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---

### Executive summary — overview, method, and key functional shocks
- Average inflation expectations do not properly summarize heterogeneous beliefs; the entire distribution of inflation expectations should be considered.
- Method and data:
  - Functional approach applied to University of Michigan’s Survey of Consumers monthly data.
  - Monthly cross-sectional responses summarized into density functions representing the expected inflation distribution (EID).
  - Functional autoregression (FAR) approximated by finite-dimensional VAR via projection onto a functional principal component basis.
- Three distinctive functional shocks drive dynamics of expected inflation distributions through:
  - disagreement,
  - level shift, and
  - degree of inflationary ambiguity.

### Major empirical findings on EIDs and sample details
- Sample windows and response counts:
  - Baseline one-year horizon: January 1983 to December 2021.
  - Medium-run horizon: April 1990 to December 2021.
  - Number of recorded responses per month varies from 408 to 1205 (about 1200 on average in early 1980s, about 600 in late 1980s, and around 450 afterwards).
- Distributional changes can be large while means move little:
  - One-year ahead average inflation expectations: 2.7 percent (pre-pandemic months) to 2.8 percent (early pandemic months).
  - Medium-run average: 2.4 percent (pre-pandemic) to 2.5 percent (early pandemic).
- Functional principal component analysis (FPCA) and truncation:
  - First FPC explains more than 64 percent of total variance for one-year expectations.
  - First three components account for more than 93 percent of total variance (one-year) and more than 98 percent (medium-run).
  - Truncation number chosen: m = 3, which explains EID variations over 90 percent.
  - Lag order used: two (second-order FAR / VAR).

### Functional principal components — interpretations and moment effects
- First principal component (disagreement / dispersion):
  - Explains more than 64 percent of total variance (one-year).
  - Accentuates bi-modal distribution: increases densities for negative expectations between −5 and −1 percent and for high expectations above 5 percent; decreases densities in the middle between −1 and 5 percent.
  - Associated with clear increases in standard deviation (cross-sectional dispersion) as loadings rise.
- Second principal component (shift / level):
  - Explains around 25 percent of variation.
  - Positive loading shifts the EID to the right; negative loading shifts it to the left.
  - Strongly associated with increases in the first moment (mean).
- Third principal component (ambiguity):
  - Explains about 3 percent of variation.
  - Increases frequency of moderate expectations around 2 percent and very high expectations above 7 percent.
  - Most associated with changes in kurtosis (fourth moment); labeled "ambiguity".

### Mapping external economic shocks to functional shocks — correlations and norms (one-year ahead)
- Mapping: estimate x_t = ρ_x′ e_t with e_t = (e1t,e2t,e3t) the three functional shocks; h_x = ρx1 h1 + ρx2 h2 + ρx3 h3 gives EID response to one-standard-deviation external shock (external shocks normalized to unit variance).
- Correlations ρx (point estimate with 68% bootstrap confidence bands in parentheses):
  - Monetary Policy (ρ_mp)
    - Disagreement: −0.029 (0.097)
    - Shifting: 0.244 (0.082)
    - Ambiguity: −0.040 (0.092)
    - Norm: 0.249 (0.075)
  - Government Spending (ρ_gs)
    - Disagreement: 0.019 (0.114)
    - Shifting: 0.166 (0.109)
    - Ambiguity: 0.110 (0.106)
    - Norm: 0.200 (0.089)
  - Personal Income Tax (ρ_it)
    - Disagreement: −0.021 (0.092)
    - Shifting: −0.043 (0.112)
    - Ambiguity: 0.003 (0.094)
    - Norm: 0.048 (0.070)
  - Gasoline Prices (ρ_gp)
    - Disagreement: 0.107 (0.119)
    - Shifting: 0.463 (0.082)
    - Ambiguity: 0.310 (0.077)
    - Norm: 0.567 (0.063)
- One-year-ahead interpretations:
  - Monetary policy effects mainly on the shifting component.
  - Gasoline price changes have strong positive relationships with shifting and disagreement components.
  - Percent of external shock transmitted to EID (norm) ranges from 0.048 (personal income tax) to 0.567 (gasoline price).

### Correlations (medium-run) — point estimates and 90% bootstrap standard errors
- Monetary Policy (ρ_mp)
  - Disagreement: −0.035 (0.116)
  - Shifting: 0.028 (0.139)
  - Ambiguity: −0.089 (0.102)
  - Norm: 0.100 (0.094)
- Government Spending (ρ_gs)
  - Disagreement: 0.218 (0.123)
  - Shifting: 0.025 (0.153)
  - Ambiguity: −0.075 (0.118)
  - Norm: 0.231 (0.107)
- Personal Income Tax (ρ_it)
  - Disagreement: 0.110 (0.125)
  - Shifting: 0.294 (0.102)
  - Ambiguity: −0.130 (0.096)
  - Norm: 0.340 (0.083)
- Gasoline Prices (ρ_gp)
  - Disagreement: 0.210 (0.087)
  - Shifting: 0.043 (0.095)
  - Ambiguity: −0.018 (0.124)
  - Norm: 0.216 (0.078)
- Medium-run interpretations:
  - Norm for government spending increases from 0.200 (one-year) to 0.231 (medium-run).
  - Norm for personal income tax increases from 0.048 (one-year) to 0.340 (medium-run).
  - Monetary and gasoline shock correlations with functional shocks are lower for medium-run EID than for one-year ahead EID.

