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---

### Introduction and research question
- Focus: whether distributional characteristics of household one-year-ahead inflation expectations contain information about future inflation.
- Motivation: Recent sharp rise in inflation since 2021 increased interest in inflation expectations of households and firms; deviations from full information rational expectations (FIRE) models are empirically important.
- Key premise: Household expectations often differ from professional forecasters and markets and exhibit relatively wide dispersion across households.
- Scope: Microdata from Canada, Germany, the United Kingdom (UK), and the United States (US); econometric focus on US.

### Main findings (summary)
- Distributions of one-year expectations shifted rightward in all four countries during the inflation surge since 2021, with more households expecting higher inflation outcomes.
- While expectations have begun to come down recently, in each country:
  - the median household continues to expect higher inflation one year into the future, and
  - a significant share of households are expecting inflation greater than 10 percent one year out.
- Econometric results (US):
  - The shape of the distribution of household expectations helps predict one-year-ahead CPI inflation.
  - Variance and skewness of household expectations’ distributions add predictive power beyond and above the median.
  - A decrease in the skewness is predictive of higher inflation one year out.
  - Higher variance signals a change in inflationary regime.
  - These qualitative results continue to hold when including market-based measures and moments of the distribution of professional forecasts.
  - Including the first three moments of the household inflation expectations distribution improves out-of-sample forecasts, relative to the median-only model, for the most recent inflationary period if the model is estimated on the 1970s/80s inflationary period.

### Data description and construction (country-specific)
- General:
  - Focus is on one-year-ahead expectations (expectations formed in month t about expected change in prices during the next 12 months).
  - For US, UK, and Germany a kernel density method following Reis (2021) is used to construct densities; for Canada, published percentiles are used to fit a skewed-t distribution.
  - Functional Principal Component Analysis (FPCA) is used as a robustness check to capture the distribution of household inflation expectations in forecasting exercises.
- US (University of Michigan Survey of Consumers):
  - Monthly data available from January 1978.
  - Survey: rotating panel, every edition includes a minimum of 600 interviews.
  - Question structure: (1) direction (Higher/Stay the same/Lower/Don’t Know), (2) percentage change expectation (or Don’t Know).
  - Construction details:
    - Use binned data published by the University of Michigan.
    - Assign lower bound of -5 and upper bound of 20 to the bottom and uppermost bins respectively.
    - Assign frequencies to mid-points of inflation bins.
    - Exclude missing data points (“NA” and “DK whether up or down”); code 96 (“DK how much up”) assigned the average of positive inflation expectations.
    - Fit a kernel density function of bandwidth 1.3.
- UK (Bank of England/Ipsos Inflation Attitudes Survey):
  - Quarterly survey conducted since 2001; minimum of 2000 respondents each quarter, random location sample.
  - Respondents choose from ranges (e.g., up by 1 percent or less, up by 1 percent but less than 2 percent, etc.).
  - Survey changes: highest buckets evolved to “up by 15 percent or more” by 2022; switch to Ipsos online format in February 2022.
  - Construction details:
    - For the largest bucket (“10 percent or more”) assume upper bound of 15 percent and mid-point of 12 percent.
    - Allocate deflation expectations in the category “down, i.e., smaller or equal 0 percent” and use -2 percent as the mid-point.
    - Fit a kernel density function of bandwidth 1.3.
- Germany:
  - Consumer Expectations Survey (CES) by the European Central Bank, monthly since April 2020; Bundesbank Survey of Consumer Expectations data used (Annex I).
  - Bundesbank: approximately 2000 households per wave; two-stage question; bins composed like US bins; Reis’ method and bandwidth 1.3 used.
  - Non-integer values outside Michigan buckets rounded to nearest integer.
- Canada:
  - Bank of Canada provided percentiles of point forecasts; authors fit a skewed t-distribution following Azzalini and Capitanio (2003).

### Econometric approach
- Forecasting model:
  - Regress one-year-ahead CPI inflation on current inflation and moments of the distribution of one-year-ahead household inflation expectations.
  - Also include moments of professional forecasts and market-based measures in robustness checks.
- Moments used:
  - Median, variance, skewness (and first three moments collectively in some specifications).
- Baseline samples and frequencies:
  - Baseline monthly models: January 1978 - June 2023 (households only); October 1989 - June 2023 for models including professionals.
  - Quarterly robustness: 1966Q2 - 2023Q2.
- Additional methods:
  - Functional Principal Component Analysis (FPCA) used as robustness; several control variables and lags tested (unemployment rate, capacity utilization, commodity inflation); Newey-West errors in reported regressions.

