## G3MOD: A Multi-Country Global Forecasting Model

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---

### 1. Introduction — purpose, design, and intended use
- Purpose and motivation:
  - Semi-structural gap-trend Quarterly Projection Models (QPMs) are widely used by central banks and ministries of finance in emerging market and developing countries.
  - Critical challenge: incorporate external developments—trading partner demand, financial markets, and commodity prices—into baseline predictions and scenario analysis when institutions have nascent modeling capacity or limited staff resources.
  - Many institutions rely on available global forecasts (for example, IMF World Economic Outlook (WEO)), which are typically annual and not expressed in gap-trend (QPM) terms.
- G3MOD objectives and design:
  - Straightforward, quickly implemented variant of standard global projection models to support inclusion of global forecasts in countries’ economic analysis and forecasting.
  - Key objectives:
    - Regularly update external assumptions about foreign growth, commodity prices, and other factors using external forecasts (for example, WEO and other global models).
    - Facilitate mapping of available global forecasts into quarterly gap-trend projections and production of user-defined scenarios, updates, and alternatives.
    - Make codes and infrastructure publicly available so central banks and ministries of finance can prepare QPM-consistent external assumptions in-house.
- Coverage and structure:
  - Covers 80 percent of global GDP, including the US, Euro Area (EA), China, and a “rest of the world” (RW) block aggregating ten large economies: Australia, Brazil, Canada, Indonesia, India, Japan, South Korea, Mexico, Russia, and the United Kingdom (aggregated into a single nominal GDP-weighted average series).
  - Model type: calibrated semi-structural model where each economy features an IS curve, a decomposed Phillips curve (core and non-core), an uncovered interest parity (UIP) condition, and a monetary policy rule.
  - Policy-regime flexibility:
    - US and EA: inflation-targeting regimes with a policy rule setting the nominal interest rate based on inflation and output.
    - China: a managed exchange rate regime reflecting limited autonomy in interest rate policy and exchange rate practices.
  - Captures trade and financial linkages and key spillover channels: financial spillovers proxied by Bank Lending Tightening (BLT) and demand spillovers via trade linkages.
- Theoretical stance:
  - Semi-structural open-economy New Keynesian framework, parsimonious and operational for quarterly policymaking and forecasting.
  - Distinguishes short-term cyclical position (gaps) and long-term potential growth (trends); trends treated as primarily exogenous assumptions.
  - Hybrid expectations: model-consistent expectations blending forward-looking (rational) and adaptive (backward-looking) components, inducing inertia.
- Mapping to QPMs and intended users:
  - Practical tool for central banks with semi-structural gap-trend QPMs to provide consistent and timely external assumptions.
  - Parsimonious quarterly projections and simulations suit policy simulations and economic analysis, especially for institutions with limited resources.
  - Codes and procedures support translation of external forecasts (for example, WEO-based forecasts) into QPM language.
- Model lineage and benchmarking:
  - Closely follows Carabenciov et al. (2008, 2013) and leverages a large body of global projection models beyond IMF Research (RES).
  - Benchmarked against GPM6 (Carabenciov et al., 2013), RES’s Flexible System of Global Models (FSGM), ECB-Global, and NiGEM.
  - Parameters updated from Carabenciov et al. (2013) to reflect economic changes in the past decade.
- Notation and decomposition:
  - Variables expressed in log terms and decomposed into gap (hat notation, e.g., ŷ) and trend (bar notation, e.g., y¯) components.
  - Trends captured as autoregressive processes and treated as exogenous for simplification.
- Paper roadmap (sections):
  - Section 2: model structure.
  - Section 3: model calibration, parameter categorization, and iterative process.
  - Section 4: impulse responses to stylized shocks and model comparisons.
  - Section 5: historical interpretation and country-specific analyses.
  - Section 6: in-sample simulations and forecasting accuracy.
  - Section 7: translating external forecasts into QPM language using WEO-based forecasts as example.
  - Section 8: summary and avenues for future enhancements.

