## 3.1    Basic VAR Structure

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---

### VAR setup and structural identification
- Model variables:
  - Vector: (f^h_t, π_t, y_t)′ where:
    - π_t denotes inflation in period t.
    - f^h_t denotes the period t forecast of inflation over the following h periods.
    - y_t denotes an n×1 vector of other macroeconomic time series.
- VAR representation (equation numbering preserved):
  - (f^h_t, π_t, y_t)′ = B (f^h_{t−1}, π_{t−1}, y_{t−1})′ + A ε_t  (5)
  - ε_t ∼ N(0,I) is an i.i.d. (n+2)×1 vector of shocks.
- Reduced-form relation:
  - Reduced-form innovations u_t relate to structural shocks by u_t = A ε_t.
- Shock partitioning:
  - ε_t = (ε^S_t, ε^F_t)′ composed of:
    - ε^S_t: one “sentiment” shock that independently affects expectations.
    - ε^F_t: n+1 “fundamental” shocks that determine inflation, output, etc.
- Identification prerequisites:
  - Σ = A A′, where Σ is the autocovariance matrix of forecast errors.
  - Σ has (n+2)(n+3)/2 unique entries.
  - A has (n+2)^2 entries; at least (n+2)(n+1)/2 independent restrictions are needed.

### Identifying assumption (sentiment vs. fundamentals)
- Main identifying assumption:
  - The sentiment shock ε^S_t is the only shock that causes contemporaneous forecasts f^h_t to deviate from rational expectations.
  - Fundamental shocks may only affect contemporaneous forecasts through their average effects on future inflation (i.e., through the rational expectation component).
- Impulse response notation and rational-expectation link:
  - Effect of reduced-form innovation u_t on inflation k periods ahead: φ^π(k) = e_π B^k.
  - Effect on horizon-h inflation: φ^h_π = ∑_{k=1}^h e_π B^k  (6).
  - Rational expectation of horizon-h inflation after u_t: E[π^h_{t+h} | u_t] = φ^h_π u_t.
  - Conditional on structural shocks: E[π^h_{t+h} | ε_t] = φ^h_π A ε_t.
- Partition of A consistent with ε_t dimensions:
  - A = [ A^S_f  A^F_f ; A^S_c  A^F_c ] with scalar A^S_f and row A^F_f.
- Identifying restriction for fundamentals on contemporaneous forecasts:
  - φ^h_π [ A^F_f ; A^F_c ] = A^F_f.
  - Inverted implication:
    - (1 − φ^h_{π,f})^{−1} φ^h_{π,c} A^F_c = A^F_f   (7)
    - φ^h_{π,f} denotes the first entry in φ^h_π; φ^h_{π,c} denotes the remaining entries.
- Implied structure for A under these restrictions:
  - A = [ *(1 − φ^h_{π,f})^{−1} φ^h_{π,c} A^F_c ; * A^F_c ]  (8)
  - “*” denotes unrestricted entries; there are n+2 unrestricted entries in the first column (the contemporaneous impact of the inflation sentiment shock).
- Treatment of A^F_c:
  - Block A^F_c is generally not identified; implementation sets A^F_c to be lower triangular (causal ordering).
  - This choice does not affect ε^S_t because any valid A^F_c is a unitary transformation of the fundamental shocks alone and yields the same sentiment shocks.
- Validation:
  - Section 5.2 reports tests on simulated data showing the method consistently identifies the sentiment shock, with accuracy even in realistically small samples.

### Implementation steps (practical algorithm)
- Stepwise procedure:
  1. Estimate the reduced-form VAR: (f_t, π_t, y_t)′ = B (f_{t−1}, π_{t−1}, y_{t−1})′ + u_t to recover B and u_t.
  2. Calculate Σ as the variance matrix corresponding to u_t.
  3. Construct φ^h_π for the chosen horizon using equation (6).
  4. Calculate A using restrictions from φ^h_π and equation (8) (note: this calculation is typically nonlinear, depending on assumptions used to identify A^F_c).
  5. Recover structural shocks by ε_t = A^{−1} u_t.

### Source attribution
*Source: wpiea2022072-print-pdf - Section 3.1*

---

### Baseline data and VAR specification (Section 4, baseline)
- Baseline VAR variables (six monthly series):
  - CPI inflation (log change in the CPI).
  - Commodity price index from the PPI.
  - Industrial production index.
  - Unemployment rate.
  - Nominal interest rate: Federal Funds Rate (FFR).
  - Expectations: median 12-month-ahead inflation forecast from the Michigan Survey of Consumers.
- Rationale:
  - Michigan measure is monthly and represents expectations of ordinary households.
- Sample and preprocessing:
  - Baseline sample: 1982:M1 - 2021:M12.
  - Variables are deseasonalized and stationarized by removing common monthly (or quarterly) components and a linear time trend.
- Alternative expectation measures considered:
  - Cleveland Fed’s 12-month-ahead market forecasts (monthly).
  - Survey of Professional Forecasters (SPF) consensus (quarterly; substitute quarterly GDP for industrial production).
  - Federal Reserve Greenbook inflation forecasts (last available observation in each quarter used).
- Baseline lag selection:
  - Akaike information criterion selects a three-lag specification.

