## 2.1–4.4 Excerpts: Monetary Policy Surprises; Dispersion of Inflation Expectations; Role of Market Surprises; Constant and State-dependent Information Rigidity

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### 2.1 Monetary Policy Surprises — Data, construction, and patterns
- Data and timeline
  - Daily inflation and federal funds rate forecasts for the United States at the analyst level between November 18, 2002 and December 18, 2018 obtained from Bloomberg.
  - Forecast panel includes 496 analysts; panel is unbalanced.
  - Analysts submit federal funds rate forecast p^e_{t−1+γ|t−1} during [t−1, t−1+γ); realized decision p^r_{t−1+γ|t−1+γ} announced at t−1+γ. Inflation forecasts π^e_{t+h|t} submitted during (t−1+γ, t] for h = {−1,3,6,9,12}; actual inflation π^r_{t+h|t+h+θ} released at t+h+θ.
  - Inflation horizon −1 is a backcast; releases occur within 15 days after the end of the month.
  - Analysts can revise and re-submit estimates at any time; on average analysts submit their latest federal funds rate forecast 9 days before the meeting.
- Construction of individual monetary policy surprises
  - Individual surprise: s^i_{e i,t−1+γ|t−1} = || p^e_{i,t−1+γ|t−1} − p^r_{t−1+γ|t−1+γ} || (Equation (1)).
  - Pre-meeting window set to a maximum of 45 days (broadly corresponds to days between federal funds rate announcements).
- Empirical patterns and caveats
  - Average surprises: small and frequent between 2002 and 2007; larger leading up to the global financial crisis.
  - Post-crisis accommodative policy and proximity to the zero lower bound made Fed decisions more predictable.
  - Since 2015 average surprises increased as expectations of normalization proved premature.
  - Identification assumption: information available at the beginning of the 45-day window is incorporated in forecasts; validity requires analysts re-submit forecasts or that new information during 45 days is not systematically correlated with forecasts or the decision.
  - Surprise computed in absolute value to study whether surprises—regardless of sign—lead to increased dispersion.
- Implication
  - Analyst-level monetary policy surprises concentrated around the run-up to the global financial crisis, during the crisis, and post-2015; used to identify causal effects on dispersion of inflation expectations measured from analysts’ post-meeting inflation projections.

### 2.2 Dispersion of Inflation Expectations — Measure, patterns, and baseline empirical strategy
- Construction of dispersion
  - Dispersion for analyst i at horizon h: dπe i,t+h|t = ∣∣∣πe i,t+h|t − 1/(N−1) (Σi=1^N πe i,t+h|t − πe i,t+h|t)∣∣∣.
  - Post-meeting window maximum: 30 days in baseline; absolute deviations used so positive and negative deviations do not cancel.
- Descriptive statistics
  - Standard deviation of dispersion:
    - h = −1 (past): 27 basis points.
    - h = 12: 50 basis points.
  - Cross-country median aggregated monthly:
    - Median dispersion h = −1: average of the median over time = 0.17 percentage points.
    - Median dispersion h = 12: average = 0.35 percentage points.
  - Sample sizes across horizons vary between 3,981 observations at h = −1 and 5,594 observations at h = 12.
- Empirical strategy and identification
  - Panel regression: dπe i,t+h|t = β s^i_{e i,t−1+γ|t−1} + φ_i + μ_i,t+h.
  - Surprise variable lagged by one period so surprise precedes inflation forecast even for h = −1.
  - Baseline post-meeting window maximum 30 days; average submission occurs 16 days after the announcement in resulting sample.
  - Robustness: post-meeting windows shortened to 25, 20, 15, 10 days; pre-meeting windows ceilings set to 30, 20, 10, 5 days; narrowest pre/post windows used in robustness: 5 and 10 days, producing average pre = 2.5 days and post = 5.9 days.
  - Time fixed effects not included to avoid absorbing shock variation.
  - Interpretation: h = −1 isolates informational effects revealed by Fed actions on the unpublished prior-month inflation; h ∈ {3,6,9,12} reflect revealed information plus revised expectations about future Fed actions.
- Baseline empirical results (effects per 100-basis point surprise; Table 2)
  - h = −1 (Past):
    - Baseline estimate: 0.965*** (0.126).
    - Observations: 2,263; Analysts: 127; R-squared: 0.061.
    - Interpretation: A 100-basis point surprise leads, on average, to an increase of 96 basis points in dispersion for the month that just finished.
  - h = 3 months:
    - Baseline estimate: 0.536** (0.223).
    - Observations: 1,390; Analysts: 126; R-squared: 0.008.
  - h = 6 months:
    - Baseline estimate: 0.688*** (0.242).
    - Observations: 1,356; Analysts: 160; R-squared: 0.011.
  - h = 9 months:
    - Baseline estimate: 0.611** (0.264).
    - Observations: 1,294; Analysts: 58; R-squared: 0.008.
  - h = 12 months:
    - Baseline estimate: 0.061 (0.221) — not statistically different from zero.
    - Observations: 1,175; R-squared: 0.000.
- Exclusion robustness (effects on h = −1, per 100-basis point surprise)
  - Excluding Global Financial Crisis (July 2008–December 2009): 1.046*** (0.174); Observations: 2,166; Analysts: 125; R-squared: 0.057.
  - Excluding zero lower bound period (January 2010–December 2014): 0.946*** (0.123); Observations: 1,538; Analysts: 118; R-squared: 0.070.
  - Excluding monetary policy normalization: 1.159*** (0.164); Observations: 2,008; Analysts: 124; R-squared: 0.073.
- Summary interpretation
  - Monetary policy surprises increase analysts’ disagreement about inflation prospects up to nine months after announcement; no effect at the one-year horizon. Effect strongest at the backward-looking horizon h = −1 (informational effect).

