## 2.1 Overview and timing ## Source details **Canonical URL:** [2.1 Overview and timing](https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026173-source-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2026/english/wpiea2026173-source-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2026/english/wpiea2026173-source-pdf.pdf.json) --- ### Dataset construction and indexing - Daily market forecasts compiled for nine year-end variables plus two constructed series (real interest rate and depreciation of the real exchange rate). - Data indexed by τ (day forecast made) and by t+h (calendar year to which the forecast applies); h = 0 denotes the calendar year of τ. - Forecast horizons collected: - For each τ: h = −1, 0, 1, 2, 3, 4. - For January and February τ: h = −2, −1, 0, 1, 2, 3. - Coverage: - Forecasts collected for the current year and the next four years for every one of the 4,120 working days from 4 January 2010 to 29 May 2026. - Adding one lag of real-time data for each series (model is dynamic with one lag) yields, for each series in Table 1, a total of 24,720 day-horizon observations. ### Series compiled (notation preserved) - Survey: IPCA (CPI) Inflation — Focus — π_{τ h} - Survey: Log output — Focus — y_{τ h} - Survey: SELIC (policy) rate — Focus — i_{τ h} - Survey: Log USD exchange rate — Focus — z_{τ h} - Survey: Primary surplus/GDP ratio — Focus — s_{τ h} - Survey: Net Debt/GDP ratio — Focus — d_{τ h} - U.S. Market data: Fed Funds rate — Christensen et al. (2011) — f_{τ h} - U.S. Market data: Oil price inflation (USD) — Futures market — π_{oil,τ h} - Estimated: US annual inflation — ARIMA forecasting model — π_{US,τ h} - Constructed: Real interest rate — r_{τ h} (see text for formula) - Constructed: Real exchange rate depreciation — q_{τ h} (see text for formula) ### Constructed definitions (preserved exactly) - Real interest rate: - r_{τ h} = 1/2 ( i_{τ h} + i_{τ h−1} ) − π_{τ h} - Real exchange rate depreciation: - q_{τ h} − q_{τ h−1} = ( z_{τ h} − z_{τ h−1} ) − π_{τ h} + π_{US,τ h} ### Data notes on vintages and interpolation - For GDP data, “real-time” vintages with exact release dates are available back to 2013Q1; prior to this the 2013 vintage is used. - Daily year-end US interest rates and dollar oil prices computed using Christensen et al. (2011) estimates and daily futures prices; raw data in relative time are converted to year-end values via quadratic interpolation. - For horizons longer than 2 years, oil futures prices are assumed constant. - Daily year-end US inflation forecasts are estimated with a simple ARIMA model (approximate; errors relatively minor). --- ### 2.2 The Focus survey ### Survey design and reporting - Backbone: Focus survey conducted by Banco Central do Brasil (BCB). - Respondents: professional market participants (banks, asset managers, consultancies, other financial institutions). - Participation: self-select; BCB requires respondents “have a research-dedicated unit and chief economist whose CV has to be accepted by BCB.” - Frequency: daily; participants may update forecasts at any time; BCB aggregates up to a fixed cutoff and publishes summary statistics each business day. - Eligible institutions: some 165 currently eligible (up from around 50 at inception). - Summary statistics reported (for each variable–horizon pair): mean, median, cross-sectional standard deviation. - This paper focuses on the cross-sectional unweighted mean forecast (mean chosen because the interpretative model is linear and the unweighted mean provides the longest available time series). ### Caveats motivating modeling choices - Participation is voluntary and panel composition varies over time; composition changes may affect aggregate forecasts independently of underlying beliefs. - Individual forecasters may update intermittently (BCB indicates most participants update weekly). - Survey forecasts may reflect strategic considerations, reputational concerns, or herding; these motives are accommodated in the intermittent-updating belief model (introduced in Section 3.1). --- ### 2.4 The forecasts — descriptive observations - Empirical patterns from daily forecasts (examples): - During 2015–2016, near-term forecasts were much more sensitive to news: inflation and growth forecasts for 2015 moved earlier and more than forecasts for later years. - Some outcomes exhibit durable effects: markets revised up persistent deficits and down long-run growth. - Late-2024 (Panel 1b): near-term forecasts for inflation and interest rates increased much more than longer-term beliefs. - Units in figures (preserved exactly): - twelve-month percentage growth rates in December (for domestic and US inflation); - percent growth, annual (for GDP); - percent of GDP (for primary balance and public debt ratios); - US dollars per Real (Exchange Rate); - percent, annualized (Fed Funds Rate); - US Dollars per Barrel (Oil Price). --- ### 3 Method — overview linking forecasts to evolving structural beliefs ### Aim and estimation approach - Aim: interpret private sector forecasts as reflecting a collective, time-varying view about macroeconomic structure, represented by an extended New Keynesian model with parameters indexed by τ. - Estimation: Bayesian methods recover the distribution of the time-varying parameter vector θ_{τ} day-by-day. - Particle filter used to sample from the posterior distribution because model nonlinearity prevents use of simpler filters. --- ### 3.1 Model (structure preserved) - Forecasts on day τ relate via equations (valid for h = 0,...,4; January/February τ extends range to h = −1,...,3 in specified equations). - Key model equations (preserved): - Phillips curve (equation (3)): - π_{τ h} − ̄π_{τ} = β_{τ}(1−η_{τ})( π_{τ h+1} − ̄π_{τ} ) + κ_{τ}(y_{τ h} − ̄y_{h,τ}) + ζ_{τ}π_{oil,τ h} + η_{τ}( π_{τ h−1} − ̄π_{τ} ) + ω_{τ}( z_{τ h} − z_{τ h−1} ) + e^{cost}_{h,τ} - IS curve (equation (4)): - y_{τ h} − ̄y_{h,τ} = (1−γ_{τ})( y_{τ h+1} − ̄y_{h+1,τ} ) − ξ_{τ}( r_{τ h} − ̄r_{τ} ) + χ_{τ}( q_{τ h} − q_{τ h+1} ) + γ_{τ}( y_{τ h−1} − ̄y_{h−1,τ} ) − λ_{τ} e^{gov}_{h,τ} + e^{d}_{h,τ} - Potential output: - ̄y_{h,τ} = g_{τ} + ̄y_{h−1,τ} + e^{s}_{h,τ} - UIP: - z_{τ h} − z_{τ h−1} = μ_{τ} + 1/2( i_{τ h−1} − f_{τ h−1} ) + 1/2( i_{τ h} − f_{τ h} ) + e^{z}_{h,τ} - Taylor rule: - i_{τ h} = (1−ρ_{τ})( ̄r_{τ} + ̄π_{τ} + φ^{π}_{τ}( π_{τ h} − ̄π_{τ} ) + φ^{y}_{τ}( y_{τ h} − ̄y_{h,τ} ) ) + ρ_{τ} i_{τ h−1} + e^{i}_{h,τ} - Fiscal rule: - s_{τ h} = ν_{τ}( d_{τ h−1} − ̄d_{τ} ) + δ_{τ}( y_{τ h} − ̄y_{h,τ} ) + e^{gov}_{h,τ} - Debt evolution: - d_{τ h} = −s_{τ h} + [ 1 + i_{τ h} + υ_{τ} / ( 1 + ( y_{τ h} − y_{τ h−1} ) + π_{τ h} ) ] d_{τ h−1} + e^{debt}_{h,τ} - Initial state: - x^{0}_{τ} = y_{τ−1} − ̄y_{−1} - Shocks e^{cost}_{h,τ}, e^{d}_{h,τ}, e^{s}_{h,τ}, e^{z}_{h,τ}, e^{i}_{h,τ}, e^{gov}_{h,τ}, e^{debt}_{h,τ} are independent normal with fixed standard deviations σ_{cost}, σ_{d}, σ_{s}, σ_{z}, σ_{i}, σ_{gov}, σ_{debt}. - Time-varying parameter vector: - θ_{τ} = ( κ_{τ}, β_{τ}, η_{τ}, ω_{τ}, ξ_{τ}, χ_{τ}, γ_{τ}, λ_{τ}, g_{τ}, μ_{τ}, ρ_{τ}, φ^{π}_{τ}, φ^{y}_{τ}, ν_{τ}, δ_{τ}, x^{0}_{τ}, ̄π_{τ}, ̄r_{τ}, ̄d ) --- ### 3.2 Inferring the evolution of structural beliefs ### Two-step intermittent-update evolution - Step 1: single Bernoulli draw with fixed probability p determines whether parameters change at all (u_t ∼ Bernoulli(p)). - Step 2: conditional on update, each parameter evolves independently as one of: - random walk, - random walk in logs (constrains parameter to be positive), - logit random walk (constrains parameter to [0,1]), - unit random walk (stretched logit to lie on [−1,1]). - Table 2 evolution categories (parameters and assigned transforms preserved exactly), example entries: - κ_{τ} — Log random walk - β_{τ} — Logit random walk - η_{τ} — Random walk - ω_{τ} — Unit random walk - ... (full mapping preserved in source) ### Conditional evolution formulation - θ_{τ+1} ∼ π(θ | θ_{τ}) = { g(θ | θ_{τ}) with probability p; θ_{τ} with probability 1−p } (equation (11)) - Rationale: captures intermittent updating observed in Focus forecasts—parameter estimates evolve sluggishly and only change when stated forecasts differ sufficiently from τ−1 as measured by the likelihood. --- ### 3.2.2 Computing the likelihood (preserved formulation) - Given forecasts X_{τ} = { π_{τ h}, y_{τ h}, i_{τ h}, z_{τ h}, s_{τ h}, d_{τ h}, r_{τ h}, f_{τ h}, q_{τ h} }_{h=−1}^{4}, eliminate unknown potential output using ̄y_{h,τ} = x^{0}_{τ} + (h+1) g_{τ} + Σ_{k=0}^{h} e^{s}_{t+k}. - Linear restriction on shocks: - A_{τ}(θ) e_{τ} = b_{τ}(θ) (equation (12)) - e_{τ} = ( e^{cost}_{τ}′, e^{d}_{τ}′, e^{s}_{τ}′, e^{z}_{τ}′, e^{i}_{τ}′, e^{gov}_{τ}′, e^{debt}_{τ}′ )′ with each shock vector for h = 0,...,4. - e_{τ} ∼ N(0, Σ) where Σ is diagonal with 5 copies of (σ_{cost})^2, (σ_{d})^2, (σ_{s})^2, (σ_{z})^2, (σ_{i})^2, (σ_{gov})^2, (σ_{debt})^2. - Likelihood integrates over shocks satisfying A_{τ} e = b_{τ}; closed-form Gaussian solution: - ℓ(θ | X_{τ}) = φ( b_{τ}(θ); 0, A_{τ}(θ) Σ A_{τ}(θ)′ ) (equation (13)) - Conditional shocks given data: - Conditional mean: ̄e_{τ} = Σ A_{τ}′ ( A_{τ} Σ A_{τ}′ )^{−1} b_{τ} (equation (14)) - Conditional variance: Var(e_{τ} | θ) = Σ − Σ A_{τ}′ ( A_{τ} Σ A_{τ}′ )^{−1} A_{τ} Σ (equation (15)) - Equations (13)–(15) enable quick likelihood evaluation and drawing of perceived shocks consistent with forecasts conditional on θ. --- ### 3.3 Drawing from the dynamic distribution of beliefs ### Particle filter implementation and draws - Recursive computation of p_τ(θ) via Bayes’ rule using a particle filter (model nonlinearity). - To avoid particle degeneracy: - 25 separate filter runs with 2000 particles each, totaling 50,000 draws of the parameter distribution on each day. - Any draw of θ produces a set of shocks consistent with forecasts, allowing recovery of the private sector’s evolving view about distribution of present and future shocks. ### Technical extensions and calibration choices - Heteroskedastic forecast variance by horizon: structural shock variances indexed by h (σ_j → σ_{jh} for h = 0,1,2,3,4). - Weak prior on initial output gap x0: very weak Gaussian prior informed by a filter on real-time GDP to prevent implausible estimates. - Shock standard deviations and random walk innovations for θ_t are set to fixed values (not jointly estimated) due to computational dimensionality; exception: p (probability of updates) is included in an extended θτ in the particle filter. - Ex post checks performed to ensure fixed assumptions consistent with simulated draws. ### Validation: calibrated vs simulated structural shock standard deviations (Table 3) - Table reports 100× average standard deviation of errors on each equation, averaged over periods, for h = 0,1,2,3,4. Selected entries (Value; Simulation Avg.; Abs. Diff): - Cost Push σcost: - Value: 0.73, 0.30, 0.14, 0.08, 0.08 - Simulation Avg.: 0.77, 0.32, 0.16, 0.10, 0.11 - Abs. Diff: 0.04, 0.02, 0.02, 0.02, 0.03 - Debt valuation σdebt: - Value: 0.92, 0.60, 0.55, 0.50, 0.50 - Simulation Avg.: 0.91, 0.57, 0.52, 0.51, 0.53 - Abs. Diff: 0.01, 0.03, 0.03, 0.01, 0.03 - Demand σd: - Value: 0.01, 0.01, 0.01, 0.01, 0.01 - Simulation Avg.: 0.01, 0.01, 0.01, 0.01, 0.01 - Abs. Diff: 0.00, 0.00, 0.00, 0.00, 0.00 - FX σz: - Value: 9.99, 1.90, 1.59, 0.93, 1.05 - Simulation Avg.: 9.90, 1.81, 1.54, 0.94, 1.09 - Abs. Diff: 0.09, 0.09, 0.05, 0.01, 0.04 - Government surplus σgov: - Value: 1.38, 0.57, 0.28, 0.10, 0.10 - Simulation Avg.: 1.33, 0.52, 0.27, 0.10, 0.13 - Abs. Diff: 0.05, 0.05, 0.01, 0.00, 0.03 - Monetary Policy σi: - Value: 1.27, 0.72, 0.24, 0.16, 0.14 - Simulation Avg.: 1.29, 0.73, 0.26, 0.18, 0.16 - Abs. Diff: 0.02, 0.01, 0.02, 0.02, 0.02 - Supply σs: - Value: 1.90, 0.64, 0.20, 0.09, 0.13 - Simulation Avg.: 1.82, 0.63, 0.20, 0.11, 0.14 - Abs. Diff: 0.08, 0.01, 0.00, 0.02, 0.01 - Summary: absolute differences below one tenth of one percentage point in all cases; many differences only a few hundredths of a percentage point or less. - Similar validation for parameter random walk innovations performed via backwards-smoothing and the difference-of-draws estimator (Appendix B, Table A1); agreement generally good, with particularly small innovation variances for slow-moving variables such as long run growth ̄r and perceived inflation target ̄π_{τ}. --- ### 4.3 Beliefs about fiscal policy ### Estimated perceived fiscal rule dynamics - Perceived fiscal stance has become gradually more tolerant of debt: - Early 2010s: government expected to raise surpluses whenever net debt exceeded threshold ̄d_{τ} of around 30 percent of GDP. - ̄d_{τ} rose in 2016 and again at the onset of the COVID-19 pandemic; the mean estimate is now a little below 70 percent of GDP. - Aggressiveness of surplus response ν_{τ} decreased over the long run: - Early 2010s: mean ν_{τ} ≈ 0.15 (increase in primary surplus of around 0.15 percent of GDP per one percentage point increase in debt ratio). - By end of sample: ν_{τ} had more than halved to less than 0.1. - Exception: pandemic period saw steepening of reaction function and ̄d_{τ} jumped by over 10 percentage points (implying negative d_{τ} − ̄d_{τ} and a large ν_{τ} reflecting expected exceptional deficits). ### Debt stability condition and adjusted interest-growth differential - Debt stabilization condition (equation (16)): - ν_{τ} > 1 + ̄r_{τ} + ̄π_{τ} + υ_{τ} / ( 1 + g_{τ} + ̄π_{τ} − 1 ) ≡ Adjusted r−g - Perceived adjusted interest-growth differential: - Increased from 2011 to reach around 5 percent in mid 2014, then dropped and stabilized at around 2.5 percent. - Perceived long-run debt stability coefficient: - ν_{τ} − (r − g): mean estimate always positive but confidence intervals often straddle zero; more recently mean estimate has come very close to zero. - Profile resembles time series for surplus response ν_{τ}, as changes in perceived r − g are relatively small versus changes in ν_{τ}. - Policy implication: perceived risks to debt stability could be addressed effectively with a credible fiscal reaction function featuring a strong response to higher debt levels. ### Fiscal impulse multiplier λ_{τ} - Figure 6 findings: - Pre-pandemic: market participants viewed the effect of deficits as mildly stimulatory. - Post-pandemic: point estimate for λ_{τ} was negative for an extended period (possible causes: concerns about debt stability or shift in deficit composition toward transfers rather than investment). - Confidence intervals include both positive and negative values throughout. - Since 2024: mean perceived multiplier has been close to zero again. ### Other fiscal-related beliefs (summary) - Pass-through of oil prices and FX depreciation to inflation typically positive but variable; oil pass-through shows salience around large persistent oil price movements (e.g., 2015 drop, COVID). - Implied dynamic structure of Phillips and IS curves relatively stable; roles of lags and expected future inflation broadly stable (expected future inflation has large confidence intervals). - Market participants view long-run growth as having declined most sharply between 2010 and 2015, with a small rebound during 2017. --- ### 5.4 The effect of policies on structural beliefs ### Method for measuring policy-innovation effects - Policy innovations fixed at end-of-year dates; information horizon of a shock changes day-by-day. - Impact parameterized as flexible polynomial spline: Σ_{m=1}^M a_m B_m(T) with B_m(T) polynomial spline basis functions. - To reduce noise from daily data: - Outcomes estimated on averages over intervals after shocks (e.g., average impact 21-40 working days). - Rolling averages ȳ_{τ+l}^{τ} = 1/(l+1) Σ_{j=0}^l y_{τ+j}. - Estimated regression (equation (18), preserved) is linear in innovations and estimated by OLS; functional form includes pre-determined covariates z_{τ−1} and controls (day-of-week, month-of-year fixed effects). - Order and number of nodes for polynomial basis chosen by the Akaike information criterion. - Confidence intervals computed by block bootstrap using 2000 replications. ### Estimation choices to improve power - Smoothing across shock horizons via polynomial spline basis (analogous to smoothed local projection). - Averaging responses over intervals rather than estimating each day separately. - Reference point comparisons use average over the five working days before the shock. ### Main empirical findings — monetary policy innovations - Monetary tightening raises market perceptions of policymakers’ responsiveness to inflation φ^{π}_{τ}: - An immediate surprise monetary tightening of 100 basis points is followed by markets revising up their view of φ^{π}_{τ} by around 0.25 during 41-80 working days (2-4 months) after the tightening. - The same 100 basis point tightening is followed by a revision of around 0.18 in the long run. - Announcements about monetary tightening at 12- and 24-month ahead horizons have long-run impacts on φ^{π}_{τ} nearer 0.05. - Effects larger and faster when the horizon of the monetary policy shock is near, though horizon differences not strongly statistically significant. - Markets’ beliefs about the implicit inflation target ̄π_{τ} are barely affected by monetary policy innovations and statistically indistinguishable from zero in presented panels. - Interpretation: - φ^{π}_{τ} captures credibility of policymakers (sensitive to conduct). - ̄π_{τ} reflects credibility of the institutional regime (less affected by conduct). - Policy innovations represent changes in conduct; signaling operates via φ^{π}_{τ} rather than ̄π_{τ}. ### Fiscal policy innovation results - Evidence for fiscal innovations less statistically robust (Appendix D Figure A6). - Intuitive sign in one dimension: fiscal tightening reduces need for monetary response (implying negative sign on inflation response coefficient). ### Implication for mapping forecasts to structural beliefs - Distinct responses of φ^{π}_{τ} and ̄π_{τ} support interpreting forecast-derived belief changes as reflecting parameter (conduct) versus regime credibility distinctions. - Methodology allows mapping changes in forecasts into changes in beliefs about structural parameters rather than treating forecasts as reduced-form objects. --- ### Appendices — calibration, matrices, and additional figures ### Calibrating innovation variances from posterior particle paths (Appendix B) - Sporadic updates with Gaussian random-walk increments: - u_t ∼ Bernoulli(p); if u_t = 1 then θ_t = θ_{t−1} + ε_t, ε_t | (u_t = 1) ∼ N(0,Q); if u_t = 0 then θ_t = θ_{t−1}. - To estimate Q from posterior draws, a difference-of-draws estimator based on pairwise differences across posterior trajectories is used to mitigate overstatement from learning and particle variability. - Empirical estimator (equation (27)) for per-period σ̂^2_{t,k} and mean of per-period standard deviations (equation (28)) detailed; draws classified as updated when |∆θ^{(i)}_{t,k}| > tol. ### Random-walk innovation calibration (Table A1) — selected entries (Assumed SD; Sample Avg.; Abs. Diff) - β_τ: 0.0540 — 0.0540 — 0.00023 - χ_τ: 0.010 — 0.0085 — 0.0015 - ̄d_τ: 0.0075 — 0.0070 — 0.00054 - δ_τ: 0.0250 — 0.0230 — 0.0023 - η_τ: 0.0480 — 0.0480 — 0.0000081 - g_τ: 0.0003 — 0.00028 — 0.00002 - ... (full table preserved in source) ### Matrix form of the model (Appendix C) - Model restated in matrix form with A_{τ} e_{τ} = b_{τ} and construction of A_{τ} from blocks A_0, M_0, M_{+1}, M_{−1}; explicit forms for predictable potential output vectors P_0_τ, P_{+1}_τ, P_{−1}_τ and b_{τ} provided in the source. ### Additional perceived parameter estimates and impulse-response figures (Appendix D and E) - Taylor rule perceptions, Phillips curve parameters, IS curve parameters, UIP intercept μ_{τ}, trend growth g_{τ}, and government debt valuation wedge υ_{τ} plotted and summarized (means and 90 percent confidence intervals) over 2011–2027 horizons in Appendix figures (A2–A5). - Responses of perceived Taylor rule parameters to fiscal shocks shown in Figure A6 across horizons (Contemporaneous, 1 month, 4 months, 12 months, 24 months) and binned working-day intervals; confidence intervals computed by block bootstrap using 2000 replications. *Italic source attribution: Content unit: wpiea2026173-source-pdf - 2.1 Overview and timing* ### 2.1 Overview and timing ### 2.1 Overview and timing ### Dataset construction and indexing - I compile daily market forecasts for nine year-end variables, supplemented with two constructed series (real interest rate and depreciation of the real exchange rate), summarized in Table 1. - Data are indexed by τ (the day on which the forecast is made) and by t+h (the calendar year to which the forecast applies, with h = 0 signifying the calendar year in which τ). - For each τ, forecasts are collected for horizons h = −1, 0, 1, 2, 3, 4 (and for January and February τ, horizons h = −2, −1, 0, 1, 2, 3). - I collect forecasts for the current year and the next four years for every one of the 4,120 working days from 4 January 2010 to 29 May 2026. - Adding one lag of real-time data for each series (model is dynamic with one lag) yields, for each series in Table 1, a total of 24,720 day-horizon observations. - For GDP