Expectations frictions module

Sticky information and under-reactive expectations:

• The module
  • This module relaxes FIRE and implements a semi-structural version of sticky-information or under-reactive expectations building on Mankiw and Reis (2001, 2003) and recent sequence-space implementations in Auclert et al. (2021), Lenney and Rosso (2026).
  • It uses a version closest to Carroll (2003), which avoids need to keep track of forecast vintages through a recursive formulation. The approximation to canonic sticky expectations by Carroll holds exactly under some technical conditions.
• Stickiness in transmission
  • Sticky informations restricts FIRE expectation formation on the one-period ahead input to a probabilistic event, governed by a time-dependent Calvo-Mankiw-Reis rigidity \(\theta \in (0,1)\). With \(\theta \rightarrow 0\), we revert to FIRE.
  • Each period, a fraction \((1-\theta)\) of agents (e.g. households) freely resets their information and concurrently form new expectations. The remaining fraction with size \(\theta\) sticks to prior average one-period ahead expectations carried from the last period. \[y_t^e = (1-\theta) y_t^{\text{FIRE}} + \theta y^e_{t-1}\]
  • Because average expectations in the economy react less than one-to-one to public news/changes in FIRE expectations, this introduces sluggishness in macroeconomic and policy transmission, allowing to capture and generate humps noted in empirical series (Auclert et al. 2020).

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Simulate dynamics in response to shocks:

• Canonic RANK Model in Pilot Version.

• Examine impact of different degrees of stickiness in households and firms forecats of future inflation.

• Toggle with monetary policy parameters and examine impact via expectations channel.

• Initial configuration with sticky households expectations only calibrated to yield hump-shaped IRF to 25 bp monetary policy shock at set persistence.

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• Technical and theoretical details
  • The average law of motion for the forecast is \[y_t^e = (1-\theta) y_t^{\text{FIRE}} + \theta y^e_{t-1}\]
  • Here \(y_t^{\text{FIRE}}=\mathbb E_t y_{t+1}\), and \(y^e_{t-s} = \mathcal{F}_{t-s} y_{t+1-s}\) with \(\mathcal{F}_t\) the average sticky expectations operator at time \(t\), defined recursively by the above equation.
  • This approximation to truly sticky expectations is exact under the technical assumptions in Carroll (2003). Particularly, where the law of motion of the forecast with truly under-reactive expectations is: \[\mathcal{F}_t y_{t+1} = (1-\theta) y_t^{\text{FIRE}} + \theta \mathcal{F}_{t-1} y_{t+1}\]
  • The above exact implementation requires tracking forecast vintages \(\mathcal{F}_{t-1} y_{t+1}\) for all \(t\). This requires either truncations in the state-space representation or moving to sequence-space methods as in (Auclert et al. 2020; Bardoczy and Guerreiro, 2024; Lenney and Rosso, 2026)
  • The approximation used here swaps \(\mathcal{F}_{t-1} y_{t+1} = \mathcal{F}_{t-1} y_t = y^e_{t-1}\), which holds:
    • Exactly and in a micro-founded sense as Carroll (2003) points out under the assumption agents believe a random-walk model for the forecast variable
    • Approximately for sufficiently persistent process.