## Introduction (wp18154)

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### Importance of long-term inflation expectations
- Long-term inflation expectations are “a crucial element of modern monetary policy.”
- Strongly anchored long-term inflation expectations improve the effectiveness of the monetary transmission mechanism.
- The estimated level and variability of trend inflation provide direct information on the degree of anchoring of inflation expectations.
- The paper builds on a literature combining unobserved components models with stochastic volatility (UCSV).

### Two main methodological contributions
- Incorporation of market-based inflation expectations into UCSV models:
  - Adapts Chan, Clark and Koop (2018) framework to use market-based measures (market-based measures declined significantly to historical low levels in recent years).
  - Benchmark UCSV model jointly estimates trend inflation consistent with market-based inflation expectations.
  - Uses the level and variation of trend inflation to assess whether the protracted period of below-target inflation since 2013 affected euro area trend inflation.
- Decomposition of observed long-term inflation compensation into:
  - trend inflation,
  - an inflation risk premium component (implicitly estimated without modeling the full term structure),
  - and an additional MA(1) error term capturing other premia or temporary disturbances.

### Data choice: euro area IL swap market and f5y5y
- Market measures considered: euro-area inflation-linked (IL) swap market and BEIRs from nominal vs IL bonds.
- Rationale for focusing on IL swaps:
  - Euro area IL bond issuance limited to a few sovereigns; significant sovereign/liquidity risk and market segmentation in IL bond prices after the Global Financial Crisis and euro area debt crisis.
  - IL swaps are net exchange contracts at maturity and “should not incorporate a liquidity premium,” providing a cleaner measure than bond-based BEIRs.
- Common metric emphasized:
  - The “five-year IL forward swap rate five years ahead” (f5y5y) captures inflation compensation between 5 and 10 years ahead and is the most widely used euro-area long-term inflation compensation measure.
- Formal relation used:
  - (1 + f5y5y_t) = (1 + s10y_t)^10 / (1 + s5y_t)^5
  - where s10y_t is the 10-year spot zero-coupon swap rate and s5y_t is the 5-year spot zero-coupon swap rate.

### Main findings (overview)
- Anchoring and trend dynamics:
  - Trend inflation was “relatively well anchored around the 2% level between 2004-2012.”
  - Significant decline in trend inflation estimates since 2013, indicating weakening anchoring in the euro area.
- Historical low market measures:
  - Actual inflation remained below the 2% level since January 2013.
  - f5y5y priced below 2% in the second-half of August 2014 for the first time in euro area history.
  - f5y5y reached a historical minimum “below 1.3% in the summer of 2016.”
- Quantified movement in trend:
  - Decline in compensation reflected a gradual but persistent decline in trend inflation towards levels around 1.5% by early 2015.
  - Trend inflation declined by around 70 basis points over the two years following early 2013, reaching levels slightly above 1.3% by early 2015.
- Decomposition of late-2016 / early-2017 rebound:
  - Rebound in long-term inflation compensation between November 2016 and January 2017: about 40 basis points.
  - Decomposition: about 15 basis points due to higher trend inflation and about 25 basis points due to higher premia.
  - Trend inflation after the rebound: around 1.6%.

*Source: wp18154 - Introduction*

---

### An overview of euro area long-term inflation expectations (Section 2.2)

### Behavior of market and survey measures
- Two empirical features over the last decade:
  - Long-term forward IL swap rate (f5y5y) tended to be significantly above survey measures (Consensus Economics: 6 to 10 years ahead; ECB SPF: five-years ahead), typically attributed to inflation risk premia.
  - Since 2013 long-term forward inflation compensation declined substantially while survey measures remained relatively more stable (survey measures moved away from 2% since mid-2013 but were less volatile).

### Model structure and interpretation
- Observed inflation decomposition:
  - π_t = π^*_t + c_t
    - π^*_t = permanent component (trend inflation = optimal conditional long-term inflation forecast).
    - c_t = deviation from trend (inflation gap).
- Key model elements (UCSV; Bayesian MCMC):
  - Measurement links current inflation and trend inflation with time-varying persistence parameter b_t (0 < b_t < 1 enforced).
  - Long-run inflation compensation ILS_t depends on trend inflation π^*_t with time-varying slope d_1,t and intercept d_0,t plus an MA(1) error to capture premia and persistence not explained by trend.
  - Stochastic volatility specified for variances h_{i,t}, i = v; n.
- Interpretation of equation for ILS_t:
  - Residual in observed inflation compensation reflects investor premia (inflation risk premium, liquidity premium); allowing d_0,t and an MA(1) term captures their size and time variation.

### Trend inflation level and uncertainty (empirical summary)
- Stable anchoring pre-2013:
  - By end-2004 a long-term inflation level of 2% was priced in and remained close to that level until 2012; anchoring broadly unchanged through the GFC and onset of the European debt crisis.
- Significant decline since 2013:
  - Actual inflation below 2% from January 2013 onward.
  - Trend inflation declined by around 70 basis points over the two years from early 2013; reached slightly above 1.3% by early 2015.
  - f5y5y priced below 2% in second-half of August 2014; historical minima below 1.3% in summer of 2016.
- Policy actions and partial rebound:
  - Announcement of QE and direct purchases in January 2015 attenuated the decline but trend inflation and long-term compensation remained significantly below historical averages.
  - Temporary rebound in actual inflation in late 2016 (base effects on energy) coincided with partial rebound in trend inflation and compensation but not a return to historical averages.
- Uncertainty around trend:
  - Despite increased volatility in actual inflation, posterior uncertainty for trend inflation did not increase; decline in trend inflation since 2013 accompanied by a slight reduction in posterior uncertainty, indicating statistical significance.

