## 2.1 Demographic Differences (and linked empirical evidence) ## Source details **Canonical URL:** [2.1 Demographic Differences (and linked empirical evidence)](https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024025-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2024/english/wpiea2024025-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2024/english/wpiea2024025-print-pdf.pdf.json) --- ### Data and methodology - Monthly University of Michigan Surveys of Consumers, January 1978 through April 2023. - Two survey questions: - (i) short-term (one-year) expectations: "By what amount do you expect prices to go up or down, on average, during the next twelve months?’" - (ii) medium-term (five-ten year) expectations: "By what percent per year do you expect prices to go up or down, on average, during the next five to ten years." - Figures referenced: B.4 and B.5 show mean one- and five-ten year expectations disaggregated by education, income, gender and income. - Subsample of respondents surveyed twice (six months apart) identified by unique participant IDs used to analyze updating. ### Key empirical findings — education - Since 2022, mean one-year ahead inflation expectation for those without a high school diploma has been over 5 percentage points higher than the mean expectation of those with college degrees at most times since 2022. - Inflation expectations of those without high school diplomas show more variance than those with higher education; divergence became apparent during 2022. - Five-ten year expectations show a similar pattern, but differences across education levels are smaller. ### Key empirical findings — income - Differences across the income distribution widened since 2021. - The mean and standard deviation of inflation expectations reported by respondents in the bottom quartile have been over five percentage points above those of respondents in the top income quartile at various times during 2022-2023. - Lower-income respondents display both a higher average level and wider dispersion of expectations. - Differences by income are present at the five-ten year horizon, though smaller. - Table B.6 (referenced) and distributions show sample variance is wider for lower incomes, with variance increasing since 2021. ### Key empirical findings — gender and age - Mean inflation expectations for women have been around one percentage point higher than those of men over the past five years. - The gender discrepancy has widened somewhat since 2021. - Sample variance of female inflation expectations has increased more than for men, but only since 2021. - No substantial differences across age groups. ### Forecast errors and updating behavior - Forecast errors defined as: reported expectations one and five years previously less actual headline CPI inflation. - Over the four decades after 1980, mean forecast errors at one and five-year horizons have been mostly positive (consumers over-estimate inflation on average), except after 2021 when mean inflation forecasts under-estimated the surge. - Sample variance of forecast errors declined from 1980-2000 and has been relatively low since, except around 2010 and after the surge following 2021. - Respondents surveyed twice (six months apart) update expectations between surveys based on information received, improving forecast accuracy; evidence consistent with adaptive learning. ### Regression evidence on forecast errors (following Madeira and Zafar (2015)) - OLS regressions include constant and dummy variables for gender, age, income, education. - Coefficients indicate: - Low-income and/or female consumers have higher forecast errors on average, all else equal. - Forecast errors are likely to decline between the first and second survey for the same respondent. - Age has a smaller impact on forecast error. - Additional years of schooling reduce forecast error on average, all else equal. - Interpretation: consumers update expectations over the six-month interval, consistent with adaptive learning. --- ### 3.2 Simple learning rules — estimation and fit ### Estimation procedure and learning algorithms - Sample split: - Pre-forecasting: 1961M6-1979M11. - In-sample: 1979M12-2011M2. - Out-of-sample forecasting: 2011M3-2022M12 (MSEs computed up to 2020M2 in some comparisons). - Household inflation expectation formation: bπ_t|t−12 = bβ′_{t−12} x_t (notation preserved). - Learning algorithms: - Constant gain least squares (CGLS): γ_t replaced by constant γ. - Recursive least squares (RLS): γ_t = t^−1. - Optimal in-sample constant gains found by minimizing MSE_IN(π) over γ ∈ (0,1) with grid distance 0.0001. - Four model specifications for agents’ inflation expectation formation: - Model 1: AR(1): bπ_t|t−12 = b1,t−12 + b2,t−12 π_t−12 - Model 2: AR(2): bπ_t|t−12 = b1,t−12 + b2,t−12 π_t−12 + b3,t−12 π_t−13 - Model 3: constant, lagged inflation and lagged output growth (z_t = industrial production growth) - Model 4: Model 3 plus changes in interest rates (w_t = Fed Funds Rate) - AR(1) and AR(2) outperformed higher-order AR models in out-of-sample MSE and MSCE. ### Results — actual inflation (Table 1 summary) - Out-of-sample MSEs (CGLS and RLS): - MSE (pre-Covid): - CGLS: Model 1 = 0.25; Model 2 = 0.13; Model 3 = 0.16; Model 4 = 0.39 - RLS: Model 1 = 1.17; Model 2 = 1.20; Model 3 = 1.23; Model 4 = 1.21 - MSE (full sample): - CGLS: Model 1 = 1.85; Model 2 = 2.12; Model 3 = 4.15; Model 4 = 1.51 - RLS: