## 2.1 The New Keynesian Model with Monetary Policy News (and related empirical sections) ## Source details **Canonical URL:** [2.1 The New Keynesian Model with Monetary Policy News (and related empirical sections)](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023208-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2023/english/wpiea2023208-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2023/english/wpiea2023208-print-pdf.pdf.json) --- ### Model setup and calibration - Model: three-equation New Keynesian model - New Keynesian Phillips curve: π_t = β E_t[π_{t+1}] + κ y_t + x_t - Euler equation: 0 = E_t[z_t + γ(y_t − y_{t+1}) + i_t − π_{t+1}] - Taylor rule: i_t = φ_π π_t + h_t - Shocks and definitions: - x_t: i.i.d. cost-push shock - z_t: i.i.d. demand shock - Policy residual: h_t = ρ h_{t−1} + u_t + v_{t−1} - u_t: i.i.d. surprise (unanticipated at t) - v_{t−1}: i.i.d. news shock known in period t−1 (anticipated) - Interpretation of policy news versus surprise: - v_{t−1} captures pre-announced policy changes (anticipation effects). - Example: communicated rate rise in t of 25 basis points ⇒ v_{t−1} = 0.25; actual 50 basis points in t implies u_t = 0.25 and total u_t + v_{t−1} = 0.5. - Calibration used for illustration (monthly, adapted from Galí (2008)): - β = 0.997 - κ = 0.2 - γ = 1 - φ_π = 1.5 - persistence ρ = 0.6 - all shock variances set to one ### Reduced-form representation and qualitative IRFs - Linear reduced-form representation: - π_t = b^π_h h_t + b^π_v v_t + b^π_x x_t + b^π_z z_t - y_t = b^y_h h_t + b^y_v v_t + b^y_x x_t + b^y_z z_t - i_t = b^i_h h_t + b^i_v v_t + b^i_x x_t + b^i_z z_t - Qualitative impulse-response differences: - News of an interest rate rise tomorrow (v_{t−1} > 0): agents anticipate a recession next period, reduce spending today → lower output and prices today. - Central bank mitigates anticipated downturn by cutting interest rates today under the Taylor rule. ### Identification in the simple example (constructive) - Identification challenge: - Structural shocks: u_t, v_t, x_t, z_t (four shocks) vs. observables π_t, y_t, i_t (three series) → insufficient without additional information. - Solution using forecasts: - Observe inflation forecast f^π_t ≡ E_t[π_{t+1}] with f^π_t = b^π_h (ρ h_t + v_t). - Recover h_t from observables: h_t = i_t − φ_π π_t - Identify policy news: v_t = f^π_t / b^π_h − ρ h_t - Identify policy surprise: u_t = h_t − ρ h_{t−1} − v_{t−1} - Identify x_t and z_t by inverting: - [x_t; z_t] = [ [b^π_x, b^π_z]; [b^y_x, b^y_z] ]^{-1} [ π_t − b^π_h h_t − b^π_v v_t ; y_t − b^y_h h_t − b^y_v v_t ] - Caveat: inversion fails when demand and cost-push shocks have colinear effects. ### General structural-VAR framework with news (inclusive form) - Structural decomposition: ε_t = u_t + v_{t−1}, with u_t ⟂ v_{t−1}; Var(u_t) = D^2_u; Var(v_t) = D^2_v; normalize Var(ε_t) = D^2_u + D^2_v = I. - Data-generating process allowing for news: - x_t = Σ_{j=1}^m B_j x_{t−j} + A ε_t + C v_t (Equation (1)) - A: contemporaneous response to surprises u_t - C: response to anticipated news v_t - Inclusive condition and Theorem 1: if the model is inclusive (exogenous states either appear in x_t or are linear combinations of x_t and lags), then x_t follows form (1). ### Identification via VAR with forecasts (constructive estimator) - Observed forecasts: f_t = Σ_{j=1}^m B_j x_{t+1−j} + A v_t (Equation (6)) - Stack (f_t, x_t) into VAR(m−1): [f_t; x_t] = Σ_{j=1}^{m−1} B_j [f_{t−j}; x_{t−j}] + A [v_t; u_t] (Equation (7)) - Reduced-form residual covariance Σ satisfies: Σ = A [ D^2_v 0 ; 0 D^2_u ] A′ - Identification (Theorem 2) conditions: - If A and D^2_v are full rank, then A, C, D^2_u, D^2_v are determined (up to sign and column order) by Σ and B_1. - Constructive estimator steps (key formulas preserved): - Partition Σ into blocks Σ_{11}, Σ_{21}, Σ_{22}. - Define φ = Σ_{11} − B_1 Σ_{21} − Σ′_{21} B′_1 + B_1 Σ_{22} B′_1 = A D^2_v A′ - Define ψ = Σ_{22} − (Σ_{21} − Σ_{22} B′_1) φ^{−1} (Σ_{21} − Σ_{22} B′_1)' = A D^2_u A′ - φ + ψ = A A′; perform SVD to recover A (up to column order and sign). - Recover D^2_v from eigenstructure; D^2_u = I − D^2_v. - Recover C: - C = (Σ_{21} − Σ_{22} B′_1) (D^2_v A′)^{−1} - Note: shocks identified only up to column order and sign; only variances of shock components are determined. ### Forecast cleaning (from empirical ̃f_t to rational f_t) - Empirical ̃f_t may include biases/noise; cleaning aims to recover rational expectations f_t. - Procedure: estimate VAR(k) with k ≥ m on [̃f_t; z_t; x_t]; denote final n rows of G_j as G_{x,j}. - Cleaned rational forecast: - f_t = Σ_{j=1}^k G_{x,j} [̃f_{t+1−j}; z_{t+1−j}; x_{t+1−j}] (Equation (8)) - Theorem 3: if H^f(L) is causally invertible, then rational forecast f_t is given by (8). - Caveats: requires observable confounders z_t and invertibility; if conditions fail cleaned forecasts are best unbiased forecasts given observables and reported forecasts (Appendix A.2). ### IRFs with news and surprises (key formulas) - Surprise-u_t IRF: φ_u(h) = B_h A - News-v_{t−1} IRF: - φ_v(0) = C - φ_v(h) = B_h C + B_{h−1} A for h > 0 - IRF to fundamental shock ε_t = u_t + v_{t−1}: - φ_ε(h) = E[x_{t+h} | ε_t = 1] = φ_u(h) D^2_u + φ_v(h+1) D^2_v - Timing cases: - h = −1: φ_ε(−1) = C D^2_v - h ≥ 0: φ_ε(h) = B_{h+1} C D^2_v + B_h A (D^2_u + D^2_v) ### Generalizations and limitations - Multiple news horizons: generalizable by including forecasts at additional horizons (Appendix D); data requirements grow rapidly; k > 1 can lead to overidentification. - Hidden/unobserved state variables: Appendix F derives SVARMA restrictions; identification more computationally intensive and may lack closed-form Σ decomposition. - Applicability: approach applies when model can be written in inclusive form and Blanchard-Kahn conditions hold. --- ### 4 Data, estimation, and shock labeling (empirical implementation) ### Data sources and forecast measures - Main forecast source: Survey of Professional Forecasters (SPF), quarterly, median forecasts used; SPF since 1968, ~40 forecasters. - SPF limitations and supplements: - SPF only collects real government consumption and investment since 1981:III; for 1968:IV–1981:II authors use Federal Reserve Greenbook forecasts (transcribed when necessary). - Federal budget receipts and surpluses: dataset by Croushore and van Norden (2018), extended to 2016:IV by transcribing Greenbooks. - Interest-rate forecasts: - Measured from yield curve; one-period-ahead forecast computed as E[r^h_{t+1}] = r^{h+1}_t − r^1_t. - Yield-curve-implied forecasts correlate closely with SPF for 3-month T-bills: correlation coefficient = 0.996. - Oil prices and exchange rates: 3-month-ahead futures contracts used. ### Variable transformations and baseline sample (Table 1 summarized exactly) - Transformations: - NIPA and federal budget variables: divided by estimated quadratic time trend in real GDP. - GDP deflator: log differences and annualized for inflation. - Regularly growing variables: logs. - Variables not clearly nonstationary (unemployment, interest and exchange rates): left in levels. - Finally remove quadratic trend and linear seasonal factors from all variables. - Baseline variables and sample ranges (source for empirical forecast ̃f_t): - Real GDP: 1968:IV - 2022:II — SPF - Federal tax receipts: 1968:IV - 2016:IV — Fed Greenbooks - Real government spending: 1968:IV - 2022:II — Fed Greenbooks (1968:IV - 1981:II); SPF (1981:III - 2022:II) - GDP deflator: 1968:IV - 2022:II — SPF - 3-month Treasury rate: 1968:IV - 2022:II — Yield curve - Housing starts: 1968:IV - 2022:II — SPF - Additional variables include Unemployment Rate, Industrial production, Federal budget surpluses, USD/CAD exchange rate, Real oil price (1983:I - 2022:II), and 1–5 year Treasury rates. ### Constructing cleaned forecasts f_t (two-step ML-enhanced approach) 1. Build z_t via machine learning: - For each of six baseline VAR variables, fit three models separately: elastic net, regression tree, simple linear projection. - Predict one-period-ahead using up to eight lags of data and outcomes for all 16 variables (≈256 predictors). - Rolling cross-validation selects tuning parameters; choose model with lowest out-of-sample average RMSE. - Fitted predictions form z_t (N entries = ML predictions for each element of x_{t+1}). 2. Include z_t in cleaning VAR (Section 3.5) to form f_t via equation (8). - Rationale: z_t incorporates information not in VAR; method limits overfitting by bottlenecking information into z_t. - Limitations: residual risk of overfitting; bias-variance tradeoff acknowledged. ### Estimation challenges and solutions - Main issues: 1. Small-sample bias of OLS in VARs. 2. Non-uniqueness across bootstrap simulations due to sign and permutation indeterminacy. - Bias correction via Kilian (1998) bootstrap: - Approximate bias at point estimate with average bias in bootstrapped samples; adjust point estimate to offset bias. - Generate bias-corrected bootstrap of reduced-form estimates B(j), Σ(j) for simulations j = 1,...,N_sim. - Apply identification (Theorem 2) to each bootstrap draw to obtain structural parameters A(j), C(j), D_u^(j), D_v^(j). - Reordering and sign indeterminacy: - For N = 6 variables there are 2^N × N! = 46,080 possible sign/ordering combinations. - Solve via a modified Quadratic Assignment Problem: compute cost matrix ψ(i,j) = min_d sum_{h=1}^H ∥φ_pt_x(i)(h) − (−1)^d φ_n_x(j)(h)∥^2 and pass to solver. - This ordering minimizes a continuous loss and does not affect asymptotic distribution (Lewis (2021) Theorem 4). - Bootstrap features captured: - Re-estimating D_u and D_v per simulation captures uncertainty over shock sizes. - Uses estimated residuals (permits non-normality). - Applies exact identification per simulation (avoids delta-method