### 2.2 Robustness: narrower time windows and high-frequency surprise measures
- Narrower windows
  - Pre-meeting windows reduced to as low as 5 days; post-meeting windows reduced to as low as 10 days.
  - Backward-looking horizon (h = −1): point estimates always statistically significant, ranging between 48 and 110 basis points across window choices.
  - h = 3 and h = 6: estimates remain positive and significant for most specifications or indistinguishable from zero.
  - h = 9 and h = 12: estimates can turn negative and statistically significant when pre-meeting window = 5 days and post-meeting window = 15 or 10 days.
  - Overall: surprises generally increase dispersion at short horizons and have no effect or reduce dispersion at longer horizons.
- High-frequency (Nakamura and Steinsson (2018)) measures
  - Federal funds rate shocks: unexpected changes in a 30-minute window around announcements.
  - Policy news shocks: first principal component of unanticipated changes in several short- and medium-term interest rate measures immediately following meetings.
  - Trade-off: minimize confounding factors but lack cross-sectional variation (same surprise across analysts).

### High-frequency shocks results (Table 3; effects per 100-basis point shock)
- Federal funds rate shock (top panel)
  - h = −1 (Past): 1.567*** (0.250).
  - h = 3: 1.994*** (0.605).
  - h = 6: 2.201*** (0.766).
  - h = 9: 2.524*** (0.719).
  - h = 12: 1.865*** (0.675).
  - Observations by horizon: 2,088; 1,308; 1,278; 1,223; 1,115.
  - Analysts by horizon: 106; 50; 50; 50; 49.
  - R-squared by horizon: 0.024; 0.010; 0.011; 0.011; 0.006.
  - Interpretation: A 100-basis point federal funds rate shock leads to a 157 basis points increase in dispersion for h = −1; effects remain statistically significant and can be larger at forward horizons (largest at h = 9: 220 basis points).
- Policy news shock (bottom panel)
  - h = −1: 1.335*** (0.253).
  - h = 3: 2.258* (1.148).
  - h = 6: 2.578*** (0.790).
  - h = 9: 3.148*** (1.140).
  - h = 12: 1.888 (1.347) — not statistically distinguishable from zero.
  - Interpretation: Policy news shocks produce somewhat larger estimates than federal funds rate shocks for most horizons, possibly reflecting forward guidance effects, except for h = −1 (smaller by 24 basis points) and h = 12 (not statistically significant).

### 3.3 The Role of Market Surprises — Specification and findings
- Specification and market surprise measurement
  - Augmented regression: dπe i,t+h|t = β s^i_{pe i,t−1+γ|t−1} + θ_m s^m_{pe i,t−1+γ|t−1} + φ_i + μ_i,t+h.
  - Market surprises: ms^pe_{i,t−1+γ|t−1} = ∣∣∣ 1/(N−1) (∑_{i=1}^N p^e_{i,t−1+γ|t−1} − p^e_{i,t−1+γ|t−1}) − p^r_{t−1+γ|t−1} ∣∣∣ with γ <45 days. Captures absolute deviation of the consensus forecast (excluding the individual’s forecast) from the actual federal funds rate.
- Empirical findings (baseline augmentation; coefficients per 100-basis point surprise)
  - h = −1 (Past; interpreted as past 3 months ahead in tables)
    - Individual surprises coefficient: 0.526*** (0.124).
    - Market surprises coefficient: 0.680*** (0.104).
    - Interpretation: A 100-basis point individual (market) surprise leads to an increase in inflation expectation dispersion by 53 (68) basis points.
  - Other horizons (6, 9, 12 months): individual surprise coefficients generally positive but not always significant; market surprises show mixed significance:
    - h = 9: market surprises 0.524* (0.298).
    - h = 12: market surprises 0.723* (0.381).
  - Table 4 summary:
    - Observations: 2,262; 1,347; 1,356; 1,294; 1,175 (Past/3/6/9/12).
    - Analysts: 126; 61; 61; 60; 58.
    - R-squared: 0.094; 0.013; 0.018; 0.016; 0.012.
    - Wald test p-values: 0.407; 0.913; 0.794; 0.569; 0.100.
- Robustness: NS-based market surprises (Table 5)
  - Federal funds rate shock proxy (top panel)
    - Past: Individual surprises 0.887*** (0.145); Federal funds rate shock 0.656** (0.257).
    - h = 12: Federal funds rate shock 2.161** (0.916).
    - Observations: 2,088; 1,308; 1,278; 1,223; 1,115.
    - Analysts: 106; 50; 50; 50; 49.
    - R-squared: 0.066; 0.013; 0.015; 0.014; 0.007.
  - Policy news shock proxy (bottom panel)
    - Past: Individual surprise 0.918*** (0.147); Policy news shock 0.171 (0.281).
    - h = 9: Policy news shock 1.811* (1.025).
    - Observations and analysts same as top panel.
    - R-squared: 0.062; 0.011; 0.013; 0.013; 0.003.
  - Interpretation
    - When market surprises proxied by federal funds rate shocks, both individual and market surprises raise dispersion.
    - When proxied by policy news shocks, individual surprises are significant at short horizons while policy news shocks are less consistently significant.
    - Conclusion: Individual surprises unequivocally lead to larger disagreement about inflation projections above and beyond market surprises.

