gain process per PLM equation
cfg = a1_learn(); cfg.id = 'a1_koz_pervar';
cfg.gain.koz.structure = 'per_variable';

% One common process driven by inflation and wage errors
cfg = a1_learn(); cfg.id = 'a1_koz_master';
cfg.expectations.variables = {'pinf', 'w'};
cfg.gain.koz.master = {'pinf', 'w'};

% KOZ's initial moment matrix R0 = 0.001*I (AR(1): two coefficients)
cfg = a1_learn(); cfg.id = 'a1_koz_R0';
cfg.beliefs.init.method = 'user';
cfg.beliefs.init.values.pinf.R = 0.001*eye(2);

Inspect the gain after a run:

res = hetlearn.run(a1_learn(), 'save', false);
g = res.learning.pinf.gain;
g(qq(2022, 2))                       % 0.0499
[min(g(qq(1966,1):qq(2022,2))), max(g(qq(1966,1):qq(2022,2)))]

In the project¶

The baseline KOZ gain starts at 0.05 in 1966Q1 and stays within 0.047-0.052 over 1966Q1-2026Q2 (mean 0.049); it never hits a bound. It is 0.0499 at the 2022Q2 origin and averages 0.0489 over the forecast. Model variant 4 holds it at 0.05:

	`a1_learn` (KOZ)	`v4_const` (0.05)
Average inflation 2022Q3-2026Q2 (%)	3.45	3.50
Learners' share at the end of 2022Q2	0.591	0.595
Largest difference in \(\beta\), 1966-2026		0.012
Largest difference in the share, 1966-2026		0.004

With mu = 0.001 and errors in percent per quarter the gain moves little, so the endogenous and the constant gain give nearly the same results on SW07.