### Impulse responses and at-impact distributional effects — by shock
- Monetary policy shocks (baseline sample January 1983–December 2021):
  - Contractionary monetary shocks:
    - Increase average one-year ahead inflation expectations by decreasing the frequency of negative expectations and increasing the frequency of positive expectations by 2 to 7 percent.
    - Reduce disagreement (in the post-Volcker shorter sample) and increase the share of households expecting inflation between 2 to 4 percent (anchoring effect).
  - If sample extended to include January 1978–December 2021 (pre-Volcker high-inflation period):
    - Monetary shocks produce a fatter right tail (more households expecting inflation 7% and above) and increase dispersion.
    - Mean responses larger and more persistent in the longer sample.
  - Distributional moments:
    - Longer sample: standard deviation increases and effects persist; skewness and kurtosis changes larger than in shorter sample.
    - Shorter sample (post-Volcker): monetary shocks reduce disagreement and anchoring effects are observed; effects on dispersion disappear within one year.
- Government spending shocks:
  - One-year ahead: decrease frequencies below 1 percent and increase frequencies between 7 to 12 percent.
  - Medium-run: increase frequencies above 5 percent; increase mean for both horizons.
  - Dispersion: decrease dispersion (one-year), increase dispersion (medium-run).
  - Frequency at 2 percent: negative impact for medium-run expectations.
- Personal income tax shocks:
  - One-year ahead: no statistically significant effect.
  - Medium-run: inflationary — lower frequencies in 0 to 3 percent range and increases in expectations beyond 3 percent; persistent increase in average medium-run expectations above 2 percent and a persistent decrease in frequency at 2 percent.
- Gasoline price shocks:
  - One-year ahead: surprise hikes decrease frequencies below 3 percent and increase frequencies above 3 percent.
  - Medium-run: increase densities for higher inflation levels and for negative values (increased disagreement).
  - Mean responses: one-year ahead response at impact is nearly three times larger than medium-run response at impact.
  - Standard deviation:
    - One-year ahead: disagreement decreases for nearly one year.
    - Medium-run: standard deviation increases persistently.
  - Frequency at 2 percent: decreases for both horizons.

### State-dependent IRFs (recession vs. non-recession)
- Fiscal policy shocks: effects on EID distribution do not depend significantly on recession state; confidence bands widen during recessions.
- Contractionary monetary policy and gasoline price shocks:
  - Inflationary effects stronger during recessions.
  - Monetary policy peak coefficient during recession almost twice non-recession peak.
  - Gasoline price effects: during recessions, dramatic fall in negative-expectation frequency accompanied by similar rise in positive-expectation frequency.

### Choice of basis, empirical comparisons, and estimator variance
- Basis comparison (selected FR2 values for representative m):
  - m = 10: FPC Basis 0.7419, Histogram Basis 0.0095, Quantile Basis 0.0322, Moment Basis 0.0048.
  - m = 30: FPC Basis 0.9513, Histogram Basis 0.0400, Quantile Basis 0.5573, Moment Basis 0.0172.
  - m = 100: FPC Basis 0.9946, Histogram Basis 0.4142, Quantile Basis 0.7849, Moment Basis 0.0878.
- Variance measure trace(Q Λ Q)́ (×10^4) highlights explosive variances for non-FPC bases with larger m:
  - Example: m = 30 trace values — FPC Basis 3.4709; Histogram Basis 890.5797; Moment Basis 3916.064.
- Empirical implication and choice:
  - FPC basis effectively represents temporal variation even with small m; for m = 3 the paper reports explaining more than 95 percent of total temporal variation.
  - Authors use m = 3 (explains EID variations over 90 percent) and second-order FAR approximation.

### Policy implications, interpretations, and open questions
- Economic policies affect inflation expectations via distinct transmission channels that differ across shock types and horizons; each channel should be studied individually and jointly.
- Monetary policy:
  - Two tentative channels for why contractionary shocks raise household inflation expectations:
    - Information channel: tightening signals higher underlying inflationary pressures.
    - Perception channel: tightening signals worse economic conditions which households may associate with higher future inflation.
  - Findings are consistent with literature documenting “price puzzles” in response to contractionary monetary policy.
- Fiscal policy:
  - Government spending shocks consistent with inter-temporal budget constraint logic and delayed-realization channels; medium-run disagreement may reflect heterogeneous beliefs about fiscal multipliers.
  - Personal income tax shocks raising medium-run inflation expectations require further validation; perfect-foresight explanations are strong assumptions.
- Gasoline prices:
  - Effects on household inflation expectations are well-understood relative to monetary and fiscal channels; surprise hikes raise both short- and medium-run expectations and increase medium-run disagreement.
- Recommendations and caveats:
  - Results invite reconsideration of prior policy recommendations that downplay economic policy effects on inflation expectations (except gasoline shocks).
  - Central bank credibility and clear communication remain important for anchoring expectations; further research needed to validate channels and understand household perception and updating of shocks.

*Source: wpiea2022132-print-pdf — Executive summary7; Conclusion and Appendices (IMF Working Paper No. WP/2022/132).*

### Executive summary7

### Executive summary

### Overview
- Average inflation expectations do not properly summarize heterogeneous beliefs; the entire distribution of inflation expectations should be considered.
- The paper applies a functional approach to a survey of household inflation expectations using data from the University of Michigan’s Survey of Consumers to document how economic shocks affect the distribution of household inflation expectations.

### Method and data
- Functional approach applied to household survey data (University of Michigan’s Survey of Consumers).
- Dynamics of expected inflation distributions are driven by three distinctive functional shocks.

### Key functional shocks
- The three functional shocks collectively influence the expected inflation distribution through:
  - disagreement
  - level shift
  - degree of inflationary ambiguity

### Effects of economic shocks on expected inflation distributions (EIDs)
- Contractionary monetary shocks:
  - increase the average level of inflation expectation
  - produce withanchoring effects, with a reduction in disagreement
  - increase the share of households expecting future inflation to be between 2 to 4 percent
  - Suchanchoring effects are not observed when the periods of high inflation prior to the Volcker disinflation are included
- Expansionary government spending shocks:
  - have inflationary effects on both short and medium-run inflation expectations
- Personal income tax shocks:
  - an increase in personal income tax shocks is inflationary only for medium-run inflation expectations
- Gasoline price shocks:
  - a surprise increase in gasoline prices increases the level of inflation expectations
  - lowers the share of households with 2 percent inflation expectations

### Policy implications
- Economic policies play a direct role in affecting inflation expectations.
- Each transmission channel should be studied carefully, individually and jointly.