### Baseline impulse response findings (Figure 3 summary)
- Definition:
  - Sentiment shock: one-standard-deviation structural inflation sentiment shock.
- On impact:
  - Expected inflation rises by around 25 basis points.
  - Inflation itself increases on impact by more than 40 basis points at an annualized rate (not statistically significant); thereafter the impulse is persistently below zero.
- Rational component of one-year-ahead inflation expectations:
  - Declines by about 10 basis points on impact.
  - Average monthly annualized inflation declines by close to 0.1 percent over periods 1 to 12.
- Sentiment component (inflation expectation minus rational component):
  - Typically around 40 basis points in magnitude.
  - Limited persistence: decays almost to zero within a year.
- Other macro dynamics:
  - Real activity (industrial production) declines.
  - Interest rates decline by around 15 to 20 basis points on impact; effect is very persistent, lasting almost two years before becoming statistically insignificant.
  - Real activity remains well below its starting level for several years.
- Overall characterization:
  - Despite a small positive impact on measured inflation initially, the overall dynamic is deflationary.
  - Dynamics contrast with standard New Keynesian model predictions: sentiments cause deflation and recession while monetary policy eases.

### Variance decomposition (Figure 4 summary)
- Decomposition separates variation into sentiment vs. fundamental shocks.
- Key shares reported:
  - Inflation expectations: sentiment shocks drive a relatively large share at short horizons but less at longer ones.
  - Long run: around 80 percent of the variation in inflation expectations is due to fundamental factors.
  - Realized inflation: inflation sentiments drive between around 10 and 20 percent of the variation in most specifications.
  - Industrial production: sentiment shocks are of significant and growing importance at long horizons.
- Interpretation:
  - Sentiment shocks may be an important driver of real macroeconomic fluctuations, even if they are less dominant for long-run inflation expectations.

### Time-series of estimated sentiment (Figure 5)
- Series displayed:
  - Estimated sentiments time series from baseline model (light gray) and 12-month moving average (dark line).
- Notable peaks:
  - Largest during the financial crisis (household inflation expectations persistently high).
  - High at the end of 2021 after a year of resurgent inflation.

### Robustness: alternative expectation measures (Figure 6 and Table 1)
- Alternative VARs: Cleveland (market), SPF (professionals), Greenbook (Fed).
- Impulse responses across measures:
  - One-standard-deviation inflation sentiment shocks are almost identical on impact across measures.
  - Time profiles of departures from rationality for households, professionals, and the Fed are near-identical; Cleveland market measure is less persistent.
  - All sentiment shocks are associated with deflation and recession, broadly in line with baseline results.
  - Interest rates fall in all cases except for the Fed (Greenbook) sentiment shock, where interest rates rise.
- Long-run variance shares attributed to the sentiment shock (Table 1; bootstrapped 90 percent confidence intervals in parentheses):
  - 100 * Log activity:
    - Michigan: 0.363 (0.169, 0.512)
    - Cleveland: 0.111 (0.022, 0.257)
    - SPF: 20412.583 (0.461, 0.786)
    - Fed Greenbook: 0.265 (0.076, 0.691)
  - Federal Funds Rate:
    - Michigan: 0.205 (0.032, 0.405)
    - Cleveland: 0.105 (0.007, 0.276)
    - SPF: 14715.964 (0.078, 0.747)
    - Fed Greenbook: 0.030 (0.046, 0.427)
  - Realized inflation:
    - Michigan: 0.056 (0.035, 0.176)
    - Cleveland: 0.154 (0.086, 0.248)
    - SPF: 11048.476 (0.532, 0.833)
    - Fed Greenbook: 0.072 (0.090, 0.596)
  - Year-ahead inflation expectations:
    - Michigan: 0.187 (0.084, 0.389)
    - Cleveland: 0.106 (0.061, 0.213)
    - SPF: 1107.642 (0.209, 0.721)
    - Fed Greenbook: 0.111 (0.035, 0.384)
- Notes:
  - In all cases, sentiment shocks drive 10% or more of the variance in real activity.
  - Effects on nominal series vary across expectation measures.
  - SPF and Greenbook sentiments contribute little to long-run variance in interest rates.
  - Cleveland and SPF sentiment shocks contribute large shares to realized inflation variance in some specifications.
- Conclusion from robustness:
  - The puzzling result—sentiment shocks being deflationary and recessionary while associated with falling interest rates—is robust to different measures of inflation expectations.
  - Only the Fed’s sentiment shock is consistent with a policy-tightening-like response (higher interest rates).