Model 1 = 3.70; Model 2 = 3.74; Model 3 = 3.21; Model 4 = 3.20 - Optimal in-sample constant gains for actual inflation (CGLS): - Model 1 = 0.19; Model 2 = 0.14; Model 3 = 0.13; Model 4 = 0.04 - Key findings: - CGLS dominates RLS in forecast accuracy (modified Diebold/Mariano tests reject equal accuracy at the 5% level for the tested comparison). - No single model best across all periods: - Pre-Covid: Model 1 performs well. - Full sample (including post-2020M2): Model 4 performs better. - Constant gain learning fits actual inflation especially well in pre-Covid period. ### Results — household inflation expectations (Table 2 summary) - Out-of-sample MSCEs (CGLS and RLS): - MSCE (pre-Covid): - CGLS: Model 1 = 4.14; Model 2 = 3.82; Model 3 = 3.83; Model 4 = 1.79 - RLS: Model 1 = 2.22; Model 2 = 2.52; Model 3 = 2.38; Model 4 = 2.04 - MSCE (full sample): - CGLS: Model 1 = 5.81; Model 2 = 7.60; Model 3 = 8.21; Model 4 = 3.53 - RLS: Model 1 = 4.11; Model 2 = 4.35; Model 3 = 3.83; Model 4 = 3.99 - Best fitting constant gains for overall household expectations (CGLS): - Model 1 = 0.21; Model 2 = 0.15; Model 3 = 0.13; Model 4 = 0.01 - Key findings: - Best fitting gains to match mean overall household expectations broadly align with optimal gains for fitting actual inflation. - Model 4 yields the lowest forecast errors for both pre-Covid and full samples when fitting household expectations. - Generated expectation series predict direction well (especially pre-Covid) but actual household expectations are more volatile than generated series. ### Heterogeneity by demographic characteristics (Table 3) - Best-fitting model, best-fitting gain, and MSCEs by subgroup: - Low income: Model 1; best fitting gain = 0.10; MSCE (pre-Covid) = 13.06; MSCE (with-Covid) = 5.04 - High income: Model 4; best fitting gain = 0.40; MSCE (pre-Covid) = 10.65; MSCE (with-Covid) = 1.74 - With college degree: Model 4; best fitting gain = 0.40; MSCE (pre-Covid) = 12.09; MSCE (with-Covid) = 1.80 - HS or less: Model 1; best fitting gain = 0.10; MSCE (pre-Covid) = 52.70; MSCE (with-Covid) = 5.58 - Female: Model 1; best fitting gain = 0.10; MSCE (pre-Covid) = 12.19; MSCE (with-Covid) = 3.74 - Male: Model 4; best fitting gain = 0.40; MSCE (pre-Covid) = 11.02; MSCE (with-Covid) = 2.40 - Interpretation: - Considerable heterogeneity in expectation formation across demographics. - More educated, wealthier, male households tend to use Model 4 with best-fitting gains that closely match optimal gains for Model 4. - Low income, less educated, female households tend to use Model 1 with best-fitting gains significantly lower than optimal gains for Model 1. - Possible explanations: greater awareness of structural breaks and readiness to use larger indicator sets among higher income/educated households; informational/updating costs lead low-income/less-educated households to update less frequently. - Additional numeric note: - Time in months for the weight on an observation to fall to 1/2 is t_half = ln 2 / γ. ### Results — using food and gasoline inflation (Table 4) - Best fitting gains and MSCEs using composite CPI for food and gasoline: - Low income: Model 1; best fitting gain = 0.10; MSCE (pre-Covid) = 41.37; MSCE (with-Covid) = 3.53 - High income: Model 4; best fitting gain = 0.40; MSCE (pre-Covid) = 8.10; MSCE (with-Covid) = 1.87 - With college degree: Model 4; best fitting gain = 0.40; MSCE (pre-Covid) = 5.10; MSCE (with-Covid) = 3.09 - HS or less: Model 1; best fitting gain = 0.10; MSCE (pre-Covid) = 5.17; MSCE (with-Covid) = 1.58 - Female: Model 1; best fitting gain = 0.10; MSCE (pre-Covid) = 5.12; MSCE (with-Covid) = 1.36 - Male: Model 4; best fitting gain = 0.40; MSCE (pre-Covid) = 7.08; MSCE (with-Covid) = 1.08 - Key finding: - Low-income and less-educated household expectations can be fit better using food and energy inflation (lower MSCE) than using overall CPI; this improvement is not observed for more educated, higher income households. ### Robustness and tests - Different data splits tried; qualitative results robust though parameters change. - Modified Diebold/Mariano tests comparing CGLS and RLS: null of equal forecast accuracy rejected at the 5% level for the tested comparison. - Additional specifications and subgroup results available from the authors as reported in the source. --- ### 4.4 Changing Inflation Dynamics over Time — model interpretation and policy implications ### Surge of inflation interpreted as price mark-up shock - Recent surge since 2021 "appears to have been estimated to be a large and short-lived price mark-up shock" in both rational expectations and adaptive learning model versions. - Price mark-up shocks in the adaptive learning model are "not estimated to be inherently persistent" based on ARMA(1,1) coefficients. - Figures B.19 and B.20 show estimated innovations to exogenous processes w_t; surge interpreted as large price mark-up shock given comparative stability of prior three decades. - Despite being short-lived, the shock has a prolonged impact in the adaptive learning model because of the propagation mechanism introduced by adaptive learning. ### Estimated persistence and consumers' Perceived Law of Motion (PLM) - Sum of estimated coefficients on first and second lags of inflation in the PLM has "been estimated to have been persistent over the whole sample, with a coefficient of around 0.9." - Estimated PLM showed "substantial volatility in the late 