linearization). - Matches observed sample length to include small-sample variation. - Lag selection: - Akaike Information Criterion selects a one-lag specification fairly conclusively. ### Shock labeling approach and qualitative labels - Structural shocks recovered are labeled heuristically using responses to surprises (unanticipated shocks) for interpretability. - Responses measured as percent deviation from trend associated with a unit standard deviation structural shock (impulse responses scaled by element of D_u). Dashed lines show 10th and 90th bootstrap percentiles. - Labels and qualitative features: - Shock 1: “fiscal stimulus” - Immediate contraction in government tax revenues; prolonged increase in government spending (delayed); output and housing starts increase with a lag; fiscal expansion induces a decline in inflation. - Shock 2: “monetary policy” - Immediate increase in short-term interest rates; subsequent decline in output over next year and later reduction in inflation. - Shock 3: “demand” - Immediate output response; more long-lasting increase in inflation; delayed interest-rate response; spending down and taxes up. - Shock 4: “supply” - Opposing responses of inflation and output on impact. - Shocks 5 and 6: unlabeled (do not pass validation tests; second unlabeled resembles monetary policy but lacks validation). --- ### 4.4 Validation exercises (fiscal and monetary) ### 4.4.1 Fiscal shock: multiplier-consistency test - Definitions: - μ_h_G = (Σ_{s=0}^h E_t ΔY_{t+s}) / (Σ_{s=0}^h E_t ΔG_{t+s}) - μ_h_T = (Σ_{s=0}^h E_t ΔY_{t+s}) / (Σ_{s=0}^h E_t ΔT_{t+s}) - For combined tax and spending responses: μ_h_Y = μ_h_G Σ_{s=0}^h E_t ΔG_{t+s} + μ_h_T Σ_{s=0}^h E_t ΔT_{t+s} (equation (10)) - Multipliers used (preserve horizons and values exactly): - Lewis 2021 - h = 2: μ_h_G = 0.56, μ_h_T = 0.03 - h = 4: μ_h_G = 0.57, μ_h_T = -0.09 - h = 8: μ_h_G = 0.57, μ_h_T = -0.71 - h = 12: μ_h_G = 0.64, μ_h_T = -1.33 - h = 16: μ_h_G = 0.76, μ_h_T = -1.77 - h = 20: μ_h_G = 0.87, μ_h_T = -2.06 - Blanchard and Perotti 2002 - h = 2: μ_h_G = 0.61, μ_h_T = -0.64 - h = 4: μ_h_G = 0.60, μ_h_T = -1.01 - h = 8: μ_h_G = 0.56, μ_h_T = -2.28 - h = 12: μ_h_G = 0.60, μ_h_T = -3.63 - h = 16: μ_h_G = 0.70, μ_h_T = -4.69 - h = 20: μ_h_G = 0.80, μ_h_T = -5.41 - Ramey 2011 / Romer and Romer 2010 - h = 20: μ_h_G = 1.20, μ_h_T = -2.60 - Caldara and Kamps 2017, penalty function - h = 2: μ_h_G = 0.05, μ_h_T = -1.05 - h = 6: μ_h_G = 0.35, μ_h_T = -1.20 - h = 12: μ_h_G = 0.55, μ_h_T = -0.80 - h = 20: μ_h_G = 0.25, μ_h_T = -0.45 - Ben Zeev and Pappa 2017 / Romer and Romer 2010 - h = 6: μ_h_G = 2.40, μ_h_T = -1.25 - Ricco 2015 / Romer and Romer 2010 - h = 16: μ_h_G = 1.50, μ_h_T = -2.60 - Findings: - Literature-implied μ_h_Y plotted against model’s bootstrapped median cumulative output response. - Remarkably close agreement with Lewis (2021) multipliers. - Most other multipliers imply larger output responses than model estimates; Blanchard and Perotti (2002) implies substantially larger response. - Satisfying multiplier consistency is necessary but not sufficient for fiscal interpretation. - Additional checks (Appendix G.3.1): supply, demand, and second unlabeled shock definitively rejected as fiscal; monetary shock shows some similarity but government spending and taxes respond insignificantly; first unlabeled shock not wildly different from fiscal-consistent responses. ### 4.4.2 Monetary shock: external-shock comparisons - External monetary shocks assembled (Table 3 exact entries): - Bauer-Swanson: Orig. Freq.: M; Start: 1988-03-01; End: 2019-12-01; N = 128 - Bauer-Swanson (orthogonalized): Orig. Freq.: M; Start: 1988-03-01; End: 2019-12-01; N = 128 - Jarocinski-Karadi, HFI from Fed Funds: Orig. Freq.: M; Start: 1990-03-01; End: 2016-12-01; N = 108 - Romer-Romer 2023: Orig. Freq.: Q; Start: 1969-03-01; End: 2019-12-01; N = 204 - Romer-Romer 2004 (up to 2007): Orig. Freq.: Q; Start: 1969-03-01; End: 2007-12-01; N = 156 - Methodology: - Common multivariate dynamic framework (one-lag VAR headline). - Rescale external shocks to match cumulative interest rate response in first 8 quarters (cumulative 100 basis points normalization used in Figure 5). - Findings (comparative qualitative/quantitative highlights): - Baseline interest-rate shock raises interest rates by about 20 basis points. - Output declines around 0.2 percent for baseline and literature-identified shocks after rescaling. - Inflation: small positive liquidity effect short run then decline at longer horizons; baseline effect on inflation generally a little larger but other estimates often lie within baseline confidence interval and peak at two to three years. - Other variables broadly agree across sources. - Robustness: Appendix G.3.2 confirms robustness to specification, estimation method, and normalization. - Conclusion: monetary shock recovered is qualitatively and quantitatively similar to standard externally-identified monetary shocks after accounting for magnitude and propagation differences. --- ### 4.6 Importance of news versus surprises (quantitative decomposition) ### Overview and qualitative findings - Structural shocks are u_t + v_{t−1} combinations; relative contributions depend on D_u, D_v and A, C matrices. - “Average” impulse responses (Figure 8) average news and surprise impulses weighted by their standard deviations. - Overall: surprises tend to be more important than news for driving macro fluctuations, but: - News generally more important for inflation (especially short horizons). - Taxes generally more driven by surprises. - For h = 24 (24 quarters), news typically accounts for between one quarter and one fifth of the variance for all but one variable; inflation is an exception where news accounts for almost two fifths of fluctuations 6 years ahead. ### Forecast-error variance decomposition (h = 24) — Table 4 (preserve numbers exactly) - Decomposition reports percent of overall forecast error variance attributable to each shock split into News and Surprise components. Totals sum to 100.0 percent for each variable. - Government spending - News: 4.5, 3.6, 1.6, 5.6, 2.2, 1.1, 24.3 - Surprise: 20.4, 10.0, 3.8, 2.3, 14.8, 12.3, 75.7 - Total: 25.6, 15.8, 6.4, 9.7, 18.5, 14.5, 100.0 - Output - News: 7.0, 2.2, 3.7, 4.3, 2.1, 2.3, 26.3 - Surprise: 8.0, 6.7, 19.5, 23.9, 4.6, 4.3, 73.7 - Total: 15.9, 9.9, 24.4, 28.3, 6.7, 6.7, 100.0 - Taxes - News: 4.9, 3.3, 1.9, 1.7, 1.8, 2.3, 19.4 - Surprise: 12.5, 4.7, 11.6, 30.3, 7.6, 7.4, 80.6 - Total: 18.2, 8.6, 14.5, 32.0, 10.1, 10.9, 100.0 - 3-month interest rate - News: 5.6, 2.2, 3.7, 5.8, 2.1, 2.3, 25.9 - Surprise: 2.8, 8.0, 16.9, 18.1, 4.9, 17.4, 74.1 - Total: 9.2, 11.2, 22.6, 24.8, 7.0, 19.7, 100.0 - Housing starts - News: 5.1, 2.4, 2.2, 2.0, 1.8, 1.8, 19.0 - Surprise: 13.8, 18.4, 17.7, 8.3, 6.0, 9.6, 81.0 - Total: 19.3, 21.7, 20.8, 11.4, 8.3, 11.4, 100.0 - Inflation - News: 4.1, 1.9, 4.3, 17.0, 1.5, 2.9, 37.8 - Surprise: 5.2, 4.0, 12.6, 21.9, 2.4, 7.0, 62.2 - Total: 10.0, 7.9, 17.0, 38.8, 3.9, 9.9, 100.0 - Unweighted average (across variables) - News: 5.2, 2.6, 2.9, 6.1, 1.9, 2.1, 25.5 - Surprise: 10.5, 8.6, 13.7, 17.5, 6.7, 9.7, 74.5 - Total: 16.4, 12.4, 16.6, 24.6, 8.6, 11.8, 100.0 ### Time-series highlights and robustness - Time series of estimated shocks (Figure 9): variance of the news component relatively larger for fiscal stimulus and supply shocks. - During 2008–2009 Global Financial Crisis: news and surprise components of monetary policy shock were similar in magnitude (news standard deviation around half that of surprise). - Robustness: alternative specifications (lags, variables, forecast cleaning, sample lengths) leave role of news vs. surprises very similar to baseline. --- ### 5.2 Counterfactual exercises (stabilization and passive policies) ### Design and approach - Consider active policies (moderate business cycles) and passive policies (hold policy instruments fixed). - For each instrument, choose linear combination of news and surprise components to minimize variance in one of three objectives: (1) output, (2) inflation, (3) dual mandate (weighted average). - Optimal policy implies responses for remaining 10 shocks (5 structural shocks × news/surprise). - Bootstrapped counterfactual impulses computed by applying median structural estimate to bootstrapped IRFs; N_sim = 1000 replications. ### Fiscal-policy counterfactuals (main findings) - Fiscal policy to minimize detrended output variance: - Effective for some shocks (e.g., after a surprise supply shock baseline output expansion nearly entirely eliminated via lower spending and higher taxes). - Effective for demand news but less so for demand surprises or supply news (those have quick decay while fiscal shocks are persistent). - Short-run moderation limited for rapidly decaying shocks; medium-run moderation stronger. - Fiscal policy to minimize inflation variance: - Slow fiscal passthrough and transitory inflation responses mean policymaker centers inflation around zero rather than damping short-term fluctuations; requires much larger swings in taxes and spending. - Fiscal policy not very effective at offsetting inflation fluctuations. - Dual mandate yields intermediate policies/outcomes. ### Monetary-policy counterfactuals (main findings) - Monetary policy effective at stabilizing output responses to certain shocks: - For demand surprises, immediate rate hikes reduce demand booms but create deflation. - Supply news triggers similar rate response with opposite sign. - Less effective for shocks with delayed output effects: immediate rate hikes mainly reduce medium-run output variance. - Inflation stabilization: monetary policy more consistently effective than fiscal policy; e.g., demand surprises produce immediate inflation in baseline, monetary inflation-targeting hikes rates and