*Source: wpiea2022132-print-pdf - Executive summary7*

### conclusion invites us to reconsider the policy recommendations derived from previous findings in

### wpiea2022132-print-pdf - conclusion invites us to reconsider the policy recommendations derived from previous findings in

### Major findings on the distribution of household inflation expectations
- Using survey data from the University of Michigan’s Survey of Consumers from January 1983 to December 2021, monthly cross-sectional responses are summarized into density functions representing the expected inflation distribution (EID).
- The estimated EIDs are driven by three distinctive functional shocks that collectively influence the expected inflation distribution through:
  - disagreement,
  - level shift, and
  - degree of inflationary ambiguity.
- Changes in the shape of the distribution can be large while the mean moves little:
  - one-year ahead average inflation expectations moved from 2.7 percent (pre-pandemic months) to 2.8 percent (early pandemic months) despite noticeable flattening and decline in modal concentration;
  - medium-run average moved from 2.4 percent (pre-pandemic) to 2.5 percent (early pandemic).
- The distributional response matters economically and is not fully captured by a small set of moments (mean, standard deviation) alone.

### Effects of identified economic shocks on EID
- Monetary policy shocks
  - In the main sample (post-Volcker era), a contractionary monetary policy shock:
    - increases the average expected inflation for one-year ahead expectations;
    - reduces expected inflation disagreement;
    - increases the share of households expecting inflation to be in the range of 2 to 4 percent (anchoring effect).
  - When the sample is extended to include the late 1970s to early 1980s (high inflation period), monetary policy shocks:
    - lead to a fatter right tail (more households believing inflation will be at high levels);
    - increase dispersion among expectations.
  - This supports documented shifts in monetary policy conduct since 1979.
- Gasoline price shocks
  - A surprise hike in gasoline prices:
    - increases the level of inflation expectations for both the one-year and the medium-run horizons;
    - decreases the standard deviation (disagreement) of one-year ahead inflation expectations;
    - widens disagreement for medium-run expectations (reflecting uncertainty about future gasoline prices);
    - lowers the share of households with inflation expectations at 2 percent, suggesting potential implications for a central bank’s inflation targeting objective.
- Fiscal policy shocks
  - Government spending shocks are inflationary for both one-year and medium-run inflation expectations, with larger disagreement among households on medium-run effects (possibly reflecting heterogeneous perceptions of fiscal multipliers or future fiscal conditions).
  - Shocks to personal income tax expectations:
    - have no effect on one-year ahead inflation expectations;
    - have long-term effects on medium-run inflation expectations.

### Methodology and empirical implementation
- Functional approach
  - Monthly survey responses converted to estimated density functions using kernel density estimation (Epanechnikov kernel with rule-of-thumb time-varying bandwidth).
  - Functional autoregression (FAR) model for the EID is approximated by a finite-dimensional vector autoregression (VAR) via projection onto a functional principal component basis.
  - The mapping from functional objects to finite-dimensional vectors uses the first m functional principal components; m is chosen to capture the bulk of variation.
- Functional principal component analysis (FPCA) results and choices
  - The first functional principal component explains more than 64 percent of the total variance in one-year inflation expectations.
  - The first three components account for more than 93 percent of the total variance in one-year inflation expectations and more than 98 percent of the variance in medium-run inflation expectations.
  - Based on this evidence, m is chosen as 3, which explains EID variations over 90 percent.
- Sample and data details
  - Monthly data from January 1983 to December 2021 are used as the primary sample.
  - The number of recorded responses per month varies from 408 to 1205 (about 1200 on average in early 1980s, about 600 in late 1980s, and around 450 afterwards).
- Structural analysis
  - Recursive identification on the VAR of principal component scores yields recursive functional shocks and baseline functional impulse responses.
  - Correlations between functional shocks and externally identified economic shocks (monetary policy, government spending, personal income tax, gasoline price) are used as weights to construct EID responses to economic shocks.

### Empirical correlations and moments vs. functional basis
- Regression evidence shows different macroeconomic indicators correlate with different features of the distribution (mean, standard deviation, skewness, kurtosis, frequency at 0 percent, frequency below 0 percent).
  - Example: mean inflation expectation is not always selected among EID properties explaining largest portion of variance in macroeconomic variables (e.g., real GDP growth).
  - Deflation expectations show relatively strong correlation with the unemployment rate.
- Econometric comparison
  - Using a pre-specified set of moments is inferior to using the functional principal component basis for explaining variation in EIDs.
  - The first functional principal component alone explains about 74 percent of EID variations (reported for the chosen FPCA basis); choosing m = 3 explains over 90 percent.
  - Alternative basis choices (e.g., moment basis) explain far less (example: moment basis with m = 10 fails to explain even 10 percent of EID variations in the comparison described).

### Policy implications and recommendations
- Economic policies have a non-negligible role in affecting inflation expectations; transmission channels differ across types of shocks and horizons and should be studied individually and jointly.
- The conclusion invites reconsideration of prior policy recommendations that de-emphasize effects of economic policy on inflation expectations (apart from gasoline price shocks).
- Credible and clear communication by central banks remains important to anchor inflation expectations; findings do not contradict this role.
- Emphasis on the need to better understand:
  - how households perceive economic shocks,
  - how they update their expectations for future inflation based on these shocks,
  - whether economic policies can be designed to ensure anchoring of inflation expectations.