### Key empirical findings — baseline and subsample results (US)
- Informativeness:
  - Shape of the household distribution is informative about future inflation; three-moment specifications yield statistically significant coefficients.
- Breakpoint analysis:
  - Bai and Perron (2003) breakpoint test (maximum 2 breakpoints) partitions the sample into regimes including 1978M1-1990M7 and 2015M11-2022M6.
  - Models perform best in high-inflation periods: adjusted R-squared highest during the first inflationary period and next highest during the most recent inflationary period.
- Coefficient signs:
  - Median coefficient: positive.
  - Skewness coefficient: negative.
  - Variance coefficient: sign depends on inflation regime:
    - Negative in the first subsample (peak and aftermath 1978-83).
    - Positive in the most recent subsample (lead-up to an inflation peak).
- Representative adjusted R-squared observations:
  - Adjusted R-squared values reported include 0.61, 0.62, 0.62, 0.81, 0.85, 0.87 across different specifications and subsamples.
- Quarterly results:
  - Quarterly full-sample results consistent with monthly findings: median positive; variance and skewness negative in full sample and significant.
  - Subsample dynamics mirror monthly patterns with variance and skewness switching signs/significance across regimes.

### Regime-dependent behavior and interaction tests
- Two exercises tested variance sign-dependence:
  - Inclusion of regime dummies (post-min/post-max).
  - Inclusion of dummies for substantially increasing vs decreasing inflation.
- Findings:
  - Variance coefficient significantly higher after inflation has reached a minimum and/or when inflation has substantially increased over the past year.
  - Variance coefficient significantly lower after inflation has reached a maximum and/or when inflation has substantially decreased over the past year.
  - Models including “increasing / decreasing inflation” dummies produce more significant results and higher adjusted R-squares.
- Definitions used:
  - Increasing (decreasing) inflation monthly: 3-month-average 1-year-ahead greater than the 3-month average one year ago plus 1 percentage point.
  - Quarterly definition: 1-year-ahead inflation greater than the inflation one year ago plus 1 percentage point.
  - Post min (max): 5-year minimum (maximum) in previous 6 months (or previous 2 quarters for quarterly).

### Horse races: households vs professionals and markets
- Approach:
  - Compare household moments with professional forecasts (Consensus Forecasts) and market-based expectations (Cleveland Fed market-implied expectations) in joint regressions (Consensus available October 1989 onward).
- Main result:
  - Moments of household expectations add informational value beyond market-based and professional forecasts, particularly in the most recent period.
  - In the most recent subsample, household median and variance coefficients remain positive and significant (except median in one specification). Median of professional forecasters is significant but negative in that subsample.

### Out-of-sample forecasting performance
- Procedure:
  - Estimate models excluding the last three or five years and compute RMSEs comparing predicted and actual inflation in the excluded period.
- Summary RMSEs (three- and five-year evaluations):
  - 3 years: 3.35, 3.02, 3.17, 3.12, 2.13
  - 5 years: 2.66, 2.41, 2.57, 2.52, 1.69
- Findings:
  - Including distributional characteristics improves out-of-sample forecasting only when accounting for different inflation regimes (interaction terms with variance*increasing and variance*decreasing inflation).
- Restricted-estimation exercise (estimate on 1978m1-1980m12, evaluate on 2020m7-2022m6):
  - RMSEs shown: 7.27, 5.74, 3.17, 2.19 for progressively richer specifications; including more moments consistently decreases RMSEs when estimated on a prior inflationary episode.

### Robustness
- Robustness checks and variants reported in Annex III include:
  - Different subsamples, three breakpoints; inclusion of lags (Inflation(-12), Inflation(-24), Inflation(-36)); controls for Unempl. rate, Capacity utilization, Commodity infl.; first-difference specifications; published vs micro moments; Interquartile range (IQR) instead of Variance; quarterly data.
- Representative reported statistics in Annex III:
  - Sample periods include 1978M1-2022M6, 1978M1-1990M7, 1990M8-2007M10, 2007M11-2015M10, 2015M11-2022M6.
  - N values: 534, 151, 303, 80, 498, 115, 30, 174, 533, 150.
  - Adj. R-squared values appearing include: 0.29, 0.54, 0.61, 0.85, 0.04, 0.62, 0.87, 0.81, 0.46, 0.61, 0.67, 0.69, 0.14, 0.19, 0.35, 0.44, 0.78, 0.80, 0.76, 0.64, 0.65.
  - Example coefficient magnitudes reported: Median coefficients 1.07***, 1.08***, 1.12***, 1.17***, 1.13***; Variance / IQR examples -0.04, -0.07**, -0.17***; IQR -0.16, -0.46***, -0.53***; Skewness examples -1.72**, -3.99***, -2.65***, -5.30***.
- FPCA robustness:
  - FPCA approximates distributions and three components explain around 94 percent of variation; FPCA-based scores are predictive though less directly interpretable than moments.