### Robustness and complementary evidence
- Robustness to alternative compensation measures:
  - Alternative forward rates tested include one-year forward in nine years and five-year forward in ten years; key finding (protracted decline since 2013) robust across measures with small quantitative differences within standard confidence bands.
- Survey-based evidence:
  - Trend inflation estimated using long-term Consensus Economics forecasts lies within benchmark confidence intervals.
  - Both market- and survey-based estimates show similar dynamics and identify the start of the decline in late 2012.
- Role of incorporating long-term information:
  - Omitting long-term information (no equation for ILS_t) yields trend estimates that decline much more sharply since 2012 and fall significantly below benchmark estimates.
  - Incorporating long-term inflation expectations is crucial for correct trend estimation during disinflationary episodes.
- Historical episodes:
  - 2009 post-Lehman: temporary deflation then sharp rebound in 2010; strong anchoring of expectations (trend insensitive to actual inflation).
  - Since late 2012: protracted below-target inflation with strong disinflationary pressures and significant weakening (partial) of anchoring; benchmark trend remains above the alternative without long-term expectations.

*Source: wp18154 - 2.2 An overview of euro area long-term inflation expectations*

---

### Survey expectations, inflation compensation, and risk premia (Sections 6.1–6.2, 7, 8)

### Survey expectations versus trend inflation (Section 6.1)
- Model extension: replace ILS_t with SUR_t in equation (3):
  - SUR_t = d_sur0,t + d_sur1,t ϕ_t + ε_z,t + ε_z,t-1
  - Decomposes discrepancy into level bias (d_sur0,t ≠ 0) and slope deviation (d_sur1,t ≠ 1).
- Empirical findings:
  - d_1t estimates are not statistically different from 1; main source of discrepancy is a significant level bias d_sur0,t.
  - Level bias evolution:
    - statistically insignificant before 2011-12,
    - increased significantly since 2011-12, reaching almost 40 basis points during most of the below-target period,
    - despite recent decreases, survey bias remains sizable.
- Implication:
  - Survey measures may be of limited reliability in a low-inflation environment and can provide biased econometric trend estimates if taken at face value.
- Potential explanations:
  - Information rigidities and sluggish survey adjustment (Coibion and Gorodnichenko, 2015).
  - Time-varying parameters and MA error terms can capture effects of information rigidities.
  - Asymmetries in adjustment when inflation stays below target for prolonged periods warrant further research (Mertens and Nason, 2015 noted).

### Inflation compensation decomposition and inflation risk premia (Section 6.2)
- Decomposition formula used:
  - ILS_t = d_0,t + d_1,t ϕ_t + ε_z,t + ε_z,t-1
    - trend component: d_1,t ϕ_t
    - inflation risk premium (IRP): captured by d_0,t estimates
    - MA(1) error: liquidity premia or temporary market disturbances
- Empirical patterns for euro area IRP:
  - Pre-2013:
    - IRP time-varying and substantial relative to trend.
    - Estimated average IRP for benchmark: around 30 basis points on average.
    - IRP around 20 basis points before 2008.
    - Significant increase in spring 2008 amid oil price surge.
    - Crisis period: high volatility with peaks around late 2010; stabilization around 30 basis points from early 2012.
  - Post-2012 / disinflation:
    - IRP fell by around 30 basis points over 2014.
    - IRP hovered around zero for most of 2015 and turned negative over most of 2016.
    - Occasional negative IRP periods: mid-2010, mid-2011, early 2015, second-half of 2016 — aligning with higher implied probability of inflation being below zero in long-term forward RNDs.
- Interpretation:
  - Time variation and sign changes in IRP consistent with changing correlations between stock and nominal bond returns; negative IRP arises if nominal bonds serve as deflation hedges and investors accept lower returns for hedging benefits.
  - Framework provides a flexible alternative to standard term-structure models for accommodating sign changes in IRP.
- Specific policy-relevant episode:
  - November 2016–January 2017 rebound: about 40 basis points total; decomposition ~25 basis points from higher premia and ~15 basis points from higher trend inflation (trend ≈ 1.6% after rebound).
- Link to option- and swap-based RND evidence:
  - IRP dynamics align with implicit forward Risk Neutral Densities (RNDs) from IL swaps and inflation options; negative IRP episodes align with higher implied probability mass of long-term inflation below zero.

### Additional model results and robustness (Section 7)
- Robustness:
  - Degree of inflation persistence, coefficient of trend in long-term expectations, and stochastic volatility estimates broadly robust across specifications and within benchmark uncertainty bands.
- Inflation persistence:
  - Sustained upward trend in persistence; parameter b_t ≈ 0.4 at end of sample.
  - Higher persistence helps explain slow rebound of inflation after the late-2012 disinflation.
- Impact of trend on expectations:
  - d_1t estimates for market measures slightly higher than for survey measures.
  - For survey measures, d_1t is below 1 but statistically not different from 1; restricted models fixing d_1 = 1 yield qualitatively similar results within uncertainty bands.
- Stochastic volatility:
  - Significant stochastic volatility in trend and especially in inflation gap.
  - Surge in inflation gap shock volatility after Lehman collapse; partial decline and stabilization until late 2013; second surge and persistently high volatility since then.
  - Longer-maturity compensation (5y forward in 10y) shows modestly higher shock volatility but within confidence bands.