### 4.2 Constant Degree of Information Rigidity — Theoretical setup, aggregation, and quantitative findings
- Setup and firm optimization
  - Information rigidity parameter 0<λ<1; λ = 0 corresponds to full information; λ = 1 to firms that never change expectations.
  - Fraction λ of firms does not update information and keeps P^j_t = P^j_{t−1}; fraction 1−λ updates.
  - Firms choose price P^j_t to maximize E_t ∑_{τ=0}^∞ λ^τ R_{t+τ|t} (P^j_t Y^j_{t+τ} − mc^j_{t+τ} Y^j_{t+τ}) (equation (22)).
  - Intertemporal discount factor: R_{t+τ|t} = β^τ E_t C_{t+τ}^{−σ} C_t^{σ} (equation (25)); utility u(C_t)=c^{1−σ}/(1−σ).
  - Optimal reset price expressions in equations (23), (24), (26).
- Aggregation and price-level dynamics
  - Aggregate reset price P^*_t given by equation (27).
  - Aggregate price level: P_t = λ P_{t−1} + (1−λ) P^*_t (equation (28)).
  - Iterated expected price: E_t[P_{t+1}] = λ P_t + (1−λ) E_t[P^*_{t+1}] (equation (29)); further expressions in (30)–(33).
- Forecast behavior and dispersion
  - Non-updaters produce constant price forecasts: E_t P^{−*}_{t+τ} = P_{t−1}; their inflation forecast π^{e,−*}_{t+τ} = 0 for all τ.
  - Updaters produce π^{e,*}_{t+τ}; dispersion at horizon t+τ: d_{t+τ|t} = λ |π^{e,*}_{t+τ}| (equation (34)).
  - Dispersion moves with |π^{e,*}_{t+τ}|; both expected inflation and dispersion are state contingent.
- Numerical experiments and calibration
  - Baseline λ = 0.78 (about 22 percent of firms change prices in any given period).
  - Simulation procedure: simulate 500 time series with 3,000 periods each; discard first 1,500 periods; extract 12-month sequences.
  - Average absolute interest rate shock |i_t − i_{t−1}| = 77 basis points in the simulated calibration and measure average response of inflation expectation dispersion.
- Simulation findings
  - Under i.i.d. aggregate state: updaters’ price expectations across horizons nearly identical; dispersion increases mainly at t and t+1 due to difference between updaters and sticky firms.
  - Under persistent aggregate state (example ρ_z = 0.9 and ρ_i = 0.5): π^{e,*}_{t+τ} changes slowly and can increase for some horizons before moving to zero.
  - Model-implied dispersion decreases with horizon, consistent with empirical pattern, but model magnitude is smaller than observed (Figure 8).
  - Varying λ (tested λ = 0.92, λ = 0.50, λ = 0.08): magnitude of dispersion falls with λ in the tested calibration; for λ = 0 and λ = 1 there is no dispersion.
- Mechanisms and conclusion
  - Two opposing effects of larger λ on dispersion at a given horizon:
    - Consensus forecast bias toward zero inflation (increasing distance between consensus and updaters).
    - Updaters weight future more heavily, stabilizing prices and dampening dispersion.
  - Net effect ambiguous analytically; numerical methods required.
  - Conclusion: constant-information-stickiness models qualitatively match time path of dispersion but cannot generate the magnitude observed in the data for λ values consistent with literature.

### 4.4 Firm State-dependent Degree of Information Rigidity — Setup, aggregation, and policy implications
- Setup and intuition
  - Degree of information stickiness depends on firms’ idiosyncratic states θ_jt.
  - For θ_jt above threshold ̃θ, information stickiness is lower than constant λ and decreases with distance between realized shock and mean.
  - Firms with θ_jt > ̃θ are in "extreme circumstances" and place relatively more weight on current conditions (lower stickiness).
  - Share of firms with lower λ_t+τ is constant in distribution, but the share in extreme conditions in a period affects the average price level.
- Optimal price for updating firms
  - Optimal price expression includes λ_j(t+τ) = λ_z(∆θ_j(t+τ)); functional form used: λ_jt = ̄λ/[ (1 + ψ_3 |θ_jt − E[θ]|) ] with ψ_3 = 3.
- Aggregation and decomposition
  - Price index decomposed into sub-indexes for:
    - Moderate firms (θ ≤ ̃θ): weight μ_{̃θ}; within these, share 1−λ change prices and share λ keep previous price.
    - Extreme firms (θ > ̃θ): weight 1−μ_{̃θ}; within these, share 1−λ^* E[z(∆θ_jt) | (θ_j > ̃θ)] change prices and share λ^* E[z(∆θ_jt) | (θ_j > ̃θ)] keep previous price.
  - Price index equation:
    - P_t = λ ( μ_{̃θ} + (1−μ_{̃θ}) E[z(∆θ_jt) | (θ_j > ̃θ)] ) P_{t−1}
      + μ_{̃θ} (1−λ) P^∗_{̃θ,t}
      + (1−μ_{̃θ}) (1−λ ̃z) P^∗_{−̃θ,t}
    - where ̃z = E[z(∆θ_jt) | (θ_j > ̃θ)] and concavity implies P^∗_{̃θ,t} > P^∗_{−̃θ,t}.
- Forecasts and dispersion
  - Forecasters processing new information have Et[P_{t+1}] and Et[P_{t+τ}] given by forward iterations combining the weighted contributions of sub-index optimal prices.
  - Updating firms’ inflation forecast at t+τ: π^{∗,e}_{t+τ} equals the ratio of the weighted expected price terms to Et[P_{t+τ−1}] and includes the term λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ).
  - Non-updaters forecast zero inflation; dispersion:
    - d_{t+τ|t} = λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ) || π^{e,∗}_{t+τ} ||.
- Comparative statics and quantitative implications
  - Lower threshold ̃θ:
    - Increases weight of Et[P^∗_{−̃θ,t+τ}]/Et[P_{t+τ−1}] in π^{e,∗}_{t+τ}, raising |π^{e,∗}_{t+τ}| and thus dispersion.
    - Hence lower ̃θ (more firms with lower stickiness) amplifies firm heterogeneity effects on aggregate variables and dispersion.
  - Heterogeneity affects magnitude but not term structure of inflation expectations because distribution of idiosyncratic shocks is stationary.
  - Quantitative results (figures and simulations):
    - Heterogeneity in information stickiness across firms can increase the magnitude of dispersion substantially — about four times larger than in other model cases and about half of what is observed in the data for some aggregate states away from their mean.
- Policy implications
  - Firm-level state-dependent information rigidity can materially increase short-horizon dispersion of inflation expectations.
  - Since short-horizon dispersion can lead to price dispersion and inflation persistence, monetary policy communication should be refined to better align information across agents.
  - Better-aligned information can help keep inflation expectations anchored at shorter horizons, when the credit channel is less effective.