### Functional Principal Component Analysis (Annex IV) — key points
- Data: US household one-year ahead expectations, January 1978 to April 2023.
- Variance explained:
  - Three components explain around 94 percent of variation.
- Models:
  - Uncentered model A.1 and centered model A.2 (centered includes mean expected inflation π̅_{t}^{e}).
- Findings:
  - Uncentered FPCA: shape informative regardless of number of scores; with three scores, score 1 and score 3 significant (full sample); one-score specifications typically find score 1 significant.
  - Centered FPCA: mean and score 2 highly significant for full period; score 1 significant at 10 percent.
  - The first principal component from uncentered FPCA is highly negatively correlated with the mean; mean tends to have positive significant coefficient in centered specification.
  - R-squared highest for earliest sub-sample, next highest for most recent sub-sample.

### Policy-relevant conclusions and research agenda
- Policy-relevant conclusions:
  - Distributions of household one-year-ahead inflation expectations contain information complementary to professional forecasts and market-implied expectations.
  - Household expectations, especially distributional features (median, variance, skewness), have predictive power for short-term inflation dynamics, but the sign and strength of that power depend on the inflation regime.
  - Policymakers should monitor household expectations closely, particularly around turning points (transitions between low- and high-inflation regimes).
- Open questions and suggested future research:
  - Extend similar quantitative investigations to other countries with long micro-data series.
  - Study the role of household expectations around turning points (attention/inattention dynamics).
  - Examine heterogeneity: how different population segments react; role in wage negotiations; impact on consumption and saving decisions across population segments.

*Source: wpiea2023224-print-pdf - References (IMF Working Paper "Do Household Expectations Help Predict Inflation?").*

### REFERENCES ______________________________________________________________________ 18

### wpiea2023224-print-pdf - REFERENCES ______________________________________________________________________ 18

### Introduction and research question
- Focus: whether distributional characteristics of household one-year-ahead inflation expectations contain information about future inflation.
- Motivation: Recent sharp rise in inflation since 2021 increased interest in inflation expectations of households and firms; deviations from full information rational expectations (FIRE) models are empirically important.
- Key premise: Household expectations often differ from professional forecasters and markets and exhibit relatively wide dispersion across households.
- Scope: Microdata from Canada, Germany, the United Kingdom (UK), and the United States (US); econometric focus on US.

### Main findings (summary)
- Distributions of one-year expectations shifted rightward in all four countries during the inflation surge since 2021, with more households expecting higher inflation outcomes.
- While expectations have begun to come down recently, in each country:
  - the median household continues to expect higher inflation one year into the future, and
  - a significant share of households are expecting inflation greater than 10 percent one year out.
- Econometric results (US):
  - The shape of the distribution of household expectations helps predict one-year-ahead CPI inflation.
  - Variance and skewness of household expectations’ distributions add predictive power beyond and above the median.
  - A decrease in the skewness is predictive of higher inflation one year out.
  - Higher variance signals a change in inflationary regime.
  - These qualitative results continue to hold when including market-based measures and moments of the distribution of professional forecasts.
  - Including the first three moments of the household inflation expectations distribution improves out-of-sample forecasts, relative to the median-only model, for the most recent inflationary period if the model is estimated on the 1970s/80s inflationary period.

### Data description and construction (country-specific)
- General:
  - Focus is on one-year-ahead expectations (expectations formed in month t about expected change in prices during the next 12 months).
  - For US, UK, and Germany a kernel density method following Reis (2021) is used to construct densities; for Canada, published percentiles are used to fit a skewed-t distribution.
  - Functional Principal Component Analysis (FPCA) is used as a robustness check to capture the distribution of household inflation expectations in forecasting exercises.
- US (University of Michigan Survey of Consumers):
  - Monthly data available from January 1978.
  - Survey: rotating panel, every edition includes a minimum of 600 interviews.
  - Question structure: (1) direction (Higher/Stay the same/Lower/Don’t Know), (2) percentage change expectation (or Don’t Know).
  - Construction details:
    - Use binned data published by the University of Michigan.
    - Assign lower bound of -5 and upper bound of 20 to the bottom and uppermost bins respectively.
    - Assign frequencies to mid-points of inflation bins.
    - Exclude missing data points (“NA” and “DK whether up or down”); code 96 (“DK how much up”) assigned the average of positive inflation expectations.
    - Fit a kernel density function of bandwidth 1.3.
- UK (Bank of England/Ipsos Inflation Attitudes Survey):
  - Quarterly survey conducted since 2001; minimum of 2000 respondents each quarter, random location sample.
  - Respondents choose from ranges (e.g., up by 1 percent or less, up by 1 percent but less than 2 percent, etc.).
  - Survey changes:
    - Prior to 2008 highest bucket was “up by 5 percent or more”.
    - Since 2008, follow-up allowed with highest possible range “up by 10 percent or more”.
    - In 2022, a third follow-up allowed with highest option “up by 15 percent or more”.
    - Prior to February 2022 Kantar conducted the survey; since February 2022 Ipsos has been conducting the survey, permanently switching to an online format.
  - Construction details:
    - For the largest bucket (“10 percent or more”) assume upper bound of 15 percent and mid-point of 12 percent.
    - Allocate deflation expectations in the category “down, i.e., smaller or equal 0 percent” in a similar way as Michigan survey and use -2 percent as the mid-point.
    - Fit a kernel density function of bandwidth 1.3.
- German data:
  - Consumer Expectations Survey (CES) is an online panel by the European Central Bank conducted monthly since April 2020.
  - Initial target sample: 2000 monthly respondents from Germany until June 2021; sample size increased as of July 2021.
  - For Germany the authors also received direct Bundesbank data on its Survey of Consumer Expectations (results reported in Annex I).