### 2. Aggregate Supply and Inflation — decomposition and transmission
- Headline decomposition:
  - Headline inflation (πi,t) = wπ,c πi,t c + (1−wπ,c) πi,t nc + επ i,t π    (equation 7).
  - Time-varying weight shock (επ i,t π) accommodates changes in the relative importance of core vs non-core.
  - Empirical alignment: core inflation correlates with real economic fluctuations; non-core (food and energy) is more influenced by commodity price volatility and is inherently more volatile.
- Core Phillips curve (quarter-on-quarter):
  - πi,t c = α1 πi,t−1 c + (1− α1) Et πi,t+1 c + α2 rmc i,t c + επ i,t πc    (equation 8).
  - Drivers: backward- and forward-looking expectations, real marginal costs (rmc), and a core cost-push shock (επ πc).
- Non-core Phillips curve:
  - πi,t nc = α5 πi,t−1 nc + (1− α5) Et πi,t+1 nc + α6 rmc i,t nc + επ i,t πnc    (equation 10).
  - Drivers: adaptive and rational expectations, non-core real marginal cost, and non-core cost shocks.
- Real marginal costs:
  - Core real marginal cost: rmc i,t c = α3 ŷi,t + α4 (1/(1−wc)) (−rp̂ i,t) + (1− α3 − α4) rmc̄ i,t m    (equation 9).
    - Components: output gap (ŷ), pass-through from non-core via relative price gap (rp̂), and import-weighted REER gap (rmc̄ m).
    - Policy implication: under floating regimes, monetary policy can affect nominal exchange rate and thus short-term core inflation through exchange rate volatility.
  - Non-core real marginal cost: rmc i,t nc = α7 ŷi,t + α8 q̂i,t nc + α9 q̂i,t ncd + α10 (1/(1−wc)) (rp̂ i,t) + (1− α7 − α8 − α9 − α10) rmc̄ i,t m    (equation 11).
    - Emphasis: depends importantly on global food (q̂ ncd) and oil (q̂) real prices in domestic currency deflated by domestic non-core CPI.
- Commodity price pass-through and gap decomposition:
  - Relative price gap (rp̂ i,t) links core to non-core inflation and generates second-round effects when volatile commodity prices shift non-core relative to core prices.
  - Conversion of global commodity prices into local currency and non-core terms via equations 11a–11c, with final simplified expression:
    - rrqrd_i_qy_i,t nc = rrqrd_i_qy_i,t udd + di,t − rppiic_nc    (equation 11b, final line).
  - Global commodity prices decomposed into trend and gap components; cyclical movements in global commodity price gaps are influenced by world demand pressures (equation 11d).
- Modeling simplifications and forecasting use:
  - Global food and oil prices modeled with trend and gap components as a simplification relative to more sophisticated forecasting tools.
  - Nominal US dollar commodity prices adjusted by US CPI to obtain real US dollar prices (consistent with GPM practices).
  - Commodity forecasts serve as external assumptions for users of G3MOD to facilitate analysis for small open economies.

