## wp18152 ## Source details **Canonical URL:** [wp18152](https://www.imf.org/-/media/files/publications/wp/2018/wp18152.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2018/wp18152.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2018/wp18152.pdf.json) --- ### Introduction - Purpose: Improve estimation of potential output and the output gap by extending a structural multivariate filter to include a monetary policy block (MPMOD). - Builds on Alichi and others (2017) combining a Phillips curve, dynamic Okun’s law, and an equation linking the output gap to capacity utilization. - Claimed advantage: the structural multivariate filter incorporates structural relationships between macroeconomic fundamentals and is flexible to incorporate additional information (short- and long-term interest rates and a monetary policy reaction function). ### MPMOD: model structure and key equations - Core macro variables included: real GDP, the unemployment rate, consumer price inflation, the Fed’s survey of manufacturing capacity utilization, the one-year government bond yield, and the 10-year government bond yield. - Output gap definition (equation numbering preserved): - (1) 푦푦�푡푡 = 푦푦푡푡 − 푦푦푡푡. - Real GDP stochastic process and shocks: - Equations (2)–(4) describe GDP with a persistent cyclical component and two permanent-type shocks (level shift and persistent deviation from long-term growth). - Output gap dynamics (equation (4)) depend on contemporaneous and lagged one-year real interest rate gap (푟푟푟푟�푡푡1푌푌), shocks to trend potential growth (휖휖푔푔푦푦�,푡푡), and shocks to level of potential (휖휖푦푦�,푡푡). - Monetary policy reaction function (equation (5)): - One-year nominal interest rate (푟푟푟푟푡푡1푌푌) responds to lagged one-year rate, perceived long-run rate, expected inflation, inflation deviation from perceived target, and the output gap. - Two shocks in the reaction function: 휖휖푇푇푠푠1푌푌,푡푡 (unexplained interest rate movements) and −훼훼4휖휖휋휋푇푇푇푇푇푇,푡푡 (captures transition from a high to low inflation regime in the early 1980’s and 1990’s). - Real one-year rate defined (equation (6)): 푟푟푟푟푡푡1푌푌 = 푟푟푟푟푡푡1푌푌 − 휋휋푡푡𝑒𝑒. - Expected inflation modeled (equation (7)) as a linear combination of model-consistent expected inflation and lagged actual inflation. - Equilibrium real interest rate follows slow AR process (equations (8)–(9)) reverting to long-run steady-state 푟푟푟푟���𝑆𝑆𝑆𝑆 and subject to shock 휖휖푇푇푇푇1푌푌,푡푡. - Ten-year yield decomposition (equation (10)): sum of average expected future one-year rates, a term premium (휎휎푡푡푇푇푒푒 푇푇푇푇), and a transitory shock (휖휖푇푇푠푠10 푌푌). - Term premium dynamics (equation (11)): slow autoregressive process toward long-run steady σ푇푒푇푇,푆푆푆푆, subject to shock 휖휖휎휎푡푡푇푇푇푇푇푇,푡푡. - Use of Consensus forecasts: Consensus CPI and GDP forecasts are modeled as MPMOD forecasts plus measurement error; variances of measurement errors are based on historical Consensus forecast errors. - Data and estimation sample: annual data covering the period from 1980 to 2016; 5-year-ahead CPI forecasts used as perceived long-term inflation target; MPMOD results presented through 2026 in figures. ### Estimation results and comparisons - MPMOD versus CBO (Figure 1): - MPMOD and CBO output gaps are highly correlated; MPMOD estimates are larger on average by about 0.8 percentage points. - Potential growth estimates are highly correlated; CBO implies higher potential growth in the early 1980s and late 1990s. - NAIRU and unemployment: - Both models’ NAIRU estimates trend down; CBO’s estimates slightly lower at end of sample and include a temporary increase during the global financial crisis. - Differences have limited effect on recent potential growth or gaps. - Sensitivity (Figure 2): - Using CBO NAIRU as observable leaves MPMOD output gap and potential growth almost unchanged. - Real-time performance (Figure 3): - MPMOD produces more robust real-time