### Contribution to the literature
- Empirical contribution: uses survey data on inflation expectations to model the full distribution (EID) rather than focusing only on means or a few moments.
- Drivers of expectations: provides evidence that monetary, fiscal, and gasoline price shocks each have distinct distributional effects on household inflation expectations.
- Methodological contribution: applies a novel functional autoregression framework (transformable into a VAR via FPCA) to analyze the dynamics of the entire distribution of inflation expectations; the methodology is broadly applicable to other functional dynamics in economics.

*Italic source attribution: Conclusion and supporting sections from wpiea2022132-print-pdf (University of Michigan Survey of Consumers data and methodology described in the provided excerpt).*

### Appendix D, the results for medium-run inflation expectations are presented, where the results

### Appendix D — Functional principal components, recursive functional shocks, and effects of economic shocks on expected inflation distributions (EID)

### Functional principal components (one-year-ahead EID)
- Methodology and visualization:
  - Loadings plotted over time for three principal components; functional principal components (FPCs) v̊1, v̊2, v̊3 shown as functions (integrate to 0).
  - For each FPC k, loadings αkt = 〈f t, v̊k〉 plotted with blue and red stars indicating min(αkt) and max(αkt) over t = 1, . . . , T.
  - Combined effects shown by adding each FPC scaled by its minimum and maximum loading to the average EID density f̄ = (1/T) ΣT t=1 f t; black solid = f̄, blue dotted = f̄ + min(αkt) v̊k, red dashed = f̄ + max(αkt) v̊k.

- Main components and interpretations:
  - First principal component (disagreement component)
    - Explains more than 64 percent of total variance of EID, alone.
    - Accentuates bi-modal distribution: increases densities for negative expectations between -5 and -1 percent and for high expectations above 5 percent; decreases densities in the middle range between -1 and 5 percent.
    - Maximum loading (red dashed) increases dispersion; minimum loading (blue dotted) concentrates the EID.
    - Loadings show cyclical features overlapping recessions (gray shaded areas); high positive factor loadings observed in the early 1980s — disagreement high when inflation level is high (poorly anchored expectations).
  - Second principal component (shift component)
    - Explains around 25 percent of the variations in EID over time, alone.
    - Shifts the EID: positive loading shifts EID to the right; negative loading shifts EID to the left.
  - Third principal component (ambiguity component)
    - Explains about 3 percent of the variation in EID over time, alone.
    - Increases frequency of moderate inflation expectations around 2 percent and also very high expectations above 7 percent.
    - Positive loading generates fat right tails (multiple bumps at very high expectations); negative loading smooths the tail.
    - Loadings increased between 2010 and 2019 — both the share of households with well-anchored expectations at 2 percent and the fraction projecting very high inflation increased.

- Moments and component effects (summary)
  - First FPC: clear increase in standard deviation with factor loadings → cross-sectional dispersion.
  - Second FPC: clear increase in the first moment (mean).
  - Third FPC: first three moments (mean, standard deviation, skewness) relatively flat; fourth moment (kurtosis) changes most significantly.

### From functional principal components to recursive functional shocks
- Approximation and model selection:
  - Truncation number m set to 3 (m = 3) based on scree-plot; approximate FAR by a three-dimensional VAR.
  - Standard lag selection (AIC and BIC) suggest optimal lag = two; use a second-order FAR approximated by a three-dimensional VAR.

- Identification of recursive functional shocks:
  - Estimate x(Σ) = (1/T) ΣT t=1 x(ε t) x(ε t)' from fitted reduced-form VAR errors.
  - Cholesky decomposition x(Σ) = L L' with lower-triangular L so x(ε t) = L e t.
  - Each column i of L (ℓ1i, ℓ2i, ℓ3i)' defines baseline EID response h i = π−1 L(:, i) (h i called baseline EID response to functional shock i).
  - Figure 7 presents at-impact baseline EID responses h i to each of the three functional shocks.

### Representation of external economic shocks using functional shocks
- External shocks considered (definitions and expectations):
  - Monetary policy (mp): series constructed by Miranda-Agrippino and Ricco (2021); available monthly and quarterly from March 1963 to December 2007. Effect on inflation expectations a priori ambiguous (demand channel vs. signaling channel).
  - Government spending (gs): own estimation of Auerbach and Gorodnichenko (2012) recursive VAR shocks using real government spending, receipts, taxes, transfers, and real GDP (chained 2000 dollars). Positive gs shock expected to boost aggregate demand and possibly increase disagreement.
  - Personal income tax (it): shock from Mertens and Ravn (2013) (narrative approach). Tax increases generally contractionary; direction of effect on household inflation expectations ambiguous.
  - Gasoline price (gp): percent change in nominal retail gasoline price (FRED CPI for All Urban Consumers: Gasoline (All Types) in the U.S. City Average). Consumers attend to gasoline price; treat real gasoline price approximately as random walk.

- Mapping external shocks to functional shocks:
  - Let e1t, e2t, e3t be the three functional shocks and x t an external shock.
  - Estimate x t = ρx′ e t + η x t where ρx = (ρx1, ρx2, ρx3)' and η x t is i.i.d. innovation.
  - Because e kt are uncorrelated, the norm ‖ρx‖ (Euclidean norm) can be interpreted as the percentage of external shock x transmitted to the EID.
  - h x = ρx1 h1 + ρx2 h2 + ρx3 h3 is the response of EID to a one-standard-deviation external shock x t (external shocks normalized to have unit variance).
  - When quarterly external shocks are only available, cumulative sum of monthly functional shocks used to obtain quarterly series.

### Samples and horizons
- Baseline samples:
  - One-year horizon: January 1983 to December 2021.
  - Medium-run horizon: April 1990 to December 2021.
- Lag order used: two (second-order FAR / VAR).
- Truncation number used: m = 3.