### 3. Model Properties — shocks, responses, and cross-model comparison
- Overview and shock-selection rationale:
  - Analyze structural adequacy via comparison with other global projection models and establish basis for validation.
  - Focus on four contractionary shocks that significantly affected the global economy since 2020: US monetary policy tightening, reductions in US and Chinese demand, real oil price increases, and a negative shift in investor risk appetite.
  - G3MOD does not model Integrated Policy Framework (IPF) policy questions (FXI, CFMs, MPMs).
- US monetary policy shock (Section 4.1):
  - Shock: unexpected one-year 100 basis point increase in the US policy rate.
  - Quantitative effects:
    - US output gap contracts by about 0.6 percent over five quarters.
    - Headline inflation in the US declines by 0.4 percent.
    - EA and China output gaps narrow by about 0.1 percent.
  - Transmission: raises real rates, US dollar appreciation, tightens global financial conditions; ECB and PBC implement expansionary policies to mitigate losses.
- Temporary decrease in US domestic demand (Section 4.2):
  - Shock: one-quarter 1 percent reduction in the US output gap.
  - Immediate effects:
    - Headline inflation decreases by 0.2 percent.
    - US Fed cuts interest rates by about 25 basis points.
  - Global spillovers:
    - EA and China output gaps contract by approximately 0.1 to 0.15 percent.
    - Inflation decelerates by about 0.05 to 0.1 percent in EA and China.
- Temporary decrease in China’s domestic demand (Section 4.3):
  - Shock: one-quarter 1 percent reduction in China’s output gap.
  - Effects:
    - Inflation in China declines by 0.1 percent.
    - PBC eases monetary policy by approximately 3 basis points.
    - China’s REER weakens; PBC can intervene to strengthen the currency without sacrificing price competitiveness.
  - Note: spillovers from China’s slowdown are significantly smaller than US-originated shocks in the model calibration.
- Temporary increase in global oil price (Section 4.4):
  - Shock: 10 percent increase in the real price of oil, lasting eight quarters.
  - Quantitative regional effects:
    - US headline inflation rises by 0.4 percent.
    - US Fed increases interest rates by over 25 basis points.
    - US output gap narrows by approximately 0.05 percent within eight quarters.
    - ECB raises policy rate by around 3 basis points.
  - Transmission asymmetry: impact and policy response particularly pronounced in the US due to high energy consumption and a low fuel tax wedge.
- Shift in Euro Area risk appetite (Section 4.5):
  - Shock: one-quarter negative shock to EA risk premium of 1 percentage point.
  - Immediate dynamics:
    - Euro depreciates → initial rise in EA inflation.
    - In absence of banking linkages, temporary risk premium increase appears to stimulate EA activity via improved price competitiveness.
  - Policy and medium-term effects:
    - ECB raises policy rate by approximately 50 basis points and maintains elevated rates → euro appreciation and decline in EA inflation.
    - Medium-term euro appreciation mildly increases US inflation, prompting marginal Fed rate increase; US impact remains modest.
- Comparison with other global models (Section 4.6):
  - Shock set for comparison:
    - Temporary 25 basis points increase in US monetary policy rates.
    - Temporary 1 percent reduction in US GDP (domestic demand).
    - Temporary 1 percent reduction in China’s GDP (domestic demand).
    - Temporary 10 percent rise in real oil prices.
  - Key comparative findings:
    - US monetary policy shock (25 basis points): US GDP declines between 0.1 and 0.6 percent across models. G3MOD responses for EA, China, and global economy closely align with FSGM and NiGEM; ECB-G shows more pronounced responses.
    - Temporary 1 percent negative US demand shock: reduces real global GDP by between 0.3 and 0.4 percent; impacts relatively consistent across models.
    - Temporary 1 percent negative Chinese demand shock: slightly smaller negative impact on world GDP than US demand shock.
    - Temporary 10 percent oil price rise: world GDP contracts by 0.02 to 0.15 percent; G3MOD elasticities are smaller than other models due to calibration to a 20-year macroeconomic dataset.
  - Overall conclusion: reasonable consistency among global models despite structural and calibration differences; ECB-G possibly captures stronger financial spillovers.