data, “real-time” vintages with exact release dates are available back to 2013Q1; prior to this I use the 2013 vintage. ### Series compiled (notation preserved exactly as in Table 1) - Survey: IPCA (CPI) Inflation — Focus — π_{τ h} - Survey: Log output — Focus — y_{τ h} - Survey: SELIC (policy) rate — Focus — i_{τ h} - Survey: Log USD exchange rate — Focus — z_{τ h} - Survey: Primary surplus/GDP ratio — Focus — s_{τ h} - Survey: Net Debt/GDP ratio — Focus — d_{τ h} - U.S. Market data: Fed Funds rate — Christensen et al. (2011) — f_{τ h} - U.S. Market data: Oil price inflation (USD) — Futures market — π_{oil,τ h} - Estimated: US annual inflation — ARIMA forecasting model — π_{US,τ h} - Constructed: Real interest rate — r_{τ h} (see text for formula) - Constructed: Real exchange rate depreciation — q_{τ h} (see text for formula) ### Constructed definitions (preserved exactly) - Real interest rate: - r_{τ h} = 1/2 ( i_{τ h} + i_{τ h−1} ) − π_{τ h} - Real exchange rate depreciation: - q_{τ h} − q_{τ h−1} = ( z_{τ h} − z_{τ h−1} ) − π_{τ h} + π_{US,τ h} *Italic source attribution: Content unit: wpiea2026173-source-pdf - 2.1 Overview and timing* ### 2.2 The Focus survey - Backbone of the dataset is the Focus survey conducted by Banco Central do Brasil (BCB). - Survey details: - Respondents: professional market participants (banks, asset managers, consultancies, other financial institutions). - Participation: self-select; BCB requires respondents “have a research-dedicated unit and chief economist whose CV has to be accepted by BCB.” - Frequency: daily; participants may update forecasts at any time; BCB aggregates up to a fixed cutoff and publishes summary statistics each business day. - Eligible institutions: some 165 currently eligible (up from around 50 at inception). - Summary statistics reported by BCB (for each variable–horizon pair): mean, median, cross-sectional standard deviation of forecasts. - This paper focuses on the cross-sectional unweighted mean forecast (mean chosen because the interpretative model is linear and the unweighted mean provides the longest available time series). - Caveats noted: - Participation is voluntary and panel composition varies over time; changes in composition may affect aggregate forecasts independently of underlying beliefs. - Individual forecasters may update intermittently (often around data releases or policy announcements); BCB indicates most participants update weekly. - These motivate the intermittent-updating model of stated beliefs (introduced in Section 3.1). - Survey forecasts may reflect strategic considerations, reputational concerns, or herding, and need not equal true subjective beliefs; such motives are reflected in the estimated belief model. ### 2.3 Other data - Daily year-end estimates of US interest rates and dollar oil prices computed using Christensen et al. (2011)’s estimates and daily futures prices respectively. - Raw data are in relative time (e.g., 12 months ahead); quadratic interpolation is used to construct equivalent year-end values. - For horizons longer than 2 years, oil futures prices are assumed constant. - Daily year-end inflation forecasts for the United States are estimated using a simple ARIMA model (approximate; errors relatively minor because real exchange rate variation is overwhelmingly due to nominal exchange rates and Brazilian inflation). - Two derived forecast series computed from definitions: real interest rate and depreciation of the real exchange rate (definitions repeated above in Constructed definitions). ### 2.4 The Forecasts (descriptive observations) - Figure 1 illustrates collected forecasts for two subsamples of years (2015–2019 and 2025–2029). - Empirical observations from daily forecasts: - During 2015–2016 (sharp contraction in activity, rising inflation, deteriorating fiscal position), near-term forecasts were typically more sensitive to news: inflation and growth forecasts for 2015 moved much more and earlier than forecasts for later years. - Some outcomes exhibit durable effects: markets came to expect persistent deficits and revised down long-run growth. - Panel 1b (late-2024) shows near-term forecasts for inflation and interest rates increasing much more than longer-term beliefs. - Figure A1 (Appendix A) shows equivalent series for 2010–2014 and 2020–2024 (not reproduced here). - Units in Figure 1 (preserved exactly): - twelve-month percentage growth rates in December (for domestic and US inflation); - percent growth, annual (for GDP); - percent of GDP (for primary balance and public debt ratios); - US dollars per Real (Exchange Rate); - percent, annualized (Fed Funds Rate); - US Dollars per Barrel (Oil Price). ### 3 Method — overview linking forecasts to evolving structural beliefs - Aim: interpret private sector forecasts as reflecting a collective, time-varying view about macroeconomic structure, represented by an extended New Keynesian model with parameters indexed by τ. - Estimation approach: Bayesian methods to recover the distribution of the time-varying parameter vector θ_{τ} day-by-day. ### 3.1 Model (structure preserved) - On day τ, forecasts relate via the following equations (valid for h = 0,...,4; note when τ is in January or February, equations (3) to (9) hold for h = −1,...,3 and the equivalent of equation (10) holds for h = −1): - Phillips curve: - π_{τ h} − ̄π_{τ} = β_{τ}(1−η_{τ})( π_{τ h+1} − ̄π_{τ} ) + κ_{τ}(y_{τ h} − ̄y_{h,τ}) + ζ_{τ}π_{oil,τ h} + η_{τ}( π_{τ h−1} − ̄π_{τ} ) + ω_{τ}( z_{τ h} − z_{τ h−1} ) + e^{cost}_{h,τ} - IS curve: - y_{τ h} − ̄y_{h,τ} = (1−γ_{τ})( y_{τ h+1} − ̄y_{h+1,τ} ) − ξ_{τ}( r_{τ h} − ̄r_{τ} ) + χ_{τ}( q_{τ h} − q_{τ h+1} ) + γ_{τ}( y_{τ h−1} − ̄y_{h−1,τ} ) − λ_{τ} e^{gov}_{h,τ} + e^{d}_{h,τ} - Potential output: - ̄y_{h,τ} = g_{τ} + ̄y_{h−1,τ} + e^{s}_{h,τ} - UIP: - z_{τ h} − z_{τ h−1} = μ_{τ} + 1/2( i_{τ h−1} − f_{τ h−1} ) + 1/2( i_{τ h} − f_{τ h} ) + e^{z}_{h,τ} - Taylor rule: - i_{τ h} = (1−ρ_{τ})( ̄r_{τ} + ̄π_{τ} + φ^{π}_{τ}( π_{τ h} − ̄π_{τ} ) + φ^{y}_{τ}( y_{τ h} − ̄y_{h,τ} ) ) + ρ_{τ} i_{τ h−1} + e^{i}_{h,τ} - Fiscal rule: - s_{τ h} = ν_{τ}( d_{τ h−1} − ̄d_{τ} ) + δ_{τ}( y_{τ h} − ̄y_{h,τ} ) + e^{gov}_{h,τ} - Debt evolution: - d_{τ h} = −s_{τ h} + [ 1 + i_{τ h} + υ_{τ} / ( 1 + ( y_{τ h} − y_{τ h−1} ) + π_{τ h} ) ] d_{τ h−1} + e^{debt}_{h,τ} - Initial state: - x^{0}_{τ} = y_{τ−1} − ̄y_{−1} - Shocks e^{cost}_{h,τ}, e^{d}_{h,τ}, e^{s}_{h,τ}, e^{z}_{h,τ}, e^{i}_{h,τ}, e^{gov}_{h,τ}, e^{debt}_{h,τ} are independent normal shocks with fixed standard deviations σ_{cost}, σ_{d}, σ_{s}, σ_{z}, σ_{i}, σ_{gov}, σ_{debt}. - The evolving view is parameterized by time-varying vector: - θ_{τ} = ( κ_{τ}, β_{τ}, η_{τ}, ω_{τ}, ξ_{τ}, χ_{τ}, γ_{τ}, λ_{τ}, g_{τ}, μ_{τ}, ρ_{τ}, φ^{π}_{τ}, φ^{y}_{τ}, ν_{τ}, δ_{τ}, x^{0}_{τ}, ̄π_{τ}, ̄r_{τ}, ̄d ) - Fixed vector of shock standard deviations: - σ = ( σ_{cost}, σ_{d}, σ_{s}, σ_{z}, σ_{i}, σ_{gov}, σ_{debt} )′ - Interpretation highlights (preserved): - Phillips curve links inflation deviations to output gap, expected future inflation, lagged inflation, oil and exchange rate