---

### Identification without the Benefit of Hindsight (Section 4.4)

- Motivation:
  - Baseline identification uses full-sample rational-expectation forecasts (equation (6)).
  - Concern: slow learning implies agents at earlier dates may use different expectations; full-sample identification may misstate sentiment effects.
- Learning-robust estimation:
  - Re-estimate procedure at every time period τ, beginning in 1992 when 10 years of data are available, using only information available up to τ.
  - At each τ the VAR has time-varying coefficient matrices B_τ and A_τ and reduced-form shock u_τ (equation (9) in source).
  - For each τ:
    - Calculate u_τ from the τ-period VAR.
    - Identify coefficient matrix A_τ.
    - Calculate structural shock vector ε_τ = A_τ^{-1} u_τ.
  - Average responses to ε_τ across periods produce learning-robust estimates of sentiment effects.
- Results relative to baseline:
  - Learning-robust estimates are qualitatively similar to baseline overall.
  - Main exception: stronger contraction in real activity in learning-robust results:
    - Industrial production falls by more than in baseline.
    - Unemployment rises on impact (instead of only rising over time in baseline).
- Inference details:
  - Plotted first column of A_τ (instantaneous effect of a sentiment shock) over time.
  - Blue dashed line: sample average (learning-robust instantaneous impact).
  - Black dotted line: endpoint (baseline using entire sample).
  - Shaded ranges show a 90 percent confidence interval computed from bootstrap with 500 replications.
- Robustness to exclusion of the ZLB period:
  - Time-varying plotted values for periods before 2009 exclude the ZLB period.
  - Estimates using data up to 2008 have the same sign as baseline results.
  - Excluding the ZLB period implies larger contemporaneous declines in real activity after a sentiment shock.
  - Truncating the sample before the ZLB period implies positive sentiment shocks increase inflation on impact in those truncated samples.

*Source: wpiea2022072-print-pdf, section 4.4, "Identification without the Benefit of Hindsight".*

---

### Validating our Identification (Section 5.2)

- Validation strategy:
  - Long-sample test: simulate the model for 100,000 periods and estimate shocks to inflation sentiments using the semi-structural identification procedure applied to a VAR with inflation, output, interest rate and year-ahead inflation expectations.
  - Short-sample (finite-sample) test: repeatedly simulate 39 year samples to mimic the baseline data and evaluate the distribution of estimated impulse responses.
- Long-sample results:
  - Long-sample estimates (green triangles in Figure 3) coincide almost exactly with the true impulse responses to the sentiment shock (red circles).
  - Interpretation: given enough data, the method precisely identifies sentiment shocks distinct from preferences, news, noise, and other structural shocks; the method is asymptotically valid.
- Short-sample (finite-sample) results:
  - Procedure: 500 shorter simulations, each of 39 years.
  - The blue squares in Figure 12 plot the median estimated impulse response from these simulations; the gray shaded area is the bootstrapped 90 percent confidence interval.
  - Findings:
    - Short-sample median is very close to the true shock and its impact, with perhaps a small amount of attenuation bias but qualitatively similar to the true response.
    - Confidence intervals, at least on impact, are tight enough to convincingly reject that empirical results—disinflation, lower interest rates, and a hump-shaped output loss—could be generated by a dynamic New Keynesian model of this sort.
- Key numeric details from validation:
  - Simulation lengths: 100,000 periods (long sample); 39 year samples (short samples).
  - Number of shorter simulations: 500.
  - Reported confidence interval: 90 percent.
- Conclusion:
  - Long-sample and short-sample exercises together serve as an acid test: the identification strategy recovers the true structural impulse responses and does not produce counterfactual impulse responses even with other confounding shocks.

---

### Partial resolution of the puzzle: how sentiments can be deflationary

- Puzzle:
  - Empirically, a positive shock to inflation expectations is contractionary, deflationary, and induces monetary loosening—contradicting the canonical New Keynesian model where such shocks are inflationary and only cause recession if policy tightens.
- Proposed partial resolution:
  - Aggregate demand need not be downward-sloping; alternative parameter values can flip its sign.
  - A sentiment shock can be deflationary if monetary policy is sufficiently passive so that inflation no longer raises the real interest rate.
- Illustrative AR(1) persistence assumption:
  - E_t[y_{t+1}] = θ_y y_t
  - E_t[π_{t+1}] = θ_π π_t
- Two-variable AS-AD system under this assumption:
  - (1−βθ_π)π_t = βζ_t + κ y_t    [AS]
  - (φ_π − θ_π)π_t = (γ(θ_y − 1) − φ_y) y_t + ζ_t    [AD]
- Condition for upward-sloping AD:
  - If θ_π > φ_π (persistence of inflation exceeds monetary policy response), then AD slopes upward and sentiment shocks shift it downward, producing deflation and recession.
- Economic interpretation:
  - θ_π > φ_π implies monetary policy is sufficiently passive that inflation today raises future inflation by more than the central bank raises nominal interest rates, so inflation lowers average real interest rates.
- Caveats:
  - Resolution is partial because it requires violating the Taylor principle (φ_π > 1).
  - If θ_π > φ_π and expectations are stable, then φ_π < 1 and the Taylor principle is violated, raising determinacy and multiplicity issues.