### Concluding quantitative statements (Section 8)
- Key quantitative findings:
  - Trend inflation estimates declined significantly below 2% to historically low levels around 1.3% by mid-2016.
  - Findings robust across market-based and survey measures.
- Policy implications:
  - Results support expansion of ECB unconventional monetary policy (UMP) measures since early 2015.
  - Market-based trend inflation estimation supplies a metric to assess the evolution of long-term inflation expectations and the likelihood of a sustained return of inflation toward levels below, but close to, 2% over the medium term.
- Further applications:
  - Extendable to international evidence, including emerging markets with developed inflation-linked markets.
  - Approach yields IRP estimates without imposing cross-maturity pricing restrictions common in macro-finance term-structure literature.

*Source: wp18154 - Sections 6.1, 6.2, 7, 8*

---

### Estimation approach, priors, and computation (Appendix and Section 9)

### State initialization and priors (as specified)
- Initialization:
  - ϕ_1 ~ N(ϕ_0; V_ϕ e h n,1)
  - b_1 ~ N(b_0; V_b)
  - d_i,1 ~ N(μ_d;i; σ^2_d;i (1 - ζ^2_d;i)); i = 0,1
  - h_i,1 ~ N(h_i,0; V_h_i); i = v,n
  - where ϕ_0 = b_0 = h_i,0 = 0 and V_ϕ = V_b = V_h_i = 100
- Independent priors for parameters:
  - μ_d,0 ~ (a_0; V_μ)
  - μ_d,1 ~ (a_1; V_μ)
  - ζ_d;i ~ TN(0,1)(a_2; V_ζ); i = 0,1
  - Prior hyperparameters: a_0 = 0, a_1 = 1, a_2 = 0.95, and V_μ = V_ζ = 0:1 2
  - Prior for MA(1) coefficient: θ ~ TN(-1,1)(0; V_θ) with V_θ = 0:25 2
- Variance priors:
  - σ^2_d;0, σ^2_w, σ^2_hv, σ^2_hn ~ IG(ν_j; S_j) with ν_j = 5 and S_j = 0:04
  - σ^2_d;1, σ^2_b ~ IG(ν_g; S_g) with S_g = 0:004

### Gibbs sampler and sampling strategy
- Nine-block Gibbs sampler draws sequentially from conditional posteriors for:
  1. p(ϕ_1:T | Data; b; d; h_v; h_n; Θ)
  2. p(b_1:T | Data; ϕ; d; h_v; h_n; Θ)
  3. p(d | Data; ϕ; b; h_v; h_n; Θ)
  4. p(h_v; h_n | Data; ϕ; b; d; Θ)
  5. p(μ_d;0; μ_d;1 | ...)
  6. p(σ^2_d;0; σ^2_d;1 | ...)
  7. p(ζ_d;0; ζ_d;1 | ...)
  8. p(θ | ...)
  9. final implicit block completing full conditional draws as in Chan et al. (2018)
- Key computational methods:
  - Precision-sampler technique of Chan and Jeliazkov (2009) for banded precision matrices.
  - Band approximations: replace matrix elements with absolute value < 10^-6 with zero to exploit sparsity.
  - Metropolis-Hastings for non-Gaussian conditional posteriors (e.g., b_t with 0 < b_t < 1 constraints; 2_b).
  - Kim, Shephard and Chib (1998) auxiliary mixture for stochastic volatility with a seven-component Gaussian mixture (used in Chan and Hsiao (2014) implementation).

### Conditional posteriors and special treatments (high level)
- Conditional posterior of trend path ϕ_1:T is multivariate normal with precision matrix K^{-1}_ϕ (band matrix) sampled using precision sampler.
- Conditional posterior of b (inflation-gap loading) requires Metropolis-Hastings due to truncation 0 < b_t < 1; Gaussian proposal built from approximate conditional used for MH acceptance.
- Conditional posterior of d (premia components) is Gaussian and sampled via precision sampler with band approximations.
- Stochastic volatilities h_v and h_n sampled using precision-sampler plus auxiliary mixture approximations.
- Variance parameters mostly conjugate inverse-Gamma draws; 2_b drawn by Metropolis-Hastings with IG proposal.

### Prior sensitivity and substantive effects
- Hyperparameter of interest: V_ (variance in state equation for d_i,t).
- Benchmark prior: V_ = 0:1 2.
- Alternative priors compared:
  - Tighter prior: V_ = 0:025 2.
  - Very non-informative prior: V_ = 1.
- Empirical effects:
  - V_ = 0:025 2 (tighter) → prior mean of d0;t closer to 0 and d1;t near 1; trend inflation tracks IL swap rates more closely and premia are smaller.
  - V_ = 1 (loose) → greater data weight; trend inflation estimates are lower and collapse sharply to implausibly low levels from 2013.
  - Across priors, estimates decline since 2013, corroborating main finding of deterioration in anchoring since mid-2012 and significant decline since 2013.

*Source: wp18154 - Appendix and Section 9*

---

*Source: wp18154 (IMF working paper PDF)*

### Introduction ...........................................................................................................

### Introduction

### Importance of long-term inflation expectations
- Long-term inflation expectations are “a crucial element of modern monetary policy.”
- Strongly anchored long-term inflation expectations improve the effectiveness of the monetary transmission mechanism (see Bernanke 2007, Draghi, 2014, 2015; Yellen, 2015).
- The estimated level and variability of trend inflation provide direct information on the degree of anchoring of inflation expectations.
- A literature combining unobserved components models with stochastic volatility (UCSV) to estimate long-term trend inflation has emerged (examples: Stock and Watson, 2015; Chan, Koop and Potter, 2013; Bednar and Clark, 2014; Garnier, Mertens, and Nelson, 2015; Mertens, 2015).