### Correlations between functional shocks and external shocks (one-year-ahead EID)
- Monetary Policy (ρm p)
  - Disagreement: −0.029 (0.097)
  - Shifting: 0.244 (0.082)
  - Ambiguity: −0.040 (0.092)
  - Norm: 0.249 (0.075)
- Government Spending (ρf p)
  - Disagreement: 0.019 (0.114)
  - Shifting: 0.166 (0.109)
  - Ambiguity: 0.110 (0.106)
  - Norm: 0.200 (0.089)
- Personal Income Tax (ρit)
  - Disagreement: −0.021 (0.092)
  - Shifting: −0.043 (0.112)
  - Ambiguity: 0.003 (0.094)
  - Norm: 0.048 (0.070)
- Gasoline Prices (ρg p)
  - Disagreement: 0.107 (0.119)
  - Shifting: 0.463 (0.082)
  - Ambiguity: 0.310 (0.077)
  - Norm: 0.567 (0.063)
- Notes: numbers in parentheses are the 68% bootstrap confidence bands.

- Key one-year-ahead interpretations:
  - Monetary policy effects mainly focused on the shifting component.
  - Fiscal shocks (government spending and personal income tax) show relatively weak correlations with one-year-ahead EID functional shocks.
  - Gasoline price changes have a positive relationship with both shifting and disagreement functional shocks.
  - Correlations across shocks range from about 5 percent (personal income tax shocks, small norm) to more than 56 percent (gasoline price shocks, large norm) in terms of percent of external shock transmitted to EID.

### Correlations between functional shocks and external shocks (medium-run EID)
- Monetary Policy (ρm p)
  - Disagreement: −0.035 (0.116)
  - Shifting: 0.028 (0.139)
  - Ambiguity: −0.089 (0.102)
  - Norm: 0.100 (0.094)
- Government Spending (ρf p)
  - Disagreement: 0.218 (0.123)
  - Shifting: 0.025 (0.153)
  - Ambiguity: −0.075 (0.118)
  - Norm: 0.231 (0.107)
- Personal Income Tax (ρit)
  - Disagreement: 0.110 (0.125)
  - Shifting: 0.294 (0.102)
  - Ambiguity: −0.130 (0.096)
  - Norm: 0.340 (0.083)
- Gasoline Prices (ρg p)
  - Disagreement: 0.210 (0.087)
  - Shifting: 0.043 (0.095)
  - Ambiguity: −0.018 (0.124)
  - Norm: 0.216 (0.078)
- Notes: numbers in parentheses are the 90% bootstrap standard errors.

- Key medium-run interpretations:
  - Norm for government spending shocks increases from 0.200 (one-year horizon) to 0.231 (medium-run).
  - Norm for personal income tax shocks increases from 0.048 (one-year horizon) to 0.340 (medium-run).
  - Correlations between monetary and gasoline shocks and functional shocks are lower for medium-run EID than for one-year ahead EID.
  - Government spending shocks correlated with the disagreement component at medium-run horizon.
  - Personal income tax shocks correlated with the shifting component with statistical significance at medium-run horizon.
  - Correlation between gasoline price shocks and shifting component weakens at medium-run horizon while correlation with disagreement component strengthens.

### Impulse response functions and dynamics
- Three-dimensional IRFs:
  - Figure 8 shows three-dimensional IRF surfaces for one-year-ahead (top row) and medium-run (bottom row) EIDs to monetary (blue), government spending (red), personal income tax (green), and gasoline price (black) shocks.
  - Responses generally phase out over time.
- Analysis focus:
  - Focus on at-impact responses of EID; multi-horizon effects examined for single statistics only.

_Italic: Source — Appendix D text excerpt from wpiea2022132-print-pdf (IMF working paper PDF). _

### 5.2    Monetary Policy Shocks and EIDs

### 5.2 Monetary Policy Shocks and EIDs

### Key findings on household inflation expectations
- The early 1980s mark a watershed in how household inflation expectations respond to monetary policy shocks.
- Baseline sample: January 1983 to December 2021.
- At-impact response of one-year ahead inflation expectations to contractionary monetary policy shocks:
  - Monetary policy shocks raise the level of average inflation expectations by decreasing the frequency of negative inflation expectations and increasing the frequency of positive inflation expectations by 2 to 7 percent.
- Sample comparisons (IRFs using different sample windows):
  - Longest sample: January 1978 to December 2021 (includes pre-Volcker high-inflation periods).
  - Other samples: from January 1983; from April 1990; from January 1997.
  - Using the longest sample, the response increases the density of positive inflation expectations from 7% and above, creating a fatter right tail of EID.
  - For the more recent sample running from January 1997, the level of inflation expectations corresponding to the peak of the distribution declines to a range of 2 to 3 percent (compared with around 10 percent for the baseline sample).

### Distributional effects (selected features and moments)
- Mean, standard deviation, skewness, kurtosis (Figure 11):
  - Mean: Average inflation expectations increase in response to monetary policy shocks in both the long (1978:1–2021:12) and short (1983:1–2021:12) samples; coefficients are statistically significant.
    - The coefficient using the longer sample is at least two times larger than that for the shorter sample.
    - The impact is more persistent in the longer sample than in the shorter sample.
  - Standard deviation:
    - During high inflation periods (longer sample), monetary policy shocks increased disagreement and effects were persistent.
    - Excluding high inflation periods (shorter sample), monetary policy shocks reduced disagreement and effects disappear within one year.
  - Skewness: Monetary policy shocks lead to a more asymmetric shape of inflation expectations; at-impact magnitudes are much larger in the longer sample.
  - Kurtosis: A decrease in kurtosis is observed for the longer sample but not for the shorter sample, indicating a reduction in tail behavior in the longer sample.
- Other distributional features (Figure 12):
  - Share of households with inflation expectations above 5 percent (right tail):
    - Positive and persistent increase for the longer sample.
    - Effects become statistically insignificant for the shorter sample.
    - Interpretation: Monetary policy shocks previously un-anchored expectations (led households to expect high future inflation); such un-anchoring effects have disappeared since the early 1980s.
  - Share of households with inflation expectations exactly at 2 percent (in line with central bank target):
    - Longer sample: monetary policy shocks reduce the share with 2 percent expectations.
    - Shorter sample: positive coefficient at impact but not statistically significant.
  - Share of households with inflation expectations between 2 to 4 percent:
    - Longer sample: persistent and negative impact on the share.
    - Shorter sample: clear but temporary increase in response to monetary policy shocks (lasting less than one year).
- Medium-run inflation expectations:
  - At-impact densities increase for positive values with peak around 2 percent, but these medium-run results are not statistically significant.