estimates than a univariate HP filter with Λ=6.25 (smoothing parameter of 6.25 in annual data is equivalent to using 1600 for quarterly data). - MPMOD with monetary policy block shows generally smaller real-time revisions than MPMOD without the monetary policy block. - HP filter end-point problem: shifts in output growth are treated as permanent by the HP filter, leading to large revisions unless manually adjusted. - Comparison with multivariate filter without monetary block (Figure 4): - MPMOD with monetary block estimates somewhat deeper and more prolonged recessions, especially after the global financial crisis. - The monetary block (reaction function and term structure) provides information about the output gap; large policy easing is explained by a large estimated output gap. ### Properties and historical dynamics (Figure 5 highlights) - Perceived inflation target: - Based on Consensus Economics 5-year-ahead expectations available 1989–2017; 1980 uses 5-year-ahead CBO inflation forecast; 1981–1988 treated as unobservable and computed via Kalman filter. - Perceived inflation target decreased in the 1980’s and 1990’s reflecting Volker disinflation. - Historical narrative: - Pre-global financial crisis: U.S. operated above potential, inflation rose, monetary policy tightened moderately. - Global financial crisis (late 2008): deep recession, unemployment rose sharply, capacity utilization dropped, monetary and fiscal policy became markedly expansionary. - Post-great recession: GDP grew faster than potential and the negative output gap narrowed; inflation remained below target; monetary policy remained accommodative. - MPMOD estimates indicate the U.S. output gap is nearly closed and unemployment almost at NAIRU, with output gap projected to go positive and unemployment to fall below NAIRU in the next few years. ### Impulse response findings (Figures 6–9) - Demand shock (Figure 6): one-standard-deviation positive demand shock to output (휖휖푦푦,푡푡) - By construction, no change in potential output. - Inflation increases by nearly 0.2 percentage points. - Short-term interest rate rises by more than 0.3 percentage points. - Long-term interest rate increases by 0.1 percentage points. - Level-of-potential shock (Figure 7): one-standard-deviation positive shock to level of potential output - Initially supply rises more than demand; output gap becomes negative. - Inflation falls (higher productivity lowers marginal costs). - Monetary policy loosens; short-term interest rate decreases by 0.04 percentage points. - Growth-rate-of-potential shock (Figure 8): one-standard-deviation positive shock to 휖휖푔푔푦푦�,푡푡 - Potential output growth increases for several periods before fading. - Consumers and firms raise consumption and investment; a positive output gap opens up. - Inflation rises and monetary policy tightens. - Inflation-target shock (Figure 9): one-standard-deviation negative shock to the inflation target by one percentage point - Monetary policy tightens. - Tighter policy reduces real output temporarily; output eventually returns to unchanged potential and inflation falls to the new target. ### Conclusions and extensions - Main conclusion: adding a monetary policy block (MPMOD) to the multivariate filter provides a more structural description of output gap dynamics, allows monetary policy to interact with demand and supply shocks, and yields deeper and more prolonged estimates of the post-global-financial-crisis recession compared with models without the monetary block. - Real-time reliability: MPMOD produces more reliable real-time estimates than an HP filter; the monetary policy block further improves reliability. - Ongoing extensions: variants adding a production function and labor disaggregation (participation rate, NAIRU, labor force population), partial hysteresis in the labor market, and incorporation of the financial cycle are underway (Alichi and others, 2018a–2018c). ### Appendix A — Model details (selected equations and links) - Output and potential dynamics: - Equation (16): 푦푦� 푡푡 = 푦푦 푡푡 − 푦푦 푡푡 (output gap is deviation of log real GDP from potential). - Equations (17)–(19): real GDP dynamics, potential growth, and output gap as functions of interest rate gaps and three shock types: level shocks (휖휖 푦푦 � ,푡푡), potential-growth shocks (휖휖 푔푔 푦푦 � ,푡푡), and demand shocks (휖휖 푦푦 � ,푡푡). - Equations (20)–(21): realized growth measures 푔푔 푡푡 = 푦푦 푡푡 − 푦푦 푡푡−1 and 푔푔 푦푦 � ,푡푡 = 푦푦 푡푡 − 푦푦 푡푡−1. - Inflation and expectations: - Phillips curve (22): 휋휋 푡푡 = 휆휆 1 퐸퐸 푡푡 휋휋 푡푡+1 + (1−휆휆 1 )휋휋 푡푡−1 + 휆휆 3 푦푦� 푡푡 + 휖휖 휋휋,푡푡 − 휆휆 4 휖휖 푦푦 � ,푡푡. - Time-varying inflation target (23): 휋휋 푡푡 푇푇푇푇푇푇 = 휋휋 푡푡−1 푇푇푇푇푇푇 + 휖휖 휋휋 푇푇푇푇푇푇 ,푡푡. - Expectations measure (24): 휋휋 푡푡 푒푒 = 훽훽 1 퐸퐸 푡푡 휋휋 푡푡+1 + (1−훽훽 1 )휋휋 푡푡−1. - Real one-year interest rate (25): 푟푟푟푟 푡푡 1푌푌 = 푟푟푟푟 푡푡 1푌푌 − 휋휋 푡푡 푒푒. - Policy rule (26): 푟푟푟푟 푡푡 1푌푌 = 훼훼 1 푟푟푟푟 푡푡−1 1푌푌 + (1−훼훼 1 )[푟푟푟푟��� 푡푡 1푌푌 + 휋휋 푡푡 푒푒 + 훼훼 2 (휋휋 푡푡 − 휋휋 푡푡 푇푇푇푇푇푇) + 훼훼 3 푦푦� 푡푡] + 휖휖 푇푇푠푠 1푌푌 ,푡푡 − 훼훼 4 휖휖 휋휋 푇푇푇푇푇푇 ,푡푡. - Equilibrium real interest rate AR(1) reversion to 푟푟푟푟��� 푆푆푆푆 (equations (27)–(28)). - Term structure and capacity utilization: - 10-year rate (29): 푟푟푟푟 푡푡 10 푌푌 = ∑ 푇푇푠푠 푖푖 1푌푌푡푡+9 푖푖=푡푡 10 + 휎휎 푡푡 푇푇푒푒 푇푇푇푇 + 휖휖 푇푇푠푠 10 푌푌. - Term premium AR(1) (30): 휎휎 푡푡 푇푇푒푒 푇푇푇푇 = 휌휌 휎휎 푇푇푇푇푇푇푇푇 휎휎 푡푡−1 푇푇푒푒 푇푇푇푇 + (1−휌휌 휎휎 푇푇푇푇푇푇푇푇) 휎휎 푇푇푒푒 푇푇푇푇,푆푆푆푆 + 휖휖 휎휎 푡푡푇푇푇푇푇푇 ,푡푡. - Unemployment and Okun’s law (31)–(34): equilibrium unemployment 푢푢 푡푡, time variation, shocks, and linking unemployment gap 푢푢� 푡푡 to output gap 푦푦� 푡푡 (equation (34) explicitly specifies Okun’s law). - Capacity utilization (35)–(38): capacity utilization gap 푐푐̂ 푡푡 = 푐푐 푡푡 − 푐푐 푡푡 and equilibrium capacity utilization dynamics; (38) links capacity utilization gap to output gap: 푐푐̂ 푡푡 = 휅휅 푦푦� 푡푡 + 휖휖 푐푐 ̂ ,푡푡. ### Appendix A — Use of Consensus forecasts to improve end-of-sample identification - Forecast-augmentation: - Equation (39): 휋휋 푡푡+푗푗 퐶퐶 = 휋휋 푡푡+푗푗 + 휖휖 휋휋 푐푐 ,푡푡+푗푗 , j = 5. - Equation (40): 푔푔 푐푐 ,푡푡+푗푗 = 푔푔 푡푡+푗푗 + 휖휖 푔푔 푐푐 ,푡푡 , j = 1,...,5. - Practical role: Consensus Economics forecasts for up to five years help identify shocks and improve end-of-sample estimates; non-zero variances of the error terms allow survey data to influence but not fully override model expectations. ### Appendix B — Parameters, shock distributions, and data sources - Tables provided: - Table B1: Estimation Results for Structural Model Parameters (prior and posterior columns; Mode and Standard Error reported). - Table B2: Estimation Results for Shock Distributions (prior and posterior columns; Mode and Standard Error reported). Note: “휎휎 denotes standard deviation” and “cf denotes consesus forecast”. - Table B3: Data Sources lists: - Inflation Expectations: Consensus Economics; CBO - Gross Domestic Product Growth Expectations (Constant Prices): Consensus Economics - Gross Domestic Product (Constant Prices): U.S Bureau of Economic Analysis - CPI Inflation: U.S Bureau of Labor Statistics - Unemployment Rate: U.S Bureau of Labor Statistics - Capacity Utilization: Federal Reserve - One-Year Treasury Constant Maturity Rate: Federal Reserve - 10-Year Treasury Constant Maturity Rate: Federal Reserve ### Appendix C — Forecast accuracy diagnostics - Figure C1: RMSE of Consensus Forecasts (figure provided; Source: Authors’ estimates). ### Major cited works (selected) - Working papers and articles cited include multiple IMF Working Papers by Alichi et al. (2015, 2017, 2018a, 2018b, 2018c), Benes et al. (2010), Blagrave et al. (2015), Borio, Disyatat and Juselius (2016), CBO (2011, 2014), Clark, Laxton and Rose (1995, 2001), Clinton et al. (2015), Debelle and Laxton (1997), Freedman et al. (2009), Hodrick and Prescott (1997), Isard et al. (1999), Juillard et al. (2007), Laxton et al. (1993, 1993a, 1993b), Laxton and Tetlow (1992), Okun (1962), Turner et al. (2016), Argov et al. (2007), among others listed in the REFERENCES section. *Source: wp18152 - References (pdf).