1970’s and 1980’s," dissipated over subsequent three decades, and "has since re-emerged during the surge of inflation during 2022." - Estimated intercept term in the inflation forecasting equation "has been close to zero over the sample," suggesting little drift in inflation. - Comparison: Slobodyan and Wouters (2012a) found autoregressive coefficients declined more sharply during the Great Moderation under a different shock specification; in this longer sample, "wage mark-up shocks still follow a persistent ARMA(1,1), even though price mark-up shocks are closer to white noise under adaptive learning." ### Impulse Response Functions (IRFs) and differences between adaptive learning (AL) and rational expectations (RE) - IRFs change over the sample under adaptive learning because ALM and model solution are time-varying; estimated PLM coefficients have been stable so IRFs change little over time. - Key quantitative IRF comparisons: - A one standard deviation price mark-up shock (equal to around 0.1 percentage points) has an initial impact on inflation in both models of "around 0.3 percentage points." - Impact is substantially more persistent under adaptive learning: "still being around 0.1 percentage points above steady state after 12 quarters." - Reduction in output under adaptive learning is "only around one third as deep as in the rational expectations model, although more persistent." - Increase in the Fed Funds Rate required to return inflation to the two percent objective is "more than 50 percent larger in the model with adaptive learning than in the model with rational expectations." - Fed Funds Rate is "significantly more persistent" under adaptive learning, "remaining around 0.1 percentage points above steady state after 20 quarters." - Wage mark-up shocks "also have a more persistent impact in the model with adaptive learning" compared with rational expectations. ### Empirical and policy-relevant conclusions - The widening of the distribution of inflation expectations since 2021 is "striking," with sharp rise in short-term expectations. - Micro data: - Lower income and less educated consumers, as well as women, "appear to have higher and more heterogeneous expectations during this period." - Survey evidence shows lower income and female consumers make larger inflation forecast errors, producing a higher mean and variance within these groups. - Modeling and forecasting: - Reduced form adaptive learning is consistent with behavior of groups that "place less weight on more recent data and have more persistent inflation expectations." - A theoretical DSGE model estimated on a US sample including the recent surge "suggests adaptive learning better matches how consumer inflation expectations are formed, producing consumer expectations more in line with consumer survey measures." - "The model with adaptive learning has better overall forecasting performance for inflation than one with rational expectations." - "Adaptive learning makes the inflation caused by cost push shocks more persistent (interpreting the recent surge as a cost push shock)." - Policy implication: - The model implies that "higher interest rates, maintained for longer, are required to return inflation to target in response to these shocks." ### Open questions and further research - A precise, micro-founded explanation for large differences in inflation expectations across demographic groups within an adaptive learning framework is "beyond the scope of this paper." - Further research needed on whether different information sets, different consumption baskets, or different ways of forming and updating expectations drive disparities across groups. --- *Source: wpiea2024025-print-pdf — sections 2.1, 3.2, and 4.4 of IMF Working Paper No. WP/2024/025* ### 2.1 Demographic Differences ### 2.1 Demographic Differences ### Data and methodology - Monthly data from the University of Michigan Surveys of Consumers spanning January 1978 through April 2023. - Two survey questions used: - (i) "By what amount do you expect prices to go up or down, on average, during the next twelve months?’" — interpreted as short-term (one-year) inflation expectations. - (ii) "By what percent per year do you expect prices to go up or down, on average, during the next five to ten years." — interpreted as medium-term (five-ten year) inflation expectations. - Figures referenced: B.4 and B.5 show mean inflation expectations at one and five-ten year horizons disaggregated by education, income, gender and income. - Subsample of respondents surveyed twice, with each survey separated by six months, identified via unique participant identification numbers; used to analyze updating of expectations. ### Key empirical findings — education - Since 2022, the mean one-year ahead inflation expectation for those without a high school diploma has been over 5 percentage points higher than the mean inflation expectation of those with college degrees at most times since 2022. - Inflation expectations of those without high school diplomas have shown more variance than those of respondents with higher education levels, with this divergence becoming apparent during 2022. - Similar pattern for five-ten year expectations, but differences across education levels are smaller. ### Key empirical findings — income - Differences in inflation expectations have widened across the income distribution since 2021. - The mean and standard