### Two contributions of the paper
- Incorporation of market-based inflation expectations into UCSV models for trend inflation estimation:
  - Adapts Chan, Clark and Koop (2018) framework to use market-based inflation expectations (market-based measures declined significantly to historical low levels in recent years).
  - The benchmark UCSV model jointly estimates trend inflation consistent with market-based inflation expectations.
  - Uses level and variation of trend inflation to assess whether the protracted period of below-target inflation since 2013 affected euro area trend inflation.
- Decomposition of observed long-term inflation compensation into inflation expectations and an inflation risk premium:
  - Model provides an implicit estimation of long-term inflation risk premia without modeling the whole term structure.
  - Observed inflation compensation reflects the sum of trend inflation, an inflation risk premia component, and an additional error term associated with other premia.

### Data choice: market-based measures and the euro area IL swap market
- Market measures considered: inflation-linked (IL) swap market in the euro area and BEIRs (break-even inflation rates) from nominal vs IL bonds.
- Rationale for using IL swaps instead of BEIRs:
  - Euro area IL bond issuance has remained relatively limited (only France, Italy, Germany, Spain and Greece have issued some IL bonds).
  - Significant market segmentation and sovereign/liquidity risk in IL bond prices after the Global Financial Crisis and euro area debt crisis.
  - IL swaps are net exchange contracts at maturity and “should not incorporate a liquidity premium,” providing a cleaner measure of inflation compensation than bond-based BEIRs.
- Market focus and common metric:
  - Euro-area IL swaps market is “the most mature and largest IL swaps market in the world in terms of trading volumes,” with contracts from 1 to 30 years, with concentrated liquidity around five- and ten-year maturities.
  - The “five-year IL forward swap rate five years ahead” (f5y5y) has become the most widely used measure to assess euro area long-term inflation compensation (capture inflation compensation between 5 and 10 years ahead).
- Zero-coupon IL swaps:
  - Zero-coupon contracts exchange a fixed inflation rate for actual inflation at maturity.
  - The fixed leg reflects expected inflation plus an inflation risk premium.
- Formal relation for forward IL swap rates (as in the source):
  - (1 + f5y5y_t) = (1 + s10y_t)^10 / (1 + s5y_t)^5
  - where s10y_t is the 10-year spot zero-coupon swap rate and s5y_t is the 5-year spot zero-coupon swap rate.

### Main findings
- Anchoring and trend inflation dynamics:
  - Trend inflation was “relatively well anchored around the 2% level between 2004-2012.”
  - Anchoring “was broadly unchanged during most of the Global Financial Crisis period,” including the Lehman collapse and the onset of the European debt crisis in 2010.
  - A “significant decline in trend inflation estimates since 2013,” indicating a weakening in the anchoring of long-term inflation expectations in the euro area.
- Market measures and historical lows:
  - Actual inflation has remained below the 2% level since January 2013.
  - Benchmark long-term inflation compensation measures (five-year forward IL swap rate in five years) began being priced below 2% in the second-half of August 2014 for the first time in euro area history.
  - The five-year forward IL swap rate in five years reached a historical minimum “below 1.3% in the summer of 2016.”
- Quantified trend movement:
  - The decline in compensation reflected a “gradual but persistent decline in trend inflation levels over the following two years towards levels around 1.5% by early 2015.”
- Decomposition around late-2016 / early-2017:
  - The rebound in long-term inflation compensation between November 2016 and January 2017 reflected:
    - A slight recovery of long-term inflation expectations (about 15 basis points).
    - Mainly an increase in inflation risk premia (about 25 basis points).

### Robustness and complementary evidence
- Findings are robust to:
  - Using different measures of long-term inflation compensation.
  - Alternative trend inflation estimates using survey data on long-term inflation expectations, which show very similar dynamics and corroborate the timing of the decline starting in late 2012.
- Survey bias:
  - Evidence of a “sizable and statistically significant bias in euro area survey inflation expectations,” consistent with U.S. evidence reported in Chan, Clark and Koop (2018).

### Policy implications and relevance
- Support for ECB policy actions:
  - The protracted decline in trend inflation since 2013 “provides substantial support for the expansion of the ECB’s unconventional monetary policy UMP measures to direct purchases of sovereign bonds (QE), among other assets, since early 2015.”
  - ECB’s QE “seems to have just managed to attenuate the decline in long-term inflation compensation measures and trend inflation,” but both have remained significantly below their historical averages since then.
- Monitoring return to target:
  - The paper’s framework provides a metric to monitor the likelihood of a sustained return of inflation towards (levels below but close to) 2% over the medium term (Draghi, 2018).

### Structure of the paper (as organized in the source)
- Section 2: overview of long-term inflation expectations and euro area IL swap market compensation.
- Section 3: empirical model.
- Section 4: main results from benchmark specification.
- Section 5: robustness checks and discussion of inflation risk premia estimates.
- Section 6: concluding remarks.