### Multiple equilibria example and selection
- Example: set φ_π = 0.2 (central bank raises rates but less than one-for-one).
- Result:
  - Multiple equilibria arise; one equilibrium qualitatively resembles empirical estimates:
    - Inflation jumps up before prolonged deflation.
    - Forecasts of annual inflation rise immediately (driven by sentiment), but 12-month-ahead inflation is negative.
    - Output falls on impact and remains below trend.
    - Central bank lowers interest rates in response to recession and deflation.
- Selection issue:
  - Equilibrium selection likely depends on mechanisms outside the three-equation New Keynesian model (e.g., fiscal policy).
  - Cochrane (2017) is cited: for any initial inflation response, some fiscal policy can rationalize it; empirical dynamics may help discipline equilibrium selection.
- Footnote caution:
  - θ_π > φ_π alone is not sufficient: monetary policy must make AD upward-sloping but not so steep that its slope exceeds AS; relaxing the Taylor principle is not straightforward.

---

### Implications and avenues for future work
- Main conclusions:
  - The identification reliably recovers sentiment shocks, and estimates are clearly inconsistent with the canonical New Keynesian model.
  - A partial tentative resolution highlights fiscal policy's potential importance in interpreting results.
- Two suggested research avenues:
  1. What explains the puzzle? Why are inflation sentiments deflationary? Does it matter who receives the sentiment? The paper finds similar responses across households, markets, and professional forecasters; the role of firms is an open question.
  2. Apply the sentiment-identification method to expectations of other variables (GDP, interest and exchange rates, income, wages) to test implications from information-frictions literature and other mechanisms.

---

### Appendix highlight — Whose sentiment?
- Sector-specific forecast notation:
  - π^{e,1}_{f,t}: one-year-ahead inflation forecasts for firms.
  - π^{e,1}_{hh,t}: for households.
  - π^{e,1}_{cb,t}: for the central bank.
- Modified three-equation model where Taylor rule depends on expected inflation:
  - New Keynesian Phillips curve: π_t = β π^{e,1}_{f,t} + κ y_t
  - Fisher equation: i_t = E_t[γ(y_{t+1} − y_t)] + π^{e,1}_{hh,t}
  - Modified Taylor rule: i_t = φ_y y_t + φ_π π_t + φ_e π^{e,1}_{cb,t}
- Static two-equation reduction when sentiments are i.i.d. and shocks differ by sector:
  - π_t = β ζ_{f,t} + κ y_t    [AS]
  - φ_π π_t = −(φ_y + γ) y_t + ζ_{hh,t} − φ_e ζ_{cb,t}    [AD]
  - ζ_{f,t}, ζ_{hh,t}, ζ_{cb,t} denote sentiment shocks to firms, households, and the central bank respectively.
- Comparative static responses (Figure 15 descriptions):
  - Firms' sentiment shock (ζ_{f,t}): AS shifts up → inflation rises, policy tightens, real interest rate increases, output contracts.
  - Households' sentiment shock (ζ_{hh,t}): households expect higher inflation, perceive real rates to decline, increase consumption → move up along Phillips curve, contemporaneous inflation and real output increase despite higher policy rates.
  - Central bank's sentiment shock (ζ_{cb,t}): central bank preemptively raises interest rates (φ_e > 0) → reduces current inflation and creates recession.
- Crucial conclusion for static model:
  - All three sectoral sentiments result in monetary policy tightening in the static model. There is no way in this static model for a sentiment shock to result in a decrease in interest rates.
  - The only way sentiments produce deflation in the static model is if they overwhelmingly affect the central bank.
  - Empirical comparison: only sentiment shocks to the Fed’s forecasts are consistent with these static-model effects.

*Source: wpiea2022072-print-pdf — Sections 4, 4.4, 5.2, Appendix excerpts*

### 3.1    Basic VAR Structure

### 3.1    Basic VAR Structure

### VAR setup and structural identification
- The VAR includes the vector (f^h_t, π_t, y_t)′ where:
  - π_t denotes inflation in period t.
  - f^h_t denotes the period t forecast of inflation over the following h periods.
  - y_t denotes an n×1 vector of other macroeconomic time series.
- The VAR is written as (equation numbering preserved):
  - (f^h_t, π_t, y_t)′ = B (f^h_{t−1}, π_{t−1}, y_{t−1})′ + A ε_t  (5)
  - ε_t ∼ N(0,I) is an i.i.d. (n+2)×1 vector of shocks.
- Reduced-form innovations u_t relate to structural shocks by u_t = A ε_t.
- Shocks are partitioned as ε_t = (ε^S_t, ε^F_t)′ into:
  - one “sentiment” shock ε^S_t that independently affects expectations, and
  - n+1 “fundamental” shocks ε^F_t that determine inflation, output, etc.
- Identification prerequisites:
  - Σ = A A′, where Σ is the autocovariance matrix of forecast errors.
  - Σ has (n+2)(n+3)/2 unique entries.
  - A has (n+2)^2 entries to identify; at least (n+2)(n+1)/2 independent restrictions are needed.