* ### References ............................................................................................................. ### References ### Appendix - A. MPMOD Equations ........................................................................................................ 23 - B. MPMOD Parameters, Shock Distribution, and Data Source ......................................... 28 - C. RMSE of Consensus Forecasts ...................................................................................... 31 ### Figures - Figure 1. Comparison of MPMOD Estimates with CBO Estimates ......................................... 8 - Figure 2. Sensitivity Analysis ................................................................................................... 9 - Figure 3. Real-time Estimates and Rolling Forecasts ............................................................. 10 - Figure 4. Comparison of Multivariate Filter Results with and without the Monetary Policy Block ....................................................................................................................................... 13 - Figure 5. MPMOD Detailed Results ....................................................................................... 14 - Figure 6. Impulse Response for a 1-Standard-Deviation Positive Shock to the Output Gap . 17 - Figure 7. Impulse Response for a 1-Standard-Deviation Positive Shock to the Level of Potential Output ...................................................................................................................... 17 - Figure 8. Impulse Response for a 1-Standard-Deviation Positive Shock to the Growth Rate of Potential Output ...................................................................................................................... 18 - Figure 9. Impulse Response for a 1-Standard-Deviation Negative Shock to the Inflation Target ...................................................................................................................................... 18 *Source: wp18152 - References (pdf).* ### Appendix Figures ### wp18152 - Appendix Figures ### Introduction - Purpose: Improve estimation of potential output and the output gap by extending a structural multivariate filter to include a monetary policy block (MPMOD). - Builds on Alichi and others (2017) which combined a Phillips curve, dynamic Okun’s law, and an equation linking the output gap to capacity utilization. - Key claimed advantage: the structural multivariate filter incorporates structural relationships between macroeconomic fundamentals and is flexible to incorporate additional information (here, short- and long-term interest rates and a monetary policy reaction function). ### MPMOD: model structure and key equations - Core macro variables included: real GDP, the unemployment rate, consumer price inflation, the Fed’s survey of manufacturing capacity utilization, the one-year government bond yield, and the 10-year government bond yield. - Output gap definition (equation numbering preserved): (1) 푦푦�푡푡 = 푦푦푡푡 − 푦푦푡푡. - Real GDP stochastic process and shocks: equations (2)–(4) describe GDP with a persistent cyclical component and two permanent-type shocks (level shift and persistent deviation from long-term growth). - Output gap dynamics (equation (4)) depend on contemporaneous and lagged one-year real interest rate gap (푟푟푟푟�푡푡1푌푌), shocks to trend potential growth (휖휖푔푔푦푦�,푡푡), and shocks to level of potential (휖휖푦푦�,푡푡). - Monetary policy reaction function (equation (5)): - One-year nominal interest rate (푟푟푟푟푡푡1푌푌) responds to lagged one-year rate, perceived long-run rate, expected inflation, inflation deviation from perceived target, and the