*Source: wpiea2020252-print-pdf — sections 2.1, 2.2, 3.3, 4.2, and 4.4 (authors’ calculations and IMF working paper excerpts).*

### 2.1    Monetary Policy Surprises  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 

### 2.1    Monetary Policy Surprises

### Data and timeline
- Daily inflation and federal funds rate forecasts for the United States at the analyst level between November 18, 2002 and December 18, 2018 obtained from Bloomberg.
- The dataset includes forecasts of 496 analysts from major financial institutions worldwide; the panel is unbalanced.
- Analysts submit a federal funds rate forecast p^e_{t−1+γ|t−1} during [t−1, t−1+γ). The realized federal funds rate decision p^r_{t−1+γ|t−1+γ} is announced at t−1+γ. Analysts then submit inflation forecasts π^e_{t+h|t} during (t−1+γ, t] for horizons h = {−,3,6,9,12}. Actual inflation π^r_{t+h|t+h+θ} is released at t+h+θ.
- For inflation horizon − the analyst submits a backcast; for h = {3,6,9,12} the analyst submits a forecast. Inflation releases occur within 15 days after the end of the month for which the forecast is made.
- Analysts can revise and re-submit estimates at any time.

### Construction of individual monetary policy surprises
- Monetary policy surprises at the individual level are computed as the absolute deviation of an analyst’s expected federal funds rate from the actual rate announced at the policy meeting:
  - s^i_{e i,t−1+γ|t−1} = || p^e_{i,t−1+γ|t−1} − p^r_{t−1+γ|t−1+γ} ||
  - (Equation labelled (1) in source.)
- The pre-meeting window (time between the analyst’s latest forecast at t−1 and the announcement at t−1+γ) is set to a maximum of 45 days. This broadly corresponds to the number of days between federal funds rate announcements.
- On average analysts submit their latest federal funds rate forecast 9 days before the meeting.

### Empirical patterns and descriptive observations
- Average surprises were quite frequent but small between 2002 and 2007; they became larger in the period leading up to the global financial crisis.
- After the crisis, accommodative policy and proximity to the zero lower bound made Fed decisions more predictable.
- Since 2015 average surprises increased again as agents expected an acceleration in normalization of monetary policy that turned out to be slower.
- Figure reported: monthly average of federal funds rate surprises across analysts (labelled “Monetary Policy Surprises (Percentage points)”).

### Identification assumptions and caveats
- The identification assumption: information available at the beginning of the 45-day window is incorporated in the analyst’s forecast and does not show up as spurious variation in the shock variable.
- New information arriving during the 45 days could make an analyst’s forecast outdated; validity requires analysts re-submit forecasts or that new information is not systematically correlated with the forecast or the federal funds rate decision.
- The surprise is computed in absolute value because the analysis asks whether monetary surprises—regardless of sign—lead to an increase in dispersion of analysts’ forecasts.

### Implications highlighted in this section
- Monetary policy surprises concentrated in the run-up to the global financial crisis, during the crisis, and post-2015 when expectations of normalization proved premature.
- These analyst-level surprises are used to identify causal effects of monetary policy surprises on dispersion of inflation expectations measured from analysts’ inflation projections produced right after Fed meetings.

*Source: wpiea2020252-print-pdf - 2.1    Monetary Policy Surprises*

### 2.2    Dispersion of Inflation Expectations

### 2.2    Dispersion of Inflation Expectations

### Construction of the dispersion measure
- Dispersion for analyst i at horizon h is defined as the absolute deviation of the individual’s inflation forecast from the average forecast across all other analysts:
  - dπe i,t+h|t = ∣∣∣πe i,t+h|t − 1/(N−1) (Σi=1^N πe i,t+h|t − πe i,t+h|t)∣∣∣
- Post-meeting window (time between the federal funds rate announcement, t−1+γ, and the analyst’s inflation forecast submission, t) is set to a maximum of 30 days in baseline analyses.
- Absolute values are used so that positive and negative deviations occurring simultaneously do not cancel out.

### Descriptive patterns and key statistics
- Cumulative distribution and box plots (Figure 3) for the United States:
  - Shorter horizons show higher frequency around zero; uncertainty increases with horizon length.
  - Standard deviation of inflation expectation dispersion:
    - For the month that just finished (h = −1 [past]), 27 basis points.
    - For h = 12 months, 50 basis points.
- Cross-country median aggregated monthly (Figure 4):
  - Median dispersion hovers between 0 and 0.5 percentage points for most of the sample across horizons.
  - Average of the median dispersion over time:
    - h = −1: 0.17 percentage points.
    - h = 12: 0.35 percentage points.
  - Dispersion spikes around the global financial crisis (2008–09) and at the taper tantrum announcement (May 2013).
  - Interquartile range around the median is wider for larger horizons and around peaks of median dispersion.
- Sample sizes across horizons (note on analyst sample):
  - Number of observations varies between 3,981 at h = −1 and 5,594 at h = 12.
  - Reducing the sample to analysts with forecasts for all horizons does not affect descriptive statistics.