### Identifying assumption (sentiment vs. fundamentals)
- Main identifying assumption:
  - The sentiment shock ε^S_t is the only shock that causes contemporaneous forecasts f^h_t to deviate from rational expectations.
  - Fundamental shocks may only affect contemporaneous forecasts through their average effects on future inflation (i.e., through the rational expectation component).
- Impulse response notation and rational expectation link:
  - The effect of reduced-form innovation u_t on inflation k periods ahead is φ^π(k) = e_π B^k.
  - The effect on horizon-h inflation is φ^h_π = ∑_{k=1}^h e_π B^k  (6).
  - Rational expectation of horizon-h inflation after u_t: E[π^h_{t+h} | u_t] = φ^h_π u_t.
  - Conditional on structural shocks: E[π^h_{t+h} | ε_t] = φ^h_π A ε_t.
- Partition A consistent with ε_t dimensions:
  - A = [ A^S_f  A^F_f ; A^S_c  A^F_c ] with scalar A^S_f (sentiment → forecasts) and row A^F_f (fundamentals → forecasts).
- Identifying restriction for fundamental shocks on contemporaneous forecasts:
  - φ^h_π [ A^F_f ; A^F_c ] = A^F_f.
  - This implies (inverted form):
    - (1 − φ^h_{π,f})^{−1} φ^h_{π,c} A^F_c = A^F_f   (7)
    - where φ^h_{π,f} denotes the first entry in φ^h_π and φ^h_{π,c} denotes the remaining entries.
- Implied structure for A under these restrictions:
  - A = [ *(1 − φ^h_{π,f})^{−1} φ^h_{π,c} A^F_c ; * A^F_c ]  (8)
  - “*” denotes unrestricted entries; there are n+2 unrestricted entries in the first column (the contemporaneous impact of the inflation sentiment shock).
- The block A^F_c is generally not identified; the implementation sets A^F_c to be lower triangular (causal ordering), which does not affect ε^S_t because any valid A^F_c is a unitary transformation of the fundamental shocks alone and yields the same sentiment shocks.
- Validation:
  - Section 5.2 (referenced) reports tests on simulated data showing the method consistently identifies the sentiment shock, with accuracy even in realistically small samples.

### Implementation steps (practical algorithm)
1. Estimate the reduced-form VAR:
   - (f_t, π_t, y_t)′ = B (f_{t−1}, π_{t−1}, y_{t−1})′ + u_t to recover B and u_t.
2. Calculate Σ as the variance matrix corresponding to u_t.
3. Construct φ^h_π for the chosen horizon using equation (6).
4. Calculate A using restrictions from φ^h_π and equation (8) (note: this calculation is typically nonlinear, depending on assumptions used to identify A^F_c).
5. Recover structural shocks by ε_t = A^{−1} u_t.

---

### 4    Estimated inflation sentiments (baseline results and robustness)

### Data (baseline specification)
- Baseline VAR includes six variables (monthly):
  - CPI inflation (log change in the CPI).
  - Commodity price index from the PPI.
  - Industrial production index.
  - Unemployment rate.
  - Nominal interest rate: Federal Funds Rate (FFR).
  - Expectations: median 12-month-ahead inflation forecast from the Michigan Survey of Consumers.
- Reasons for data choices:
  - Michigan measure is monthly and represents expectations of ordinary households.
  - Baseline sample: 1982:M1 - 2021:M12.
  - Variables are deseasonalized and stationarized by removing common monthly (or quarterly) components and a linear time trend.
- Alternative expectation measures considered:
  - Cleveland Fed’s 12-month-ahead market forecasts (monthly).
  - Survey of Professional Forecasters (SPF) consensus (quarterly; substitute quarterly GDP for industrial production).
  - Federal Reserve Greenbook inflation forecasts (last available observation in each quarter used).
- Baseline lag selection:
  - Akaike information criterion selects a three-lag specification.

### Baseline impulse response findings (Figure 3 summary)
- Sentiment shock: one-standard-deviation structural inflation sentiment shock.
- On impact:
  - Expected inflation rises by around 25 basis points.
  - Inflation itself increases on impact by more than 40 basis points at an annualized rate (not statistically significant); thereafter the impulse is persistently below zero.
- Rational component of one-year-ahead inflation expectations:
  - Declines by about 10 basis points on impact.
  - Average monthly annualized inflation declines by close to 0.1 percent over periods 1 to 12.
- Sentiment component (inflation expectation minus rational component):
  - Typically around 40 basis points in magnitude.
  - Limited persistence: decays almost to zero within a year.
- Dynamics of other macro variables:
  - Real activity (industrial production) declines.
  - Interest rates decline by around 15 to 20 basis points on impact; effect is very persistent, lasting almost two years before becoming statistically insignificant.
  - Real activity remains well below its starting level for several years.
- Overall characterization:
  - Despite a small positive impact on measured inflation initially, the overall dynamic is deflationary.
  - The dynamic pattern is at odds with the standard New Keynesian model predictions (Section 2): sentiments cause deflation and recession while monetary policy eases (contrary to standard theory).