output gap. - Two shocks in the reaction function: 휖휖푇푇푠푠1푌푌,푡푡 (unexplained interest rate movements) and −훼훼4휖휖휋휋푇푇푇푇푇푇,푡푡 (captures transition from a high to low inflation regime in the early 1980’s and 1990’s). - Real one-year rate defined (equation (6)): 푟푟푟푟푡푡1푌푌 = 푟푟푟푟푡푡1푌푌 − 휋휋푡푡𝑒𝑒. - Expected inflation modeled (equation (7)) as a linear combination of model-consistent expected inflation and lagged actual inflation. - Equilibrium real interest rate follows slow AR process (equations (8)–(9)) reverting to long-run steady-state 푟푟푟푟���𝑆𝑆𝑆𝑆 and subject to shock 휖휖푇푇푇푇1푌푌,푡푡. - Ten-year yield decomposition (equation (10)): sum of average expected future one-year rates, a term premium (휎휎푡푡푇푇푒푒 푇푇푇푇), and a transitory shock (휖휖푇푇푠푠10 푌푌). - Term premium dynamics (equation (11)): slow autoregressive process toward long-run steady σ푇푒푇푇,푆푆푆푆, subject to shock 휖휖휎휎푡푡푇푇푇푇푇푇,푡푡. - Use of Consensus forecasts: Consensus CPI and GDP forecasts are modeled as MPMOD forecasts plus measurement error; variances of measurement errors are based on historical Consensus forecast errors. - Data and estimation sample: annual data covering the period from 1980 to 2016; 5-year-ahead CPI forecasts used as perceived long-term inflation target; MPMOD results presented through 2026 in figures. ### Estimation results and comparisons - Figure 1 comparison (MPMOD vs CBO): - MPMOD and CBO output gaps are highly correlated; MPMOD estimates are larger on average by about 0.8 percentage points. - Both models’ potential growth estimates are highly correlated; CBO implies higher potential growth in the early 1980s and late 1990s. - NAIRU: both models’ estimates trend down; CBO’s estimates slightly lower at end of sample and include a temporary increase during the global financial crisis; differences have limited effect on recent potential growth or gaps. - Sensitivity: using CBO NAIRU as observable leaves MPMOD output gap and potential growth almost unchanged (Figure 2). - Real-time performance (Figure 3): - MPMOD produces more robust real-time estimates than a univariate HP filter with Λ=6.25 (smoothing parameter of 6.25 in annual data is equivalent to using 1600 for quarterly data). - MPMOD with monetary policy block shows generally smaller real-time revisions than MPMOD without the monetary policy block. - HP filter exhibits the end-point problem: when there is a shift in output growth the HP filter treats it as permanent and forecasts accordingly, leading to large revisions unless manually adjusted. - Comparison with multivariate filter without monetary block (Figure 4): - MPMOD with monetary block estimates somewhat deeper and more prolonged recessions, especially after the global financial crisis. - The monetary block (reaction function and term structure) provides information about the output gap; large policy easing is explained by a large estimated output gap. ### Properties and historical dynamics (Figure 5 highlights) - Perceived inflation target: based on Consensus Economics 5-year-ahead expectations available 1989–2017; 1980 uses 5-year-ahead CBO inflation forecast; 1981–1988 treated as unobservable and computed via Kalman filter; perceived inflation target decreased in the 1980’s and 1990’s reflecting Volker disinflation. - Historical narrative: - Pre-global financial crisis: U.S. operated above potential, inflation rose, monetary policy tightened moderately. - Global financial crisis (late 2008): deep recession, unemployment rose sharply, capacity utilization dropped, monetary and fiscal policy became markedly expansionary. - Post-great recession: GDP grew faster than potential and the negative output gap narrowed; inflation remained below target; monetary policy remained accommodative. - MPMOD estimates indicate the U.S. output gap is nearly closed and unemployment almost at NAIRU, with output gap projected to go positive and unemployment to fall below