deviation of inflation expectations reported by respondents in the bottom quartile of the sample income distribution have been over five percentage points above those of respondents in the top income quartile, at various times during 2022-2023. - This implies both a higher average level of inflation expectations and a wider dispersion of expectations for lower-income respondents. - Differences by income are present at the five-ten year horizon as well, though smaller. - Table B.6 (referenced) plots distributions over time for bottom and top income quartiles at one and five-ten year horizons; sample variance is wider for lower incomes, with variance increasing since 2021. ### Key empirical findings — gender and age - Mean inflation expectations for women have been around one percentage point higher than those of men over the past five years. - The gender discrepancy has widened somewhat since the inflation surge of 2021. - The sample variance of female inflation expectations has increased by more than that for men, although only since 2021. - No substantial differences in inflation expectations are found across age groups. ### Forecast errors and updating behavior - Forecast errors defined as: one-year ahead and five-ten year ahead inflation expectations reported one and five years previously less actual headline CPI inflation. - Over the four decades after 1980, mean forecast errors at one and five-year horizons have been mostly positive, suggesting consumers over-estimate future inflation on average. - Exception after 2021, when mean inflation forecasts under-estimated the surge of inflation. - Sample variance of forecast errors was in trend decline from 1980-2000 and has been relatively low since, except around the post-crisis period in 2010 and following the surge after 2021. - Comparing respondents surveyed twice (six months apart) shows consumers update inflation expectations between surveys based on information received, improving forecast accuracy — evidence consistent with adaptive learning. ### Regression evidence on forecast errors (following Madeira and Zafar (2015)) - OLS regressions of inflation expectations and forecast errors (absolute terms and changes between first and second surveys) include a constant and dummy variables for gender, age, income and education levels. - Coefficients (illustrated in Figure B.8) indicate: - Low-income and/or female consumers have higher forecast errors on average, all else equal. - Forecast errors are likely to decline between the first and second survey for the same respondent. - Age has a smaller impact on forecast error. - Additional years of schooling reduce forecast error on average, all else equal. - Interpretation: Consumers update expectations over the six-month interval, consistent with adaptive learning — consumers adjust expectations as more information becomes available. *Source: wpiea2024025-print-pdf - 2.1 Demographic Differences* ### 3.2 Simple learning rules ### 3.2 Simple learning rules ### 3.2.1 Estimation procedure - Sample split: - Pre-forecasting period: 1961M6-1979M11 (used to compute initial estimates). - In-sample period: 1979M12-2011M2 (used to determine optimal constant gains or update recursive gains). - Out-of-sample forecasting period: 2011M3-2022M12. Mean-square forecast errors also computed up to 2020M2. - Monthly household inflation expectation formation: - Inflation expectation by households in period t−12 for period t is bπ_t|t−12 = bβ′_{t−12} x_t (equation (9) notation preserved). - Agents use the best estimate of coefficients in period t−12 and update estimates as new data arrive. - Learning algorithms: - Constant gain least squares (CGLS): γ_t in the recursion is replaced by constant γ. - Recursive least squares (RLS): γ_t = t^−1 and the gain sequence is updated with t^−1. - Optimal in-sample constant gain for actual inflation: - Computed by minimizing MSE_IN(π) = (1/T) ∑_{t=t0}^T (π_t − bπ_t|t−12)^2 over γ ∈ (0,1) with t0 = 1979M12 and T = 2011M2. Grid distances set at 0.0001. - bπ_t|t−12 denotes hypothetical forecast made in period t−12 for t using actual inflation and optimal constant gains, with recursions (7) and (8) started at γ_t = γ and initial values from pre-sample period. - Out-of-sample MSE: - MSE_OUT(π) = (1/T) ∑_{t=1}^T (π_t − bπ_t|t−12)^2 where t ranges from 2011M3 to 2020M2 or 2022M12. - Best-fitting gains for household expectations: - Computed by minimizing MSCE_IN(π) = (1/T) ∑_{t=t0}^T (π^F_t − bπ_t|t−12)^2 over γ ∈ (0,1) with t0 = 1979M12 and T = 2011M2. Grid distances set at 0.0001. - Out-of-sample comparison error: MSCE_OUT(π) = (1/T) ∑_{t=1}^T (π^F_t − bπ_t|t−12)^2 where t ranges from 2011M3 to 2020M2 or 2022M12. - For RLS learning, recursions updated with t^−1. - Four model specifications for agents’ inflation expectation formation: - Model 1: AR(1): bπ_t|t−12 = b1,t−12 + b2,t−12 π_t−12 - Model 2: AR(2): bπ_t|t−12 = b1,t−12 + b2,t−12 π_t−12 + b3,t−12 π_t−13 - Model 3: constant, lagged inflation and lagged output growth (z_t = industrial production growth): bπ_t|t−12 = b1,t−12 + b2,t−12 π_t−12 + b3,t−12 z_t−12 - Model 4: Model 3 plus changes in interest rates (w_t = Fed Funds Rate): bπ_t|t−12 = b1,t−12 + b2,t−12 π_t−12 + b3,t−12 z_t−12 + b4,t−12 w_t−12 - Higher-order AR models were computed but AR(1) and AR(2) outperformed them in out-of-sample MSE and MSCE. ### 3.2.2 Results — actual inflation (Table 1 summary) - Out-of-sample MSEs (CGLS and RLS reported side-by-side as in the source table): - MSE (pre-Covid): - CGLS: Model 1 = 0.25; Model 2 = 0.13; Model 3 = 0.16; Model 4 = 0.39 - RLS: Model 1 = 1.17; Model 2 = 1.20; Model 3 = 1.23; Model 4 = 1.21 - MSE (full sample): - CGLS: Model 1 = 1.85; Model 2 = 2.12; Model 3 = 4.15; Model 4 = 1.51 - RLS: Model 1 = 3.70; Model 2 = 3.74; Model 3 = 3.21; Model 4 = 3.20 - Optimal constant gains (in-sample) for actual inflation (CGLS only; RLS N/A): - Model 1 = 0.19 - Model 2 = 0.14 - Model 3 = 0.13 - Model 4 = 0.04 - Key findings on actual inflation: - Constant gain least squares (CGLS) dominates recursive least squares (RLS) in terms of forecast accuracy (modified Diebold/Mariano tests reject equal accuracy at the 5% level for the comparison described in the source). - No single model is best across all periods: - Pre-Covid out-of-sample MSE: simple Model 1 (constant + lagged inflation) performs well. - Including post-2020M2 (full sample): Model 4 performs better. - Constant gain learning fits actual inflation especially well in the pre-Covid period. ### 3.2.2 Results — household inflation expectations (Table 2 summary) - Out-of-sample MSCEs (CGLS and RLS reported side-by-side): - MSCE (pre-Covid): - CGLS: Model 1 = 4.14; Model 2 = 3.82; Model 3 = 3.83; Model 4 = 1.79 - RLS: Model 1 = 2.22; Model 2 = 2.52; Model 3 = 2.38; Model 4 = 2.04 - MSCE (full sample): - CGLS: Model 1 = 5.81; Model 2 = 7.60; Model 3 = 8.21; Model 4 = 3.53 - RLS: Model 1 = 4.11; Model 2 = 4.35; Model 3 = 3.83; Model 4 = 3.99 - Best fitting constant gains for overall household expectations (CGLS): - Model 1 = 0.21 - Model 2 = 0.15 - Model 3 = 0.13 - Model 4 = 0.01 - RLS values N/A in source table. - Key findings on household expectations: - Best fitting gains to match mean overall household inflation expectations are broadly in line with optimal gains for fitting actual inflation. - Model 4 yields the lowest forecast errors for both pre-Covid and full sample when fitting household expectations. - Generated expectation series predict direction well (especially pre-Covid) but actual household expectations are more volatile than the generated series. ### 3.2.2 Results — heterogeneity by demographic characteristics (Table 3) - Table 3: best fitting model (implicitly reported as Model 1 or Model 4 in the source table), best fitting gain, and MSCEs: - Low income: - Model specification reported as Model 1 - Best fitting gain = 0.10 - MSCE (pre-Covid) = 13.06 - MSCE (with-Covid) = 5.04 - High income: - Model specification reported as Model 4 - Best fitting gain = 0.40 - MSCE (pre-Covid) = 10.65 - MSCE (with-Covid) = 1.74 - With college degree: - Model specification reported as Model 4 - Best fitting gain = 0.40 - MSCE (pre-Covid) = 12.09 - MSCE (with-Covid) = 1.80 - HS or less: - Model specification reported as Model 1 - Best fitting gain = 0.10 - MSCE (pre-Covid) = 52.70 - MSCE (with-Covid) = 5.58 - Female: - Model specification reported as Model 1 - Best fitting gain = 0.10 - MSCE (pre-Covid) = 12.19 - MSCE (with-Covid) = 3.74 - Male: - Model specification reported as Model 4 - Best fitting gain = 0.40 - MSCE (pre-Covid) = 11.02 - MSCE (with-Covid) = 2.40 - Interpretation: - Considerable heterogeneity in expectation formation across demographics. - More educated, wealthier, male households tend to use Model 4 (more complex) and best-fitting gains that closely match optimal gains for Model 4. - Low income, less educated, female households tend to use Model 1 (simple AR(1)) with best-fitting gains significantly lower than optimal gains for Model 1. - Possible explanations: greater awareness of structural breaks and readiness to use larger indicator sets among higher income/educated households; informational/updating costs leading low-income/less-educated households to update less frequently. - Additional numeric note from source: - The time (in months) it takes for the weight given to an observation to fall to 1/2 is t_half = ln 2 / γ. ### 3.2.2 Results — using food and gasoline inflation (Table 4) - Best fitting gains and MSCEs when fitting expectations using composite CPI for food and gasoline: - Low income: - Model specification: Model 1 - Best fitting gain = 0.10 - MSCE (pre-Covid) = 41.37 - MSCE (with-Covid) = 3.53 - High income: - Model specification: Model 4 - Best fitting gain = 0.40 - MSCE (pre-Covid) = 8.10 - MSCE (with-Covid) = 1.87 - With college degree: - Model specification: Model 4 - Best fitting gain = 0.40 - MSCE (pre-Covid) = 5.10 - MSCE (with-Covid) = 3.09 - HS or less: - Model specification: Model 1 - Best fitting gain = 0.10 - MSCE (pre-Covid) = 5.17 - MSCE (with-Covid) = 1.58 - Female: - Model specification: Model 1 - Best fitting gain = 0.10 - MSCE (pre-Covid) = 5.12 - MSCE (with-Covid) = 1.36 - Male: - Model specification: Model 4 - Best fitting gain = 0.40 - MSCE (pre-Covid) = 7.08 - MSCE (with-Covid) = 1.08 - Key finding: - Low-income and less-educated household expectations can be fit better using food and energy inflation (lower MSCE) than using overall CPI; this improvement is not observed for more educated, higher income households. ### Additional methodological and robustness notes (as stated in source) - Different splits of the data were tried; qualitative results remained robust though estimation parameters changed somewhat. - Modified Diebold/Mariano tests were performed comparing CGLS and RLS; the null of equal forecast accuracy can be rejected at the 5% level for the tested comparison. - Results for alternative model specifications and subgroup runs are available from the authors upon request (as reported in the source). *Source: IMF Working Paper content unit "3.2 Simple learning rules" (wpiea2024025-print-pdf).