### Variance decomposition (Figure 4 summary)
- Variance decomposition attributes variation in each variable to sentiment vs. fundamental shocks.
- Key shares:
  - Inflation expectations: sentiment shocks drive a relatively large share at short horizons but less at longer ones.
  - Long run: around 80 percent of the variation in inflation expectations is due to fundamental factors.
  - Inflation (realized): inflation sentiments drive between around 10 and 20 percent of the variation in most specifications.
  - Industrial production: sentiment shocks are of significant and growing importance at long horizons.
- Interpretation:
  - Sentiment shocks may be an important driver of real macroeconomic fluctuations, even if they are less dominant for long-run inflation expectations.

### Time-series of estimated sentiment (Figure 5)
- Estimated sentiments time series from baseline model (light gray) and 12-month moving average (dark line).
- Notable peaks:
  - Largest during the financial crisis (household inflation expectations persistently high).
  - High at the end of 2021 after a year of resurgent inflation.

### Robustness: alternative expectation measures (Figure 6 and Table 1)
- Three alternative VARs estimated: Cleveland (market), SPF (professionals), Greenbook (Fed).
- Impulse responses across measures:
  - One-standard-deviation inflation sentiment shocks are almost identical on impact across measures.
  - Time profiles of departures from rationality for households, professionals, and the Fed are near-identical; Cleveland market measure is less persistent.
  - All sentiment shocks are associated with deflation and recession, broadly in line with baseline results.
  - Interest rates fall in all cases except for the Fed (Greenbook) sentiment shock, where interest rates rise (consistent with the Fed acting on a mistaken inflation expectation, producing a policy-tightening-like outcome).
- Long-run variance shares attributed to the sentiment shock (Table 1; bootstrapped 90 percent confidence intervals in parentheses):
  - 100 * Log activity:
    - Michigan: 0.363 (0.169, 0.512)
    - Cleveland: 0.111 (0.022, 0.257)
    - SPF: 20412.583 (0.461, 0.786)  [table formatting preserved exactly as in source]
    - Fed Greenbook: 0.265 (0.076, 0.691)
  - Federal Funds Rate:
    - Michigan: 0.205 (0.032, 0.405)
    - Cleveland: 0.105 (0.007, 0.276)
    - SPF: 14715.964 (0.078, 0.747)
    - Fed Greenbook: 0.030 (0.046, 0.427)
  - Realized inflation:
    - Michigan: 0.056 (0.035, 0.176)
    - Cleveland: 0.154 (0.086, 0.248)
    - SPF: 11048.476 (0.532, 0.833)
    - Fed Greenbook: 0.072 (0.090, 0.596)
  - Year-ahead inflation expectations:
    - Michigan: 0.187 (0.084, 0.389)
    - Cleveland: 0.106 (0.061, 0.213)
    - SPF: 1107.642 (0.209, 0.721)
    - Fed Greenbook: 0.111 (0.035, 0.384)
- Notes on interpretation:
  - In all cases, sentiment shocks drive 10% or more of the variance in real activity.
  - Effects on nominal series vary across expectation measures.
  - SPF and Greenbook sentiments contribute little to long-run variance in interest rates.
  - Cleveland and SPF sentiment shocks contribute large shares to realized inflation variance in some specifications.
- Conclusion from robustness checks:
  - The puzzling result—sentiment shocks being deflationary and recessionary while associated with falling interest rates— is robust to different measures of inflation expectations.
  - Only the Fed’s sentiment shock is consistent with a policy-tightening-like response (higher interest rates) and thus can be rationalized by controlling which agents receive a sentiment shock (Appendix A).

*Source: wpiea2022072-print-pdf - 3.1    Basic VAR Structure*

### 4.4    Identification without the Benefit of Hindsight

### 4.4    Identification without the Benefit of Hindsight

### Approach and motivation
- Baseline identification assumes agents form rational expectations using the true statistical model: the conditional forecast (equation (6)) estimated from the entire sample.
- Concern: if learning is slow, rational agents at an earlier date (e.g., 1992) may form different expectations than agents in 2022, so the full-sample rational-expectation identification may misstate sentiment effects.
- To test robustness to slow learning, the authors re-estimate the entire procedure at every time period τ, beginning in 1992 when 10 years of data are available, so that the identified sentiment in each period uses only information available up to that date.