### Econometric approach
- Forecasting model:
  - Regress one-year-ahead CPI inflation on current inflation and moments of the distribution of one-year-ahead household inflation expectations.
  - Also include moments of professional forecasts and market-based measures in robustness checks.
- Moments used:
  - Median, variance, skewness (and first three moments collectively in some specifications).
- Robustness and supplementary analyses:
  - Use Functional Principal Component Analysis (FPCA) to capture distributional information.
  - Multiple robustness checks reported in annexes: differing subsamples, alternative dummies, quarterly data, inclusion of lags, controls (unemployment rate, capacity utilization, commodity inflation), first differences, interquartile range instead of variance, micro data-based moments, and models with published vs. micro moments.
  - Out-of-sample forecast performance evaluated, with improvements noted for specific estimation/evaluation setups (e.g., estimating on the 1970s/80s inflationary period).

### Figures and tables (as listed in the content)
- Figures include distributions and time-series of one-year expectations across US, UK, Germany, and Canada and annex figures on German expectations, moments, FPCA variance explained, and centered/uncentered distributions.
- Tables include dataset characteristics, moments model results across subsamples and dummies, horse races with professional forecasters and market expectations, out-of-sample RMSEs (full sample and restricted periods), and numerous annex tables for robustness checks and FPCA/score models.

*Source: wpiea2023224-print-pdf - REFERENCES ______________________________________________________________________ 18*

### 3000. The survey enquires about one-year-ahead inflation expectations in two stages. Firstly, respondents are

### wpiea2023224-print-pdf - 3000. The survey enquires about one-year-ahead inflation expectations in two stages. Firstly, respondents are

### Data and construction of distributions
- Survey design:
  - One-year-ahead inflation expectations elicited in two stages: qualitative response on direction over 12 months and a point estimate in the range of -100% to 100%.
  - A third question (not used here) asks probability allocations across categories with probabilities summing to 100.
- Binning and density construction:
  - German bins composed using answers to the point-forecast question and constructed similarly to US bins.
  - Reis’ method used: assign bin frequency to bin mid-point and fit a kernel density function with bandwidth 1.3.
  - For Canada, Bank of Canada provided percentiles of point forecasts; authors fit a skewed t-distribution following Azzalini and Capitanio (2003).
- Key data sets and sample coverage:
  - Michigan Survey (household expectations), monthly January 1978 – April 2023; minimum of 600 interviews; one year ahead; distribution available.
  - Michigan Survey, quarterly Q2 1966 – Q2 2023; minimum of 600 interviews; one year ahead; distribution available.
  - Consensus Forecasts (professionals), October 1989 – June 2023; between 19 and 32 respondents; current and next year forecasts; distributions constructed from those series.
  - Cleveland Fed market-implied expectations, January 1982 – June 2023; one year ahead; no distribution.

### Recent shifts in household expectation distributions (US, UK, Germany, Canada)
- Common patterns across the four countries since 2019:
  - Distributions shifted rightward; more households expect higher inflation one year ahead.
  - The median household expects higher inflation one year in the future than in 2019-20.
  - A significant share of households expect inflation greater than 10 percent one year out.
- Changes in dispersion and shape:
  - Increases in variance (more disagreement) and fattening of the right tail.
  - US, UK, and Germany: distributions became more bi- or trimodal; secondary modes at:
    - UK and Germany: 10-15 percent.
    - US: 15-20 percent.
- Recent dynamics:
  - Most recent surveys suggest expectations have slowly started to come down, but medians remain substantially above target except in Canada.
  - Five-year horizon expectations have shifted much less than one-year expectations.

### Econometric framework for predictive analysis (US focus)
- Forecasting equation:
  - One-year-ahead inflation regressed on current inflation and the first three moments (mean/median, variance, skewness) of household expectation distributions:
    - π_{t+12} = c + α π_t + Σ_{k=1}^K β_k m_{k,t} + ε_t
- Frequency and samples used:
  - Baseline monthly models: January 1978 - June 2023 (households only); October 1989 - June 2023 for models including professionals.
  - Quarterly robustness: 1966Q2 - 2023Q2.
- Additional methods:
  - Functional principal components analysis (FPCA) used in robustness checks to describe distributional shape.
  - Several control variables and lags of inflation tested for robustness: monthly unemployment rate, monthly producer price inflation (all commodities), monthly capacity utilization.