### Empirical strategy and identification
- Panel regression estimated to assess impact of analysts’ surprises about policy rate decisions (si e i,t−1+γ|t−1) on inflation expectation dispersion (dπe i,t+h|t):
  - dπe i,t+h|t = β si e i,t−1+γ|t−1 + φi + μi,t+h
  - φi: analyst fixed effects; μi,t+h: i.i.d. error term.
- Surprise variable is lagged by one period so the surprise precedes the inflation forecast even for h = −1.
- Identification relies on narrow windows around Fed meetings:
  - Post-meeting window baseline maximum: 30 days; average submission occurs 16 days after the federal funds rate announcement in resulting sample.
  - Robustness checks shorten post-meeting windows to 25, 20, 15, and 10 days.
  - Pre-meeting window ceilings set to 30, 20, 10, and 5 days in alternative surprise constructions.
  - Narrowest pre- and post-meeting windows used in robustness: 5 and 10 days, respectively; average time spans then are 2.5 days (pre) and 5.9 days (post).
- Time fixed effects are not included because common events affecting all forecasters should be reflected in the latest pre-meeting federal funds forecast; including time effects could absorb shock variation.
- Interpretation distinction:
  - Horizon h = −1 (backward-looking) isolates informational effects revealed by Fed actions on the unpublished inflation of the preceding month.
  - Horizons h ∈ {3,6,9,12} are forward-looking and reflect both revealed information and revised expectations about future Fed actions.

### Baseline empirical results (Table 2; effects per 100-basis point surprise)
- Column (1) — Past (h = −1):
  - Baseline estimate: 0.965*** (0.126)
  - Observations: 2,263
  - Analysts: 127
  - R-squared: 0.061
  - Interpretation: A 100-basis point surprise leads, on average, to an increase of 96 basis points in dispersion for the month that just finished.
- Column (2) — 3 months ahead:
  - Baseline estimate: 0.536** (0.223)
  - Observations: 1,390
  - Analysts: 126
  - R-squared: 0.008
- Column (3) — 6 months ahead:
  - Baseline estimate: 0.688*** (0.242)
  - Observations: 1,356
  - Analysts: 160
  - R-squared: 0.011
- Column (4) — 9 months ahead:
  - Baseline estimate: 0.611** (0.264)
  - Observations: 1,294
  - Analysts: 58
  - R-squared: 0.008
- Column (5) — 12 months ahead:
  - Baseline estimate: 0.061 (0.221) — not statistically different from zero
  - Observations: 1,175
  - Analysts: (implicit as above)
  - R-squared: 0.000
- Exclusion robustness (GFC, zero lower bound, monetary policy normalization):
  - Excluding Global Financial Crisis (July 2008–December 2009):
    - Past: 1.046*** (0.174); Observations: 2,166; Analysts: 125; R-squared: 0.057
  - Excluding zero lower bound period (January 2010–December 2014):
    - Past: 0.946*** (0.123); Observations: 1,538; Analysts: 118; R-squared: 0.070
  - Excluding monetary policy normalization:
    - Past: 1.159*** (0.164); Observations: 2,008; Analysts: 124; R-squared: 0.073
- Summary interpretation:
  - Monetary policy surprises increase analysts’ disagreement about inflation prospects up to nine months after the announcement; no effect at the one-year horizon.
  - Effect strongest for the backward-looking horizon (h = −1), interpreted as the informational effect of the Fed decision.

### Robustness: narrower time windows
- Concerns addressed:
  - Analysts may not re-submit forecasts even if they receive new information, biasing measured surprises.
  - Analysts may delay resubmission after the Fed meeting; longer post-meeting windows may capture other developments.
- Time windows varied:
  - Pre-meeting windows reduced from maximum 45 days to 5 days.
  - Post-meeting windows reduced from maximum 30 days to 10 days.
- Findings (Figure 6 summary):
  - Backward-looking horizon (h = −1): point estimates always statistically significant, ranging between 48 and 110 basis points across window choices.
  - Horizons h = 3 and h = 6: estimates remain positive and significant for most specifications or indistinguishable from zero.
  - Horizons h = 9 and h = 12: estimates can turn negative and statistically significant when pre-meeting window is 5 days and post-meeting window is 15 or 10 days.
  - Overall: monetary policy surprises generally increase dispersion at short horizons and have no effect or reduce dispersion at longer horizons.
- Additional robustness using high-frequency surprise measures from Nakamura and Steinsson (2018):
  - Federal funds rate shocks: unexpected changes in a 30-minute window around announcements.
  - Policy news shocks: first principal component of unanticipated changes in several short- and medium-term interest rate measures immediately following meetings.
  - Trade-off: these aggregate high-frequency measures minimize confounding factors but lack cross-sectional variation (same surprise imposed across analysts).

### High-frequency shocks results (Table 3; effects per 100-basis point shock)
- Federal funds rate shock (top panel):
  - Past (h = −1): 1.567*** (0.250)
  - 3 months ahead: 1.994*** (0.605)
  - 6 months ahead: 2.201*** (0.766)
  - 9 months ahead: 2.524*** (0.719)
  - 12 months ahead: 1.865*** (0.675)
  - Observations by horizon: 2,088; 1,308; 1,278; 1,223; 1,115
  - Analysts by horizon: 106; 50; 50; 50; 49
  - R-squared by horizon: 0.024; 0.010; 0.011; 0.011; 0.006
  - Interpretation: A 100-basis point federal funds rate shock leads to a 157 basis points increase in dispersion for h = −1; effects remain statistically significant and are larger at some forward horizons (largest at h = 9: 220 basis points).
- Policy news shock (bottom panel):
  - Past (h = −1): 1.335*** (0.253)
  - 3 months ahead: 2.258* (1.148)
  - 6 months ahead: 2.578*** (0.790)
  - 9 months ahead: 3.148*** (1.140)
  - 12 months ahead: 1.888 (1.347) — not statistically distinguishable from zero
  - Observations and analysts identical to federal funds rate shock rows.
  - R-squared by horizon: 0.009; 0.005; 0.006; 0.008; 0.003
  - Interpretation: Policy news shocks produce somewhat larger estimates than federal funds rate shocks for most horizons, possibly reflecting forward guidance effects, except for h = −1 (smaller by 24 basis points) and h = 12 (not statistically significant).