*Source: wpiea2022132-print-pdf - 5.2    Monetary Policy Shocks and EIDs*

### 5.3 Fiscal Policy Shocks and EIDs

#### Government spending shocks
- At-impact responses:
  - One-year ahead inflation expectations: government spending shocks are inflationary by decreasing frequencies of inflation expectations below 1 percent and increasing frequencies of high-level inflation expectations between 7 to 12 percent.
  - Medium-run inflation expectations: inflationary effect is more pronounced, with increased frequencies of high inflation expectations above 5 percent.
- Moments and dispersion (Figure 15):
  - Mean: Government spending shocks increase average inflation expectations for both one-year ahead and medium-run horizons.
  - Agreement: Stronger agreement among respondents that government spending shocks are inflationary for the short-run than for the medium-run.
  - Standard deviation / dispersion:
    - One-year ahead: government spending shocks decrease dispersion among respondents.
    - Medium-run: government spending shocks increase dispersion among respondents (increase in negative-value densities).
  - Frequency at 2 percent: government spending shocks have a negative impact on the frequency of responses at 2 percent for medium-run inflation expectations.

#### Personal income tax shocks
- Significance and direction:
  - Significant effects are observed only for medium-run inflation expectations.
  - One-year ahead: personal income tax shocks appear disinflationary by lowering frequencies of high-level inflation expectations, but results are statistically insignificant.
  - Medium-run: personal income tax shocks are inflationary by lowering frequencies of small positive inflation expectations ranging between 0 to 3 percent and increasing frequencies of expected inflations beyond 3 percent.
- Moments (Figure 17):
  - Personal income tax shocks lead to a persistent increase in the average inflation expectations at levels higher than 2 percent for medium-run expectations.
  - This is associated with a persistent decrease in the frequency of inflation expectations at 2 percent.

*Source: wpiea2022132-print-pdf - 5.2    Monetary Policy Shocks and EIDs*

### 5.4 Gasoline Price Shocks and EIDs

#### At-impact distributional responses
- Gasoline price shocks are inflationary.
- One-year ahead inflation expectations:
  - A surprise hike in gasoline prices decreases the frequencies of inflation expectations below 3 percent and increases the frequencies of inflation expectations higher than 3 percent.
- Medium-run inflation expectations:
  - A surprise hike in gasoline prices increases densities for higher levels of inflation expectations and also increases densities for negative values of inflation expectations, implying increased disagreement among households about medium-run inflation outcomes.

#### Moments and disagreement (Figure 19)
- Mean:
  - A surprise increase in gasoline prices increases the average level of inflation expectations for both one-year ahead and medium-run horizons.
  - Magnitude: the response of one-year ahead inflation expectations at impact is nearly three times larger than that of medium-run inflation expectations.
- Standard deviation / disagreement:
  - One-year ahead: disagreement decreases for nearly one year, implying greater consensus that gasoline shocks will raise short-run inflation expectations.
  - Medium-run: standard deviation increases persistently, implying growing disagreement on how current gasoline price shocks will affect medium-run inflation.
- Frequency at 2 percent:
  - Gasoline price shocks decrease the frequency at 2 percent for both one-year ahead and medium-run inflation expectations, indicating these shocks do not support the central bank’s inflation targeting.

*Source: wpiea2022132-print-pdf - 5.2    Monetary Policy Shocks and EIDs*

### 5.5    Economic Interpretation of Findings

### 5.5    Economic Interpretation of Findings

### Summary of empirical findings
- Contractionary monetary policy shocks have inflationary effects on household inflation expectations, but only for one-year ahead inflation expectations.
- Anchoring effects, with a reduction in disagreement with an increased frequency of responses between 2 to 4 percent, are observed, but only for samples that exclude the period prior to Volcker disinflation.
- Government spending shocks are inflationary for both one-year ahead and medium-run inflation expectations, while increasing the disagreement for medium-run inflation expectations.
- Personal income tax shocks are inflationary, but only for medium-run inflation expectations.
- A surprise increase in gasoline prices increases the level of inflation expectations for both one-year ahead and medium-run inflation expectations, while disagreement increases for medium-run inflation expectations.
- Interpretations provided below are tentative and "need to be validated with a more rigorous analysis."

### Interpretation of monetary policy effects
- Two plausible channels for why contractionary monetary policy shocks lead to an increase in household inflation expectations:
  - Information channel (Nakamura and Steinsson (2018)):
    - Monetary tightening may signal that the central bank has superior information about the projection of inflation.
    - If this channel dominates, households may interpret contractionary monetary policy as indicating that ’actual’ underlying inflationary pressures are higher than households previously believed, prompting upward revisions in expectations.
  - Perception of inflation as negative news about economic conditions (Coibion and Gorodnichenko (2015)):
    - Households may view tightening monetary policy as forecasting a slowdown in economic activity.
    - If households associate projected economic downturns with higher future inflation, they may raise inflation expectations.
- Empirical literature has documented contractionary monetary policy shocks leading to an increase in inflation through price puzzles (Sims, 1992; Hanson, 2004; Rusnák, Havranek, and Horváth, 2013) and related findings (Uribe, 2018).