pass-through (κ_{τ}, β_{τ}, η_{τ}, ζ_{τ}, ω_{τ}). - IS curve links output gap to expected future gap, real interest rate gap, real exchange rate effects, fiscal impulse, and lagged output (ξ_{τ}, χ_{τ}, λ_{τ}, γ_{τ}). - Potential output ̄y_{h,τ} is a random walk with drift g_{τ} and supply shocks e^{s}_{h,τ}. - UIP center around μ_{τ} (interpretable as the inverse exchange rate risk premium). - Monetary policy perceived as a Taylor rule with smoothing ρ_{τ} and responses φ^{π}_{τ}, φ^{y}_{τ}. - Fiscal policy as a Bohn-style rule with debt threshold ̄d_{τ} and responsiveness ν_{τ}, δ_{τ}. - Initial output gap parameter x^{0}_{τ}. - Model purpose: not to be the “correct” economy model but to flexibly capture how market participants view the macroeconomy; the model has 19 time-varying parameters and seven structural shocks. ### 3.2 Inferring the evolution of structural beliefs - Goal: infer θ_{τ} day-by-day using Bayesian methods to recover the distribution of market beliefs about the macro structure. - Structural beliefs evolution (two-step process): - Step 1: single Bernoulli draw with fixed probability p determines whether parameters change at all. - Step 2: transformed random walk for each parameter, evolving independently as one of: - random walk, - random walk in logs (constrains parameter to be positive), - logit random walk (constrains parameter to [0,1]), - unit random walk (stretched logit to lie on [−1,1], useful for persistence parameters). - Table 2 (parameter evolution categories) — parameters and assumed evolution (preserved exactly): - κ_{τ} — Slope of the Phillips curve — Log random walk - β_{τ} — Pass-through of expected future inflation — Logit random walk - η_{τ} — Inflation persistence — Random walk - ω_{τ} — FX inflation pass-through — Unit random walk - ζ_{τ} — Oil price inflation pass-through — Random walk - ξ_{τ} — Slope of IS curve — Log random walk - γ_{τ} — Output gap persistence — Unit random walk - χ_{τ} — Real exchange rate expenditure switching effect — Log random walk - λ_{τ} — Fiscal impulse multiplier — Log random walk - g_{τ} — Long run potential growth — Random walk - μ_{τ} — UIP intercept — Random walk - ρ_{τ} — Interest rate smoothing — Random walk - φ^{π}_{τ} — Long run monetary inflation response — Random walk - φ^{y}_{τ} — Long run monetary output gap response — Random walk - ν_{τ} — Fiscal debt response — Log random walk - δ_{τ} — Fiscal output gap response — Random walk - ̄π_{τ} — Long run inflation — Random walk - ̄r_{τ} — Natural rate — Random walk - ̄d_{τ} — Long run debt level — Random walk - υ_{τ} — Debt valuation wedge — Random walk - x^{0}_{τ} — Initial output gap — Random walk - Conditional evolution of θ: - θ_{τ+1} ∼ π(θ | θ_{τ}) = { g(θ | θ_{τ}) with probability p; θ_{τ} with probability 1−p } (equation (11)) - g(θ | θ_{τ}) is the joint distribution of the transformed random walks in Table 2. - Rationale: the two-step process captures intermittent updating in the Focus forecasts — estimates for θ_{τ} evolve sluggishly and change only if stated forecasts at τ differ enough from τ−1 as measured by the likelihood. ### 3.2.2 Computing the likelihood (preserved formulation) - Given forecasts X_{τ} = { π_{τ h}, y_{τ h}, i_{τ h}, z_{τ h}, s_{τ h}, d_{τ h}, r_{τ h}, f_{τ h}, q_{τ h} }_{h=−1}^{4}, the likelihood that candidate θ represents market views must account for missing potential output. - Eliminating unknown potential output using ̄y_{h,τ} = x^{0}_{τ} + (h+1) g_{τ} + Σ_{k=0}^{h} e^{s}_{t+k}, equations (3)–(9) are a linear restriction on shocks: - A_{τ}(θ) e_{τ} = b_{τ}(θ) (equation (12)) - e_{τ} is the vector of shocks at each horizon for period τ: e_{τ} = ( e^{cost}_{τ}′, e^{d}_{τ}′, e^{s}_{τ}′, e^{z}_{τ}′, e^{i}_{τ}′, e^{gov}_{τ}′, e^{debt}_{τ}′ )′ where each shock vector contains shocks for h = 0,...,4. - e_{τ} ∼ N(0, Σ) where Σ is the diagonal matrix of 5 copies each of (σ_{cost})^2, (σ_{d})^2, (σ_{s})^2, (σ_{z})^2, (σ_{i})^2, (σ_{gov})^2, (σ_{debt})^2. - Because b_{τ}(θ) has 5 elements fewer than e_{τ}, there is no unique solution to A_{τ} e = b_{τ}; the likelihood integrates over the set of shocks satisfying the restriction. - The restricted integral has an explicit Gaussian solution: - ℓ(θ | X_{τ}) = ∫_{ {e: A_{τ}(θ)e = b_{τ}(θ)} } φ(e; 0, Σ) de = φ( b_{τ}(θ); 0, A_{τ}(θ) Σ A_{τ}(θ)′ ) (equation (13)) - φ(x; μ, Σ) is the multivariate normal density evaluated at x with mean μ and covariance Σ. - Conditional distribution of shocks given data (within A_{τ} e = b_{τ}) is joint normal with: - Conditional mean: - ̄e_{τ} = Σ A_{τ}′ ( A_{τ} Σ A_{τ}′ )^{−1} b_{τ} (equation (14)) - Conditional variance (Schur complement): - Var(e_{τ} | θ) = Σ − Σ A_{τ}′ ( A_{τ} Σ A_{τ}′ )^{−1} A_{τ} Σ (equation (15)) - Equations (13)–(15) provide a computationally tractable recipe: quick evaluation of the likelihood for any θ via a multivariate normal density and a way to draw perceived shocks consistent with forecasts conditional on θ. ### 3.3 Drawing from the dynamic distribution of beliefs ### 3.3 Drawing from the dynamic distribution of beliefs ### Method: inference of evolving structural beliefs - Given the likelihood function and any initial prior p0(θ), the evolution of the distribution of structural beliefs consistent with private sector forecastspτ(θ) is computed recursively via Bayes’ rule. - A particle filter is used to sample from the posterior likelihood of the distribution of structural beliefs because model nonlinearity (e.g. ζτ × ̄rτ, φyτ × ̄yτ) makes simpler approaches such as the Kalman filter infeasible. - To avoid particle degeneracy: - 25 separate runs of a filter are used, with 2000 particles each, for a total of 50,000 draws of the parameter distribution on each day. - Any draw of the parameter vector θ immediately produces a set of shocks consistent with the forecasts, allowing recovery of the private sector’s evolving view about the distribution of present and future shocks. ### Technical extensions applied to the model - Heteroskedastic forecast variance by horizon: - Structural shock variances are indexed by forecast horizon h, replacing σj with σjh for h = 0,1,2,3,4 for all structural shocks j. - Weak prior on initial output gap x0: - A very weak Gaussian prior on x0 is imposed, with the prior center informed by a filter applied to real-time GDP data, to prevent implausible unconstrained estimates due to limited information in the short forecast horizon and a flat Phillips curve. ### Calibration choices and dimensionality - Shock standard deviations and random walk innovations for θt are set to fixed values rather than estimated jointly, due to the computational curse of dimensionality (the parameter space already includes 19 model parameters). - Exception: p, the probability of parameter updating, is included in an extended version of θτ in the particle filter (adding one extra dimension). - Ex post checks are performed to ensure fixed assumptions are consistent with simulated draws from the particle filter. ### Validation: calibrated vs simulated structural shock standard deviations (Table 3) - Table shows 100× the average standard deviation of the errors on each equation, averaged over periods. Units interpretable as percentage points. - Calibrated and simulation comparison (h = 0,1,2,3,4): - Cost Push σcost - Value: 0.73, 0.30, 0.14, 0.08, 0.08 - Simulation Avg.: 0.77, 0.32, 0.16, 0.10, 0.11 - Abs. Diff: 0.04, 0.02, 0.02, 0.02, 0.03 - Debt valuation σdebt - Value: 0.92, 0.60, 0.55, 0.50, 0.50 - Simulation Avg.: 0.91, 0.57, 0.52, 0.51, 0.53 - Abs. Diff: 0.01, 0.03, 0.03, 0.01, 0.03 - Demand σd - Value: 0.01, 0.01, 0.01, 0.01, 0.01 - Simulation Avg.: 0.01, 0.01, 0.01, 0.01, 0.01 - Abs. Diff: 0.00, 0.00, 0.00, 0.00, 0.00 - FX σz - Value: 9.99, 1.90, 1.59, 0.93, 1.05 - Simulation Avg.: 9.90, 1.81, 1.54, 0.94, 1.09 - Abs. Diff: 0.09, 0.09, 0.05, 0.01, 0.04 - Government surplus σgov - Value: 1.38, 0.57, 0.28, 0.10, 0.10 - Simulation Avg.: 1.33, 0.52, 0.27, 0.10, 0.13 - Abs. Diff: 0.05, 0.05, 0.01, 0.00, 0.03 - Monetary Policy σi - Value: 1.27, 0.72, 0.24, 0.16, 0.14 - Simulation Avg.: 1.29, 0.73, 0.26, 0.18, 0.16 - Abs. Diff: 0.02, 0.01, 0.02, 0.02, 0.02 - Supply σs - Value: 1.90, 0.64, 0.20, 0.09, 0.13 - Simulation Avg.: 1.82, 0.63, 0.20, 0.11, 0.14 - Abs. Diff: 0.08, 0.01, 0.00, 0.02, 0.01 - Summary of validation results: - Absolute differences are below one tenth of one percentage point in all cases, and in many cases are only a few hundredths of a percentage point or less. - Similar validation for parameter random walk innovations is performed by backwards-smoothing the particle filter distributions and calculating the standard deviation of resulting innovations conditional on them being strictly positive (details in Appendix B, Table A1). - Agreement between assumed and estimated random walk innovation variances is generally good; slow-moving variables such as long run growth r̄ and perceived inflation target ̄πτ have particularly small innovation variances. *Source: wpiea2026173-source-pdf - 3.3 Drawing from the dynamic distribution of beliefs* ### 4.3 Beliefs about fiscal policy ### 4.3 Beliefs about fiscal policy ### Estimated parameters of the perceived fiscal policy rule - Fiscal policy has been seen as becoming gradually more tolerant of debt. - In the early 2010s, the government was expected to systematically raise surpluses whenever net debt exceeded the threshold, ̄dτ, of around 30 percent of GDP. - This threshold rose in 2016 and again at the onset of the COVID-19 pandemic and the mean estimate is now a little below 70 percent of GDP. - The aggressiveness with which the government is expected to respond to higher debt, ντ, has decreased over the long run: - In the early 2010s, the government was expected on average to increase the primary surplus by around 0.15 percent of GDP for every one percentage point increase in the debt ratio. - By the end of the sample, this number had more than halved to less than 0.1. - One exception was the pandemic, when the government’s fiscal reaction function steepened; this coincided with a jump up in the intercept of the fiscal reaction function: ̄dτ jumped by over 10 percentage points. As a result, dτ − ̄dτ was negative, and so the large value of ντ reflected a belief that the government would run exceptional deficits during this period. ### Debt stability condition and adjusted interest-growth differential - A government issuing debt following equations (8) and (9) will be able to stabilize its debt ratio at some finite level if and only if: ντ > 1 + ̄rτ + ̄πτ + υτ 1 + gτ + ̄πτ − 1 |{z} Adjusted r−g (16) - Intuition: if the debt ratio increases by one percentage point, the rate at which the government raises the primary balance, ντ, must exceed the rate at which the debt ratio grows in the long run (the nominal natural rate plus the valuation wedge, less the nominal growth rate of GDP). - The adjusted interest-growth differential (the right-hand side of (16)) was perceived as increasing from 2011 to reach around 5 percent in mid 2014 before dropping, and stabilizing at around 2.5 percent. - The perceived long-run debt stability coefficient, ντ − (r − g), is the difference between the left and right hand sides of equation (16): - The mean estimate is always positive, but the confidence intervals often straddle zero. - More recently, even the mean estimate of the stability coefficient has come very close to zero. - The profile for the stability coefficient looks very similar to the time series for the surplus response to debt, reflecting that changes in beliefs about natural debt dynamics, r − g, are relatively small compared with changes in beliefs about the government’s fiscal reaction parameter, ντ. - Implication: perceived risks to the stability of public debt could be addressed effectively with a credible fiscal reaction function featuring a strong response to higher debt levels. ### Fiscal impulse multiplier and perceived fiscal effects on output - Figure 6 shows the estimated multiplier on the fiscal impulse, λτ, summarizing the (immediate) increase in the output gap as a function of public spending. - Findings on λτ: - Pre-pandemic, market participants thought that the effect of deficits was mildly stimulatory. - Since then, the point estimate for the perceived multiplier was negative for an extended period, which could be due to concerns about debt stability or the changing nature of deficit spending (e.g., more transfers and less investment). - Confidence intervals throughout include both positive and negative values. - The mean perceived multiplier has been close to zero again since 2024. ### Other beliefs (summary of Appendix D.1 findings) - Markets’ views about the pass-through of oil prices and FX depreciation to inflation are typically positive, but tend to vary considerably. - For oil prices, variation may reflect salience: oil prices appear more important in perceived Phillips curves shortly after periods where oil prices change significantly and persistently (notably following the drop in prices in 2015 and during COVID). - The implied dynamic structure of both the Phillips and IS curves remains relatively stable, with the role of lags and expected future inflation remaining broadly stable (although expected future inflation has very large confidence intervals). - Market participants view long-run growth as having declined, most sharply between 2010 and 2015, although with a small rebound during 2017. *Source: 4.3 Beliefs about fiscal policy, wpiea2026173-source-pdf.