*Source: wp18154 - Introduction ...........................................................................................................*

### 2.2  An overview of euro area long-term ináation expectations

### 2.2  An overview of euro area long-term inflation expectations

### Behavior of market and survey measures
- Long-term forward IL swap rate (five-year forward in five years) and two survey measures (Consensus Economics: 6 to 10 years ahead; ECB SPF: five-years ahead) are compared.
- Two important features over the last decade:
  - Long-term forward inflation compensation tended to be significantly above survey measures. This discrepancy has generally been attributed to inflation risk premia embedded in inflation compensation.
  - Since 2013 long-term forward inflation compensation declined substantially while survey measures remained relatively more stable (though they have moved away from the 2% reference level since mid-2013).
- Implication: either markets may have overpriced risks of severe deflation, or surveys may have become disconnected from actual inflation dynamics in a low-inflation environment.

### Model for trend inflation (structure and interpretation)
- Observed inflation π_t decomposed as π_t = π^*_t + c_t where:
  - π^*_t is the permanent component (trend inflation = optimal conditional long-term inflation forecast).
  - c_t is the deviation from trend (inflation gap).
- Key model elements (unobserved components with stochastic volatility; Bayesian MCMC estimation):
  - Equation (2): measurement linking current inflation and trend inflation with time-varying persistence parameter b_t; inflation gap is stationary at each point in time by imposing 0 < b_t < 1 via a truncated normal on variance of b_t.
  - Equation (3): long-run inflation compensation ILS_t depends on trend inflation π^*_t with time-varying slope d_1,t and time-varying intercept d_0,t (captures premia), plus an MA(1) error term to capture persistence not explained by trend.
  - Equation (4): transition for trend inflation π^*_t.
  - Equation (5): transition for b_t (time-varying persistence).
  - Equation (6): dynamics for d_i,t around mean μ_{d_i}.
  - Equation (7): stochastic volatility for variances h_{i,t}, i = v; n.
- Interpretation of equation (3):
  - Once the level of long-term inflation expectations and its market pricing are pinned down, the residual in observed inflation compensation reflects investor premia (inflation risk premium, liquidity premium). These premia are unobservable but vary significantly over time; allowing d_0,t and an MA(1) term helps capture their size and variation.

### Trend inflation: estimated level and uncertainty
- Trend inflation was relatively stable around the 2% level for most of the sample:
  - Early sample low levels reflect early development of euro area IL swap market in 2004; by end-2004 a long-term inflation level of 2% was priced in and remained close to that level until 2012.
  - Anchoring broadly unchanged during the Global Financial Crisis (GFC) and European debt crisis onset since early 2010.
- Significant decline since 2013:
  - Actual inflation went below the 2% mark in January 2013 and remained below target thereafter.
  - Trend inflation declined by around 70 basis points over the following two years from a level close to 2% at the beginning of 2013, reaching levels slightly above 1.3% by early 2015.
  - Benchmark long-term inflation compensation (five-year forward IL swap rate in five years) priced below 2% in the second-half of August 2014 for the first time and reached a historical minima below 1.3% in the summer of 2016.
- Policy actions and partial rebound:
  - Announcement of QE and direct purchases in January 2015 attenuated the decline but trend inflation and long-term compensation remained significantly below historical averages.
  - Almost two years after bond purchases and with a temporary rebound in actual inflation in late 2016 (base effects on energy), trend inflation and long-term inflation compensation partially rebounded but stayed far from historical average.
- Uncertainty:
  - Despite increased volatility in actual inflation, uncertainty surrounding posterior estimates of trend inflation did not increase; the decline in trend inflation since 2013 was accompanied by a slight reduction in posterior uncertainty, indicating statistical significance of the decline.

### Robustness checks and additional evidence
- Robustness to different inflation compensation measures:
  - Alternative long-term forward rates used: one-year forward in nine years and five-year forward in ten years (ten-year maturity and long institutional-maturity segment).
  - Key findings (protracted decline in trend inflation since 2013) are robust across these measures; quantitative differences are relatively small and remain within standard confidence bands.
- Evidence from survey data:
  - Trend inflation estimated using long-term (6-10 years) Consensus Economics forecasts.
  - Survey-based trend lies within standard statistical confidence intervals of the benchmark (five-year forward in five years) over the whole sample.
  - Both market- and survey-based estimates show similar dynamics and identify the start of decline in late 2012.
  - Conclusion: decline in trend inflation is not an artifact of using market-based compensation measures; both sources corroborate weakening inflation dynamics.
- Role of long-term information:
  - Alternative specification that omits equation (3) (no long-term information) resembles UCSV model without bounds on trend inflation.
  - Before 2012, trend estimates are similar with or without long-term information; since 2012, the model without long-term information declines much more sharply and moves significantly below benchmark estimates.
  - Evidence indicates incorporating long-term inflation expectations is crucial for correct trend inflation estimation, especially during disinflationary episodes.
  - Historical episodes:
    - 2009 (post-Lehman): temporary deflation followed by sharp rebound in 2010; strong anchoring of expectations (trend insensitive to actual inflation) likely important.
    - Since late 2012: protracted below-target inflation. Results point to strong disinflationary pressures plus a significant weakening (but only partial) of the anchoring of long-term inflation expectations; benchmark trend remains significantly above the alternative without long-term expectations.

### Relationship between trend inflation and inflation expectations
- Financial indicators are advocated for measuring long-term inflation expectations in this framework because:
  - Over the protracted disinflationary period since late 2012, survey measures reacted mutedly and appeared somewhat disconnected from actual developments, whereas long-term forward inflation compensation declined markedly.
- Empirical conclusion:
  - There has been a significant decline in euro area trend inflation in recent years (start of decline identified in late 2012).
  - Long-term inflation expectations are a crucial input for estimating trend inflation and for understanding the anchoring (or weakening of anchoring) of expectations in the euro area.