* ### 4.4 Changing Inflation Dynamics over Time ### 4.4 Changing Inflation Dynamics over Time ### Surge of inflation interpreted as price mark-up shock - The recent surge of inflation since 2021 "appears to have been estimated to be a large and short-lived price mark-up shock" in both the rational expectations and adaptive learning versions of the model. - Price mark-up shocks in the adaptive learning model are "not estimated to be inherently persistent" based on the estimated coefficients of the relevant ARMA(1,1) process. - Figures B.19 and B.20 (referenced in the source) show the estimated innovations to the exogenous processes w_t over the sample, illustrating the surge being interpreted as a large price mark-up shock given the comparative stability of inflation over the preceding three decades. - Despite being short-lived, the shock has a prolonged impact in the adaptive learning model because of the propagation mechanism that adaptive learning introduces. ### Estimated persistence and consumers' Perceived Law of Motion (PLM) - The sum of the estimated coefficients of the first and second lags of inflation in the PLM used by consumers has "been estimated to have been persistent over the whole sample, with a coefficient of around 0.9." - The estimated PLM showed "substantial volatility in the late 1970’s and 1980’s," which dissipated over the subsequent three decades but "has since re-emerged during the surge of inflation during 2022." - The estimated intercept term in the inflation forecasting equation "has been close to zero over the sample," suggesting little drift in inflation over the sample. - Comparison to Slobodyan and Wouters (2012a): - Slobodyan and Wouters (2012a) found the estimated autoregressive coefficients in the inflation forecasting equation declined more sharply during the Great Moderation in a model where price and wage mark-up shocks were modeled as white noise rather than ARMA(1,1). - In this paper's longer sample including recent data, "wage mark-up shocks still follow a persistent ARMA(1,1), even though price mark-up shocks are closer to white noise under adaptive learning." ### Impulse Response Functions (IRFs) and differences between adaptive learning and rational expectations - IRFs change over the sample in the model with adaptive learning because the ALM and model solution are time-varying. - The source compares IRFs from the adaptive learning model in three dates: 1981, 2019, and 2022, and also shows IRFs from the model with rational expectations. Given the estimated PLM coefficients have been stable, "the IRFs in the model with adaptive learning change little over time." - Key quantitative IRF findings: - A one standard deviation price mark-up shock (equal to around 0.1 percentage points) has a similar initial impact on inflation in both models of "around 0.3 percentage points." - The impact is substantially more persistent under adaptive learning, "still being around 0.1 percentage points above steady state after 12 quarters." - The reduction in output in the adaptive learning model is "only around one third as deep as in the rational expectations model, although more persistent." - The increase in the Fed Funds Rate required to return inflation to the two percent objective is "more than 50 percent larger in the model with adaptive learning than in the model with rational expectations." - The Fed Funds Rate is also "significantly more persistent" under adaptive learning, "remaining around 0.1 percentage points above steady state after 20 quarters." - Wage mark-up shocks "also have a more persistent impact in the model with adaptive learning, compared with that of rational expectations." ### Empirical and policy-relevant conclusions - The widening of the distribution of inflation expectations since 2021 is "striking," with a sharp rise in short-term inflation expectations. - Micro data findings: - Lower income and less educated consumers, as well as women, "appear to have higher and more heterogeneous expectations during this period." - Survey evidence shows lower income and female consumers make larger inflation forecast errors, producing a higher mean and variance of inflation expectations within these groups. - Modeling and forecasting conclusions: - A reduced form model of adaptive learning is consistent with behavior of groups that "place less weight on more recent data and have more persistent inflation expectations." - A theoretical DSGE model estimated on a US sample including the recent inflation surge "suggests adaptive learning better matches how consumer inflation expectations are formed, producing consumer expectations more in line with consumer survey measures." - "The model with adaptive learning has better overall forecasting performance for inflation than one with rational expectations." - The model shows "adaptive learning makes the inflation caused by cost push shocks more persistent (interpreting the recent surge as a cost push shock)." - Policy implication: - The model implies that "higher interest rates, maintained for longer, are required to return inflation to target in response to these shocks." ### Open