* ### 5.4 The effect of policies on structural beliefs ### 5.4 The effect of policies on structural beliefs ### Methodology for measuring policy-innovation effects - Policy innovations occur at fixed dates (end of each year), so the information horizon of a shock changes day-by-day. - The impact of an innovation at fixed horizon T is parameterized as a flexible polynomial spline: - Sum from m=1 to M of a_m B_m(T), where B_m(T) are polynomial spline basis functions and M is the number of nodes. - To reduce noise from daily data, outcomes are estimated on averages over intervals after the shock (e.g., average impact 21-40 working days after the policy shock) rather than daily impulse responses. - Rolling averages are defined for any series y_τ as ȳ_{τ+l}^{τ} = 1/(l+1) ∑_{j=0}^l y_{τ+j}. - Estimated regression form (equation (18)): - ȳ_{τ+l_k+1}^{τ+l_{k+1}} − ȳ_{τ−1}^{τ−5} = a_{k0} + ∑_{h=0}^3 innovation_{h,τ} ∑_{m=1}^M a_{km} B_m(T(τ,y(h,τ))) + γ_k′ z_{τ−1} + u_{kτ} - Where: - The horizons l_0, l_1, ..., l_K partition impulse response horizons in days; responses are estimated on average over each interval [l_k+1, l_{k+1}] days after the innovation. - y(h,τ) is December 31 of the year h years after date τ. - T(τ,y) measures the number of days between date τ and y. - z_{τ−1} includes pre-determined covariates, including day-of-week and month-of-year fixed effects. - The functional form is linear in the innovations and is estimated by ordinary least squares. - The order and number of nodes for the polynomial basis is chosen by the Akaike information criterion. - Confidence intervals computed by block bootstrap using 2000 replications. ### Estimation choices to improve statistical power - Smoothing across shock horizons via polynomial spline basis functions (analogous to Barnichon and Brownlees (2019) smoothed local projection but smoothing across the shock horizon). - Averaging responses over intervals (e.g., 21-40 working days) instead of estimating each day separately to reduce noise and improve power. - Reference point comparisons use the average over the five working days before the shock. ### Main empirical findings (monetary policy innovation) - A positive monetary policy innovation induces an increase in mean perceptions of the systematic responsiveness of policymakers to inflation departures (φ^π_τ). - Timing and magnitude details: - An immediate surprise monetary tightening of 100 basis points is followed by markets revising up their view of φ^π_τ by around 0.25 during 41-80 working days (2-4 months) after the tightening. - The same 100 basis point tightening is followed by a revision of around 0.18 in the long run. - Announcements about monetary tightening at 12- and 24-month ahead horizons have long-run impacts on φ^π_τ nearer 0.05. - Effects are larger and faster when the horizon of the monetary policy shock is near, although differences across horizons are not strongly statistically significant. - Markets’ beliefs about the implicit inflation target (π̄_τ) are barely affected by monetary policy innovations and are statistically indistinguishable from zero in all panels of Figure 10. - Figure annotation specifics: - Figure 10 shows responses to a one percentage point unexpected policy shock at different fixed horizons. - X axis measures working days since the innovation, averaging over days to improve power. - Y axis shows interpolated point estimates as points, with error bars and shaded ribbons for the 90 and 68 percent confidence intervals respectively. ### Interpretation of results - φ^π_τ appears to capture the credibility of policymakers (sensitive to changes in the conduct of policy). - π̄_τ appears to reflect the credibility of the institutional regime (less affected by changes in policy conduct). - Monetary policy innovations represent changes in conduct of policy, not changes in the broader monetary framework; therefore, signaling should operate via credibility of policymakers rather than the framework. - If market participants initially treat news about policy conduct as shocks and later revise beliefs about structural parameters, the defined innovations serve as proxies for the (exogenous) arrival of that information. ### Fiscal policy innovation results - Appendix D Figure A6 presents the same estimates for the fiscal policy innovation. - Evidence for fiscal innovations is generally less statistically robust. - Signs are intuitive in at least one dimension: when fiscal policy tightens, the need for the monetary policymaker to respond is lessened (implying a negative sign on the inflation response coefficient). ### Implications and links to broader analysis - The distinct responses of φ^π_τ and π̄_τ support interpreting changes in forecast-derived beliefs as reflecting parameter versus regime credibility distinctions. - The methodology enables mapping changes in forecasts into changes in beliefs about structural parameters rather than treating forecasts as reduced-form objects. - These findings feed into the broader paper conclusions about interrogating forecasts with the same discipline as realized data and about characterizing evolving beliefs for Brazilian data. *Source: 5.4 The effect of policies on structural beliefs (excerpt).* ### 4. Primary Balance Ratio5. Net Public Sector Debt Ratio6. Exchange Rate vs. USD ### 4. Primary Balance Ratio5. Net Public Sector Debt Ratio6. Exchange Rate vs. USD ### Forecast data (Figure A1) - Figures show the nine primary series collected for two subsets of forecast years. In all cases, forecasts are for year-end observations. - Units: - twelve-month percentage growth rates in December (for domestic and US inflation); - percent growth, annual (for GDP); - percent of GDP (for primary balance and public debt ratios); - US dollars per Real (Exchange Rate); - percent, annualized (Fed Funds Rate); - US Dollars per Barrel (Oil Price). ### Time series axes and ranges depicted - Years shown: 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 - Axes tick values (examples from figures): - vertical ranges include −12, −8, −4, 0 and −5.0, −2.5, 0.0, 2.5, 5.0 - other numeric markers: 0, 2, 4; 4, 5, 6; 0.0, 2.5, 5.0, 7.5, 10.0; 0, 5, 10, 15; 50, 60, 70; 40, 60, 80, 100 - Label: "Date Forecast is Made" and "value" - Forecast for end of: 2020 2021 2022 2023 2024 2025 *Italic: Forecast data (Figure A1) as presented in the source.* --- ### Calibrating innovation variances from posterior particle paths (Appendix B) - Set-up: sporadic parameter updates with Gaussian random-walk increments - Let θ_t ∈ R^K denote a vector of time-varying parameters. - Update indicator: - u_t ∼ Bernoulli(p), u_t ∈ {0,1}, (19) - u_t = 1 indicates parameters are updated at time t. - Conditional on an update, parameters follow a mean-zero Gaussian random-walk increment: - θ_t = θ_{t−1} + ε_t, ε_t | (u_t = 1) ∼ N(0,Q), (20) - Q is a diagonal covariance matrix. - If no update occurs: - θ_t = θ_{t−1} if u_t = 0. (21) - Increment ∆θ_t ≡ θ_t − θ_{t−1} is a mixture: - ∆θ_t = (0, u_t = 0; ε_t, u_t = 1). (22) - Model estimation uses a particle filter and stores N draws from the posterior distribution of parameter paths. (Footnote: these are smoothing trajectories obtained by tracing particle ancestry.) - Object of interest and naive moment-matching problem - Target conditional second moment: - E[(∆θ_{t,k})^2 | u_t = 1, y_{1:T}] = Var(∆θ_{t,k} | u_t = 1, y_{1:T}). (23) - Posterior increments can overstate structural innovation variance because: - Posterior variation in θ_t reflects both structural evolution and learning from data. - Resampling and finite-sample Monte Carlo variability in particle methods can distort