NAIRU in the next few years. ### Impulse response findings (Figures 6–9) - Demand shock (Figure 6): a one-standard-deviation positive demand shock to output (휖휖푦푦,푡푡) - By construction, no change in potential output. - Inflation increases by nearly 0.2 percentage points. - Short-term interest rate rises by more than 0.3 percentage points. - Long-term interest rate increases by 0.1 percentage points. - Level-of-potential shock (Figure 7): one-standard-deviation positive shock to level of potential output - Initially supply rises more than demand; output gap becomes negative. - Inflation falls (higher productivity lowers marginal costs). - Monetary policy loosens; short-term interest rate decreases by 0.04 percentage points. - Growth-rate-of-potential shock (Figure 8): one-standard-deviation positive shock to 휖휖푔푔푦푦�,푡푡 - Potential output growth increases for several periods before fading. - Consumers and firms raise consumption and investment; a positive output gap opens up. - Inflation rises and monetary policy tightens. - Inflation-target shock (Figure 9): one-standard-deviation negative shock to the inflation target by one percentage point - Monetary policy tightens. - Tighter policy reduces real output temporarily; output eventually returns to unchanged potential and inflation falls to the new target. ### Conclusions and extensions - Main conclusion: adding a monetary policy block (MPMOD) to the multivariate filter provides a more structural description of output gap dynamics, allows monetary policy to interact with demand and supply shocks, and yields deeper and more prolonged estimates of the post-global-financial-crisis recession compared with models without the monetary block. - Real-time reliability: MPMOD produces more reliable real-time estimates than an HP filter; the monetary policy block further improves reliability. - Ongoing extensions: variants adding a production function and labor disaggregation (participation rate, NAIRU, labor force population), partial hysteresis in the labor market, and incorporation of the financial cycle are underway (Alichi and others, 2018a–2018c). *Source: wp18152 - Appendix Figures* ### REFERENCES ### wp18152 - REFERENCES ### Major cited works - Lists working papers and articles used in the chapter, including multiple IMF Working Papers by Alichi et al. (2015, 2017, 2018a, 2018b, 2018c), Benes et al. (2010), Blagrave et al. (2015), Borio, Disyatat and Juselius (2016), CBO (2011, 2014), Clark, Laxton and Rose (1995, 2001), Clinton et al. (2015), Debelle and Laxton (1997), Freedman et al. (2009), Hodrick and Prescott (1997), Isard et al. (1999), Juillard et al. (2007), Laxton et al. (1993, 1993a, 1993b), Laxton and Tetlow (1992), Okun (1962), Turner et al. (2016), Argov et al. (2007), and others cited in the REFERENCES section. ### Appendix A — MPMOD: model structure and equations - Core data used: GDP, CPI, the unemployment rate, the capacity utilization rate in the manufacturing sector, and short- and long-term interest rates; data measured at annual frequency; Consensus forecasts of annual CPI inflation and real GDP growth are incorporated. - Output gap definition: - Equation (16): 푦푦� 푡푡 = 푦푦 푡푡 − 푦푦 푡푡 (output gap is deviation of log real GDP from potential) - Stochastic process for output and potential: - Equations (17)–(19) define real GDP dynamics, potential growth, and the output gap as functions of contemporaneous and lagged one-year real interest rate gap and three shock types: level shocks (휖휖 푦푦 � ,푡푡), potential-growth shocks (휖휖 푔푔 푦푦 � ,푡푡), and demand shocks (휖휖 푦푦 � ,푡푡). - Equations (20)–(21) define realized growth measures: 푔푔 푡푡 = 푦푦 푡푡 − 푦푦 푡푡−1 and 푔푔 푦푦 � ,푡푡 = 푦푦 푡푡 − 푦푦 푡푡−1. - Graphical interpretation: Figure A1 (Shocks to the Level and Growth Rate of Potential Output, and to the Output Gap) illustrates: - Level shock: a permanent one-off shift in potential (dashed