pushes inflation IRF nearly to zero. - Some short-term supply-news bursts hard to fully moderate due to smooth persistent effect of monetary policy. ### Cooperation and quantitative trade-offs (Table 5 exact numerical entries) - Counterfactual variances relative to baseline (norm of IRF). Columns: Target = Inflation / Output / Dual Mandate. Policies: Fisc. / Mon. / Joint. - Inflation target row: - Inflation: 0.19 0.09 0.00 - Output: 0.46 1.41 0.80 - Dual Mandate: 0.43 0.44 0.34 - Output target row: - Inflation: 1.03 1.45 1.56 - Output: 0.30 0.33 0.01 - Dual Mandate: 0.56 0.58 0.37 - Government spending row: - Inflation: 3.28 2.73 1.24 - Output: 2.77 1.08 2.47 - Dual Mandate: 2.49 1.22 1.32 - Taxes row: - Inflation: 4.56 2.22 3.06 - Output: 5.15 2.09 3.91 - Dual Mandate: 6.02 1.43 1.72 - 3-month interest rate row: - Inflation: 1.22 0.84 2.02 - Output: 1.06 1.15 2.12 - Dual Mandate: 1.69 0.82 0.43 - Housing starts row: - Inflation: 1.85 1.09 1.09 - Output: 0.92 1.06 0.89 - Dual Mandate: 0.80 0.68 0.25 - Example calculation reported: - Improvement in output variance reduction for joint policy relative to monetary policy alone = (1 − 0.58)/(1 − 0.37) = 1.5 ### Passive policies (main findings) - Making policy instruments as fixed as possible amplifies output volatility: - Fixing government spending or taxes amplifies output IRFs to demand and supply shocks — current fiscal policy already moderates cycles. - Interest-rate peg / minimizing interest-rate volatility: modestly reduces rate responses but amplifies output responses; indicates current monetary policy reduces some output volatility and passive-rate counterfactual is hard to implement given news/surprise structure. --- *Italic: Source: wpiea2023208-print-pdf (Identifying News Shocks from Forecasts — Working Paper No. WP/2023/208).* ### 2.1 The New Keyensian Model with Monetary Policy News ### 2.1 The New Keynesian Model with Monetary Policy News ### Model setup: three-equation New Keynesian model - New Keynesian Phillips curve: π_t = β E_t[π_{t+1}] + κ y_t + x_t - Euler equation: 0 = E_t[z_t + γ(y_t − y_{t+1}) + i_t − π_{t+1}] - Taylor rule: i_t = φ_π π_t + h_t - Definitions: - π_t, y_t, i_t are inflation, the output gap, and the nominal interest rate respectively. - x_t is an i.i.d. cost-push shock. - z_t is an i.i.d. demand shock. - Policy residual with separate news and surprise components: - h_t = ρ h_{t−1} + u_t + v_{t−1} - u_t is an i.i.d. surprise (unanticipated at t). - v_{t−1} is an i.i.d. news shock known in period t−1 (anticipated). - Interpretation: - v_{t−1} captures pre-announced policy changes (anticipation effects). - Example: if communicated rate rise in t of 25 basis points then v_{t−1} = 0.25; actual 50 basis points in t implies u_t = 0.25 and total u_t + v_{t−1} = 0.5. ### Reduced-form solution and impulse responses (qualitative) - Equilibrium solution (coefficients b) written as linear combinations of shocks: - π_t = b^π_h h_t + b^π_v v_t + b^π_x x_t + b^π_z z_t - y_t = b^y_h h_t + b^y_v v_t + b^y_x x_t + b^y_z z_t - i_t = b^i_h h_t + b^i_v v_t + b^i_x x_t + b^i_z z_t - Qualitative IRF differences: - News of an interest rate rise tomorrow (v_{t−1} > 0) leads agents to anticipate a recession next period: consumption smoothing causes reduced spending today → lower output and prices today. - Central bank responds via Taylor rule by cutting interest rates today to mitigate the anticipated downturn. - Calibration used in illustrative Figure 1 (monthly calibration adapted from Galí (2008)): - β = 0.997 - κ = 0.2 - γ = 1 - φ_π = 1.5 - persistence ρ = 0.6 (lower than Galí) - all shock variances set to one ### Identification in the simple example - Information counts: - Structural shocks: u_t, v_t, x_t, z_t (four shocks). - Observables: π_t, y_t, i_t (three time series) → insufficient to identify all four structural shocks. - Solution via observed forecasts: - Include agents' forecasts; for the simple example a single forecast suffices. - Define inflation forecast f^π_t ≡ E_t[π_{t+1}] with representation f^π_t = b^π_h (ρ h_t + v_t). - Identification steps (constructive): - Recover h_t from observables: h_t = i_t − φ π_t - Identify policy news shock: v_t = f^π_t / b^π_h − ρ h_t - Identify policy surprise: u_t = h_t − ρ h_{t−1} − v_{t−1} - Identify remaining shocks x_t and z_t by inverting the linear system: - [x_t; z_t] = [ [b^π_x, b^π_z]; [b^y_x, b^y_z] ]^{-1} [ π_t − b^π_h h_t − b^π_v v_t ; y_t − b^y_h h_t − b^y_v v_t ] - Caveat: inversion fails in non-invertible edge cases when demand and cost-push shocks have colinear effects. - Naive VAR (without forecasts) fails when news v_{t−1} matters: - SVAR without forecasts consistently identifies effects of u_t only in absence of news. - When news shocks matter, the Naive VAR returns a linear combination of current and past shocks and does not separately identify news and surprise. ### General identification framework (structural VAR with news) - Structural shock decomposition for n-dimensional series x_t and n structural shocks ε_t: - ε_t = u_t + v_{t−1} - u_t and v_{t−1} are orthogonal: u_t ⟂ v_{t−1} - Var(u_t) = D^2_u and Var(v_t) = D^2_v with D_u and D_v diagonal - Normalize Var(ε_t) = D^2_u + D^2_v = I - Data-generating process allowing for news: - x_t = Σ_{j=1}^m B_j x_{t−j} + A ε_t + C v_t (Equation (1)) - A measures contemporaneous response to surprises u_t; C measures response to anticipated news v_t - Inclusivity condition: - A model can be written in inclusive form if exogenous state variables either appear in x_t or can be expressed as linear combinations of entries in x_t and its lags. - Theorem 1: If model is inclusive, implied x_t follows form (1). ### VAR with forecasts and constructive identification - Observe forecasts f_t = E[x_{t+1} | {x_{t−j}}_{j=0}^{m−1}, ε_t, v_t] - From structure: f_t = Σ_{j=1}^m B_j x_{t+1−j} + A v_t (Equation (6)) - Stack (f_t, x_t) into a VAR(m−1): - [f_t; x_t] = Σ_{j=1}^{m−1} B_j [f_{t−j}; x_{t−j}] + A [v_t; u_t] (Equation (7)) - Reduced-form estimation yields B_1 and residual covariance Σ satisfying: - Σ = A [ D^2_v 0 ; 0 D^2_u ] A′ - Identification theorem (Theorem 2): - If A and D^2_v are full rank, then A, C, D^2_u, and D^2_v are determined (up to sign and column order) by Σ and B_1. - Constructive estimator summary: - Partition Σ into blocks Σ_{11}, Σ_{21}, Σ_{22}. - Define φ = Σ_{11} − B_1 Σ_{21} − Σ′_{21} B′_1 + B_1 Σ_{22} B′_1 = A D^2_v A′ - Define ψ = Σ_{22} − (Σ_{21} − Σ_{22} B′_1) φ^{−1} (Σ_{21} − Σ_{22} B′_1)' = A D^2_u A′ - φ + ψ = A A′; perform SVD to recover A (up to column order and sign), D^2_v from eigenstructure, and then D^2_u = I − D^2_v; finally recover C by: - C = (Σ_{21} − Σ_{22} B′_1) (D^2_v A′)^{−1} - Notes: shocks identified only up to column order and sign; only variances of shock components are determined. ### Forecast cleaning (empirical forecasts → rational expectations) - Empirical forecasts ̃f_t may deviate from rational expectations due to biases/noise. - Cleaning procedure: estimate VAR(k) with k ≥ m: - [̃f_t; z_t; x_t] = Σ_{j=1}^k G_j [̃f_{t−j}; z_{t−j}; x_{t−j}] + υ_t - Let G_{x,j} denote final n rows of G_j. - Cleaned rational forecast: - f_t = Σ_{j=1}^k G_{x,j} [̃f_{t+1−j}; z_{t+1−j}; x_{t+1−j}] (Equation (8)) - Model for forecast deviations (general form): - ̃f_t = Σ_{j=0}^k [ H^f_j f_{t−j} + H^x_j x_{t−j} + H^z_j z_{t−j} + H^u_j u_{t−j} + H^v_j v_{t−j} ] (Equation (9)) - Theorem 3: If H^f(L) is causally invertible, then rational forecast f_t is given by (8). - Caveats: - Requires observable confounders z_t and invertibility of H^f(L). - If unobserved confounders exist, cleaning and identification are still possible under looser assumptions, but the interpretation of news shocks changes (see Appendix A.2). ### Impulse Response Functions (IRFs) with news - Surprise-u_t IRF (standard): φ_u(h) = B_h A - News-v_{t−1} IRF includes additional terms because of timing: - φ_v(h) = C for h = 0 - φ_v(h) = B_h C + B_{h−1} A for h > 0 - Conditional expectations: - E[x_{t+h} | u_t] = φ_u(h) u_t - E[x_{t+h} | v_t] = φ_v(h) v_t - IRF to a unit fundamental shock ε_t = u_t + v_{t−1}: - φ_ε(h) = E[x_{t+h} | ε_t = 1] = φ_u(h) D^2_u + φ_v(h+1) D^2_v - Explicit timing cases: - h = −1: φ_ε(−1) = C D^2_v - h ≥ 0: φ_ε(h) = B_{h+1} C D^2_v + B_h A (D^2_u + D^2_v) ### Generalizations and limitations - News horizons: - The framework assumes news one period in advance but can be generalized to multiple horizons by including additional forecasts at each horizon (Appendix D). - Data requirements grow rapidly with additional horizons; identification may be possible when forecasts at multiple horizons are available. - Unobserved state variables: - If some state variables are unobserved, Appendix F derives SVAR restrictions and identification procedures; identification is more computationally intensive and may lack an analytical solution for Σ decomposition. - Applicability: - The approach applies when the model can be written in inclusive form and Blanchard-Kahn conditions hold; when these fail, identification is more challenging. ### Empirical application (preview) - The method is applied to US time series data to identify fiscal and monetary shocks, estimate multipliers, and compare effects of news versus surprises (detailed empirical results follow in Section 4). *Source: 2.1 The New Keynesian Model with Monetary Policy News (from wpiea2023208-print-pdf)* ### 4.1 Data ### 4.1 Data ### Data sources and forecast measures - Main source of forecast data: Survey of Professional Forecasters (SPF), run by the Federal Reserve Bank of Philadelphia, administered quarterly to roughly 40 anonymous forecasters since 1968. Median reported values are used as the measure of forecasts. - SPF limitations and supplements: - SPF only collects estimates on real government consumption and