### Key empirical findings — baseline and subsample results (US)
- Overall informativeness:
  - Shape of the household distribution is informative about future inflation; three-moment specifications yield statistically significant coefficients.
- Breakpoint analysis:
  - Bai and Perron (2003) breakpoint test (maximum 2 breakpoints) partitions the sample into three regimes, identifying inflationary regimes:
    - 1978M1-1990M7
    - 2015M11-2022M6
  - Models perform best in high-inflation periods: adjusted R-squared highest during the first inflationary period and next highest during the most recent inflationary period.
- Coefficient signs (as hypothesized):
  - Median coefficient: positive.
  - Skewness coefficient: negative.
  - Variance coefficient: sign depends on inflation regime:
    - Negative in the first subsample (peak and aftermath 1978-83).
    - Positive in the most recent subsample (lead-up to an inflation peak).
- Representative adjusted R-squared observations (from tables and descriptions):
  - Adjusted R-squared values reported include 0.61, 0.62, 0.62, 0.81, 0.85, 0.87 across different specifications and subsamples.
- Quarterly results (1966Q2 onward):
  - Quarterly full-sample results consistent with monthly findings: median positive; variance and skewness negative in full sample and significant.
  - Subsample dynamics: early sharply rising inflation period shows positive and significant variance coefficient; peak-and-decline subsample shows negative and significant variance coefficient; recent increase subsample shows positive and significant variance and negative and significant skewness.

### Regime-dependent behavior of variance and interaction tests
- Two exercises to test variance sign-dependence:
  - Inclusion of regime dummies (post-min/post-max).
  - Inclusion of dummies for substantially increasing vs decreasing inflation.
- Findings:
  - Variance coefficient significantly higher after inflation has reached a minimum and/or when inflation has substantially increased over the past year.
  - Variance coefficient significantly lower after inflation has reached a maximum and/or when inflation has substantially decreased over the past year.
  - Models including “increasing / decreasing inflation” dummies produce more significant results and higher adjusted R-squares.

### Horse race: households vs professionals and markets
- Approach:
  - Compare moments from household distributions with moments from professional forecasters and market-based expectations in joint regressions (sample restricted due to Consensus Forecasts availability from October 1989).
- Main result:
  - Moments of household expectations add informational value beyond market-based and professional forecasts, particularly in the most recent period.
  - In the most recent subsample, household median and variance coefficients remain positive and significant (except median in one specification). Median of professional forecasters is significant but negative in that subsample.

### Out-of-sample forecasting performance
- Procedure:
  - Estimate models excluding the last three or five years and compute RMSEs comparing predicted and actual inflation in the excluded period.
  - Also test including a dummy for high inflation.
- Summary RMSEs (three- and five-year evaluations):
  - 3 years: 3.35, 3.02, 3.17, 3.12, 2.13
  - 5 years: 2.66, 2.41, 2.57, 2.52, 1.69
  - Including distributional characteristics improves out-of-sample forecasting only when accounting for different inflation regimes (interaction terms with variance*increasing and variance*decreasing inflation).
- Restricted-estimation exercise (estimate on 1978m1-1980m12, evaluate on 2020m7-2022m6):
  - RMSEs shown: 7.27, 5.74, 3.17, 2.19 for progressively richer specifications (only constant; plus median; plus variance; plus skewness; plus variance*regime interactions where applicable).
  - Including more moments consistently decreases RMSEs when model is estimated on a prior inflationary episode.

### Robustness
- Robustness checks reported:
  - Models in first differences give similar coefficient sign patterns but lower significance.
  - FPCA approximates distributions better than moments but yields less-easily-interpretable components.
  - Results robust to using moments based on micro data vs published moments in many specifications; some differences in latest subsample when using published moments (variance sometimes insignificant).
  - Additional robustness tables and exercises reported in Annexes III and IV.

### Policy-relevant conclusions and research agenda
- Main policy-relevant conclusions:
  - Distributions of household one-year-ahead inflation expectations contain information complementary to professional forecasts and market-implied expectations.
  - Household expectations, especially distributional features (median, variance, skewness), have predictive power for short-term inflation dynamics, but the sign and strength of that power depend on the inflation regime.
  - Policymakers should monitor household expectations closely, particularly around turning points (transitions between low- and high-inflation regimes).
- Open questions and suggested future research:
  - Extend similar quantitative investigations to other countries with long micro-data series.
  - Study the role of household expectations around turning points (attention/inattention dynamics).
  - Examine heterogeneity: how different population segments react; role in wage negotiations; impact on consumption and saving decisions across population segments.