*Source: wp18154 - 2.2  An overview of euro area long-term inflation expectations*

### 6.1  Survey expectations and trend ináation

### wp18154 - 6.1  Survey expectations and trend ináation

### Survey expectations and trend inflation (Section 6.1)
- Framework extends Chan et al. (2018) to euro area survey data to quantify why observed (reported) surveys diverge from estimated trend inflation.
- Replacing ILS_t with SUR_t in equation (3) (SUR_t = d_sur0,t + d_sur1,t ϕ_t + ε_z,t + ε_z,t-1) allows decomposition of discrepancy into:
  - level bias: d_sur0,t ≠ 0
  - inefficiency / slope deviation: d_sur1,t ≠ 1
- Empirical finding:
  - d_1t estimates are not statistically significant from 1 (see Section 7), implying the main source of discrepancy is a significant level bias d_sur0,t.
  - The level bias in survey measures:
    - statistically insignificant before 2011-12
    - increased significantly since 2011-12, reaching almost 40 basis points during most of the below-target period
    - despite recent decreases, the survey bias remains sizable
- Implication:
  - Survey measures are of limited reliability in a low inflation environment and may provide biased estimates of econometric trend inflation if taken at face value.
- Potential explanations (literature links):
  - Information rigidities and sluggish adjustment of survey expectations (Coibion and Gorodnichenko, 2015).
  - Time-varying parameters and MA error terms in the model can flexibly capture effects of information rigidities.
  - Asymmetries in adjustment when inflation realizations remain below target for prolonged periods merit further investigation (Mertens and Nason, 2015 approach noted as promising).

### Inflation compensation, trend inflation, and risk premia (Section 6.2)
- Key methodological advance: incorporating inflation compensation (market-based measures) into trend inflation estimation for the euro area.
- Decomposition logic:
  - ILS_t = d_0,t + d_1,t ϕ_t + ε_z,t + ε_z,t-1 decomposes long-term inflation compensation into:
    - trend inflation component: d_1,t ϕ_t (long-term conditional forecast)
    - inflation risk premium (IRP): captured by d_0,t estimates
    - MA(1) error terms: attributed to liquidity premia or temporary market disturbances
- Empirical patterns for euro area IRP:
  - Pre-2013:
    - IRP exhibits substantial time variation relative to trend inflation
    - Estimated average level of IRP for benchmark specification: around 30 basis points on average
    - IRP around 20 basis points before 2008
    - Significant increase in spring 2008 amid oil price surge and inflation
    - Crisis period: high volatility with highs around late 2010, followed by severe declines, and stabilization around 30 basis points from early 2012
  - Post-2012 / protracted disinflation:
    - Declines in IRP: IRP fell by around 30 basis points over 2014
    - IRP hovered around zero for most of 2015 and turned negative over most of 2016
    - Few periods of negative IRP in sample: around mid-2010, mid-2011, early 2015, second-half of 2016 — coinciding with rise in implied probability of inflation being below zero in long-term forward RNDs
  - Interpretation:
    - Time variation and possible sign changes in IRP are consistent with literature documenting changing correlations between stock and nominal bond returns (e.g., negative correlations over recent decades)
    - Negative IRP can arise if nominal bonds act as deflation hedges, leading investors to accept lower returns for hedging benefits
    - The paper’s framework offers a flexible alternative to standard term structure models for accommodating sign changes in IRP
- Specific policy-relevant episode (late 2016 rebound):
  - Rebound in long-term inflation compensation between November 2016 and January 2017: about 40 basis points
  - Decomposition of the rebound:
    - around 25 basis points due to higher premia
    - about 15 basis points due to higher trend inflation
    - trend inflation at around 1.6% after the rebound, still far from ECB’s price stability objective
- Link to option- and swap-based RND evidence:
  - Dynamics of IRP estimates align with implicit forward Risk Neutral Densities (RNDs) based on IL swap and inflation options: negative IRP episodes align with higher implied probability mass of long-term inflation below zero.

### Additional model results (Section 7)
- Robustness across specifications:
  - Degree of inflation persistence, coefficient of trend inflation in long-term expectations, and stochastic volatility estimates are broadly robust across different model specifications and lie within uncertainty bands of benchmark estimates.
- Inflation persistence:
  - Sustained upward trend in inflation persistence over the sample; parameter b_t remains around 0.4 at end of sample.
  - Higher persistence may help explain the slow rebound of inflation after the disinflation since late 2012.
- Impact of trend inflation on expectations:
  - d_1t estimates for inflation compensation measures tend to be slightly higher than for survey measures.
  - For survey measures, d_1t is below 1 throughout the sample, though estimates are close to and statistically not different from 1.
  - Restricted models fixing trend inflation parameter to 1 produce qualitatively similar results within uncertainty bands.
- Stochastic volatility:
  - Significant stochastic volatility found in the trend inflation and particularly in the inflation gap equation.
  - Surge in inflation gap shock volatility after the Lehman Brothers collapse, partial decline and stabilization until late 2013, followed by a second surge and persistently high volatility since then.
  - Using longer maturity inflation compensation (5-year forward in ten years) shows some evidence of higher shock volatility, but differences are within standard confidence bands.