investment since 1981:III; for 1968:IV–1981:II the authors draw on the Federal Reserve’s Greenbook forecasts, transcribing values from original Greenbooks when necessary. For each quarter the most recent estimate is used. - Federal budget receipts and surpluses use the dataset collected by Croushore and van Norden (2018), extended to 2016:IV by transcribing from Greenbooks. - Interest rate forecasts: - Measured directly from the yield curve because the SPF provides forecasts for a limited number of interest rate horizons and only since 1981:III. - If rh t denotes the return from timet to t+h, the yield-curve-implied one-period-ahead forecast is computed as E[rh t+1] = rh+1 t − r1 t. - The yield-curve-implied forecasts are known to be biased (yield curve incorporates liquidity and risk premia) but track SPF forecasts closely; for 3-month T-bills the correlation coefficient is 0.996. - Oil prices and exchange rates: - Forecasts measured using 3-month-ahead futures contracts; deviations from covered interest rate parity depend on expected costs of holding oil or interest rate differences across countries. ### Variable transformations and pre-processing - Transformations applied: - NIPA variables and federal budget variables: divided by an estimated quadratic time trend in real GDP (following Ramey (2016)) to allow fiscal multipliers to be read directly from impulse response functions. - GDP deflator (price level): log differences and annualized to calculate the inflation rate. - Regularly growing variables (e.g., housing starts): take logs. - Variables not clearly nonstationary (unemployment, interest and exchange rates): left in levels. - Finally, a quadratic trend and linear seasonal factors are removed from all variables. - Forecast cleaning objective: produce forecast series ft from empirical forecasts ̃ft that (i) plausibly reflect information about xt+1 at time t, (ii) avoid overfitting, and (iii) satisfy the identifying assumption that forecasts contain all information already available to the VAR structure. ### Baseline variables and sample ranges (as reported in Table 1) - Baseline specification (variables and sample ranges; source for empirical forecast ̃ft): - Real GDP: 1968:IV - 2022:II — SPF - Federal tax receipts: 1968:IV - 2016:IV — Fed Greenbooks - Real government spending: 1968:IV - 2022:II — Fed Greenbooks for 1968:IV - 1981:II; SPF for 1981:III - 2022:II - GDP deflator: 1968:IV - 2022:II — SPF - 3-month Treasury rate: 1968:IV - 2022:II — Yield curve - Housing starts: 1968:IV - 2022:II — SPF - Additional variables: - Unemployment Rate: 1968:IV - 2022:II — SPF - Industrial production: 1968:IV - 2022:II — SPF - Federal budget surpluses: 1968:IV - 2016:IV — Fed Greenbooks - USD/CAD exchange rate: 1968:IV - 2022:II — Futures contracts - Real oil price: 1983:I - 2022:II — Futures contracts - 1, 2, 3, 4, and 5-year Treasury rates: 1968:IV - 2022:II — Yield curve ### Constructing cleaned forecasts ft - Two-step approach (based on methodology in Section 3.5): 1. Construct a vector zt intended to include as much information available at time t about future outcomes. - For each of the six variables in the baseline VAR, build three machine learning models separately: an elastic net, a regression tree, and a simple linear projection. - Each model predicts one-period-ahead outcomes using up to eight lags of both data and outcomes for all 16 variables in Table 1 (some 256 possible predictors). - Rolling cross-validation selects tuning parameters; the model with lowest out-of-sample average RMSE is chosen individually for each of the six variables. - The fitted predictions are labeled zt; the N entries of zt are the machine learning predictions for each element of xt+1. 2. Include zt in the cleaning process described in Section 3.5 to form ft. - Rationale and advantages: - If relevant predictive variables are not in the VAR specification, this method can incorporate their information via zt. - If empirical forecasts ̃ft already embody all available information, the method allows ft to fully reflect that. - Limitations: - Some residual risk of overfitting remains because forecasts are cleaned after cross-validation. - The bias-variance tradeoff means avoiding overfitting may conflict with the identifying assumption; the method mitigates this by using machine learning forecasts as a bottleneck, limiting information about future outcomes to the same dimension as the data. --- ### 4.2 Estimation ### Main estimation challenges - Two interacting issues complicate accurate estimation: 1. Small-sample bias of OLS estimates in VARs (OLS consistent asymptotically but biased in small samples; see Shaman and Stine (1988) discussion). 2. Non-uniqueness across bootstrap simulations: simulated structural matrices are only unique up to sign and re-ordering of shocks. ### Bias correction via bootstrap - Apply Kilian (1998) bootstrap-based bias-correction: - Approximate bias at the point estimate with the average bias in bootstrapped samples generated by the point estimate. - Adjust the point estimate to offset this bias, yielding reduced-form coefficients B( j )1,...,B( j )m, Σ( j ) for simulations j = 1,...,N. - The variation across simulations reflects sampling uncertainty under the null that point estimates are consistent. - Use the bootstrap sample of reduced-form estimates to compute confidence intervals for structural impulse responses and variance decompositions by applying the identification process to each simulated reduced form estimate (Theorem 2). - For each j = 1,...,N this yields structural parameter estimates A( j ), C( j ), D( j )u, D( j )v. ### Reordering and sign indeterminacy in simulations - Problem: - Simulated structural matrices may reorder shocks or flip signs relative to the point estimate; identification is not unique up to sign and permutation. - Solution: - For each simulation, search over all possible combinations of re-orderings and sign flips to minimize squared difference to the point estimates for structural impulse responses. - Combinatorial scale: with N = 6 variables there are 2^N possible sign flips and N! possible reorderings, giving 2^N × N! = 46,080 possible combinations in total. - The search is reduced to a modified Quadratic Assignment Problem, enabling relatively swift solutions (details in Appendix B.2). - The ordering procedure minimizes a continuous loss function and, per Lewis (2021) Theorem 4, does not affect the asymptotic distribution of structural matrices or implied impulse responses. ### Interaction and additional estimation features - Interaction risks: - Without bias-correcting reduced form estimates, incorrect ordering of simulated structural decompositions becomes more likely, widening error bands. - Bootstrap advantages and captured uncertainties: - Re-estimating shock variance matrices D( j )u, D( j )v for each simulation means impulse responses reflect uncertainty over shock propagation and over shock size. - Reordering and re-signing imperfections can cause variation due to one shock to be mistakenly attributed to another; this variability is included. - Using estimated residuals in the bootstrap permits non-normality. - Applying the exact identification method to each reduced form simulation captures full nonlinearity in identification (vs. delta method linearization). - Matching observed sample length includes small-sample variation and avoids reliance on large-sample approximations. - Alternative methods for assessing sample uncertainty (used as checks): - Draw reduced form samples from the asymptotic distribution of VAR coefficients (isolates small-sample portion of sampling variability). - Use the delta method (linearizes mapping from reduced form to structural coefficients, compressing distribution tails). - Investigate role of uncertainty over structural variances Du, Dv by presenting results where shock sizes are held fixed. - Lag length selection: - Akaike Information Criterion selects a one-lag specification fairly conclusively (see Figure 15 in Appendix G.2). - Rationale: stacked VAR includes rational forecasts of next-period outcomes, which incorporate much lagged information; the data generating process is VARMA, and the MA component (infinite autoregressive interpretation) shortens required lag length. - Alternative lag structures are considered in robustness checks. --- ### 4.3 Impact of Surprises and Shock Labels ### Labeling approach - The method recovers structural shocks but does not inherently label them; labels are assigned heuristically for convenience and interpretability. - Labels are based on the signs of impulse responses to surprises (unanticipated shocks), not to news or combinations, to align with commonly understood effects and standard priors about shocks. - Responses to unanticipated surprises are measured as percentage deviation from trend associated with a unit standard deviation structural shock (impulse responses scaled by the appropriate element of Du). Dashed lines show 10th and 90th percentiles of the bootstrapped distribution. ### Shock labels and qualitative features - Shock 1: “fiscal stimulus” - Features: - Immediate and statistically significant contraction in government tax revenues. - Prolonged and statistically significant increase in government spending (somewhat delayed). - Output and real activity (housing starts) increase with a lag. - Interpretation: labeled fiscal stimulus; Section 4.4.1 checks magnitude of output response vs. tax and spending multipliers in literature. - Notable observed effect: fiscal expansion induces a decline in inflation; one possible interpretation is that the tax decrease is sufficiently deflationary to offset inflationary government spending. - Shock 2: “monetary policy” - Features: - Clear, statistically significant, and immediate increase in short-term interest rates. - Followed by a decline in output over the next year or so and then a subsequent reduction in inflation (not always strongly statistically significant). - Section 4.4.2 verifies this shock against estimated monetary policy shocks in the literature. - Shock 3: “demand” - Features: - Immediate and statistically significant output response. - More long-lasting increase in inflation. - Delayed interest rate response. - Spending goes down and taxes go up, consistent with aggregate expansion not driven by the public sector. - Shock 4: “supply” - Features: - Markedly opposing responses of inflation and output on impact (basis for the supply label). - Shocks 5 and 6: unlabeled - Rationale: neither passes some quantitative validation tests; the second unlabeled shock resembles the monetary policy shock but lacks validation, so no structural interpretation is provided. *Source: wpiea2023208-print-pdf - 4.1 Data (excerpt).