### Learning-robust estimation procedure
- At each period τ the VAR structure becomes (equation (9) in the source) with time-varying coefficient matrices B_τ and A_τ and reduced-form shock u_τ.
- For each τ:
  - Calculate the reduced form shock u_τ from the τ-period VAR.
  - Identify the coefficient matrix A_τ.
  - Calculate the structural shock vector ε_τ = A_τ^{-1} u_τ.
- The authors then calculate average responses to the identified shocks ε_τ across periods to produce a learning-robust estimate of sentiment effects.

### Results: similarity to baseline and key differences
- The learning-robust estimates are qualitatively similar to the baseline results overall.
- Figure 7 (described in text) plots the first column of A_τ (the contemporaneous effect of a sentiment shock on each time series) estimated using data available in the given period.
  - For each series:
    - The blue dashed line is the average value over the sample period: the learning-robust estimate of the instantaneous impact of a sentiment shock.
    - The black dotted line is the value at the end point: the baseline result using information from the entire sample.
- Main exception:
  - The learning-robust result suggests a stronger contraction in real activity than the baseline:
    - Industrial production falls by more than in the baseline result.
    - Unemployment rises on impact (instead of only rising over time in the baseline).
- Inference details reported for Figure 7:
  - Shaded ranges show a 90 percent confidence interval.
  - Confidence intervals are computed from bootstrap with 500 replications.

### Robustness to exclusion of the zero-lower-bound (ZLB) period
- The time-varying plotted value for any particular period is the baseline estimate excluding any data after that time; thus values from before 2009 exclude the ZLB period in the United States.
- Example: estimates using data up to 2008 all have the same sign as baseline results.
- Excluding the ZLB period implies larger contemporaneous declines in real activity after a sentiment shock.
- In particular, truncating the sample before the ZLB period implies that positive sentiment shocks increase inflation on impact, consistent with the estimated effects on real activity in those truncated samples.

*Italic: Source — wpiea2022072-print-pdf, section 4.4, "Identification without the Benefit of Hindsight".*

### 5.2    Validating our Identification

### 5.2    Validating our Identification

### Validation strategy and tests
- Two tests are used to validate the semi-structural identification procedure applied to the extended New Keynesian model:
  - Long-sample test: simulate the model for 100,000 periods and estimate shocks to inflation sentiments using the semi-structural identification procedure in Section 3 applied to a VAR featuring inflation, output, interest rate and year-ahead inflation expectations.
  - Short-sample (finite-sample) test: repeatedly simulate 39 year samples to mimic the data in the baseline regressions and evaluate the distribution of estimated impulse responses.

### Long-sample results
- The long-sample estimates (green triangles in Figure 3 as reported) coincide almost exactly with the true impulse responses to the sentiment shock (red circles).
- Interpretation:
  - Given enough data, the method can precisely identify the effects of sentiment shocks distinct from preferences, news, noise, and other structural shocks in the model.
  - The method is asymptotically valid.

### Short-sample (finite-sample) results
- Procedure:
  - 500 shorter simulations, each of 39 years, as in Figure 6.
  - The blue squares in Figure 12 plot the median estimated impulse response function from these smaller simulations; the gray shaded area is the bootstrapped 90 percent confidence interval.
- Findings:
  - The short-sample median is very close to the true shock and its impact, perhaps with a small amount of attenuation bias, but qualitatively similar to the true response.
  - Confidence intervals, at least on impact, are tight enough to convincingly reject that the empirical results—disinflation, lower interest rates, and a hump-shaped output loss—could be generated by a dynamic New Keynesian model of this sort.
- Conclusion:
  - Together the long-sample and short-sample exercises serve as an acid test: the identification strategy recovers the true structural impulse responses and does not produce counterfactual impulse responses even in the presence of other potentially confounding shocks.

### Key numeric details from validation
- Simulation lengths: 100,000 periods (long sample); 39 year samples (short samples).
- Number of shorter simulations: 500.
- Reported confidence interval: 90 percent.

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### Partial resolution of the puzzle: how sentiments can be deflationary
- Puzzle restated:
  - Empirically, a positive shock to inflation expectations is contractionary, deflationary, and induces monetary loosening—this is inconsistent with the canonical New Keynesian model where such shocks are inflationary and only cause recession if policy tightens aggressively.
- Proposed partial resolution:
  - In a dynamic New Keynesian model, the aggregate demand curve need not be downward-sloping; alternative parameter values can flip the sign.
  - A sentiment shock can be deflationary if monetary policy is sufficiently passive, so that inflation no longer raises the real interest rate.
- Illustrative assumption and two-variable AS-AD system (persistence assumption):
  - Assume an equilibrium where inflation and output are both AR(1):
    - E_t[y_{t+1}] = θ_y y_t
    - E_t[π_{t+1}] = θ_π π_t
  - With this, the two-variable AS-AD system becomes:
    - (1−βθ_π)π_t = βζ_t + κ y_t    [AS]
    - (φ_π − θ_π)π_t = (γ(θ_y − 1) − φ_y) y_t + ζ_t    [AD]
- Condition for AD to slope upward:
  - If θ_π > φ_π — that is, if monetary policy response to inflation is less than the persistence of inflation itself — then the AD curve slopes upward and sentiment shocks shift it downward, producing deflation and recession.
- Economic interpretation:
  - The θ_π > φ_π condition implies monetary policy is sufficiently passive that inflation today raises future inflation by more than the central bank raises nominal interest rates; thus inflation must lower the average real interest rate.
- Caveats:
  - This resolution is partial because it requires violating the Taylor principle. The Taylor principle usually imposes φ_π > 1 to determine equilibrium in the New Keynesian model.
  - If θ_π > φ_π and inflation expectations are stable, then φ_π < 1 and the Taylor principle is violated; relaxing the principle raises questions about equilibrium determinacy and multiplicity.