*IMF WORKING PAPERS: Do Household Expectations Help Predict Inflation?*

### References

### References

### Bibliographic sources cited
- Alvarez, J., Bluedorn, J. C., Hansen, N. J. H., Huang, Y., Pugacheva, E., and Sollaci, A., (2022). “Wage-Price Spirals: What is the Historical Evidence?” IMF Working Paper No. 2022/221, (Washington, International Monetary Fund).
- Andrade, P., Gautier, E., and Mengus, E., (2020). “What Matters in Households’ Inflation Expectations?” FRBB, Research Department Working Papers No. 20-18, (Boston, Federal Reserve Bank of Boston).
- Azzalini, A., and Capitanio, A., (2003). “Distributions Generated by Perturbations of Symmetry with Emphasis on a Multivariate Skew t Distribution.” Journal of the Royal Statistical Society, Series B 65, Pp. 367–89.
- Bai, J. and Perron, P., (2003), "Computation and Analysis of Multiple Structural Change Models", Journal of Applied Econometrics, Vol. 18, Issue 1, Pp 1-22.
- Botsch, M. J., and Malmendier, U., (2021). “The Long Shadows of the Great Inflation: Evidence from Residential Mortgages” CEPR Discussion Paper No. DP14934 (London, Centre for Economic Policy Research).
- Bracha, A., and Tang, J., (2022). “Inflation Levels and (In)Attention,” FRBB, Research Department Working Papers No. 22-4, (Boston, Federal Reserve Bank of Boston).
- Candia, B., Coibion, O., and Gorodnichenko, Y., (2021). “The Inflation Expectations of US Firms: Evidence from a New Survey” NBER Working Paper No. 28836, (Cambridge, National Bureau of Economic Research).
- Chang, Y., Gómez-Rodríguez, F., and Hong, G. H., (2022). "The Effects of Economic Shocks on Heterogeneous Inflation Expectations." IMF Working Paper No. 2022/132, (Washington, International Monetary Fund).
- Chen, J., Gornycka, L., and Zdarek, V., (2022). "Biases in Survey Inflation Expectations: Evidence from the Euro Area,“ IMF Working Paper No. 2022/205, (Washington, International Monetary Fund).
- Coibion, O., Gorodnichenko, Y., and Kumar, S., (2018). “How Do Firms Form Their Expectations? New Survey Evidence,” American Economic Review, Vol. 108(9), No. 9, Pp. 2671-2713.
- D’Acunto, F., Hoang, D., and Weber, M., (2022). “Managing Households’ Expectations with Unconventional Policies,” The Review of Financial Studies, Vol. 35(4), Pp. 1597-1642.
- D’Acunto, F., Malmendier, U., and Weber, M., (2022). “What Do the Data Tell Us About Inflation Expectations?” NBER Working Paper No. 29825, (Cambridge, National Bureau of Economic Research).
- Duca-Radu, I., Kenny, G., and Reuter, A., (2021). “Inflation Expectations, Consumption and the Lower Bound: Micro Evidence from a Large Multi-Country Survey,” Journal of Monetary Economics, Vol. 118, Pp. 120-134.
- Faust, J., and Wright, J. H., (2013). Handbook of Economic Forecasting, “Forecasting Inflation,” Vol. 2, 1st ed., pp. 2-56 (Elsevier).
- Glick, R., Leduc, S. and Popper, M., (2022). “Will Workers Demand Cost-of-Living Adjustments?” FRBSF Economic Letter, 2022-21, August (San Francisco, Federal Reserve Bank of San Francisco).
- Malmendier, U., and Nagel, S., (2016). “Learning from Inflation Experiences,” The Quarterly Journal of Economics, Vol. 131(1), Pp. 53-87.
- Meeks, R., and Monti, F., (2022). “Heterogeneous Beliefs and the Phillips Curve” CAMA Working Paper No. 51/2022, September 2022 (Acton, Center for Applied Macroeconomic Analysis ANU).
- Malmendier, U., and Wellsjo, A., (2020). “DP14935 Rent or Buy? The Role of Lifetime Experiences on Homeownership Within and Across Countries,” CEPR Press Discussion Paper No. 14935, (London, Centre for Economic Policy Research).
- Mankiw N.G., Reis, R. and Wolfers, W. (2003) “Disagreement about Inflation Expectations” NBER Macroeconomics Annual 2003 18:, 209-248.
- Reis, R., (2021). “Losing the Inflation Anchor,” CEPR Discussion Paper No. DP16664, (London, Centre for Economic Policy Research).
- Verbrugge, R. J., and Zaman, S., (2021). “Whose Inflation Expectations Best Predict Inflation?” FRBC Economic Commentary, 2021-19 (Cleveland, Federal Reserve Bank of Cleveland).
- Werning, I., (2022). “Expectations and the Rate of Inflation” NBER Working Paper No. 30260, (Cambridge, National Bureau of Economic Research).