* ### 4.4 Validating the Shock Interpretations ### 4.4 Validating the Shock Interpretations ### 4.4.1 The Fiscal Policy Shock - Approach: - Fiscal multipliers defined as: - μ_h_G = (Σ_{s=0}^h E_t ΔY_{t+s}) / (Σ_{s=0}^h E_t ΔG_{t+s}) - μ_h_T = (Σ_{s=0}^h E_t ΔY_{t+s}) / (Σ_{s=0}^h E_t ΔT_{t+s}) - For a general fiscal shock that combines tax and spending changes, compute implied cumulative output response: - μ_h_Y = μ_h_G Σ_{s=0}^h E_t ΔG_{t+s} + μ_h_T Σ_{s=0}^h E_t ΔT_{t+s} (equation (10)) - Test: compare μ_h_Y (literature multipliers combined with estimated tax and spending profiles) to the model’s estimated cumulative output response Σ_{s=0}^h E_t ΔY_{t+s}. Failure would rule out a fiscal interpretation. - Multipliers used (Table 2) — preserve horizons and multiplier values exactly: - Lewis 2021 - h = 2: μ_h_G = 0.56, μ_h_T = 0.03 - h = 4: μ_h_G = 0.57, μ_h_T = -0.09 - h = 8: μ_h_G = 0.57, μ_h_T = -0.71 - h = 12: μ_h_G = 0.64, μ_h_T = -1.33 - h = 16: μ_h_G = 0.76, μ_h_T = -1.77 - h = 20: μ_h_G = 0.87, μ_h_T = -2.06 - Blanchard and Perotti 2002 - h = 2: μ_h_G = 0.61, μ_h_T = -0.64 - h = 4: μ_h_G = 0.60, μ_h_T = -1.01 - h = 8: μ_h_G = 0.56, μ_h_T = -2.28 - h = 12: μ_h_G = 0.60, μ_h_T = -3.63 - h = 16: μ_h_G = 0.70, μ_h_T = -4.69 - h = 20: μ_h_G = 0.80, μ_h_T = -5.41 - Ramey 2011 / Romer and Romer 2010 - h = 20: μ_h_G = 1.20, μ_h_T = -2.60 - Caldara and Kamps 2017, penalty function - h = 2: μ_h_G = 0.05, μ_h_T = -1.05 - h = 6: μ_h_G = 0.35, μ_h_T = -1.20 - h = 12: μ_h_G = 0.55, μ_h_T = -0.80 - h = 20: μ_h_G = 0.25, μ_h_T = -0.45 - Ben Zeev and Pappa 2017 / Romer and Romer 2010 - h = 6: μ_h_G = 2.40, μ_h_T = -1.25 - Ricco 2015 / Romer and Romer 2010 - h = 16: μ_h_G = 1.50, μ_h_T = -2.60 - Findings: - The literature-implied μ_h_Y points (using the table multipliers and estimated tax/spending IRFs) are plotted against the model’s bootstrapped median cumulative output response (Figure 4). - Agreement: - Remarkably close agreement with Lewis (2021) multipliers. - Most other literature estimates imply somewhat larger output responses than the model’s estimates. - Blanchard and Perotti (2002) multipliers imply a substantially larger output response, reflecting their larger estimated multipliers. - Interpretation caveats: - Differences across estimates can reflect different combinations of news and surprise shocks (in this paper: surprise begins contemporaneous with announcement; news starts one period after). - Satisfying the multiplier consistency is necessary but not sufficient for a fiscal interpretation. - Additional checks: - Appendix G.3.1 repeats the multiplier-consistency exercise for all other shocks. Results: - Supply shock, demand shock, and the second unlabeled shock: definitively reject a fiscal interpretation based on extant multipliers. - Monetary policy shock: shows some similarities to estimated multipliers, but government spending and taxes respond statistically insignificantly at almost all horizons, undermining a fiscal interpretation. - First unlabeled shock: cumulative output response is not wildly different from consistency with standard multipliers, suggesting it may have some fiscal aspect. ### 4.4.2 The Monetary Policy Shock - Objective: - Validate interpretation of the identified shock as a monetary policy shock by comparing model IRFs to IRFs implied by externally-identified monetary shocks from the literature. - External monetary shocks assembled and their coverage (Table 3), preserved exactly: - Bauer-Swanson - Orig. Freq.: M - Start: 1988-03-01 - End: 2019-12-01 - N = 128 - Bauer-Swanson (orthogonalized) - Orig. Freq.: M - Start: 1988-03-01 - End: 2019-12-01 - N = 128 - Jarocinski-Karadi, HFI from Fed Funds - Orig. Freq.: M - Start: 1990-03-01 - End: 2016-12-01 - N = 108 - Romer-Romer 2023 - Orig. Freq.: Q - Start: 1969-03-01 - End: 2019-12-01 - N = 204 - Romer-Romer 2004 (up to 2007) - Orig. Freq.: Q - Start: 1969-03-01 - End: 2007-12-01 - N = 156 - Methodology to ensure comparability: - Use a common multivariate dynamic framework (one-lag VAR as headline; Appendix G.3.2 reports longer-lag VAR and local projection robustness). - Rescale external shocks so impulses have equivalently-sized interest rate responses. - Two rescalings considered: initial 100 basis point increase; cumulative 100 basis point increase. Preferred: rescale to match cumulative interest rate response in first 8 quarters (cumulative 100 basis points normalization used in Figure 5). - Findings (Figure 5 comparisons, preserved quantitative points where provided): - Interest rate shock in the baseline raises interest rates by about 20 basis points. - Output response: - Output declines around 0.2 percent both for the baseline shock and for those identified elsewhere in the literature (after rescaling to match cumulative interest rate response). - Timing differences: other shocks often display longer lags (more backward-loaded impulse). - Inflation: - Almost all methods show a small positive liquidity effect in the short run and a decline at longer horizons. - Baseline estimated effect on inflation is generally a little larger, but most other estimates lie within the baseline confidence interval and agree on a peak impact on inflation at two to three years. - Other variables (government consumption, taxes, housing starts): broadly agree across sources, with some differences in dynamics. - Robustness: - Variants reported in Appendix G.3.2 confirm robustness to specification, estimation method, and normalization of shock size. - Conclusion: - The monetary shock recovered in the model is qualitatively and quantitatively similar to standard externally-identified monetary shocks once accounted for differences in magnitude and propagation. - This provides reasonable validation of the monetary interpretation. ### 4.5 The Impact of News - Scaling and interpretation: - For comparability, news impulses are scaled by the standard deviation of the surprise shock. Interpretation: in period 1 it is revealed that there will be a one standard deviation surprise shock in period 2. The plotted news impulse combines anticipation (matrix C) and realized impact (matrix A), separated from relative importance of news and surprise (matrices D_u and D_v). - Policy shocks (Figure 6) — key comparative findings: - Long run: - Anticipated shocks have much the same long-run effects as unanticipated shocks (impulses after more than 10 or 12 quarters are much the same). - Short run differences — Fiscal shock: - Taxes systematically jump up prior to an announced expansion, indicating governments raise immediate revenue to offset future spending. - Interest rates fall persistently, consistent with a tighter fiscal position. - Output response to the news shock is similar to surprise, implying slightly larger multipliers for pre-announced fiscal expansions than for surprise ones. - Housing starts decline immediately before rebounding—consistent with house builders delaying activity until stimulus arrives. - Short run differences — Monetary shock: - Anticipation effects are larger than for fiscal shock, but news shock is less well estimated (larger confidence intervals). - Anticipated monetary tightening causes a small contemporaneous increase in interest rates and a temporary expansion in housing starts (possible intertemporal substitution by interest-sensitive housebuilders). - Inflation and output drop sooner for an anticipated shock than for an unanticipated one. - The “liquidity effect” (activity and inflation increase temporarily on impact of a monetary tightening) appears to be a feature of surprises and not of news shocks. - Supply and demand shocks (summary of Figure 7 discussion): - Demand shock: - News component is very poorly estimated; news component estimated to be very small. Do not strongly rely on news impulse responses for the demand shock. - Supply shock: - Large and often countervailing anticipation effect. - When future supply increases are expected, inflation spikes today (agents expect higher future incomes without immediate expansion in supply). - As a result, gains in output and housing starts and the decline in interest rates are much mitigated when supply improvements are anticipated. _Italic: Source: wpiea2023208-print-pdf - 4.4 Validating the Shock Interpretations_ ### 4.6 The Importance of News Versus Surprises in Macroeconomic Fluc- ### 4.6 The Importance of News Versus Surprises in Macroeconomic Fluctuations ### Overview - Structural shocks are combinations of news and surprises; their relative contribution depends on the variances of news and surprise (captured by D_u and D_v) and their causal impacts (A and C matrices). - “Average” impulse responses (Figure 8) are an average of the news and surprise impulses, weighted by their respective standard deviations. - Overall, surprises