blue line). - Growth shock: a temporary increase in growth rate with effects that fade (dashed red line). - Output-gap shock: temporary deviation of output from potential (dashed green line). ### Appendix A — Inflation, expectations, and policy rule - Phillips curve linking inflation to the output gap: - Equation (22): 휋휋 푡푡 = 휆휆 1 퐸퐸 푡푡 휋휋 푡푡+1 + (1−휆휆 1 )휋휋 푡푡−1 + 휆휆 3 푦푦� 푡푡 + 휖휖 휋휋,푡푡 − 휆휆 4 휖휖 푦푦 � ,푡푡 - Last term captures productivity shocks that lower marginal cost and therefore reduce inflation. - Time-varying inflation target modeled as a random walk: - Equation (23): 휋휋 푡푡 푇푇푇푇푇푇 = 휋휋 푡푡−1 푇푇푇푇푇푇 + 휖휖 휋휋 푇푇푇푇푇푇 ,푡푡 - Measure of inflation expectations used to compute real returns: - Equation (24): 휋휋 푡푡 푒푒 = 훽훽 1 퐸퐸 푡푡 휋휋 푡푡+1 + (1−훽훽 1 )휋휋 푡푡−1 - Real one-year interest rate: - Equation (25): 푟푟푟푟 푡푡 1푌푌 = 푟푟푟푟 푡푡 1푌푌 − 휋휋 푡푡 푒푒 - Policy (interest rate reaction) function: - Equation (26): 푟푟푟푟 푡푡 1푌푌 = 훼훼 1 푟푟푟푟 푡푡−1 1푌푌 + (1−훼훼 1 )[푟푟푟푟��� 푡푡 1푌푌 + 휋휋 푡푡 푒푒 + 훼훼 2 (휋휋 푡푡 − 휋휋 푡푡 푇푇푇푇푇푇) + 훼훼 3 푦푦� 푡푡] + 휖휖 푇푇푠푠 1푌푌 ,푡푡 − 훼훼 4 휖휖 휋휋 푇푇푇푇푇푇 ,푡푡 - Equilibrium real interest rate modeled as an autoregressive process reverting to long-run steady state: - Equations (27)–(28) specify decomposition and AR(1) reversion to 푟푟푟푟��� 푆푆푆푆. ### Appendix A — Term structure and capacity utilization - 10-year government bond interest rate modeled as: - Equation (29): 푟푟푟푟 푡푡 10 푌푌 = ∑ 푇푇푠푠 푖푖 1푌푌푡푡+9 푖푖=푡푡 10 + 휎휎 푡푡 푇푇푒푒 푇푇푇푇 + 휖휖 푇푇푠푠 10 푌푌 - Time-varying term premium: - Equation (30): 휎휎 푡푡 푇푇푒푒 푇푇푇푇 = 휌휌 휎휎 푇푇푇푇푇푇푇푇 휎휎 푡푡−1 푇푇푒푒 푇푇푇푇 + (1−휌휌 휎휎 푇푇푇푇푇푇푇푇) 휎휎 푇푇푒푒 푇푇푇푇,푆푆푆푆 + 휖휖 휎휎 푡푡푇푇푇푇푇푇 ,푡푡 - Unemployment (NAIRU) dynamics and Okun’s law: - Equations (31)–(34) define equilibrium unemployment 푢푢 푡푡, its time variation, shocks, and an Okun’s law relation linking unemployment gap 푢푢� 푡푡 to the output gap 푦푦� 푡푡 (equation (34) explicitly specifies the Okun’s law relationship). - Capacity utilization incorporation: - Equations (35)–(38): capacity utilization gap 푐푐̂ 푡푡 = 푐푐 푡푡 − 푐푐 푡푡 and a time-varying equilibrium capacity utilization 푐푐̅ 푡푡 with growth 푔푔 푐푐 ̅ ,푡푡 and shocks; equation (38) links capacity utilization gap to output gap: 푐푐̂ 푡푡 = 휅휅 푦푦� 푡푡 + 휖휖 푐푐 ̂ ,푡푡. ### Appendix A — Use of Consensus forecasts to improve end-of-sample identification - Forecast-augmentation for model-consistent forward expectations: - Equation (39): 휋휋 푡푡+푗푗 퐶퐶 = 휋휋 푡푡+푗푗 + 휖휖 휋휋 푐푐 ,푡푡+푗푗 , j = 5 - Equation (40): 푔푔 푐푐 ,푡푡+푗푗 = 푔푔 푡푡+푗푗 + 휖휖 푔푔 푐푐 ,푡푡 , j = 1,...,5 - Practical role: Consensus Economics forecasts for up to five years help identify shocks and improve end-of-sample estimates; non-zero variances of the error terms allow survey data to influence but not fully override model expectations. ### Appendix B — Parameters, shock distributions, and data sources - Tables provided: - Table B1: Estimation Results for Structural Model Parameters (prior and posterior columns; Mode and Standard Error reported). - Table B2: Estimation Results for Shock Distributions (prior and posterior columns; Mode and Standard Error reported). Note: “휎휎 denotes standard deviation” and “cf denotes consesus forecast”. - Table B3: Data Sources lists: - Inflation Expectations: Consensus Economics; CBO - Gross Domestic Product Growth Expectations (Constant Prices): Consensus Economics - Gross Domestic Product (Constant Prices): U.S Bureau of Economic Analysis - CPI Inflation: U.S Bureau of Labor Statistics - Unemployment Rate: U.S Bureau of Labor Statistics - Capacity Utilization: Federal Reserve - One-Year Treasury Constant Maturity Rate: Federal Reserve - 10-Year Treasury Constant Maturity Rate: Federal Reserve ### Appendix C — Forecast accuracy diagnostics - Figure C1: RMSE of Consensus Forecasts (figure provided; Source: Authors’ estimates). *Source: wp18152 - REFERENCES (IMF Working Paper content provided).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18152.pdf_