### 4. Historical Interpretation — methodology and selected findings
- Methodology and purpose (Section 5):
  - Use Kalman filter in the QPM suite to estimate unobserved variables (gaps and trends) and shocks from observed time-series data to build coherent narratives and evaluate calibration/performance.
  - Sample and focus: decomposition performed for 2006Q1 to 2023Q4, emphasizing the GFC and the COVID-19 pandemic.
  - Uses: interpret historical developments, estimate initial conditions for forecasts, and decompose external forecasts by estimating cyclical positions based on observed inflationary pressure and monetary stance.
- US historical findings (Section 5.1):
  - Growth and potential:
    - Outside GFC and COVID-19, US GDP growth fluctuated between 2 and 3 percent; model estimates stable potential growth.
    - COVID-19 caused downturn in both actual level and potential growth (potential turned negative), with subsequent rebound.
  - Output gap decomposition:
    - Overheated cycle before GFC driven by positive domestic and spillover demand shocks.
    - Deep recession during GFC exacerbated by stringent lending conditions (negative BLT contribution); recovery aided by easing of lending conditions.
    - Post-GFC: neutral output gap 2014–2017, followed by above-trend phase pre-COVID-19 despite restrictive monetary policy.
  - COVID-19 dynamics:
    - Large negative domestic demand shocks in early 2020 from imposed restrictions.
    - Recovery driven by unprecedented fiscal stimulus (reflected as positive domestic demand shocks) and monetary easing.
    - Subsequent overheated state persisted despite policy tightening, driven by accumulated savings and delayed consumption.
  - Inflation dynamics:
    - US inflation hovered around target with core inflation stable and non-core more volatile.
    - Commodity price shocks (including Russia-Ukraine war) pushed headline inflation above medium-term target; commodity price collapses after GFC and in 2015–2016 produced decelerations.
    - Post-pandemic acceleration in core inflation linked to supply bottlenecks; inflation began to decelerate more rapidly in 2023 as the Fed tightened monetary stance.
  - Non-core inflation: higher volatility and less persistence than core; decelerated quickly once global oil and food inflation normalized.
- Euro Area historical findings (Section 5.2):
  - Output and growth trajectory:
    - Overheated pre-GFC, deep recession during and after GFC, and sovereign debt crisis effects during 2012–2014.
    - Recovery with loose bank lending conditions and rebound in foreign demand; slightly positive output gap in 2018–2019.
    - Pandemic: sharp decline in activity in early-2020; slower post-pandemic recovery than US, aided by loosening bank lending conditions and fiscal policy effects captured as domestic demand shocks.
    - From 2023: unwinding of adverse supply shocks, moderating commodity prices, and ECB tightening contributed to disinflation over the forecast horizon.
  - Inflation dynamics and decomposition:
    - EA inflation historically exhibited more deviation from its medium-term target than US inflation.
    - Core inflation prior to post-COVID recovery often below ECB’s 2 percent target.
    - Supply chain disruptions in 2021 and EUR depreciation contributed to core acceleration; core decelerated in 2023 as real marginal costs decreased and monetary tightening anchored expectations.
    - Non-core inflation mirrored global commodity cycles and the real exchange rate gap; Russia halting natural gas exports to the EU in response to sanctions after its invasion of Ukraine in 2022 exacerbated non-core acceleration.
  - Energy and commodity specifics:
    - Natural gas price increased fifteenfold in European Union between January 2022 and its peak level in September 2022.
    - Oil price increase was about 50 percent between January 2022 and its peak in June 2022.
    - EU responses included re-routing energy imports and voluntary 15 percent gas use reduction targets.
  - Model-based interpretation and calibration notes:
    - G3MOD captures EA dynamics including pre-GFC overheating, GFC trough, COVID-19 collapse, and post-pandemic inflationary pressures.
    - Larger inflation expectations persistence in the EA required more prolonged acceleration to generate inflationary pressures; increasing backward-looking weight (α1, α5) could improve fit in some episodes but worsen fit in troughs.