*Source: Bloomberg and authors’ calculations.*

### 3.3    The Role of Market Surprises

### 3.3    The Role of Market Surprises

### Specification and measurement
- Augmented baseline regression (equation (4)):
  - dπe i,t+h|t = βs pe i,t−1+γ|t−1 + θm s pe i,t−1+γ|t−1 + φi + μi,t+h
- Market surprises definition (equation (5)):
  - ms pe i,t−1+γ|t−1 = ∣∣∣∣∣ 1 N−1 (∑N i=1 pe i,t−1+γ|t−1 − pe i,t−1+γ|t−1) − pr t−1+γ|t−1 ∣∣∣∣∣
  - with γ <45 days
- Interpretation: market surprises capture the absolute deviation of the consensus forecast (mean across all analysts excluding the individual’s forecast over the 30 days prior to the announcement day) from the actual federal funds rate.

### Empirical findings (baseline augmentation)
- At horizon h = − (past 3 months ahead):
  - Individual surprises coefficient: 0.526*** (standard error 0.124)
  - Market surprises coefficient: 0.680*** (standard error 0.104)
  - Interpretation: A 100-basis point individual (market) surprise in the federal funds rate leads to a higher inflation expectation dispersion by 53 (68) basis points.
- Other horizons (6, 9, 12 months ahead):
  - Coefficients on individual surprises are not always significant but are generally positive.
  - Market surprises show mixed significance; some coefficients are significant:
    - At 9 months: market surprises 0.524* (0.298)
    - At 12 months: market surprises 0.723* (0.381)
- Table 4 summary statistics:
  - Observations: 2,262; 1,347; 1,356; 1,294; 1,175 (for columns Past/3/6/9/12 months ahead respectively)
  - Analysts: 126; 61; 61; 60; 58
  - R-squared: 0.094; 0.013; 0.018; 0.016; 0.012
  - Wald test p-values: 0.407; 0.913; 0.794; 0.569; 0.100
- Notes:
  - All regressions include analyst fixed effects.
  - Heteroskedasticity and autocorrelation consistent standard errors in parentheses.
  - Significance codes: ***p <0.01, **p <0.05, *p <0.1.

### Robustness: NS-based market surprises (Nakamura and Steinsson (2018) measures)
- Two proxies considered: federal funds rate shocks; policy news shocks.
- Results (Table 5, top panel — federal funds rate shock proxy):
  - Past (3 months) ahead:
    - Individual surprises: 0.887*** (0.145)
    - Federal funds rate shock: 0.656** (0.257)
  - 6 months ahead:
    - Federal funds rate shock: 1.483 (0.923)
  - 9 months ahead:
    - Federal funds rate shock: 1.554 (1.088)
  - 12 months ahead:
    - Federal funds rate shock: 2.161** (0.916)
  - Wald test p-values: 0.495; 0.345; 0.429; 0.266; 0.044
  - Observations: 2,088; 1,308; 1,278; 1,223; 1,115
  - Analysts: 106; 50; 50; 50; 49
  - R-squared: 0.066; 0.013; 0.015; 0.014; 0.007
- Results (Table 5, bottom panel — policy news shock proxy):
  - Past (3 months) ahead:
    - Individual surprise: 0.918*** (0.147)
    - Policy news shock: 0.171 (0.281)
  - 6 months ahead:
    - Individual surprise: 0.473* (0.270)
    - Policy news shock: 1.549 (1.466)
  - 9 months ahead:
    - Individual surprise: 0.590** (0.278)
    - Policy news shock: 1.811* (1.025)
  - 12 months ahead:
    - Individual surprise: 0.511* (0.298)
    - Policy news shock: 2.428* (1.438)
  - Wald test p-values: 0.018; 0.506; 0.331; 0.264; 0.278
  - Observations and analysts same as top panel
  - R-squared: 0.062; 0.011; 0.013; 0.013; 0.003
- Interpretation:
  - When market surprises are proxied by federal funds rate shocks, both individual and market surprises lead to higher inflation expectation dispersion.
  - When market surprises are proxied by policy news shocks, only individual surprises are statistically significant at short horizons.
  - Magnitude of the coefficient on individual surprises is similar to the baseline specification.
  - At intermediate horizons, coefficients are generally positive but often not significant.
  - At h = 12, the only significant coefficient is on the federal funds rate shock; coefficients on individual surprises turn negative (but not significant).
- Conclusion drawn in text:
  - Individual surprises unequivocally lead to a larger disagreement about inflation projections above and beyond the surprises of the market.

### Transition to next exercise
- The authors proceed to analyze deviations from the inflation target (section 3.4) using an alternative dependent variable to proxy anchoring of inflation expectations.

*Source: Authors’ calculations (from wpiea2020252-print-pdf — section 3.3 and related tables).*

### 4.2    Constant Degree of Information Rigidity

### 4.2    Constant Degree of Information Rigidity

### Setup and firm optimization
- Information rigidity parameter: 0<λ<1, with λ= 0 corresponding to full information and λ= 1 to firms that never change expectations.
- A random fraction λ of firms does not update its information set and keeps price P^j_t = P^j_{t−1}; fraction 1−λ updates.
- Firms choose price P^j_t to maximize expected discounted profits:
  - Objective function: E_t ∑_{τ=0}^∞ λ^τ R_{t+τ|t} (P^j_t Y^j_{t+τ} − mc^j_{t+τ} Y^j_{t+τ}) (equation (22)).
  - First-order condition (w.r.t. P^j_t) given by equation (23).
  - Optimal individual price:
    - P^j_t = (γ/(γ−1))
      (E_t ∑_{τ=0}^∞ λ^τ R_{t+τ|t} (P_{t+τ})^γ C_{t+τ} mc^j_{t+τ} / E_t ∑_{τ=0}^∞ λ^τ R_{t+τ|t} (P_{t+τ})^γ C_{t+τ})
      (equation (24) and (26) after substituting R_{t+τ|t}).

