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Heterogeneous information and New Keynesian forecasts

puremacro.expectations represents dated Gaussian information sets. Different groups receive different signals, retain past observations, update beliefs and forecast linear economic outcomes. models.information_nk applies these operators to a finite present-value New Keynesian system.

The information structure is an explicit processing constraint: agents use only the listed signals. Market prices, income and other groups' observations do not automatically become information. This implementation solves the conditional-forecast system under that restriction. It does not solve endogenous market learning, optimal attention, or a fully specified microeconomic information equilibrium. Its first-order paths do not identify welfare. These distinctions matter when interpreting monetary-policy results.

A dated information experiment

The primitive state contains policy news and two independent group noises. Group A receives a more precise signal than B at date zero. Both observe the policy news exactly when the rate changes at date 8.

import numpy as np
from puremacro.expectations import (
    GaussianSignal, build_gaussian_information, audit_gaussian_information,
)
from puremacro.models import solve_information_nk, audit_information_nk

signals = (
    (GaussianSignal(0, [1., .5, 0.], "initial"),
     GaussianSignal(8, [1., 0., 0.], "reveal")),
    (GaussianSignal(0, [1., 0., 2.], "initial"),
     GaussianSignal(8, [1., 0., 0.], "reveal")),
)
info = build_gaussian_information(
    np.eye(3), signals, horizon=12, weights=[.5, .5],
    group_names=("A", "B"), state_names=("news", "noise_A", "noise_B"),
)
rate_loadings = np.zeros((12, 3))
rate_loadings[8, 0] = -.0025
solution = solve_information_nk(info, rate_loadings)
assert audit_gaussian_information(info)["accepted"]
assert solution.converged and audit_information_nk(solution)["accepted"]
path = solution.evaluate([1., 0., 0.])
print(path.to_frame().head())
print(path.forecast_frame().head())
print(info.belief_frame([1., 0., 0.]).head())

Rates and inflation are per-period deviations. Here a quarterly rate change of −.0025 means −25 quarterly basis points. Output is a log deviation; multiplying by 100 is a local percentage approximation, not an exact level percentage. The realization above sets group noises to zero. In this linear independent-noise example it also equals the conditional mean response given policy news one. It is not the unconditional mean of the zero-mean Gaussian experiment.

What a signal means

Let \(z\sim N(0,\Sigma)\) contain all primitive shocks, including signal noise. A GaussianSignal(date, loading, label) reveals \(h z\) to its group at that date. The function does not add another noise draw. A separate primitive component represents an independent new measurement; reusing the same loading represents the same observation, even if its label or date differs.

Groups share observations internally. A group-specific noise is therefore common within that group and need not average to zero in one realization. The covariance can encode correlation across primitives. Its dimension must match every loading. Group weights are nonnegative and sum to one; zero weights are permitted. Dates are integer values from zero to horizon-1.

Stack each group's released signal rows in \(H_{gt}\). The conditional-mean map and posterior covariance are

\[ P_{gt}=\Sigma H_{gt}'(H_{gt}\Sigma H_{gt}')^+H_{gt},\qquad E[z\mid\mathcal I_{gt}]=P_{gt}z,\qquad V_{gt}=\Sigma-P_{gt}\Sigma. \]

For an outcome \(a z\), info.forecast(a, z, date=t, group=g) returns \(aP_{gt}z\). beliefs returns date × group × primitive conditional means. belief_frame reports means and posterior standard deviations. No observations give a zero mean and the prior covariance; full-state observation gives \(P=I\) and zero conditional variance.

The prior must be positive definite with condition number at most \(10^{12}\). The builder uses Cholesky whitening, observation-row normalization and SVD. Exact redundancy is recorded through the rank. Relative singular values in \([10^{-13},10^{-11}]\) are rejected as ambiguous around the \(10^{-12}\) rank threshold. No jitter is added. Dense projectors and covariance arrays are limited to two million entries each, and the signal count to 4,096.

The independent audit checks ranks, observed signals, idempotence, covariance orthogonality, posterior positivity and nested information. Individual conditional expectations obey the tower property. The average map \(\bar P_t=\sum_gw_gP_{gt}\) generally does not: \(\bar P_t\bar P_s\) can differ from \(\bar P_t\), even when each individual's information is nested.