### Concluding remarks (Section 8)
- Contributions:
  - Incorporates market-based inflation expectations into trend inflation estimation, providing a useful complement to survey measures.
  - Demonstrates strong evidence of significant deterioration in long-term inflation expectations in the euro area since 2013.
- Key quantitative findings:
  - Trend inflation estimates declined significantly below 2% to historically low levels around 1.3% by mid-2016.
  - Findings robust across measures of long-term inflation compensation and survey measures.
- Policy implications:
  - Results provide support for the expansion of unconventional monetary policy (UMP) measures by the ECB since early 2015.
  - Market-based trend inflation estimation supplies a useful metric to assess the evolution of long-term inflation expectations and the likelihood of a sustained return of inflation toward levels below, but close to, 2% over the medium term.
- Further applications:
  - Extending market-based trend inflation estimation can provide international evidence on global inflation trends, including in emerging markets with developed inflation-linked markets.
  - Approach yields estimates of inflation risk premia without imposing common cross-maturity pricing restrictions used in macro-finance term structure literature.

### Appendix: Estimation approach — priors and Gibbs sampler (Appendix)
- Initialization of state equations (as specified):
  - ϕ_1 ~ N(ϕ_0; V_ϕ e h n,1)
  - b_1 ~ N(b_0; V_b)
  - d_i,1 ~ N(μ_d;i; σ^2_d;i (1 - ζ^2_d;i)); i = 0,1
  - h_i,1 ~ N(h_i,0; V_h_i); i = v,n
  - where ϕ_0 = b_0 = h_i,0 = 0 and V_ϕ = V_b = V_h_i = 100
- Independent priors for model parameters:
  - μ_d,0 ~ (a_0; V_μ)
  - μ_d,1 ~ (a_1; V_μ)
  - ζ_d;i ~ TN(0,1)(a_2; V_ζ); i = 0,1
  - Prior hyperparameters set: a_0 = 0, a_1 = 1, a_2 = 0.95, and V_μ = V_ζ = 0:1 2
  - Prior for MA(1) coefficient: θ ~ TN(-1,1)(0; V_θ) with V_θ = 0:25 2
- Variance priors:
  - Independent inverse gamma priors for variance parameters:
    - σ^2_d;0, σ^2_w, σ^2_hv, σ^2_hn ~ IG(ν_j; S_j) with ν_j = 5 and S_j = 0:04
    - σ^2_d;1, σ^2_b ~ IG(ν_g; S_g) with S_g = 0:004
- Gibbs sampler (nine-block) draws sequentially from conditional posteriors for:
  1. p(ϕ_1:T | Data; b; d; h_v; h_n; Θ)
  2. p(b_1:T | Data; ϕ; d; h_v; h_n; Θ)
  3. p(d | Data; ϕ; b; h_v; h_n; Θ)
  4. p(h_v; h_n | Data; ϕ; b; d; Θ)
  5. p(μ_d;0; μ_d;1 | Data; ϕ; b; d; h_v; h_n; Θ \ {μ_d;0; μ_d;1})
  6. p(σ^2_d;0; σ^2_d;1 | Data; ϕ; b; d; h_v; h_n; Θ \ {σ^2_d;0; σ^2_d;1})
  7. p(ζ_d;0; ζ_d;1 | Data; ϕ; b; d; h_v; h_n; Θ \ {ζ_d;0; ζ_d;1})
  8. p(θ | Data; ϕ; b; d; h_v; h_n; Θ \ {θ})
  9. (implicit final block completing full conditional draws as in Chan et al. (2018))

*Source: wp18154 - 6.1  Survey expectations and trend ináation (IMF working paper PDF).*

### 9. Drawp(

### 9. Drawp(

### Measurement and state-equation reformulations
- Measurement equation (re-written): H b  = H b  + ~  + v; vN(0; v ).
- Definitions preserved from source:
  - ~  = (b1(00);0;:::;0)0.
  -  v = diag(e h v;1 ;:::;e h v;T )0.
  - v = (v1;:::;vT )0.
  - H b is the banded matrix specified in equation (19) with |H b | = 1 and therefore invertible.
- Resulting conditional: (j;b;h v )N(+ ;(H0 b1 v H b )1) as in equation (20).
- Observation equation (rewritten): z = d0 + X   + H  z ;  z N(0;2 w I T ) with H as specified in (22).
- State equation (rewritten): H  =   + n t ; n t N(0; n ) with   and  n defined and H as in (25).

### Conditional posterior of  (trend inflation path)
- Log posterior combining (19), (22), (25) yields (equations (27)-(29)):
  - log p( | Data; b; d; h v ; h n ; )/1/2 ( )0(H0 b1 v H b )1/2( )
  - terms including 1/22 w (zd0X  )0(H H0 )1(zd0X  ) and state prior ( )0(H01 n H)( ).
- Conditional posterior is multivariate normal:
  - ( | Data; b; d; h v ; h n ; )(^;K1 ) (equation (29)).
  - K  = (H0 b1 v H b + 1 2 w ^X0  ^X  + H01 n H)1 (equation (30)).
  - ^ = K1 (H0 b1 v H b ( ) + 1 2 w ^X0  ~z + H01 n H  ) (equation (31)).
- Computational approach:
  - Precision matrix K1  is a band matrix; apply the precision sampler technique of Chan and Jeliazkov (2009).
  - Band approximation: replace elements with absolute value less than 106 with zero.