- Intertemporal discount factor used by firms:
  - R_{t+τ|t} = β^τ E_t C_{t+τ}^{−σ} C_t^{σ} (equation (25)), where β is consumer discount factor and σ is coefficient of risk aversion (CRRA utility u(C_t)=c^{1−σ}/(1−σ)).

### Aggregation and price-level dynamics
- Aggregate reset price of firms updating information (P^*_t) given by equation (27):
  - P^*_t = (γ/(γ−1)) [ E_t ∑_{τ=0}^∞ λ^τ β^τ (P_{t+τ})^γ (C_{t+τ})^{1−σ} (α_1 z_{t+τ} + α_2 i_{t+τ}) (E[θ^{1−γ}])^{1/(1−γ)} / E_t ∑_{τ=0}^∞ λ^τ β^τ (P_{t+τ})^γ (C_{t+τ})^{1−σ} ].
- Aggregate price level is a weighted average:
  - P_t = λ P_{t−1} + (1−λ) P^*_t (equation (28)).
- Iterated expected next-period price:
  - E_t[P_{t+1}] = λ P_t + (1−λ) E_t[P^*_{t+1}] (equation (29)); expression for E_t[P^*_{t+1}] in equation (30).
- Forecasts for horizon t+τ for firms that update:
  - E_t[P_{t+τ}] follows recursion (31) and expanded form (32).
  - Inflation forecast for updaters:
    - π^{e,*}_{t+τ} = λ + (1−λ) E_t[P^*_{t+τ}] / E_t[P_{t+τ−1}] (equation (33)).

### Forecast behavior of sticky-information firms and implications
- Firms that do not update information produce constant price forecasts across horizons:
  - E_t P^{−*}_{t+τ} = E_t P^{−*}_t = P_{t−1}; thus their inflation forecast π^{e,−*}_{t+τ} = 0 for all τ (footnotes 18–19).
- Two groups:
  - Size 1−λ: forecast inflation consistent with expected evolution of aggregate state π^{e,*}_{t+τ}.
  - Size λ: forecast zero inflation π^{e,−*}_{t+τ} = 0.
- Dispersion of inflation expectations at horizon t+τ:
  - d_{t+τ|t} = max{ |π^{e,*}_{t+τ} − E_t[π^e_{t+τ}]|, |π^{e,−*}_{t+τ} − E_t[π^e_{t+τ}]| } = λ |π^{e,*}_{t+τ}| (equation (34)).
  - Interpretation: dispersion moves with |π^{e,*}_{t+τ}|; both expected inflation and dispersion are state contingent.

### Numerical experiments and key calibrated values
- Baseline stickiness used to be consistent with literature: λ= 0.78 (about 22 percent of firms change prices in any given period).
- Simulation procedure to match empirical measure:
  - Simulate 500 time series with 3,000 periods each; discard first 1,500 periods.
  - For each series, extract all 12-month sequences starting with i = E(i).
  - Compute average absolute interest rate shock |i_t − i_{t−1}| = 77 basis points and average response of inflation expectation dispersion.
- Findings from figures and simulations:
  - Figure 7: dispersion responses under constant λ for i.i.d. and persistent aggregate states; medium shock = one standard deviation, large = three standard deviations.
  - When state is i.i.d., updaters’ price expectations across horizons are almost identical (P^{e,*}_{t+1}≈P^{e,*}_{t+2}≈...); dispersion increases mainly at t and t+1 due to difference between updaters and sticky firms.
  - When state is persistent (example parameters ρ_z = 0.9 and ρ_i = 0.5), π^{e,*}_{t+τ} changes slowly over horizon and may increase for some horizons before moving to zero.
  - Model-implied dispersion decreases with horizon, in line with empirical pattern, but magnitude is smaller (Figure 8).
  - Varying λ (Figure 9): tested λ= 0.92 (high), λ= 0.50 (medium), λ= 0.08 (low).
    - For calibration used, magnitude of dispersion falls with the value of λ (nonlinear relationship); for λ=0 and λ=1 there is no dispersion.

### Mechanisms: how λ affects dispersion
- Two opposing effects of higher λ on dispersion at a given horizon t+τ:
  - Consensus forecast bias: larger λ biases consensus toward zero inflation, increasing distance between consensus and updaters’ forecasts.
  - Optimal price-setting response: larger λ makes updaters weight future more heavily, stabilizing price level and dampening dispersion.
- Net effect is ambiguous analytically; requires numerical methods.
- Overall conclusion for constant λ:
  - Models with information stickiness qualitatively match observed time path of dispersion in US data.
  - For values of λ consistent with literature, constant-information-stickiness assumption is insufficient to generate the magnitude of dispersion observed in the data.

*Source: IMF working paper — Section 4.2, "Constant Degree of Information Rigidity".*

### 4.4    Firm State-dependent Degree of Information Rigidity

### 4.4    Firm State-dependent Degree of Information Rigidity

### Setup and intuition
- Degree of information stickiness depends on firms’ idiosyncratic states θ_jt rather than only on aggregate states.
- For idiosyncratic shocks above a threshold ̃θ, information stickiness is lower than a constant level λ and is a decreasing function of the distance between the realized shock and the mean of those shocks.
- Firms with θ_jt > ̃θ are in "extreme circumstances" and place relatively more weight on current conditions (lower information stickiness) than firms with moderate shocks.
- Because idiosyncratic shocks are time-invariant in distribution, the share of firms with lower λ_t+τ is constant across periods, but the share of firms in extreme conditions in a given period affects the average price level.

### Optimal price chosen by updating firms
- Firms updating in period t choose optimal price:
  - P_jt = (γ/(γ−1)) [ (Et Σ_{τ=0}^∞ λ_j(t+τ)(θ_j(t+τ)) β^τ (P_{t+τ})^γ (C_{t+τ})^{1−σ} (α_1 z_{t+τ}+α_2 i_{t+τ})^{θ_j(t+τ)} ) / (Et Σ_{τ=0}^∞ λ_j(t+τ)(θ_j(t+τ)) β^τ (P_{t+τ})^γ (C_{t+τ})^{1−σ}) ]
  - where λ_j(t+τ) = λ_z(∆θ_j(t+τ)).
- The functional form used in the text: λ_jt = ̄λ/[ (1 + ψ_3 |θ_jt − E[θ]|) ] with ψ_3 = 3.

