Timing and information

Four options decide what each agent knows when it forecasts, and which estimates of the past the agents learn from.

Option	Values (default first)	Meaning
beliefs.info	'lagged', 'current'	Learners see data up to t-1, or up to t
expectations.re_info	'current', 'lagged'	RE agents use \(E_t x_{t+1}\) or \(E_{t-1} x_{t+1}\) for the learning variables. Project baseline: 'lagged'
history.re_states	'smoothed', 'filtered'	Source of the RE forecasts, the learners' data and the realised values in the fitness errors over the history: the two-sided smoother or the one-sided filter. Project baseline: 'filtered'
history.realtime	false, true	RE filter on data up to the forecast origin only; history.mode = 're_smoother' only (ignored by the inversion and the Kalman history, which are one-sided already)

What each type forecasts in quarter t

With an AR(1) PLM, beliefs \(\phi_{t-1} = (\alpha_{t-1}, \beta_{t-1})\):

	'lagged'	'current'
Learners (beliefs.info)	\(\alpha_{t-1} + \beta_{t-1}(\alpha_{t-1} + \beta_{t-1} x_{t-1})\)	\(\alpha_{t-1} + \beta_{t-1} x_t\), determined jointly with \(x_t\) (Özden 2025, eq. 12)
RE agents (expectations.re_info)	\(E_{t-1} x_{t+1}\)	\(E_t x_{t+1}\)

In both cases the learners use the beliefs formed at the end of t-1, and the belief update is the same. What changes is the forecast, and through the forecast errors, the shares.

The order of events in quarter t

1. The model with beliefs \(\phi_{t-1}\) and shares \(n_{t-1}\) is in force; \(x_t\) is observed (history) or simulated (forecast).
2. Each type's squared error on \(x_t\), of the forecast it made in t-1, updates its fitness; the discrete choice gives the shares \(n_t\).
3. The gain rule gives the gain; RLS updates the beliefs to \(\phi_t\).
4. The projection facility checks the model with \(\phi_t\) and \(n_t\), which will be in force in t+1.

So the shares and beliefs in force in t are always those computed at the end of t-1. The project checks this: in every forecast quarter of a1_learn, the learners' share in the model equals the share computed the quarter before (check C10).

Examples

% Learners see period-t data, as in Ozden (2025)
cfg = a1_learn(); cfg.id = 'a1_info_current';
cfg.beliefs.info = 'current';
res = hetlearn.run(cfg, 'save', false);

% RE agents with period-t information (and the matching RE benchmark)
cfg = a1_learn(); cfg.id = 'a1_reinfo_current';
cfg.expectations.re_info = 'current';
re  = a1_re();    re.id  = 'a1_re_reinfo_current';
re.expectations.re_info = 'current';

Compare a run with its RE benchmark under the same re_info: with 'lagged' the RE benchmark is the RE model with t-1 information for the learning set, not the original model (RE agents).

Filtered or smoothed history

Over the history (from beliefs.init.date to the forecast origin) the agents learn on the RE model's estimates of the states, because most model variables are not observed.

• 'smoothed' uses the Kalman smoother, which estimates \(x_t\) with data after t. The RE agents' forecast made in t then benefits from hindsight, which helps them in the fitness race.
• 'filtered' (the project baseline) uses the one-sided filter \(x_{t|t}\), data up to t only.

For an observed variable the two agree (SW07 inflation outside the covid quarters), so the option mainly moves the RE forecasts and, in quarters with missing data, the learners' data. In either case the forecast itself starts from the smoothed state at the origin.

cfg = a1_learn(); cfg.id = 'a1_smoothed';
cfg.history.re_states = 'smoothed';          % baseline 'filtered'
res = hetlearn.run(cfg, 'save', false);

'filtered' applies to history.mode = 're_smoother' only. The inversion and Kalman histories are one-sided already, and combining them with 'filtered' is an error; their scenarios set 'smoothed'.

Real-time filtering

By default the RE filter runs on the whole sample (to 2026Q2 for SW07), so the state at a 2022Q2 origin is estimated with data after 2022Q2. history.realtime = true clips the data at the origin:

cfg = a1_learn(); cfg.id = 'a1_realtime';
cfg.history.realtime = true;
res = hetlearn.run(cfg, 'save', false);

history.re_states = 'filtered' with history.realtime = true is the fully real-time setup. The forecast evaluation uses real-time filtering by default. Real-time filter results are cached per end date in models/<name>/generated/refilter_rt_<date>_<hash>.mat.

How much does it matter for Application 1? The largest gap between the real-time and the full-sample forecast from 2022Q2, over horizons 1-8, is 0.36 pp for RE inflation and 0.26 pp for learning inflation (annualised).