### Conditional posterior of b (inflation-gap loading)
- Measurement rewrite: ~ = X b b + v; vN(0; v ) (equation (32)).
  - ~ = (11;:::;TT )0.
  - X b = diag(00;:::;T1T1)0.
- State equation for b: H b = ~ b +  b ;  b N(0;2 b I T ) with ~ b = (b0;0;:::;0)0 and  b elements independent truncated normal (equation (33)).
- Inequality constraint: 0 < b t < 1 must hold for all t; conditional posterior is non-normal → Metropolis-Hastings required.
- Prior density for b includes truncation probabilities:
  - Pr(0< b1 <1) =(1b0 p V b )(b0 p V b ).
  - Pr(0< b t <1) =(1b t1  b )(b t1  b ) (equation (34)).
- Log conditional posterior (equation (37)):
  - log p(b | Data; ; d; h v ; h n ; )/1/2(b^ b )0K1 b (b^ b ) + g(b;2 b ).
- Gaussian proposal for Metropolis-Hastings:
  - (b | Data; ...) approx N(^ b ; K1 b ) where K b = (H01 b H + X0 b1 v X b ) and ^ b = K1 b (H01 b H b + X0 b1 v ~) (equations (38)-(40)).
  - Candidates drawn from this Gaussian are accepted/rejected via Metropolis-Hastings.

### Conditional posterior of d (inflation compensation premia components)
- Rewrite observation and state equations:
  - z = X d d + H  z ;  z N(0;2 w I T ) (equation (41)).
  - H  d = ~ d +  d ;  d N(0; d ) (equation (42)).
- Definitions:
  - ~ d,  d , X d and H  specified in equations (42)-(44).
- Conditional posterior is Gaussian:
  - (d | Data; ; b; h v ; h n ; )(^ d ; K1 d ) (equation (46)).
  - K d = (H0  1 d H  + 1 2 w ~X0 d ~X d ) (equation (47)).
  - ^ d = K1 d (H0  1 d ~ d + 1 2 w ~X0 d H1 z ) (equation (48)).
- Computational approach:
  - Construct band approximation of ~X d by setting elements < 106 to zero.
  - Use precision sampler (Chan and Jeliazkov, 2009) to sample ^ d.

### Sampling stochastic volatilities h v and h n
- Implement precision sampler technique of Chan and Hsiao (2014).
- Use Kim, Shephard and Chib (1998) auxiliary mixture sampler approximating log(2 z ) with a seven-component Gaussian mixture with fixed parameters.
- Referenced procedure: Chan and Hsiao (2014).

### Other parameter draws and conditional posteriors
- ( d;0 ;  d;1 ) | ... are Gaussian: ( d;i | ...)N(^ d;i ; K1 d;i ) (equation (49)).
- (2 d;i | ...) follow inverse-Gamma: (2 d;i | ...)IG( d;i + T/2 ; ~S d;i ) (equation (50)).
  - K d;i, ^ d;i and ~S d;i expressions given in source.
- ( d;0 ;  d;1 ) | ... non-standard; draw via Metropolis-Hastings using independence-chain with proposal N(^ d;i ; K1  d;i ) as in Chan et al. (2017).
  - K  d;i and ^ d;i definitions provided, with X  d;i and y  d;i constructed from d series.
- Draw for  (scalar vector of other parameters): independence-chain Metropolis-Hastings; evaluate log-density using band matrix routines; maximize numerically to obtain mode ^ and negative Hessian K ; candidates from N(^ ; K1 ).
  - Log posterior expression in (52)-(53): includes log p(z |  ; d ; 2 w ) + log p( ).

### Variance parameters and special treatments
- Variance parameters are conditionally independent and standard:
  - (2 w | ...)IG(2 w + T/2 ; S2 w + 1/2 Σ T t=1 ~ 2 z;t ) (equation (54)).
  - (2 h i | ...)IG(2 h i + (T1)/2 ; S2 h i + 1/2 Σ T t=2 (h i;th i;t1 )2 ) for i = v;n (equation (55)).
  - ~ z = H1 (zX d d ).
- 2 b requires Metropolis-Hastings; log conditional given in equation (56) with additive term g b (b; 2 b ).
  - Proposal: IG(2 b + (T1)/2 ; S2 b + 1/2 Σ T t=2 (b tb t1 )2 ) (equation (57)).

### Computational notes and small-threshold approximations
- Band approximations applied to design matrices ~X  and ~X d : replace elements below absolute value of 106 with zero.
- Precision sampler (Chan and Jeliazkov, 2009) and related band-matrix routines are central to efficient sampling.

### Prior sensitivity analysis and substantive findings
- Hyperparameter of interest: V (variance in state equation for d i;t).
- Benchmark prior in paper: V = 0:1 2.
- Alternative priors compared:
  - Tighter prior: V = 0:025 2.
  - Very non-informative prior: V = 1.
- Intuition and empirical effects:
  - V = 0:025 2 (tighter prior) implies prior mean of d0;t closer to 0 and d1;t very close to 1 → trend inflation estimates track IL swap rates more closely; the premia (difference between trend inflation and IL swap rates) are smaller.
  - V = 1 (loose prior) places larger weight on data → trend inflation estimates are lower across the sample and collapse sharply to implausibly low levels from 2013.
- Empirical result (Figure A1 summary):
  - A tighter prior moves trend inflation estimates closer to observed IL swap rates while remaining within benchmark uncertainty bands.
  - Across priors, estimates decline since 2013, corroborating main findings: deterioration in anchoring of long-term inflation expectations since mid-2012 and significant decline since 2013.

*Source: wp18154 - 9. Drawp( (PDF chapter) *

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18154.pdf_