The finite monetary application

Consumer weights come from info.weights; optional firm_weights can differ. There is one exogenous nominal-rate path \(i_t=R_tz\), zero natural-rate deviations, and an exact zero continuation for output, inflation and rates at every date at or beyond \(T\). Include the complete policy support in the input horizon. The model has no Taylor rule or endogenous policy choice.

Defaults are discount factor \(\beta=.99\), intertemporal elasticity \(s=1\), Calvo persistence \(\theta=.75\), and inverse Frisch elasticity \(\epsilon=1\). The implied Phillips slope is \(\kappa=(1-\theta)(1-\beta\theta)(\epsilon+1/s)/\theta\). All defaults are illustrative assumptions.

The demand and pricing present values follow the distinction in Angeletos and Lian (2018), equations 5–6. The finite boundary and maintained signal sets are explicit restrictions of this implementation; the paper's endogenous market-signal equilibrium is not reproduced. With consumer and firm averages \(\bar E^c_t\) and \(\bar E^f_t\):

\[ y_t=-s\sum_{j=t}^{T-1}\beta^{j-t}\bar E^c_t[i_j-\pi_{j+1}] +(1-\beta)\sum_{j=t+1}^{T-1}\beta^{j-t-1}\bar E^c_t[y_j], \]

\[ \pi_t=\kappa y_t+\sum_{j=t+1}^{T-1}(\beta\theta)^{j-t} \left(\kappa\bar E^f_t[y_j]+\frac{1-\theta}{\theta}\bar E^f_t[\pi_j]\right). \]

The code solves outcome loadings backwards with discounted-sum accumulators. audit_information_nk reconstructs the explicit sums for every primitive direction, independently of those accumulators. Acceptance requires information and coefficient errors at most \(10^{-9}\) by default. The result retains coefficient arrays, parameters, information and model fingerprints, plus diagnostics. evaluate returns realized paths and group forecasts of next-date output, inflation and the nominal rate; final next-date forecasts are zero. For a longer forecast horizon, apply info.forecast to the desired date's outcome loading. Tables support Markdown, LaTeX and Typst.

Under full information the equations reduce to the textbook recursions \(y_t=y_{t+1}-s(i_t-\pi_{t+1})\) and \(\pi_t=\kappa y_t+\beta\pi_{t+1}\). Tests check them directly. Under dispersed information, replacing the textbook Euler expectation with an average forecast generally gives a different model. This is why the implementation keeps the present values. Individual forecasts are Bayesian under the declared signals; conditional forecast errors in a particular realization need not be zero.

Reproduce and interpret the illustration

python -m puremacro.examples.heterogeneous_expectations --output expectations-example --lang en

The example compares full information, no advance news, heterogeneous initial signals, and a second independent signal at date 4. All groups learn the policy news at implementation. It exports coefficients, paths, beliefs, forecasts, audits, complete array files, parameters, source hashes and PNG/PDF figures. A seeded 50,000-draw experiment checks analytic posterior variances, zero unconditional forecast errors and orthogonality to observed signals, with Monte Carlo uncertainty reported and fixed acceptance criteria.

For a date-8 rate cut, initial output responses (log deviation × 100) are approximately 3.9586 under full information, 0 without advance news, 0.8028 with heterogeneous signals and 1.1838 with the second update. The initial average policy forecast in the last two cases is half the full-information forecast. The figure therefore also reports the response ratio divided by the policy-forecast ratio, separating first-order news attenuation from subsequent expectation feedback. This ratio is undefined when advance news is absent.

The horizon sweep retains large responses: at date 16 the full-information linear output response exceeds one log unit. These values are sensitivities of the linear system, not credible quantitative forecasts at that shock size. The CSV flags absolute output/inflation deviations above .05; that flag is descriptive, not a bound on approximation error. Zero-tail padding preserves earlier coefficients exactly up to numerical precision, but does not establish infinite-horizon determinacy. No empirical calibration, policy recommendation, welfare, or HANK integration is claimed by this first implementation.