serial dependence across trajectories. - To reduce sensitivity to posterior mean movements, the author uses a difference-of-draws estimator based on pairwise differences across posterior trajectories. - Difference-of-draws estimator (pairwise differences) - Let X_{t,k} ∼ p(∆θ_{t,k} | u_t = 1, y_{1:T}). (24) - For independent posterior draws X^{(i)}_{t,k} and X^{(j)}_{t,k}: - E[(X^{(i)}_{t,k} − X^{(j)}_{t,k})^2 | u_t = 1, y_{1:T}] = 2 Var(X_{t,k} | u_t = 1, y_{1:T}). (25) - In output paths, u_t is not directly observed; ∆θ^{(i)}_{t,k} = 0 whenever u_t = 0. Condition on updates by selecting draws with strictly non-zero increments. - Define I_{t,k} ⊆ {1,...,N} as indices with an update at (t,k): - I_{t,k} = { i : |∆θ^{(i)}_{t,k}| > tol }, m_{t,k} = |I_{t,k}|. (26) - Provided m_{t,k} ≥ 2, empirical analogue: - \hat{σ}^2_{t,k} = 1/2 · 1/[1/2 m_{t,k} (m_{t,k}−1)] ∑_{i tol. --- ### Calibrated and simulated values: random walk standard deviations (Table A1) - Table shows average standard deviation of the random walk innovations, averaged over periods. - Parameter — Assumed SD — Sample Avg. — Abs. Diff - β_τ: 0.0540 — 0.0540 — 0.00023 - χ_τ: 0.010 — 0.0085 — 0.0015 - ̄d_τ: 0.0075 — 0.0070 — 0.00054 - δ_τ: 0.0250 — 0.0230 — 0.0023 - η_τ: 0.0480 — 0.0480 — 0.0000081 - g_τ: 0.0003 — 0.00028 — 0.00002 - γ_τ: 0.0450 — 0.0450 — 0.00029 - κ_τ: 0.0580 — 0.0580 — 0.000036 - λ_τ: 0.0550 — 0.0510 — 0.0043 - μ_{0τ}: 0.0025 — 0.0023 — 0.00023 - ν_τ: 0.0680 — 0.0630 — 0.0049 - ω_τ: 0.0070 — 0.0067 — 0.00025 - φ_{πτ}: 0.10 — 0.0980 — 0.0015 - φ_{yτ}: 0.0670 — 0.0630 — 0.0041 - ̄π_τ: 0.00035 — 0.00033 — 0.000018 - ̄r_τ: 0.0014 — 0.0012 — 0.00019 - ρ_τ: 0.0390 — 0.0370 — 0.0018 - υ_τ: 0.0025 — 0.0022 — 0.00025 - x_{0τ}: 0.0010 — 0.00091 — 0.000089 - ξ_τ: 0.0620 — 0.0630 — 0.00068 - ζ_τ: 0.0050 — 0.0049 — 0.000074 --- ### Matrix form of the model (Appendix C) - Model equations restated: - Phillips curve: - π_{τ h} − ̄π_{τ} = β(1−η)(π_{τ h+1} − ̄π_{τ}) + κ(y_{τ h} − ̄y_{h,τ}) + ζ_{τ} π^{oil}_h + η_{τ}(π_{τ h−1} − ̄π_{τ}) + ω_{τ}(z_{τ h} − z_{τ h−1}) + e^{cost}_{h,τ} - IS curve: - y_{τ h} − ̄y_{h,τ} = (1−γ)(y_{τ h+1} − ̄y_{h+1,τ}) − ξ(r_{τ h} − ̄r_{τ}) + χ_{τ}(q_{τ h} − q_{τ h+1}) + γ_{τ}(y_{τ h−1} − ̄y_{h−1,τ}) − λ_{τ} e^{gov}_{h,τ} + e^{d}_{h,τ} - Potential: - ̄y_{h,τ} = g_{τ} + ̄y_{h−1,τ} + e^{s}_{h,τ} - UIP: - z_{τ h} − z_{τ h−1} = μ_{τ} + (i_{τ h−1} − f_{τ h−1}) + e^{z}_{h,τ} - Taylor rule: - i_{τ h} = (1−ρ_{τ})( ̄r_{τ} + ̄π_{τ} + φ_{πτ}(π_{τ h} − ̄π_{τ}) + φ_{yτ}(y_{τ h} − ̄y_{h,τ}) ) + ρ_{τ} i_{τ h−1} + e^{i}_{h,τ} - Fiscal rule: - s_{τ h} = ν_{τ}(d_{τ h−1} − ̄d_{τ}) + δ_{τ}(y_{τ h} − ̄y_{h,τ}) + e^{gov}_{h,τ} - Debt evolution: - d_{τ h} = −s_{τ h} + [ 1 + i_{τ h} + υ_{τ} ] / [ 1 + (y_{τ h} − y_{τ h−1}) + π_{τ h} ] d_{τ h−1} + e^{debt}_{h,τ} - Initial state: - x_{0τ} = y_{τ −1} − ̄y_{−1} - Shocks and notation: - There are six shocks; define e_{τ} = (e^{cost}_{τ}′, e^{d}_{τ}′, e^{s}_{τ}′, e^{z}_{τ}′, e^{i}_{τ}′, e^{gov}_{τ}′, e^{debt}_{τ}′)′ where each shock is of length 5. - Goal: find A_{τ}, b_{τ} such that A_{τ} e_{τ} = b_{τ}. - Potential output across horizons: - If x_0 denotes the output gap in the initial (data) period, then potential output in period t+h is given by: - ̄y_{t+h} = x_0 + (h − h) g + ∑_{k=h+1}^{h} e^{s}_{t+k} - Define matrix M_0 (5×5 block structure) capturing marginal effect of contemporaneous potential output; analogous matrices M_{+1} and M_{−1} capture leads and lags. - A_{τ} can be written as: - A_{τ} = A_0 + [−κ, 1, 0, −(1−ρ) φ_{y}, −δ, 0]′ ⊗ M_0 + [0, −(1−γ), 0, 0, 0, 0]′ ⊗ M_{+1} + [0, −γ, 0, 0, 0, 0]′ ⊗ M_{−1} - A_0 block structure presented with identity matrices and zero blocks (dimensions: I_{10×10}, 25×5, 5×20, −λ I_{5×5}, etc.) - Predictable part of contemporaneous potential output: - P_0_τ = x_0 + g [1 2 3 4 5]′ - P_{+1}_τ = x_0 + g [2 3 4 5 6]′ - P_{−1}_τ = x_0 + g [0 1 2 3 4]′ - Dot subscript denotes vector of forecasts for each variable at time τ (e.g., π_{τ •,0} is the vector of the five contemporaneous inflation forecasts). - b_{τ} expression (vector of right-hand side predictable terms) provided explicitly combining forecast vectors, parameters, and P_0_τ, P_{+1}_τ, P_{−1}_τ. --- ### More results: Perceived parameter estimates (Appendix D) - Taylor rule (D.1.1) - Dates shown: Feb 2017 May 2017 Aug 2017 Nov 2017 Feb 2018 May 2018 Aug 2018 Nov 2018 - Plotted series: Mean Taylor Threshold, 90 percent confidence interval, Inflation Target - Notes: Figure A2 displays estimates of market beliefs of the Taylor rule parameters ̄π_τ and φ_{πτ}. - Phillips curve (D.1.2) - Parameters plotted over 2011 2013 2015 2017 2019 2021 2023 2025 2027: - Slope κ_τ (range shown includes −0.10, −0.05, 0.00, 0.05, 0.10) - Lag coefficient η_τ (range 0.6, 0.8, 1.0) - Oil price inflation pass-through ζ_τ (range −0.10 to 0.10 illustrated) - Importance of inflation expectations β_τ - FX inflation pass-through ω_τ - Figure A3 shows mean and 90 percent confidence interval for perceived Phillips curve parameters specified in equation (3). - IS curve (D.1.3) - Parameters plotted over 2011–2027: - Slope ξ_τ (range −1 to 1) - Lag coefficient γ_τ (range −0.1 to 0.2) - FX real depreciation effect χ_τ (range 0.0 to 0.4) - Fiscal impulse multiplier λ_τ (range 0.0 to 0.4) - Figure A4 shows mean and 90 percent confidence interval for perceived IS curve parameters specified in equation (4). - Other parameters (D.1.4) - UIP intercept μ_τ (range −0.050 to 0.050) - Trend growth g_τ (range 0.00 to 0.05) - Government debt valuation wedge υ_τ (range 0.000 to 0.075) - Figure A5 shows mean and 90 percent confidence interval for long-run growth and the UIP premium. --- ### Responses of perceived Taylor rule parameters to a fiscal shock (Appendix E) - Figure A6 panels: - Horizons: Contemporaneous, 1 month, 4 months, 12 months, 24 months - Variables tracked: Perceived inflation response φ_{πτ} and Perceived inflation target ̄π_τ - Working days after policy innovation grouped into bins: - 1 - 2−5 - 6−10 - 11−20 - 21−40 - 41−80 - 81−125 - 126−250 - 251−500 - Y-axis ranges illustrated (examples): −0.8, −0.4, 0.0, 0.4; and −1.0, −0.5, 0.0, 0.5, 1.0 - Description: - Figure shows the response of mean beliefs about the Taylor rule inflation coefficient φ_{πτ} and the inflation target ̄π_τ to a one percentage point unexpected policy shock at different fixed horizons. - X-axis measures days since the innovation, averaging over days to improve power. - Y-axis shows interpolated point estimates as points, with error bars and shaded ribbon for the 90 and 68 percent confidence intervals respectively. - Confidence intervals computed by block bootstrap using 2000 replications. - Panels correspond to responses at different anticipated horizons: "Contemporaneous" (implemented immediately), "1 month" (implemented one month later), "4 months", etc. *Italic: Excerpted figures and appendices from "Inferring Structural Beliefs from Macroeconomic Expectations: Evidence from Brazil — Working Paper No. WP/2026/173".* --- _Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026173-source-pdf.pdf_