tend to be more important than news for driving macroeconomic fluctuations, but this varies substantially across shocks, variables, and horizons. - News is generally more important for inflation, especially at short horizons; taxes are generally more driven by surprise shocks. - For h = 24 (24 quarters ahead, a proxy for the long run), a variance decomposition shows news typically accounts for between one quarter and one fifth of the variance for all but one variable; inflation is an exception, where news accounts for almost two fifths of fluctuations 6 years ahead. ### Average impulse responses (Figure 8) — qualitative findings - Supply shocks are driven more by news; demand shocks are driven more by surprises. - Fiscal policy appears on average to have a larger surprise component than monetary policy; for monetary policy, surprises are generally more important. - Monetary policy shocks only drive a small amount of the variance in interest rates, consistent with policymakers following a policy rule that responds to other shocks (i.e., monetary policy not injecting noise into interest rates). - Housing starts are most affected by monetary policy, demand, and fiscal shocks. ### Variance decomposition method (key formula) - The h-step ahead forecast error variance can be written as the sum of contributions from the news and surprise components of each structural shock. For M = 1, the decomposition is given as equation (11) in the source: MSE_t x_{t+h} = sum_{j=1}^N sum_{s=1}^h B^{h-s} (A_j A'_j) (B')^{h-s} ! sigma^2_{u,j} + sum_{j=1}^N sum_{s=1}^h 1_{h>1} B^{h-s-1} [ A_j A'_j + B(C_j A'_j) + (A_j C'_j)B' + B(C_j C'_j)B' ] (B')^{h-s-1} + (C_j C'_j) ! sigma^2_{v,j} - The decomposition is linear in the variances sigma^2_{u,j} and sigma^2_{v,j}, allowing an additive attribution of total variance to each shock’s news and surprise components. - Table 4 reports this variance decomposition for h = 24 (average from a bootstrap with N_sim = 1000 replications). Totals for each variable sum to 100.0 percent. ### Forecast error variance decomposition at h = 24 (Table 4) - The table reports, in percent, the fraction of the overall forecast error variance attributable to each shock, split into news and surprise components. The shock columns are: Fiscal stimulus, Mon. policy, Demand, Supply, Unlabeled #1, Unlabeled #2, Total. - Government spending - News: 4.5, 3.6, 1.6, 5.6, 2.2, 1.1, 24.3 - Surprise: 20.4, 10.0, 3.8, 2.3, 14.8, 12.3, 75.7 - Total: 25.6, 15.8, 6.4, 9.7, 18.5, 14.5, 100.0 - Output - News: 7.0, 2.2, 3.7, 4.3, 2.1, 2.3, 26.3 - Surprise: 8.0, 6.7, 19.5, 23.9, 4.6, 4.3, 73.7 - Total: 15.9, 9.9, 24.4, 28.3, 6.7, 6.7, 100.0 - Taxes - News: 4.9, 3.3, 1.9, 1.7, 1.8, 2.3, 19.4 - Surprise: 12.5, 4.7, 11.6, 30.3, 7.6, 7.4, 80.6 - Total: 18.2, 8.6, 14.5, 32.0, 10.1, 10.9, 100.0 - 3-month interest rate - News: 5.6, 2.2, 3.7, 5.8, 2.1, 2.3, 25.9 - Surprise: 2.8, 8.0, 16.9, 18.1, 4.9, 17.4, 74.1 - Total: 9.2, 11.2, 22.6, 24.8, 7.0, 19.7, 100.0 - Housing starts - News: 5.1, 2.4, 2.2, 2.0, 1.8, 1.8, 19.0 - Surprise: 13.8, 18.4, 17.7, 8.3, 6.0, 9.6, 81.0 - Total: 19.3, 21.7, 20.8, 11.4, 8.3, 11.4, 100.0 - Inflation - News: 4.1, 1.9, 4.3, 17.0, 1.5, 2.9, 37.8 - Surprise: 5.2, 4.0, 12.6, 21.9, 2.4, 7.0, 62.2 - Total: 10.0, 7.9, 17.0, 38.8, 3.9, 9.9, 100.0 - Unweighted average (across variables) - News: 5.2, 2.6, 2.9, 6.1, 1.9, 2.1, 25.5 - Surprise: 10.5, 8.6, 13.7, 17.5, 6.7, 9.7, 74.5 - Total: 16.4, 12.4, 16.6, 24.6, 8.6, 11.8, 100.0 - Key quantitative takeaways from Table 4: - For most variables both news and surprises play an important role; news generally accounts for a smaller share but is not trivial. - For all but one variable, news shocks account for between one quarter and one fifth of the variance (reported in source text). - Inflation is the variable where news matters relatively more: news accounts for almost two fifths of fluctuations 6 years ahead, principally driven by news about supply. - Fiscal shocks are the most “newsy” on the unweighted average across variables, consistent with long implementation lags in fiscal policy. - Monetary policy shocks contribute relatively little to interest rate variance in this decomposition, consistent with a reaction-function view of policy. ### Time series of estimated shocks (Figure 9) — highlights - The variance of the news component is relatively larger for the fiscal stimulus and supply shocks. - During the Global Financial Crisis of 2008-2009, the news and surprise components of the monetary policy shock are of similar magnitudes (the standard deviation of the news component is around half that of the surprise for monetary policy), consistent with increased use of forward guidance by the Federal Reserve. ### Robustness - Appendix G reports alternative specifications (different lag structures, different variables, dropping cleaning for forecasts, different sample lengths); while shocks are not always directly comparable across specifications, the role of news and surprises remains very similar to the baseline. *Source: IMF Working Paper — Section 4.6 of the provided content unit.* ### 5.2 Counterfactual Exercises ### 5.2 Counterfactual Exercises ### Business cycle stabilization: design and approach - Two types of counterfactual policies studied: active policies (aim to moderate business cycles) and passive policies (attempt to hold policy instruments fixed). - For each policy instrument, select the linear combination of news and surprise shocks that minimize the variance in one of three objectives: (1) output, (2) inflation, and (3) a “dual mandate” weighted average. - For each objective, the optimal policy implies a different response for each of the remaining 10 shocks (5 structural shocks, each with a news and surprise component). - Plots report counterfactual impulse responses to the non-policy “supply” and “demand” structural shocks; summary numbers reported in Table 5 use all shocks. - Bootstrapped counterfactual impulses are computed by applying the counterfactual policy for the median structural estimate to the bootstrapped impulse responses; the bootstrap uses N_sim = 1000 replications (Figure 11). ### Fiscal policy counterfactuals — main findings - Using fiscal policy to minimize detrended output variance: - Fiscal policy is effective at moderating some shocks but not all. - Example: after a surprise supply shock the baseline output expansion is nearly entirely eliminated under the fiscal-output-stabilizing policy, achieved via lower government spending and raising taxes. - Fiscal policy similarly effective for demand news, but less effective for demand surprises or supply news because these have large, quickly decaying responses while fiscal shocks have highly persistent baseline effects on output. - Short-run moderation is limited for shocks with rapid decay; medium-run moderation is stronger. - Using fiscal policy to minimize inflation variance: - Slow passthrough of fiscal policy and transitory inflation responses mean the policymaker centers inflation fluctuations around zero rather than damping short-term fluctuations. - Achieving inflation stabilization via fiscal policy typically requires much larger swings in taxes and spending. - Fiscal policy is not very effective at offsetting inflation fluctuations. - Dual mandate (weighted average of output and inflation): - Produces policies and outcomes approximately halfway between inflation targeting and output stabilization. ### Monetary policy counterfactuals — main findings - Monetary policy is effective at stabilizing output responses to certain shocks: - After a demand surprise, immediate interest rate hikes reduce demand-induced output booms but create deflation. - Supply news shows similar interest-rate response with opposite sign. - Monetary policy less effective for shocks with delayed output effects: - When supply surprises create an output boom with a long delay, monetary policy mainly reduces medium-run output variance via immediate rate hikes. - Inflation stabilization via monetary policy: - Monetary policy more consistently effective at moderating inflation when the objective is to minimize inflation variance. - Example: demand surprises produce immediate inflation in baseline; inflation-targeting monetary policy hikes rates and pushes the inflation IRF nearly to zero. - Some short-term bursts of inflation from supply news are difficult to fully moderate because monetary policy affects inflation smoothly and persistently; the policymaker reduces the short-term burst only somewhat while tolerating some medium-run deflation. ### Cooperation and variance trade-offs (summary from Table 5) - Fiscal and monetary policy cooperating is more effective than either instrument alone. - When output stabilization is the goal: - Fiscal or monetary policy alone can reduce output variance by about two thirds. - Joint fiscal and monetary policy can reduce output variance to nearly zero. - Joint policy provides more degrees of freedom (four) to stabilize one target. - For inflation: - Monetary policy is more effective than fiscal policy at moderating inflation. - Together, they can nearly eliminate inflation volatility. - Trade-offs: - Almost every counterfactual policy increases the volatility of government spending and taxes. - When used alone, fiscal policy tends to increase interest rate volatility as well. - When monetary policy aims to moderate output, interest rate volatility rises; when the goal is moderating inflation, less interest rate volatility is needed. - Dual mandate is a more challenging and realistic test; coordination benefits are smaller but still meaningful. The incremental reduction in variance of joint policy is in the order of around an extra 50 percent for output and 20 percent for inflation. - Numerical