### 5. Translating External Forecasts into QPM language — procedures and institutional uses
- Purpose and scope (Section 7):
  - G3MOD manipulates and interprets projections of key external variables—GDP growth, inflation, exchange rates, and commodity prices—within a QPM-consistent framework.
  - Procedures reproduce headline values of an external forecast while decomposing those forecasts into gap and trend components, and produce quarterly versions when the original forecast is annual.
  - Example application uses IMF’s published WEO forecasts for the US, Euro Area, and China (CPI, GDP, exchange rates), and commodity prices (food, oil).
- Key procedural steps:
  - Data collection and transformation:
    - Automated data transformation script applies frequency conversion, log-linearization, and constructs a single rest of the world block as a weighted average based on nominal GDP sizes.
  - Estimation of initial conditions:
    - Kalman filter estimates initial conditions independently of future information.
    - Users may apply expert judgment to reflect assessments or overwrite filter estimates (example: level shift in the GDP trend during the COVID-19 period).
  - Forecasting and imposing external forecasts:
    - Annual-frequency projections from WEO are interpolated to quarterly frequency using the Kalman filter.
    - Filter estimates unobserved variables (gaps and trends) consistently with the forecast story, including inflationary pressures and projected monetary policy measures.
    - Expert judgment can fine-tune decompositions (example: aligning projected US policy rate with the Fed dot plot).
- Technical approach and interpretation:
  - Interpolation and decomposition: annual WEO forecast numbers interpolated to quarterly frequency by the Kalman filter; filter estimates gap and trend components consistent with projected monetary policy and inflation outlooks.
  - Information usage: inflation outlook, commodity price movements, and exchange rate fluctuations help determine strength of demand-side inflationary pressures and output gaps underpinning GDP projections.
  - Role of expert judgment: supplements G3MOD where needed—e.g., when unconditional model projection of US policy rate is not aligned with Federal Reserve expectations or when a WEO forecast is outdated.
- Forecast comparison, validation, and adjustments:
  - Compare unconditional model forecast (from estimated initial conditions without imposed assumptions) with external forecast (WEO) before imposing external forecast.
  - Significant discrepancies require revisiting estimation of initial conditions to avoid assuming low-probability shocks.
  - Use expert judgments under extraordinary circumstances (example: level shift in GDP trend during COVID-19).
- Outputs, applications, and flexibility:
  - Primary output: decomposes a given external forecast into quarterly gap-trend projections consistent with QPM frameworks; does not inherently produce its own external forecast in the example.
  - Alternative uses: can generate a pure model ‘unconditional’ forecast to demonstrate implications of estimated initial conditions.
  - Institutional applications:
    - G3MOD codes can produce in-house QPM-consistent external assumptions for IMF staff and other users.
    - IMF TA recipients can independently generate external assumptions after training; example trade-weighted guidance: Bank of Mauritius foreign demand gap weights could be 30 percent US and 70 percent Euro Area.
  - Practical benefits: maps annual institutional forecasts into quarterly gap-trend projections while maintaining the underlying narrative; supports baseline and alternative scenarios, policy-contingent forecasts, risk assessments, and capacity building.

*Source: wpiea2024254-print-pdf — selected sections from G3MOD: A Multi-Country Global Forecasting Model.*

### 1. Introduction ........................................................................................................

### 1. Introduction

### Purpose and motivation
- Semi-structural gap-trend Quarterly Projection Models (QPMs) are widely used by central banks and ministries of finance in emerging market and developing countries to support policy-making.
- A critical challenge is incorporating external developments—trading partner demand, financial markets, and commodity prices—into baseline predictions and scenario analysis, particularly where institutions have nascent modeling capacity or limited staff resources.
- Many institutions rely on available global forecasts (for example, IMF World Economic Outlook (WEO)), but these are typically annual and not expressed in gap-trend (QPM) terms, complicating direct incorporation into QPM frameworks.

### G3MOD: objectives and design
- G3MOD is presented as a straightforward, quickly implemented variant of standard global projection models to support inclusion of global forecasts in countries’ economic analysis and forecasting.
- Key objectives:
  - Regularly update external assumptions about foreign growth, commodity prices, and other factors using external forecasts (for example, WEO and other global models).
  - Facilitate mapping of available global forecasts into quarterly gap-trend projections and production of user-defined scenarios, updates, and alternatives.
  - Make codes and infrastructure publicly available so central banks and ministries of finance can prepare QPM-consistent external assumptions in-house.

### Coverage and structure
- G3MOD covers 80 percent of global GDP, including:
  - The US, Euro Area (EA), and China (the G3 economies).
  - A “rest of the world” (RW) block aggregating ten large economies: Australia, Brazil, Canada, Indonesia, India, Japan, South Korea, Mexico, Russia, and the United Kingdom. Data for these economies are aggregated into a single nominal GDP-weighted average series.
- Model type and key features:
  - Calibrated semi-structural model where each economy features:
    - An IS curve (aggregate demand).
    - A decomposed Phillips curve with core and non-core (usually food and energy) components modeled separately and linked by relative price gap terms.
    - An uncovered interest parity (UIP) condition.
    - A monetary policy rule.
  - Flexible to accommodate various policy regimes:
    - US and EA: inflation-targeting regimes with a policy rule setting the nominal interest rate based on inflation and output.
    - China: a managed exchange rate regime reflecting limited autonomy in interest rate policy and exchange rate practices.
  - Incorporates trade and financial linkages among major economies and the RW block.
  - Captures key spillover channels: financial spillovers (proxied by Bank Lending Tightening (BLT)) and demand spillovers (trade linkages).