### Interpretation of fiscal policy effects
- Government spending shocks:
  - Findings that fiscal policies have medium-run effects are consistent with inter-temporal government budget constraint perspectives (Leeper and Nason, 2010): fiscal decisions today affect the level of future prices, so households may adjust beliefs about future prices contemporaneously with fiscal shocks.
  - Delayed realization channel (Leeper, Walker, and Yang, 2010): households may believe expansionary effects of government spending are implemented with delay, leading to upward revisions of medium-run inflation expectations.
  - Increased disagreement (higher standard deviation) for medium-run expectations may reflect heterogeneous beliefs about the future multiplier of government spending.
  - Attentiveness to fiscal shocks: households who are not directly attentive to fiscal shocks may still adjust expectations to future government debt (Coibion, Gorodnichenko, and Weber, 2021); some households fearful of fiscal dominance may predict higher future inflation when government spending increases.
- Personal income tax shocks:
  - An increase in personal income tax associated with higher medium-run inflation expectations could be consistent under strong assumptions of perfect foresight, if households perceive current tax increases as linked to higher future government spending.
  - Perfect foresight is a strong assumption that does not allow agents to be uncertain about others’ beliefs and responses (Angeletos and Lian (2018); García-Schmidt and Woodford (2019)).
  - Prior evidence (Mertens and Ravn (2013)) finds a cut in personal income tax does not influence the level of inflation, indicating that the channels linking personal income tax changes to household inflation expectations require better understanding.

### Interpretation of gasoline price effects
- Surprise gasoline price increases:
  - Increase the level of both one-year ahead and medium-run household inflation expectations.
  - Increase disagreement for medium-run inflation expectations.
  - The literature relatively well understands gasoline price effects on household inflation expectations compared with monetary and fiscal policy channels.

### Implications and caveats
- Monetary and fiscal policy shocks exert distinct influences on the distribution of household inflation expectations; monetary shocks mainly affect one-year ahead expectations while fiscal spending affects both horizons and influences disagreement.
- Findings suggest that economic policies can play a role in anchoring household inflation expectations, contrary to prior literature suggesting limited influence.
- Further research is needed to:
  - Validate the proposed channels with more rigorous analysis.
  - Understand how households perceive and incorporate personal income tax changes into inflation expectations.
  - Explore how the distribution of inflation expectations interacts with other macroeconomic variables and how monetary and fiscal policies jointly shape that distribution.

*Source: wpiea2022132-print-pdf - 5.5    Economic Interpretation of Findings*

### References

### wpiea2022132-print-pdf - References

### Choice of Basis (Appendix A)
- Methodology:
  - The VAR representation in (6) may be implemented with any orthonormal basis pv_i q of H; performance depends critically on choice of basis.
  - Denote functional principal component basis used in the paper by pv̊_i q (m components) and its subspace by V̊; projection operator P̊. Arbitrary sub-basis pv_i q spans V with projection P.
  - FR^2 (functional R-squared) of a sub-basis pv_i q_{i=1}^m is FR^2 = Σ_{t=1}^T ||P f_t||^2 / Σ_{t=1}^T ||f_t||^2; FR^2̊ denotes FR^2 for pv̊_i q.
  - FR^2̊ ≥ FR^2 for any other basis; P̊ f_t has maximum temporal variation.
- Variance–bias tradeoff:
  - Increasing truncation number m increases FR^2 but can sharply increase variances of estimators pA_k for autoregressive operators A_k (k = 1,2).
  - The MSE of pA decomposes into variance and squared bias terms:
    - Variance term approximately traceΣ + trace(Q Λ Q) ́ (notation as in text).
    - Squared bias term approximately ||A(1 − Q) Δ Q (Q Δ Q) ́||^2 (notation as in text).
  - Using the FPC basis reduces the variance term E||pÅ − E pÅ||^2 relative to other bases; bias term E||pÅ − Å||^2 generally expected to be smaller because P̊ Γ(1 − P̊) = 0 while P Γ(1 − P) ≠ 0 for other bases.
  - If first-order FAR f_t = A f_{t−1} + ε_t holds, the estimator pÅ is unbiased.
- Comparison of bases (empirical results):
  - Four bases compared: functional principal component (FPC) basis, histogram basis, quantile basis, moment basis.
  - Table 4. FR2 for Four Choices of Basis (rows show m; columns show FR2 for each basis)
    - m = 10: FPC Basis 0.7419, Histogram Basis 0.0095, Quantile Basis 0.0322, Moment Basis 0.0048
    - m = 20: FPC Basis 0.8829, Histogram Basis 0.0123, Quantile Basis 0.3707, Moment Basis 0.0064
    - m = 30: FPC Basis 0.9513, Histogram Basis 0.0400, Quantile Basis 0.5573, Moment Basis 0.0172
    - m = 40: FPC Basis 0.9676, Histogram Basis 0.1103, Quantile Basis 0.5611, Moment Basis 0.0220
    - m = 50: FPC Basis 0.9785, Histogram Basis 0.0735, Quantile Basis 0.7181, Moment Basis 0.0353
    - m = 60: FPC Basis 0.9861, Histogram Basis 0.2789, Quantile Basis 0.7309, Moment Basis 0.0461
    - m = 70: FPC Basis 0.9898, Histogram Basis 0.1430, Quantile Basis 0.7580, Moment Basis 0.0561
    - m = 80: FPC Basis 0.9921, Histogram Basis 0.3938, Quantile Basis 0.7731, Moment Basis 0.0678
    - m = 90: FPC Basis 0.9936, Histogram Basis 0.2482, Quantile Basis 0.7737, Moment Basis 0.0771
    - m = 100: FPC Basis 0.9946, Histogram Basis 0.4142, Quantile Basis 0.7849, Moment Basis 0.0878
    - Note: FR2 is expected to strictly increase as m gets large; histogram basis does not satisfy this due to definition differences across m.
  - Table 5. trace(Q Λ Q) ́ for Four Choices of Basis (×10^4) (rows show m; columns show trace values)
    - m = 10: FPC Basis 0.613, Histogram Basis 16.0847, Quantile Basis 7.9906, Moment Basis 9.714
    - m = 20: FPC Basis 1.4752, Histogram Basis 230.7581, Quantile Basis 0.8676, Moment Basis 39.162
    - m = 30: FPC Basis 3.4709, Histogram Basis 890.5797, Quantile Basis 27.6424, Moment Basis 3916.064
    - m = 40: FPC Basis 4.8939, Histogram Basis 3743.1065, Quantile Basis 52.2481, Moment Basis 11926.281
    - m = 50: FPC Basis 16.9301, Histogram Basis 7148.7476, Quantile Basis 57.7802, Moment Basis 16819.808
    - m = 60: FPC Basis 24.6762, Histogram Basis 1877.4292, Quantile Basis 209.5832, Moment Basis 21002.513
    - m = 70: FPC Basis 39.7823, Histogram Basis 30720.005, Quantile Basis 161.3063, Moment Basis 27026.392
    - m = 80: FPC Basis 60.5034, Histogram Basis 41978.159, Quantile Basis 487.0560, Moment Basis 35492.106
    - m = 90: FPC Basis 88.2825, Histogram Basis 53705.816, Quantile Basis 666.3564, Moment Basis 42583.255
    - m = 100: FPC Basis 10124.7846, Histogram Basis 2145.588, Quantile Basis 1079.753, Moment Basis 59256.713
    - Notes: trace(Q Λ Q) ́ values are asymptotically proportional to variances of autoregressive operator estimators in the Hilbert-Schmidt norm.
  - Empirical implications:
    - The FPC basis effectively represents temporal variation of the EID even with small m (m = 1 or 2).
    - For m = 3, FPC captures more than 95% of total temporal variation of the EID.
    - Variance of autoregressive operator estimator using FPC increases moderately with m; other bases show much faster increases and may yield exploding variances (histogram or moment basis).
    - Moment basis performs particularly poorly: it captures only 8.7 percent of total variation for m = 10 (table shows moment basis FR2 = 0.0878 at m = 100; text notes 8.7 percent for m = 10 in narrative).
  - Choice used in empirical analysis:
    - The authors use m = 3 with the FPC basis.
    - This choice explains 95.1 percent of total EID variation and yields variance measure 4.7 for the autoregressive operator estimator.
    - If the moment basis is used, only 1.7 percent of total EID variation is explained while corresponding variance measure is 3916 for the autoregressive operator estimator.