### Aggregation: decomposition of the price index
- The overall price index P_t is the sum of sub-indexes for:
  - Firms with moderate idiosyncratic shocks (θ ≤ ̃θ): weight μ_{̃θ} in the population; of these, share 1−λ change prices in the period and share λ set P_jt = P_j(t−1).
  - Firms with extreme idiosyncratic shocks (θ > ̃θ): weight 1−μ_{̃θ} in the population; of these, share 1−λ^* E[z(∆θ_jt) | (θ_j > ̃θ)] change prices and share λ^* E[z(∆θ_jt) | (θ_j > ̃θ)] keep P_jt = P_j(t−1).
- Price index expression:
  - P_t = λ ( μ_{̃θ} + (1−μ_{̃θ}) E[z(∆θ_jt) | (θ_j > ̃θ)] ) P_{t−1}
    + μ_{̃θ} (1−λ) P^∗_{̃θ,t}
    + (1−μ_{̃θ}) (1−λ^* E[z(∆θ_jt) | (θ_j > ̃θ)]) P^∗_{−̃θ,t}

- Sub-index of optimal prices for moderate firms:
  - P^∗_{̃θ,t} = (γ/(γ−1)) { ∫_{0}^{̃θ} [ ... ]^{1−γ} dj }^{1/(1−γ)} with expectations and products over z(θ_jt+h) appearing inside the integrand.

- Sub-index of optimal prices for extreme firms:
  - P^∗_{−̃θ,t} = (γ/(γ−1)) { ∫_{̃θ}^{1} [ ... ]^{1−γ} dj }^{1/(1−γ)} with analogous structure.

- Concavity of the price index function implies P^∗_{̃θ,t} > P^∗_{−̃θ,t}.
- Define ̃z = E[z(∆θ_jt) | (θ_j > ̃θ)] to rewrite:
  - P_t = λ [ μ_{̃θ} + (1−μ_{̃θ}) ̃z ] P_{t−1}
    + μ_{̃θ} (1−λ) P^∗_{̃θ,t}
    + (1−μ_{̃θ}) (1−λ ̃z) P^∗_{−̃θ,t}

### Forecasts and dispersion of inflation expectations
- Iterating forward, forecasters processing new information have:
  - Et[P_{t+1}] = λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ) P_t
    + μ_{̃θ} (1−λ) Et[P^∗_{̃θ,t+1}]
    + (1−μ_{̃θ}) (1−λ ̃z) Et[P^∗_{−̃θ,t+1}]
- General horizon t+τ forecast:
  - Et[P_{t+τ}] = λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ) Et[P_{t+τ−1}]
    + μ_{̃θ} (1−λ) Et[P^∗_{̃θ,t+τ}]
    + (1−μ_{̃θ}) (1−λ ̃z) Et[P^∗_{−̃θ,t+τ}]
- Inflation forecast for updating firms at horizon t+τ:
  - π^{∗,e}_{t+τ} =
    [ λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ) Et[P_{t+τ−1}]
      + μ_{̃θ} (1−λ) Et[P^∗_{̃θ,t+τ}]
      + (1−μ_{̃θ}) (1−λ ̃z) Et[P^∗_{−̃θ,t+τ}] ] / Et[P_{t+τ−1}]
    = λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z )
      + μ_{̃θ} (1−λ) Et[P^∗_{̃θ,t+τ}]/Et[P_{t+τ−1}]
      + (1−μ_{̃θ}) (1−λ ̃z) Et[P^∗_{−̃θ,t+τ}]/Et[P_{t+τ−1}]

- Firms not updating their information set forecast inflation to be zero.
- Dispersion of inflation expectations:
  - d_{t+τ|t} = | π^{e,∗}_{t+τ} − [ λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ) π^{e,−∗}_{t+τ} + { 1 − λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ) } π^{e,∗}_{t+τ} ] |
    = λ ( μ_{̃θ} + (1−μ_{̃θ}) ̃z ) || π^{e,∗}_{t+τ} ||

### Comparative statics and key implications
- Lower threshold ̃θ:
  - Increases the weight of Et[P^∗_{−̃θ,t+τ}]/Et[P_{t+τ−1}] in π^{e,∗}_{t+τ}.
  - Implies a larger magnitude of Et[P^∗_{−̃θ,t+τ}]/Et[P_{t+τ−1}], and, other things given, a larger magnitude of π^{e,∗}_{t+τ}.
  - Therefore, lower ̃θ (which yields lower information stickiness for more firms) increases the impact of firm heterogeneity on aggregate variables and on the dispersion of inflation expectations.
- Heterogeneity across firms affects the magnitude of dispersion but not the term structure of inflation expectations because the distribution of idiosyncratic shocks is stationary.
- Quantitative results (from figures and simulations in the text):
  - Heterogeneity in information stickiness across firms can increase the magnitude of inflation expectations dispersion substantially — about four times larger than in other model cases and about half of what is observed in the data for some aggregate states away from their mean realization.

### Policy implications summarized in this section
- Firm-level state-dependent information rigidity can materially increase short-horizon dispersion of inflation expectations.
- Since inflation expectation dispersion at short horizons can lead to price dispersion and inflation persistence, monetary policy communication should be refined to better align information across agents.
- Better-aligned information can help keep inflation expectations anchored at shorter horizons, when the credit channel is less effective.

*Source: IMF Working Paper — section 4.4, "Firm State-dependent Degree of Information Rigidity", wpiea2020252-print-pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020252-print-pdf.pdf_