### Theoretical stance and expectations
- Semi-structural open-economy New Keynesian framework, parsimonious and operational for quarterly policymaking and forecasting.
- Distinguishes short-term cyclical position (gaps) and long-term potential growth (trends); trends treated as primarily exogenous assumptions.
- Prices fully flexible in the long term; monetary policy neutral in the long term; sticky prices in the short term allow monetary policy to influence real interest rates, output gaps, and inflation.
- Uses hybrid expectations: model-consistent expectations blending forward-looking (rational) and adaptive (backward-looking) components, inducing inertia.

### Main model architecture (high-level)
- Each region (US, EA, China, RW) includes:
  - Aggregate demand (IS curve): output gap depends on expected and past output gaps, real interest rate gap, BLT measure, export-weighted real effective exchange rate gap, foreign demand gap, direct demand spillovers, and a domestic demand shock.
  - Decomposed aggregate supply (Phillips curves): core and non-core inflation linked to past and expected future inflation, output gap, exchange rates, commodity prices (food and oil), and cross-component spillovers.
  - Nominal exchange rate: UIP for US, EA, and RW with backward-looking expectations; policy-driven managed exchange rate for China.
  - Monetary policy rule: policy nominal interest rate based on output gap and expected inflation.

### Mapping to QPMs and intended users
- Designed to be a practical tool for central banks with semi-structural gap-trend QPMs, providing consistent and timely external assumptions.
- Parsimonious quarterly projections and simulations suit policy simulations and economic analysis, especially for institutions with limited resources.
- Codes and procedures support translation of external forecasts (for example, WEO-based forecasts) into QPM language.

### Model lineage and benchmarking
- G3MOD closely follows Carabenciov et al. (2008, 2013) and leverages a large body of global projection models beyond IMF Research (RES).
- For benchmarking and calibration, comparisons are made with:
  - GPM6 (Carabenciov et al., 2013),
  - RES’s Flexible System of Global Models (FSGM),
  - ECB-Global,
  - NiGEM.
- Parameters were updated from Carabenciov et al. (2013) to reflect economic changes in the past decade.

### Notation and decomposition
- Variables expressed in log terms and decomposed into gap (hat notation, e.g., 푦푦�) and trend (bar notation, e.g., 푦푦¯) components.
- Trends are captured as autoregressive processes and treated as exogenous for simplification.

### Specific modeling details highlighted in Section 2 (preview)
- Aggregate demand IS curve (equation (1)) includes terms:
  - 훽1퐸푡푦̂i,t+1 + 훽2ŷi,t−1 − 훽3 r̂i,t + ηi,t + 훽4 rer̂i,t x + 훽5 ŷi,t f + dddi,t f + εi,t ddd + εi,t y
- BLT formulation:
  - BLLi,t = BLL̄i,t − ki Et ȳt+4 − εt bllt  (equation (2))
  - BLL̄i,t = BLL̄i,t−1 + εt bllt̄  (equation (3))
  - ηi,t = 0.04 εt−1 bllt + 0.08 εt−2 bllt + 0.12 εt−3 bllt + 0.16 εt−4 bllt + 0.20 εt−5 bllt + 0.16 εt−6 bllt + 0.12 εt−7 bllt + 0.08 εt−8 bllt + 0.04 εt−9 bllt  (equation (4))
- Real effective exchange rate construction:
  - reer i,t j = w i USD,j reer i,t USD + Σ w i −i,j (reer −i,j,t + reer i,t USD −i\{USD})  (equation (5))
  - bilateral reer: reer i,t USD = d i,t USD + p USD,t c − p i,t c  (equation (6))
- Model uses core inflation for deflating and constructing real effective exchange rates.

### Paper roadmap
- Section 2: model structure.
- Section 3: model calibration, parameter categorization, and iterative process.
- Section 4: impulse responses to stylized shocks and model comparisons.
- Section 5: historical interpretation and country-specific analyses.
- Section 6: in-sample simulations and forecasting accuracy.
- Section 7: translating external forecasts into QPM language using WEO-based forecasts as example.
- Section 8: summary and avenues for future enhancements.