### Multiple equilibria and an example
- Example modification:
  - Relax the Taylor principle by setting φ_π = 0.2 (central bank still raises rates to combat inflation but less than one-for-one).
- Result:
  - Multiple equilibria arise in the dynamic New Keynesian model with this modification.
  - One particular equilibrium qualitatively resembles empirical estimates:
    - Inflation jumps up before quickly becoming prolonged deflation.
    - Forecasts of annual inflation immediately rise (driven by sentiment), but 12-month-ahead inflation is negative.
    - Output falls on impact and remains below trend.
    - The central bank lowers interest rates in response to both the recession and the deflation.
- Selection issue:
  - The macroeconomy is probably not flush with multiplicity; selection mechanisms outside the three-equation New Keynesian model (for example, fiscal policy) may select which equilibrium emerges.
  - Cochrane (2017) is cited as arguing that for any initial inflation response, there exists some fiscal policy that rationalizes it; empirical dynamics may help discipline equilibrium selection.
- Footnote caution:
  - The θ_π > φ_π condition alone is not sufficient: monetary policy must make the AD curve upward-sloping but not so steeply that its slope exceeds that of the AS curve; and relaxing the Taylor principle is not as straightforward as the simplified system suggests.

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### Implications and avenues for future work
- Main conclusion:
  - The identification reliably recovers sentiment shocks, and estimates are clearly inconsistent with the canonical New Keynesian model.
  - A partial tentative resolution highlights that fiscal policy might be important in interpreting results.
- Two suggested avenues for additional work:
  - First: What explains the puzzle? Why are inflation sentiments deflationary? Does it matter who receives the sentiment? The paper finds similar responses to household, market, and professional forecaster sentiments. What about firms?
  - Second: The method for identifying sentiments is robust and can apply to expectations of quantities other than inflation. Questions include:
    - Do sentiments of future GDP have the effects that a long information frictions literature predicts?
    - What about sentiments for other variables with measured expectations, such as interest and exchange rates, income, wages, and so forth?

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### Appendix highlight — Whose sentiment?
- Setup for sector-specific sentiment shocks:
  - Denote one-year-ahead inflation forecasts by:
    - π^{e,1}_{f,t} for firms
    - π^{e,1}_{hh,t} for households
    - π^{e,1}_{cb,t} for the central bank
  - Modify the model so the central bank’s Taylor rule depends on expected inflation; together with the Fisher equation and the New Keynesian Phillips Curve, the three-equation model becomes:
    - New Keynesian Phillips curve: π_t = β π^{e,1}_{f,t} + κ y_t
    - Fisher equation: i_t = E_t[γ(y_{t+1} − y_t)] + π^{e,1}_{hh,t}
    - Modified Taylor rule: i_t = φ_y y_t + φ_π π_t + φ_e π^{e,1}_{cb,t}
- Static two-equation reduction when sentiments are i.i.d. and shocks differ by sector:
  - π_t = β ζ_{f,t} + κ y_t    [AS]
  - φ_π π_t = −(φ_y + γ) y_t + ζ_{hh,t} − φ_e ζ_{cb,t}    [AD]
  - ζ_{f,t}, ζ_{hh,t}, ζ_{cb,t} denote sentiment shocks to firms, households, and the central bank respectively.
- Comparative responses (Figure 15 descriptions preserved):
  - Shock to firms (ζ_{f,t}): firms set prices higher, AS shifts up → inflation rises, policy tightens, real interest rate increases, output contracts.
  - Shock to households (ζ_{hh,t}): households expect higher inflation, perceive real rates to decline, increase consumption → move up along Phillips curve, contemporaneous inflation and real output increase despite central bank raising rates.
  - Shock to the central bank (ζ_{cb,t}): central bank preemptively raises interest rates (φ_e > 0) → reduces current inflation and creates a recession.
- Crucial conclusion for the static model:
  - All three sentiments result in monetary policy tightening. There is no way in this static model for a sentiment shock to result in a decrease in interest rates.
  - The only way sentiments produce deflation in the static model is if they overwhelmingly affect the central bank.
  - Empirical comparison: only sentiment shocks to the Fed’s (central bank’s) forecasts are consistent with these static-model effects.

*Working Paper No. WP/2022/072 — Section 5.2 and related passages*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022072-print-pdf.pdf_