results from Table 5: Counterfactual variances relative to baseline (measured by the norm of the impulse response function). Columns are arranged as Target: Inflation / Output / Dual Mandate and Policy used: Fisc. / Mon. / Joint. - Inflation target row: - Inflation: 0.19 0.09 0.00 - Output: 0.46 1.41 0.80 - Dual Mandate: 0.43 0.44 0.34 - Output target row: - Inflation: 1.03 1.45 1.56 - Output: 0.30 0.33 0.01 - Dual Mandate: 0.56 0.58 0.37 - Government spending row: - Inflation: 3.28 2.73 1.24 - Output: 2.77 1.08 2.47 - Dual Mandate: 2.49 1.22 1.32 - Taxes row: - Inflation: 4.56 2.22 3.06 - Output: 5.15 2.09 3.91 - Dual Mandate: 6.02 1.43 1.72 - 3-month interest rate row: - Inflation: 1.22 0.84 2.02 - Output: 1.06 1.15 2.12 - Dual Mandate: 1.69 0.82 0.43 - Housing starts row: - Inflation: 1.85 1.09 1.09 - Output: 0.92 1.06 0.89 - Dual Mandate: 0.80 0.68 0.25 - Example calculation reported in text: - Improvement in output variance reduction for joint policy relative to monetary policy alone = (1 − 0.58)/(1 − 0.37) = 1.5 ### Passive policies — main findings - Counterfactuals constructed to make policy instruments "as fixed as possible" produce larger output volatility: - Fixing government spending (making it as passive as possible) amplifies output IRFs to demand and supply shocks — implying current government spending behavior already moderates business cycles. - Fixing taxes (tax revenues as acyclical as possible) similarly amplifies output IRFs. - Interest rate peg / minimizing interest-rate volatility: - Difficult to construct a passive interest rate counterfactual; the interest rate volatility-minimizing counterfactual only modestly reduces interest rate responses but amplifies output responses. - This suggests current monetary policy effectively reduces some output volatility; inability to achieve a passive interest rate may be due to the news/surprise structure of the data or deeper macroeconomic properties. *Source: wpiea2023208-print-pdf - 5.2 Counterfactual Exercises* ### 2015. Publisher: ECB Working Paper. ### wpiea2023208-print-pdf - 2015. Publisher: ECB Working Paper. ### Appendix A — Forecast Cleaning Properties (A.1 Proof of Theorem 3) - Starting point: causal invertibility and equation (9) imply the rational expectation can be written as f_t = H_f(L)^{-1} \tilde f_t - H_f(L)^{-1} H_x(L) x_t - H_f(L)^{-1} H_z(L) z_t - H_f(L)^{-1} H_u(L) u_t - H_f(L)^{-1} H_v(L) v_t. - Represent lags of u_t and v_t as u_t = M_u^x(L) x_t + M_u^f(L) f_t and v_t = M_v^x(L) x_t + M_v^f(L) f_t. - Aggregate into causal lag polynomials ψ_{˜f}, ψ_x, ψ_z to obtain the simplified expression (equation (14)): f_t = ψ_{˜f}(L) ˜f_t + ψ_x(L) x_t + ψ_z(L) z_t. - Using x_{t+1} = f_t + A u_{t+1} + C v_{t+1} and orthogonality of u_{t+1}, v_{t+1} to current and past observables, forecasting x_{t+1} by regressing on lags of ˜f_t, x_t, and z_t recovers the rational expectation: E[x_{t+1} | {˜f_{t-j}, x_{t-j}, z_{t-j}}_{j=0}^\infty] = E[f_t | {˜f_{t-j}, x_{t-j}, z_{t-j}}_{j=0}^\infty], which is given by equation (8). ### Appendix A.2 — Noisy Forecast Cleaning - When conditions of Theorem 3 fail, cleaned forecasts are the best unbiased forecasts given observable time series and reported forecasts, not ideal rational expectations. - Modify structural VAR to depend on E_t[ε_{t+1}] in general: x_t = sum_{j=1}^m B_j x_{t-j} + A ε_t + C E_t[ε_{t+1}]. - Empirical forecasts take the form: ˜f_t = H_x(L) x_t + H_z(L) z_t + H_u(L) u_t + H_v(L) v_t + H_ζ(L) ζ_t. - Define ξ_t ≡ H_u(L) u_t + H_v(L) v_t + H_ζ(L) ζ_t. Let H_ξ(L) w_{ξ,t} be the Wold decomposition of ξ_t with w_{ξ,t} white noise. - Cleaned forecast: f_t = E[x_{t+1} | Ω] = sum_{j=1}^m B_j x_{t+1-j} + A E[ε_{t+1} | Ω] = sum_{j=1}^m B_j x_{t+1-j} + A E[ε_{t+1} | {ξ_{t-j}}_{j=0}^\infty] = sum_{j=1}^m B_j x_{t+1-j} + A E[ε_{t+1} | w_{ξ,t}]. - Define reduced-form news ˜v_t as ˜v_t ≡ E[ε_{t+1} | w_{ξ,t}] = D_v H_{v0}′ Σ_{w_ξ}^{-1} w_{ξ,t}, where H_{v0} is the contemporaneous coefficient matrix in the H_v(L) polynomial. - Identification condition for ˜v_t via Section 3 method: H_{v0}′ Σ_{w_ξ}^{-1} is diagonal (dimensions of ˜v_t are orthogonal). Interpretation: each structural shock ε_{i,t+1} is associated one-for-one with a noise shock ζ_{i,t}; noise shocks to different dimensions cannot co-vary. - Clarification: this does not preclude agents receiving signals about multiple fundamentals with correlated signal content (e.g., GDP noisy about productivity and labor supply). The requirement is separability of noise shocks by fundamental (noise orthogonality). News-noise equivalence (Chahrour and Jurado, 2018) implies equivalence to structural assumption that news shocks are mutually orthogonal. ### Appendix B — Computational details (B.1 Computing Impulse Responses) - Data notation: [F X], where F = (f_1, f_2, ..., f_T)' are forecasts and X = (x_1, x_2, ..., x_T)' are non-forecast data. - Procedure to compute impulse responses: 1. Calculate an initial reduced-form point estimate ˜B_pt, ˜Σ_pt from a restricted VAR using specification in equation (7); Σ_pt is the variance-covariance matrix of residuals. 2. Simulate N_sim samples of the data using estimates ˜B, ˜Σ. Call these {[˜F^n, ˜X^n]}_{n=1}^{N_sim}. 3. Apply the same estimation as in step 1 to each simulated data set in step 2. Call resulting estimates {˜B_{sim,n}, ˜Σ_{sim,n}}_{n=1}^{N_sim}. 4. Define the mean simulated coefficient: ˜B_{sim,avg} = (1 / N_sim) sum_{n=1}^{N_sim} ˜B_{sim,n}. Approximate mean bias Φ = ˜B_pt − ˜B_{sim,avg}. 5. Bias-corrected reduced-form point estimate (following Kilian (1998)): ˆB_pt = ˜B_pt + Φ. 6. Compute reduced-form residuals for the point estimate: ˆE = [F X]_{-1} − [F X]_{-T} ˆB_pt, where M_{−k} means removing the k-th row from matrix M. Then compute point estimate of reduced-form variance-covariance matrix: ˆΣ_pt = (1 / T) ˆE′ ˆE. 7. Use bias-corrected reduced-form point estimate to construct N_sim samples of the data using estimates ˆB_pt, ˆΣ_pt. Call these {[ˆF^n, ˆX^n]}_{n=1}^{N_sim} (bias-corrected bootstrap of the DGP). 8. Compute a bias-corrected bootstrap of reduced-form coefficients: i. Apply estimation process from step 1 to each {[ˆF^n, ˆX^n]}_{n=1}^{N_sim}. Denote estimates {¯B_{sim,n}}_{n=1}^{N_sim}. ii. Bias-correct these estimates to get final bootstrapped estimates ˆB_{sim,n} = ¯B_{sim,n} + Φ for all n. 9. Compute bootstrapped variance-covariance estimators {ˆΣ_{sim,n}}_{n=1}^{N_sim} using residuals from {[ˆF^n, ˆX^n]}_{n=1}^{N_sim} via method in step 6. 10. To obtain a point estimate and distribution of structural parameters, apply the algorithm in Theorem 2 to ˆB, ˆΣ and each element of {(ˆB_{sim,n}, ˆΣ_{sim,n})}_{n=1}^{N_sim}. *Source: wpiea2023208-print-pdf - 2015. Publisher: ECB Working Paper.* ### 11. To compute confidence intervals for a given impulse response, compute the structural ### To compute confidence intervals for a given impulse response, compute the structural impulse separately for each element of the bootstrap (see next section for elaboration of this step). To form confidence intervals, take percentiles. ### B.2 Labelling shocks in the bootstrap - Problem statement: - Let φ_pt_x(h) denote a structural impulse response at horizon h for type x (either news or surprise) computed using the point estimate. This is an N×N matrix (columns = structural shocks, rows = series). - Let φ_n_x(h) be the equivalent object for the n^th bootstrapped simulation. - Identification is unique only up to sign and ordering. Hence φ_n_x(h) and ˜φ_n_x(h) = D P φ_n_x(h) are observationally equivalent, where P is an N×N permutation matrix and D is an N×N diagonal matrix with entries 1 and −1. - Solution approach (re-ordering and re-signing): - For each n, choose (D*, P*) = arg min_{D∈D, P∈P} sum_{h=1}^H ∥φ_pt_x(h) − D P φ_n_x(h)∥^2, where D and P are the sets of all possible D and P, and ∥·∥_2 is the entry-wise sum of squares. - This ordering procedure minimizes a continuous function of the structural parameters and satisfies Lewis (2021) Theorem 4. - Computational tractability: - Naïvely there are 2^N possible D matrices and N! possible P matrices. - Cast the re-ordering problem as a quadratic assignment (assignment of N objects to N locations with cost phi(i,j)) to reduce complexity: computing the loss of assigning shock j to position i for all i and j requires only N^2 calculations rather than N!. - To incorporate sign flips more efficiently, define for i ∈ 1,...,N; j ∈ 1,...,N; d ∈ {0,1}: - ψ(i,j,d) = sum_{h=1}^H ∥φ_n_x(i)(h) − (−1)^d φ_n_x(j)(h)∥^2 - ψ(i,j) = min(ψ(i,j,0), ψ(i,j,1)) - This requires only 2 N^2 calculations to form a cost matrix which is then passed to a quadratic assignment solver. ### C Additional proofs regarding the structural model - Lemma 1 (solution to the general model (3)): - x_t = sum_{j=1}^{k+1} β_j x_{t−j} + α u_t + γ v_t + (α − ργ) v_{t−1} (equation (15)) - Forecast: f_t = sum_{j=1}^{k+1} β_j x_{t+1−j} + (α − ργ) v_t - Definitions: - α ≡ sum_{j=0}^{∞} Ξ^j (H_z,0 + H_z,1 R_y) R_y^j - γ ≡ H_z,1 K_y + Ξ α - ρ ≡ α R_y α^{-1} - β_j given piecewise: - β_1 = Φ_1 − ρ^{j=1} (as in source) - β_j = Φ_j − ρ Φ_{j−1} for 1 < j ≤ k - β_{k+1} = −ρ Φ_k - Key steps in the proof: - Rewrite model using z_t ≡ [I − sum_{j=1}^k Φ_j L^j] x_t and definitions H_z,0 ≡ Φ_0^{-1} Ψ_{y,0}, H_z,1 ≡ Φ_0^{-1} Ψ_{y,1}. - Use expectations and infinite sums to derive α and express z_t and then x_t. - Use ρ = α R_y α^{-1} to simplify and obtain the stated solution. - Theorem 1 (inclusive form R_y = 0): - Then ρ = 0 and solution simplifies to x_t = sum_{j=1}^{k+1} β_j x_{t−j} + α u_t + γ v_t + α v_{t−1} - With ε_t = u_t + v_{t−1}, this matches equation (1) with mappings β_j = B_j, α = A, γ = C, and m = k + 1. ### D Additional news horizons - Generalization to multiple anticipated horizons: - Decompose structural shocks: - ε_t = ν_{t|t} + ν_{t|t−1} + ν_{t|t−2} + ... + ν_{t|t−k} - Mapping to one-period-ahead notation: ν_{t|t} = u_t, ν_{t|t−1} = v_{t−1}. - Linear model generalization: - x_t = sum_{j=1}^m B_j x_{t−j} + sum_{i=0}^k A_i E_t[ε_{t+i}] - This expands to x_t = sum_{j=1}^m B_j x_{t−j} + sum_{i=0}^k A_i sum_{ℓ=0}^{k−i} ν_{t+i|t−ℓ} - Stack forecasts and series into a VAR(m−1) with block structure (equation (17)): - [f_k^t ... f_1^t x_t]' = sum_{j=1}^{m−1} B_j [f_k^{t−j} ... x^{t−j}]' + A [ν_{t+k|t} ... ν_{t|t}]' - Identification and overidentification: - With longer horizons, matrix A is overidentified; additional forecast horizons help discipline estimation. - Unknowns: matrices A_0,...,A_k have (k+1) n^2 unknowns; Var(ν_{t+j|t}) adds (k+1) n unknowns. - The residual covariance matrix Σ has up to ((k+1) n)^2/2 + (k+1) n independent entries. - Variance adding up gives n additional restrictions: I_n = sum_{j=0}^k Var(ν_{t+j|t}). - Condition when independent entries and restrictions exceed unknowns: - The inequality reduces to (k−1) n^2 + (k−1) n > 0, which holds with equality for k = 1. Thus k > 1 yields overidentification potential. ### E Variance decomposition - Model restated (equation (1) variant): - x_t = sum_{j=1}^m B_j x_{t−j} + A ε_t + C v_t = sum_{j=1}^m B_j x_{t−j} + A u_t + A v_{t−1} + C v_t - Stacked representation and h-step forecast error: - Let X_t be stacked vector of m lags. Then X_t = ˆB X_{t−1} + ˆA u_t + ˆA v_{t−1} + ˆC v_t. - h−period forecast error formulas given in the text (two cases: h = 1; h > 1). - Forecast error variance (MSE) formulas (h = 1 and h > 1) provided exactly as in source. - Contribution of the j^th shock to h−period-ahead variance: - Surprise component = σ^2_{u,j} sum_{s=1}^h ˆB^{h−s} (ˆA_j ˆA'_j) (ˆB')^{h−s} - News component: - For h = 1: σ^2_{v,j} (ˆC_j ˆC'_j) - For h > 1: σ^2_{v,j} (ˆC_j ˆC'_j) + sum_{s=1}^h ˆB^{h−s−1} [ˆA_j ˆA'_j + ˆB(ˆC_j ˆA'_j) + (ˆA_j ˆC'_j) ˆB' + ˆB(ˆC_j ˆC'_j) ˆB'] (ˆB')^{h−s−1} - Here ˆA_j etc. are the j^th columns of the corresponding matrices. - Variance decomposition with confidence intervals (Table 6 summary and key properties): - Table 6: Forecast error variance decomposition, 24 quarters ahead. - The decomposition reports, for each variable (in percent), the fraction of overall forecast error variance attributable to each shock, split into news and surprise components. Totals are shown in the right-hand column. The news and surprise components sum to 100 for each variable. - Numbers in parentheses show the 10th and 90th percentiles from a bootstrap with N_sim = 1000. - Example entries (exactly as reported in the table): - Variable: Gov. spending - News: 4.5 3.6 1.6 5.6 2.2 1.1 24.3 - (0.8, 11.1) (0.5, 10.2) (0.2, 6.5) (2.0, 12.8) (0.4, 8.0) (0.2, 4.6) (16.0, 33.3) - Surprise: 20.4 10.0 3.8 2.3 14.8 12.3 75.7 - (7.3, 35.0) (2.1, 28.4) (1.3, 11.3) (0.3, 12.2) (4.1, 34.2) (3.8, 27.4) (66.7, 84.0) - Total: 25.6 15.8 6.4 9.7 18.5 14.5 100.0 - (11.5, 42.5) (5.3, 33.6) (2.6, 15.1) (3.9, 20.7) (8.4, 35.7) (5.4, 29.2) (100.0, 100.0) - Variable: Output - News: 7.0 2.2 3.7 4.3 2.1 2.3 26.3 - (2.6, 14.2) (0.4, 6.9) (0.7, 10.3) (1.5, 8.8) (0.5, 6.3) (0.4, 6.9) (18.5, 34.6) - Surprise: 8.0 6.7 19.5 23.9 4.6 4.3 73.7 - (1.9, 19.4) (2.2, 15.9) (11.5, 29.0) (10.2, 37.4) (1.8, 12.1) (1.0, 12.6) (65.4, 81.5) - Total: 15.9 9.9 24.4 28.3 8.3 7.7 100.0 - (7.6, 29.0) (4.3, 19.8) (14.9, 35.2) (14.9, 42.7) (4.0, 14.6) (2.8, 17.4) (100.0, 100.0) - Variable: Taxes - News: 4.9 3.3 1.9 1.7 1.8 2.3 19.4 - (3.0, 8.2) (1.0, 7.9) (0.5, 6.8) (0.5, 4.4) (0.5, 5.3) (0.4, 6.9) (13.7, 26.7) - Surprise: 12.5 4.7 11.6 30.3 7.6 7.4 80.6 - (7.5, 23.3) (1.2, 12.3) (5.8, 18.5) (15.9, 44.2) (2.0, 17.4) (2.7, 17.7) (73.3, 86.3) - Total: 18.2 8.6 14.5 32.1 10.1 10.9 100.0 - (12.0, 29.2) (3.7, 17.6) (8.3, 21.6) (18.8, 46.2) (3.8, 20.5) (4.8, 20.4) (100.0, 100.0) - Additional rows for 3-month interest rate, Housing starts, Inflation presented analogously in the table with exact numeric entries and percentile ranges. - Interpretation: the table decomposes forecast error variance at 24 quarters ahead across identified shocks, separately reporting news and surprise contributions and bootstrap percentile ranges. ### F Hidden states - Generalization when some dimensions of x_t are unobserved: - Observation equation: y_t = x_t + G u_t + G v_{t−1} + H v_t (equation (18)) - Normalization: hidden states normalized to obey equations (1) and (18). - Relations between observations and forecasts: - y_t = f_{t−1} + (A + G) u_t + (C + H) v_t - Forecasts f_t = E_t[y_{t+1}] = E_t[x_{t+1}] + G v_t - After algebra and stacking, one obtains a VARMA(m−1,m) (equation (19)): - [f_t; y_t] = sum_{j=1}^{m−1} B_j [f_{t−j}; y_{t−j}] + sum_{j=0}^m A_j [v_{t−j}; u_{t−j}] - Identification challenges and prospects: - A_0 now contains additional matrices G and H that complicate identification. - The MA structure introduces additional moment conditions: Σ_{ij} = A_i diag(D_v^2, 0, 0, D_u^2) A'_j. - Blockwise equations for Σ_{0,11}, Σ_{0,21}, Σ_{0,22} are provided, involving (B_1, C, A, G, H) and D_v^2, D_u^2. - Additional cross-covariances between MA terms (Σ_{01} etc.) provide further linear restrictions. - Together with D_u^2 + D_v^2 = I, these give as many linear restrictions as unknowns — sufficient conditions for identification (but not a constructive proof analogous to Theorem 2). Estimation requires numerical decomposition. ### G Robustness - Six alternative specifications considered to assess model sensitivity: - These span alternative lag structures (2 or 4 lags), alternative creation of proxy expectations series, re-running on a post-Volcker disinflationary era sample, and alternative measures of public spending and real activity. - Alternate specifications list (as introduced): 1. Baseline. The baseline specification in the text 2. 2 Lags. Baseline but with 2 lags in the VAR. - (Additional alternative specifications are listed in the source beyond item 2.) *Italicized source attribution: Excerpt from wpiea2023208-print-pdf (canonical URL provided in pipeline).* ### 3.4 Lags.Baseline but with 4 lags in the VAR. ### 3.4 Lags.Baseline but with 4 lags in the VAR. ### Robustness specifications and data variants - Alternate specifications considered: - 4 lags (baseline but with 4 lags in the VAR). - Literal forecasts: Uses the collected forecasts without any machine learning or other processing. - Short Sample: Data starts in March 1988. - Inc. Federal Spending: Replaces government spending with Federal Government Spending. - Inc. Unemployment: Replaces housing starts with unemployment. - 2 Lags (implicit via comparison). - Only common variables are shown in comparisons; specifications that replace a variable will be missing from one panel. ### Variance decomposition: main summary measure and results - Summary measure: variance in each variable attributable to news and surprise (24-quarter horizon variance decomposition). - Presentation details: - Points show 50th percentile; error bars show the 10th to 90th percentile range from a bootstrap simulation with Nsim = 1000 draws. - Main findings: - The split between news and surprise shocks in driving outcomes is very stable across specifications. - For almost all variables, a single value fits within the confidence interval for all news shocks (likewise for surprises). - Surprise shocks account for around three quarters of the variance for most variables except inflation, where news plays a more important role (and in some specifications, the dominant role). ### Lag selection (Akaike Information Criterion) - Four ways to calculate the Akaike Information Criterion were examined: - “F-form”: uses a likelihood for the stacked VAR in equation (7). - “X-form”: uses just the residuals from the errors on the non-forecast variables. - For each form, likelihood computed under: - “restricted”: zero restrictions on the Bj matrices are imposed. - “unrestricted”: restrictions are not imposed. - Note: The X-form AIC lines are equivalent across restrict/unrestrict because the restrictions do not bind on the lower half of the stacked coefficient matrices. - Result: - In all cases, the one-lag specification is strongly preferred. ### Shock validation exercises: fiscal and monetary - Fiscal shock validation (Figure 16): - The solid line is the median cumulative output response for an unanticipated surprise fiscal expansion shock from a bootstrap simulation with Nsim = 1000 replications. - Dashed and dotted lines are the 10th−90th and 25th−75th percentile ranges, respectively. - Points show cumulative output responses, μhY, implied by estimated tax and spending responses if multipliers were those in the literature. - Monetary shock validation (Figures 17–20): - Comparative impulse responses to a monetary policy shock from the baseline versus various literature sources (Bauer−Swanson, Jarocinski−Karadi, Romer−Romer variants, etc.). - To match samples and specification: - One-lag VAR, four-lag VAR, linear projection (one lag), and linear projection (four lags) versions are reported as appropriate. - Impulse responses are computed from a Cholesky decomposition with the monetary shock ordered first. - All impulses are scaled such that the initial interest rate impulse is 100 basis points. - Presentation details: - Solid line labeled “Baseline” shows median; shaded area shows 10th−90th percentile ranges from a bootstrap with Nsim = 1000 replications. - Variables shown in monetary comparisons include: 3−month interest rate, Housing starts, Inflation, Government consumption, Output, Taxes. - Comparisons include orthogonalized versions and HFI variants where applicable. *Identifying News Shocks from Forecasts — Working Paper No. WP/2023/208* --- _Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023208-print-pdf.pdf_