*Source: wpiea2024254-print-pdf - 1. Introduction ........................................................................................................*

### 2.2   Aggregate Supply and Inflation

### 2.2   Aggregate Supply and Inflation

### Decomposition of headline inflation
- Headline inflation (휋휋푖푖,푡푡) is decomposed into core (휋휋푖푖,푡푡푐푐) and non-core (food and energy primarily, 휋휋푖푖,푡푡푐푐푐푐) components:
  - 휋휋푖푖,푡푡 = 푤푤휋휋,푐푐 휋휋푖푖,푡푡푐푐 + (1−푤푤휋휋,푐푐)휋휋푖푖,푡푡푐푐푐푐 + 휀휀푖푖,푡푡 휋휋    (equation 7)
- The time-varying weight shock (휀휀푖푖,푡푡 휋휋) accommodates changes in the relative importance of core vs non-core.
- Empirical alignment:
  - Core inflation correlates with real economic fluctuations and underlying demand-side pressures.
  - Non-core inflation is more influenced by commodity price volatility and is inherently more volatile.

### New Keynesian Phillips curve structure for core and non-core
- Core inflation equation (quarter-on-quarter):
  - 휋휋푖푖,푡푡푐푐 = 훼훼1 휋휋푖푖,푡푡−1푐푐 + (1− 훼훼1) E_t 휋휋푖푖,푡푡+1푐푐 + 훼훼2 푟푟푟푟푟푟푖푖,푡푡푐푐 + 휀휀푖푖,푡푡 휋휋푐푐    (equation 8)
  - Drivers: backward- and forward-looking expectations, real marginal costs (푟푟푟푟푟푟푖푖,푡푡푐푐), and a core cost-push shock (휀휀푖푖,푡푡 휋휋푐푐).
- Non-core inflation equation:
  - 휋휋푖푖,푡푡푐푐푐푐 = 훼훼5 휋휋푖푖,푡푡−1푐푐푐푐 + (1− 훼훼5) E_t 휋휋푖푖,푡푡+1푐푐푐푐 + 훼훼6 푟푟푟푟푟푟푖푖,푡푡푐푐푐푐 + 휀휀푖푖,푡푡 휋휋푛푛푐푐    (equation 10)
  - Drivers: adaptive and rational expectations, non-core real marginal cost (푟푟푟푟푟푟푖푖,푡푡푐푐푐푐), and non-core cost shocks (휀휀푖푖,푡푡 휋휋푛푛푐푐).

### Real marginal costs and transmission channels
- Core real marginal cost specification:
  - 푟푟푟푟푟푟푖푖,푡푡푐푐 = 훼훼3 𝑦𝑦�푖푖,푡푡 + 훼훼4 (1/(1−푤푤푐푐)) (−푟푟푝푝�푖푖,푡푡) + (1− 훼훼3 − 훼훼4) 푟푟푟푟푟푟푟푟�푖푖,푡푡푚푚    (equation 9)
  - Components: output gap (𝑦𝑦�), pass-through from non-core via relative price gap (푟푟푝푝�), and import-weighted real effective exchange rate gap (푟푟푟푟푟푟푟푟�푚푚).
  - Notes: output gap proxies domestic cost pressures (wages and capital costs); import-weighted REER gap proxies imported inflationary pressure via foreign price differentials and nominal exchange rate movements.
  - Policy implication: under floating regimes, monetary policy can affect nominal exchange rate and thus short-term core inflation through exchange rate volatility.
- Non-core real marginal cost specification:
  - 푟푟푟푟푟푟푖푖,푡푡푐푐푐푐 = 훼훼7 𝑦𝑦�푖푖,푡푡 + 훼훼8 푞푞푞푞푞푞푞푞�푖푖,푡푡푐푐푐푐 + 훼훼9 푞푞푞푞푞푞푞푞𝑑𝑑�푖푖,푡푡푐푐푐푐 + 훼훼10 (1/(1−푤푤푐푐)) (푟푟푝푝�푖푖,푡푡) + (1− 훼훼7 − 훼훼8 − 훼훼9 − 훼훼10) 푟푟푟푟푟푟푟푟�푖푖,푡푡푚푚    (equation 11)
  - Emphasis: non-core marginal cost depends importantly on global food (푞푞푞푞푞푞푞푞𝑑𝑑�) and oil (푞푞푞푞푞푞푞푞�) real prices expressed in domestic currency and deflated by domestic non-core CPI.