questions and directions for further research - A precise, micro-founded explanation for the large differences in inflation expectations between demographic groups (e.g. income and education) within an adaptive learning framework is "beyond the scope of this paper." - Further research is needed into whether different information sets, different consumption baskets, or different ways of forming and updating expectations are responsible for the disparity across these groups. *Source: 4.4 Changing Inflation Dynamics over Time (wpiea2024025-print-pdf)* ### References ### References ### Key bibliographic citations - Branch, W. A. and G. W. Evans (2006): “A Simple Recursive Forecasting Model,” Economics Letters, 91, 158–66. - Branch, W. A. and B. McGough (2009): “A New Keynesian model with heterogeneous expectations,” Journal of Economic Dynamics and Control, 33, 1036–1051. - Brandao-Marques, L., G. Gelos, D. Hofman, J. Otten, G. Pasricha, and Z. Strauss (2023): “Do Household Expectations Help Predict Inflation,” IMF Working Paper, 224. - Carroll, C. D. (2003): “Macroeconomic Expectations of Households and Professional Forecasters,” Quarterly Journal of Economics, 118, 268–98. - Diebold, F. and R. Mariano (1995): “Comparing Predictive Accuracy,” Journal of Business & Economic Statistics, 13, 253–63. - Evans, G. W. and S. Honkapohja (2001): Learning and Expectations in Macroeconomics, Oxford: Princeton University Press. - Gelain, P., N. Iskrev, K. J. Lansing, and C. Mendicino (2019): “Inflation dynamics and adaptive expectations in an estimated DSGE model,” Journal of Macroeconomics, 59, 258–277. - Hamilton, J. D. (1994): Time Series Analysis, Princeton: Princeton University Press. - Malmendier, U. and S. Nagel (2016): “Learning from Inflation Experiences,” Journal of Quarterly Economics, 131, 53–88. - Mankiw, N., R. Reis, and J. Wolfers (2003): “Disagreement about Inflation Expectations,” NBER Macroeconomics Annual, 18, 209–248. - Marcet, A. and T. Sargent (1989): “Convergence of Least Squares Learning Mechanism in Self-Referential Linear Stochastic Model,” Journal of Economic Theory, 48, 337–68. - Milani, F. (2007): “Expectations, Learning and Macroeconomic Persistence,” Journal of Monetary Economics, 54, 2065–82. - Newey, W. and K. West (1987): “A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, 55(3), 703–708. - Reis, R. (2022): “The burst of high inflation in 2021-22: How and why did we get here,” Bank for International Settlements Working Paper, 1060. - Sargent, T. (1999): The Conquest of American Inflation, Princeton: Princeton University Press. - Smets, F. and R. Wouters (2003, 2007): Key DSGE model references on euro area and U.S. business cycles. - Weber, M., F. D’Acunto, Y. Gorodnichenko, and O. Coibion (2022): “The Subjective Inflation Expectations of Households and Firms: Measurement, Determinants, and Implications,” Journal of Economic Perspectives, 36, 157–184. *Additional cited works are listed in the source text.* --- ### Technical Appendix: DSGE model equations (Slobodyan and Wouters (2012a) framework) ### Consumption Euler equation - Equation (13): - ˆc_t = (γ/(γ+η)) E_t[ˆc_{t+1}] + (η/(γ+η)) ˆc_{t−1} + (γ/(γ+η))(σ_c^{−1})(wL/Cσ_c)(ˆL_t − E_t[ˆL_{t+1}]) − (γ/σ_c(γ+η))((γ−η)/γ)(ˆR_t − E_t[ˆπ_{t+1}] + ˆε^b_t) - Definitions: - ε^b_t is the AR(1) exogenous process for the risk premium. - η is the external habits parameter. - γ is the steady-state growth rate of the economy. - σ_c is the inverse of the intertemporal elasticity of consumption. ### Investment Euler and Q-equation - Equation (14): - ˆi_t = (1/(1 + βγ^{2−σ_c})) ˆi_{t−1} + ((βγ^{2−σ_c}/(1 + βγ^{2−σ_c})) ) ˆi_{t+1} + (1/(1 + βγ^{2−σ_c}) γ^{2} φ) ˆQ^k_t + ˆε^q_t - ε^q_t is AR(1) for investment-specific technology; φ is elasticity of capital adjustment cost. - Equation (15): - ˆQ^k_t = −(ˆR_t − E_t[ˆπ_{t+1}] + ˆε^b_t) + (r_k^*/(r_k^* + (1−δ))) E_t[r_k_{t+1}] + ((1−δ)/(r_k^* + (1−δ))) E_t[Q^k_{t+1}] ### Aggregate demand and supply - Equation (16) (aggregate demand): - ˆy_t = (c^*/y^*) ˆc_t + (i^*/y^*) ˆi_t + ˆε^g_t + (r_k^* k^*/y^*) ˆu_t - Equation (17) (aggregate supply): - ˆy_t = Φ_p(α ˆk_t + (1−α) ˆL_t + ˆε^a_t) - Φ_p represents the steady-state price mark-up. - Exogenous AR(1) processes: ˆε^g_t (aggregate demand) and ˆε^a_t (TFP). ### Price and wage setting - Price equation (18): - ˆπ_t − ι_p ˆπ_{t−1} = βγ^{2−σ_c}(E_t[ˆπ_{t+1}] − ι_p ˆπ_t) − z_p ˆμ^p_t + ε^p_t - z_p = (1−βγ^{2−σ_c} ξ_p)(1−ξ_p)/[ξ_p)(1 + (Φ_p −1) ε_p] - ˆμ^p_t is inverse of real marginal cost: ˆmc_t = (1−α) ˆw_t + α ˆr_k_t − ˆA_t - ξ_p is Calvo price-change probability; ι_p is indexation parameter; ε_p curvature of aggregator. - Wage equation (19): - ˆπ^w_t − ι_w ˆπ_{t−1} = βγ^{2−σ_c}(E_t[ˆπ^w_{t+1}] − ι_w ˆπ_t) − z_w ˆμ^w_t + ε^w_t - z_w = (1−βγ^{2−σ_c} ξ_w)(1−ξ_w)/[ξ_w)(1 + (φ_w −1) ε_w] - Wage markup: ˆμ^w_t = ˆw_t − (γ/(γ+η)) ˆc_t + (η/(γ+η)) ˆc_{t−1} − σ_l ˆL_t ### Capital accumulation, utilization, and input choice - Capital accumulation (20): - ˆ ̄k_t = (1−(i^*/ ̄k^*)) ˆ ̄k_{t−1} + (i^*/ ̄k^*) ˆi_t + (i^*/ ̄k^*)(1 + βγ^{2−σ_c}) γ^{2} S'' ˆε^q_t - Capital services (21): - ˆk_t = ˆμ_t + ˆ ̄k_{t−1} - Optimal capital utilization (22): - ˆμ_t = (1−ψ)/ψ ˆr_k_t - ψ is elasticity of capital utilization. - Optimal input condition (23): - ˆk_t = ˆw_t − ˆr_k_t + ˆL_t ### Monetary policy rule - Equation (24): - ˆR_t = ρ_R ˆR_{t−1} + (1−ρ_R)(r_π ˆπ_t + r_y(ˆygap_t) + r_{yΔ} Δ(ˆygap_t)) + ˆε^r_t - Output