### Annex I. Bundesbank Data

- Survey on Consumer Expectations of the Deutsche Bundesbank:
  - Conducted monthly since April 2020.
  - Preceded by a brief pilot period from April to June 2019.
  - Approximately 2000 households are asked to respond in each wave, with some respondents asked in multiple waves.
  - Survey inquires about one-year ahead inflation expectations using a two-stage question:
    - Stage 1: whether respondents “think inflation or deflation is more likely over the next twelve months.”
    - Stage 2: conditional on Stage 1, respondents assess the rate of inflation/deflation over the next twelve months; the analysis uses answers to this second question.
  - Bins are composed in the same way as the US bins and Reis’ method was used to construct the distributions.
- Data processing note:
  - Non-integer values that lie outside the Michigan Survey buckets (e.g., between 2-3 percent), are rounded up or down to the nearest integer.

### Annex II. Moments of Households’ and Professionals’ Expectations

- Contains:
  - Figure A.2. Moments of Households’ Expectations.
  - Figure A.3. Moments of Professionals’ Expectations.
- Visuals present the empirical moments (e.g., median, variance, skewness) for households and professionals over the sample periods shown in the paper.

### Annex III. Moments Model Robustness — Tables and Key Specifications

- Annex III includes Tables A.1 through A.13 reporting robustness checks and alternative specifications of the moments model for forecasting Inflation t+12 (and in some tables Inflation t+4 or Diff(Inflation t+12)), including:
  - Table A.1. Moments Model Based on Different Subsamples, Three Breakpoints.
  - Table A.2. Moments Model Including Lags Based on Different Subsamples.
  - Table A.3. Moments Model, Controlling for Unemployment Rate.
  - Table A.4. Moments Model, Controlling for Capacity Utilization.
  - Table A.5. Moments Model, Controlling for Commodity Inflation.
  - Table A.6. Moments Model in First Differences.
  - Table A.7. Moments Model, with Published Moments.
  - Table A.8. Moments Model with Increasing / Decreasing Inflation Dummies, with Published Moments.
  - Table A.9. Moments Model, with Moments Based on Micro Data.
  - Table A.10. Moments Model with Increasing / Decreasing Inflation Dummies, with Moments Based on Micro Data.
  - Table A.11. Moments Model, with Interquartile Range instead of Variance.
  - Table A.12. Moments Model with Increasing / Decreasing Inflation Dummies, with Interquartile Range instead of Variance.
  - Table A.13. Moments Model with Increasing / Decreasing Inflation Dummies, Quarterly Data.

- Reported sample periods (examples as presented):
  - 1978M1-2022M6
  - 1978M1-1990M7
  - 1990M8-2007M10
  - 2007M11-2015M10
  - 2015M11-2022M6
  - 1981M1-2022M6
  - 1981M1-1990M7
  - 1990M10-2007M10
  - 2016M5-2022M6
  - Diff sample: 1978M2-2022M6 (and subsamples 1978M2-1990M7, 1990M8-2015M10, 2015M11-2022M6)

- Reported sample sizes and fit statistics (examples as presented):
  - N values appearing include: 534, 151, 303, 80, 498, 115, 30, 174, 533, 150.
  - Adj. R-squared values appearing include: 0.29, 0.54, 0.61, 0.85, 0.04, 0.62, 0.87, 0.81, 0.46, 0.61, 0.67, 0.69, 0.14, 0.19, 0.35, 0.44, 0.78, 0.80, 0.76, 0.64, 0.65.
  - Inflation horizons shown: Inflation t+12 and Inflation t+4 (for quarterly specifications).

- Covariates and model elements reported across tables (preserved terminology and numeric indicators):
  - Core regressors include: Inflation, Median, Variance, Skewness, Diff(Inflation), Diff(Median), Diff(Variance), Diff(Skewness), IQR (interquartile range), Pearson coeff (Pearson coefficient of skewness), Capacity utilization, Unempl. rate, Commodity infl.
  - Lagged inflation terms included in some specifications: Inflation(-12), Inflation(-24), Inflation(-36).
  - Interaction terms and dummies:
    - Variance*Dummy1, Variance*Dummy2, IQR*Dummy1, IQR*Dummy2.
    - GFC dummy: takes value of 1 in years 2008-2010.
    - Increasing / decreasing inflation definitions:
      - Monthly definition: Increasing (decreasing) inflation is defined as the 3-month-average 1-year-ahead being greater than the 3-month average one year ago plus 1 percentage point.
      - Quarterly definition (Table A.13 notes): Increasing (decreasing) inflation is defined as 1-year-ahead inflation being greater than the inflation one year ago plus 1 percentage point.
    - Post min (max) definitions in IQR tables: Post min (max) is when there was a 5-year minimum (maximum) in the previous 6 months (or previous 2 quarters for quarterly tables).
  - Significance notation used uniformly:
    - *** = significant at 0.01 level, ** = significant at 0.05 level, * = significant at 0.1 level.
  - Estimation details:
    - Subsamples determined using the Bai-Perron (2003) global breakpoint test based on the median model, with a significance level of 5 percent and a maximum number of breakpoints of 2 or 3 as indicated per table.
    - Errors are Newey-West in reported specifications.
    - Lag length based on smallest AIC for the median model in lag specifications.