### Interpretation of Functional Principal Components (Appendix B)
- Objective:
  - Examine how each of the three FPCs affects the first four standardized moments (mean, standard deviation, skewness, kurtosis) computed from the EID across the entire range of historical loadings.
- Empirical labeling and justification:
  - First FPC (blue) — labeled the "dispersion" (disagreement) component:
    - The second moment (variance) varies more than the first moment across loadings.
    - Variations in third and fourth moments are implied by those in first and second moments.
    - Significant variation in kurtosis observed; movement in second moment has clear economic interpretation.
  - Second FPC (red) — labeled the "shift" component:
    - The first moment (mean) shows the strongest relationship with the loading.
  - Third FPC (green) — labeled the "ambiguity" component:
    - First three moments are relatively stable across loadings; fourth moment (kurtosis) shows clear relationship with loadings.
    - Economic meaning of fourth moment is difficult to interpret; hence "ambiguity" label.

### State-Dependent IRFs: Recession vs. Non-Recession (Appendix C)
- Exercise:
  - Compare at-impact responses of one-year ahead inflation expectations for recession vs. non-recession periods using NBER recession dates.
  - Focus limited to one-year ahead expectations due to too few matched recession episodes for medium-run expectations.
  - Figure 21 presents at-impact responses of four economic shocks in three rows: non-recession (top), entire sample (middle: 01/1983 to 12/2021), recession (bottom).
- Key findings:
  - Fiscal policy shocks:
    - Effects on distribution of household inflation expectations do not depend on state (recession vs. non-recession).
    - Confidence bands widen during recessions (likely due to fewer observations) but coefficients and IRF shapes remain relatively unchanged across states.
  - Contractionary monetary policy shocks and gasoline price increases:
    - Inflationary effects on household inflation expectations are stronger during recessions compared to non-recession periods.
    - For monetary policy: peak coefficient during recession is almost twice as large as during non-recession; increase around the peak accompanied by fall around the trough during recessions.
    - For gasoline price shocks: during recessions, a dramatic fall in frequency of negative inflation expectations is accompanied by an increase of similar size in frequency of positive inflation expectations, strengthening inflationary effects.
- Interpretation and open questions:
  - Possible channels:
    - Contractionary monetary policy in recession may worsen economic conditions; households translate pessimism into higher inflation expectations (see Coibion, Gorodnichenko, and Kamdar (2018) for discussion).
    - Households may not expect central bank to persist with contractionary policy during recession, leading to upward revision of future inflation expectations.
    - Increases in gasoline prices during recessions may be perceived as signals of forthcoming stagflation, prompting upward revisions.
  - Authors note further research needed to identify underlying mechanisms.

### Additional Figures (Appendix D)
- Figure 22 (description):
  - Functional principal components and loadings for medium-run inflation expectations:
    - Left panels: loadings over time for three principal components (top: disagreement; middle: shift; bottom: uncertainty).
    - Center panels: each principal component as a function (integrates to 0).
    - Right panels: how sample mean density function is modified with maximum and minimum loadings from left panels.

*The Effects of Economic Shocks on Heterogeneous Inflation Expectations — Working Paper No. WP/2022/132 (References and Appendices A–D).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022132-print-pdf.pdf_