gap: ygap_t = ˆy_t − Φ_p ˆε^a_t --- ### Data Sources (Section 4: seven observables used in estimation) - (i) Quarter-over-quarter growth rate of real personal consumption expenditure: - Nominal personal consumption expenditure (BEA) divided by GDP deflator (BEA), all divided by non-institutional civilian population (over age 16) (BLS). - (ii) Quarter-over-quarter growth rate of private fixed investment: - Private fixed investment (BEA) divided by GDP deflator (BEA), all divided by non-institutional civilian population (over age 16) (BLS). - (iii) Quarter-over-quarter growth rate of real gross domestic product (BEA) divided by civilian non-institutional population (BLS). - (iv) Labor input index: - Non-farm business average weekly hours multiplied by total non-civilian employment (over age 16) (both BLS), divided by civilian non-institutional population (BLS), expressed as an index. - (v) Quarter-over-quarter growth rate of the GDP deflator (BEA). - (vi) Quarter-over-quarter growth rate of non-farm business compensation per hour (BLS), divided by the GDP deflator (BEA). - (vii) Annual effective Federal Funds Rate (Federal Reserve Board), expressed as a quarterly rate by dividing by four. --- ### Figures, Tables, and Key Quantitative Results ### Figures and notes (selected) - Figure B.1: Headline and Core CPI Inflation — Sources: Bureau of Labor Statistics. - Figure B.2: Short-Term Measures of Inflation Expectations — Sources: University of Michigan and Federal Reserve Bank of Philadelphia; shows inflation expected over the next year by consumers (University of Michigan) and professional forecasters (SPF). - Figure B.3: Medium-Term Measures of Inflation Expectations — Sources: University of Michigan, Federal Reserve Bank of Philadelphia and Haver; shows 5-year consumer and professional forecasts and 5-year TIPS-implied expectations. - Figures B.4–B.7: One-Year and Five-to-Ten-Year Ahead Consumer Inflation Expectations; Distribution by Income; Forecast Errors — Sources: University of Michigan and authors’ calculations. - Figures B.9–B.12: Actual vs. fitted inflation and household mean expectations (overall, high-income, low-income) — Sources: University of Michigan, BLS, IMF Staff Calculations. - Figures B.13–B.16: Cumulative Inflation Forecast Errors and Impulse Response Functions comparing PLM, ALM, RE models and surveys — Notes specify comparisons between AR(2) PLM, structural model under rational expectations (RE) and adaptive learning (AL), SPF and UMich. - Figures B.17–B.20: Estimated PLM coefficients, intercept, and estimated shocks for RE and AL models — Sample: 228 quarters from 1965Q1-2022Q4, excluding 2020Q1-Q4. ### Selected tables and numeric summaries - Table B.1 (Intercept Coefficients: Consumers’ Inflation Expectations and Forecast Errors) - Observations: columns show 82,865; 82,858; 68,977; 82,865; 68,213. - R^2 values: 0.084, 0.070, 0.419, 0.084, 0.172. - Variables include Female, Young, Mid-age, Lowest income tercile, Middle income tercile, Education, Inflation in survey month, Realized 1-year ahead inflation, Absolute error in first survey, Actual chg. inflation bet. surveys, Constant. - Standard errors reported in parentheses; significance indicated by ***, **, *. - Table B.2 (Education & Income Quartile Observations) - HS or Less: 33,520 | 24,449 | 24,899 | 15,785 = Total 98,653 - Some College: 13,925 | 17,869 | 20,285 | 17,109 = Total 69,188 - College or Above: 8,694 | 17,581 | 31,277 | 47,974 = Total 105,526 - Total observations by quartile: 56,139 | 59,899 | 76,461 | 80,868 = Total 273,367 - Note: The data covers the period from January 1978 to December 2023. - Table B.3 (Rationality Tests for Household Expectations) - Reports α, β, χ^2, DW, Observations for subgroups (Bottom 33%, Middle 33%, Top 33%, Male, Female, education groups, age groups, All). - Example: All: α = −1.333**, β = 1.085***, χ^2 = 14.34, DW = 0.2075865539; standard errors and P-values reported. - Table B.4 (Mean Forecast Error on Constant) - Example entries: Bottom 33% α = 1.885*** Observations = 516 (0.0766); All α = 0.952*** Observations = 539 (0.0778). - Table B.5 (Prior Distributions of Estimated Parameters) - Priors include means and Std. Dev. for many parameters. Examples: - α Cobb-Douglas parameter: Normal 0.3 0.1 - ψ: Beta 0.5 0.2 - 100(β^{-1} − 1): Gamma 0.3 0.2 - 100(γ−1): Normal 0.4 0.1 - φ s.s.: Normal 4.0 1.5 - η: Beta 0.7 0.1 - σ_c: Normal 1.5 0.4 - φ_p: Normal 1.3 0.1 - ξ_p: Beta 0.5 0.1 - ρ_R: Beta 0.8 0.1 - τ_π: Normal 1.5 0.3 - Fixed parameter δ: Depreciation rate 0.0; φ_w s.s. labor market mark-up 1.6; cg 0.2; ε_p 10; ε_w 10. - Table B.6 (Posterior Distributions: Rational Expectations vs Adaptive Learning) - Posterior means and 5%–95% intervals for many parameters under both paradigms. - Examples (selected): - α: RE Posterior Mean 0.2 5% 0.1 95% 0.3; AL Posterior Mean 0.2 5% 0.1 95% 0.2 - ψ: RE 0.8 (0.7–0.9); AL 0.7 (0.6–0.9) - 100(β^{-1} − 1): RE 0.3 (0.1–0.4); AL 0.2 (0.1–0.3) - φ s.s. elasticity: RE Posterior Mean 6.1 (4.3–8.2); AL 4.3 (5.5–9.0) [table formatting preserved as in source] - ρ_a (TFP persistence): RE 0.98 (0.95–0.99); AL 0.96 (0.94–0.98) - σ_a (TFP innovation std): RE 0.6 (0.5–0.7); AL 0.6 (0.5–0.6) - Marginal Density of Data: RE −1386.8; AL −1356.1 --- *U.S. Inflation Expectations During the Pandemic Working Paper No. WP/2024/025* --- _Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024025-print-pdf.pdf_