- Representative coefficient signs and notable reported magnitudes (as presented in tables):
  - Median coefficient frequently positive and often significant (examples: 1.07***, 1.08***, 1.12***, 1.17***, 1.13***).
  - Variance / IQR coefficients often negative and sometimes significant (examples: Variance -0.04, -0.07**, -0.17***; IQR -0.16, -0.46***, -0.53***).
  - Skewness coefficients reported negative and sometimes significant (examples: Skewness -1.72**, -3.99***, -2.65***, -5.30***).
  - Inflation (contemporary) coefficients vary in sign and significance across subsamples (examples: Inflation 0.18, -0.24*, 0.24**).
  - Diff(Inflation) coefficients in first-difference specifications often negative and significant (example: Diff(Inflation) -0.45*** across multiple subsamples).

- Data sources and notes:
  - Sources: Survey of Consumers - University of Michigan and authors’ calculations.
  - Where skewness is not part of published variables, Pearson coefficient of skewness is computed from published mean, median, and variance.

*Italic — Source: wpiea2023224-print-pdf - References (IMF Working Paper "Do Household Expectations Help Predict Inflation?").*

### Annex IV.  Functional Principal Component

### Annex IV. Functional Principal Component Analysis (FPCA)

### FPCA approach and data
- FPCA is used as a robustness check to describe the shape of the monthly distributions of household inflation expectations instead of moments.
- Data: US households’ one-year ahead expectations, January 1978 to April 2023.  
- FPCA can be applied to original (uncentered) distributions or to distributions centered around their cross-sectional mean.

### Distributional patterns
- In most months, the distributions depict a mode close to, but above, the Fed’s 2-percent inflation target.
- Kernel densities plotted for January 1978 – April 2023 (grey lines) with the cross-months average (black line).
- When distributions are centered, the mean curve has a lower variance and a shorter right tail compared to the uncentered distributions.

### Variance explained by FPCA
- Three components explain around 94 percent of variation of the distribution.
- Scree plot (Figure A.5) indicates marginal gains from further components; analyses use three principal components as explanatory variables.

### Forecasting models and specifications
- Uncentered distributions model (A.1):  
  휋_{t+12} = c + α π_{t} + ∑_{k=1}^{K} β_{k} s_{k,t}^{uncent} + ε_{t}  
  - s_{k,t}^{uncent} are scores obtained from FPCA on uncentered distributions.
- Centered distributions model (A.2) (mean included):  
  휋_{t+12} = c + α π_{t} + ∑_{k=1}^{K} β_{k} s_{k,t}^{cent} + γ π̅_{t}^{e} + ε_{t}  
  - s_{k,t}^{cent} are scores from FPCA on centered distributions; π̅_{t}^{e} is mean expected inflation.

### Key empirical findings — uncentered FPCA (model A.1)
- The shape of the distribution of expectations is informative of future inflation, independently of the number of scores used (one, two or three).
- For the specification including three scores:
  - Coefficients of the first and the third score are statistically significant (full sample).
- For specifications including just one or two scores:
  - Only the first score is significant.
- Model performance:
  - Models perform best in the high-inflation periods at the beginning and end of the sample.

### Key empirical findings — centered FPCA (model A.2)
- When using scores from centered distributions, the mean expected inflation (π̅_{t}^{e}) should be included as an explanatory variable.
- For the full period:
  - The mean and score 2 are highly significant.
  - Score 1 is significant at a 10 percent level.
- The first principal component obtained from the uncentered FPCA is highly negatively correlated with the mean; consequently, the mean tends to have a positive significant coefficient in the centered specification.
- R-squared pattern across subsamples:
  - Highest R-squared obtained for the earliest sub-sample, followed by the most recent one, and lower values for the intermediate period.

### Additional notes on estimation and diagnostics
- Scores measure the extent to which a curve consists of the respective functional principal component.
- Errors are Newey-West in reported regressions.
- Subsample breakpoints are based on the Bai-Perron (2003) global breakpoint test (significance level 5 percent, maximum number of breakpoints of 2) for the centered results; subsamples for comparison use the moments model with the same breakpoint settings.

*Source: Annex IV. Functional Principal Component Analysis (FPCA), Survey of Consumers – University of Michigan, and authors’ calculations.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023224-print-pdf.pdf_