### Commodity price pass-through and gap decompositions
- Relative price gap role:
  - The relative price gap (푟푟푝푝�푖푖,푡푡) links core to non-core inflation and generates second-round effects when volatile commodity prices shift non-core relative to core prices.
  - A positive relative price gap indicates core prices rising faster than non-core, exceeding the headline index; to keep headline inflation at target, the central bank must manage deceleration in core inflation.
- Conversion of global commodity prices into local currency and non-core terms:
  - Equation 11a defines real commodity price gaps in domestic terms, combining gaps in real US dollar commodity prices, real exchange rate gaps vs USD, and relative price gaps.
  - Equation 11b expands these terms to show adjustment by domestic non-core CPI, yielding an expression that simplifies to:
    - 푟푟푞푞푟푟푟푟푞푞𝑑푑푖푖푞푞푦푦푖푖,푡푡푐푐푐푐 = 푟푟푞푞푟푟푟푟푞푞𝑑푑푖푖푞푞푦푦푖푖,푡푡𝑢𝑢𝑑𝑑𝑑𝑑 + 푑푑푖푖,푡푡 − 푟푟푝푝푖푖푖푖,푡푡푐푐푐푐    (equation 11b, final line)
- Global commodity price dynamics:
  - Real US dollar commodity prices are decomposed into trend and gap components (equation 11c).
  - Cyclical movements in global commodity price gaps (푞푞푟푟...�푢푢𝑑𝑑𝑑𝑑) are influenced by world demand pressures (푑푑푟푟푟푟푎푎푎푎푑푑 푟푟훥훥푟푟𝑏𝑏푑푑𝑡푡), which are weighted sums of innovation terms across country IS equations (BLT shock, spillover shocks, idiosyncratic output gap shocks) (equation 11d).

### Modeling simplifications and use in forecasting
- Global food and oil prices are modeled with trend and gap components to capture long-term movements and cyclical fluctuations; this is acknowledged as a simplification relative to more sophisticated forecasting tools.
- These commodity forecasts serve as external assumptions for users of G3MOD to facilitate analysis and projections for small open economies.
- Adjustments: nominal US dollar commodity prices are adjusted by US CPI to obtain real US dollar prices (consistent with GPM practices).

*Source: wpiea2024254-print-pdf — 2.2   Aggregate Supply and Inflation*

### 4.   Model Properties

### 4.   Model Properties

### Overview
- Purpose: analyze G3MOD’s structural adequacy via comparison with other global projection models and to establish basis for validation and comparative analysis in sections 5 and 6.
- Shock-selection rationale: focus on four contractionary shocks that significantly affected the global economy since 2020—US monetary policy tightening, reductions in US and Chinese demand, real oil price increases, and a negative shift in investor risk appetite.
- Alignment: each shock corresponds to idiosyncratic shocks (ε terms) in Section 2 and maps to channels: US monetary policy → global financial conditions; US/China demand shocks → global trade dynamics; temporary real oil price increase → energy-dependent cost shock; EA risk appetite shift → market sentiment.
- Scope note: G3MOD does not model Integrated Policy Framework (IPF) policy questions (FXI, CFMs, MPMs); these are left to other IMF models under development.

### US Monetary Policy Shock (Section 4.1)
- Shock: unexpected one-year 100 basis point increase in the US policy rate.
- Transmission channels:
  - Raises real interest rates → dampens domestic demand.
  - US dollar appreciation (nominal and real) → shifts consumption toward imports, reducing domestic demand.
  - Tightens global financial conditions → raises cost and restricts availability of credit.
- Quantitative effects:
  - US output gap contracts by about 0.6 percent over five quarters.
  - Headline inflation in the US declines by 0.4 percent.
  - EA and China output gaps narrow by about 0.1 percent.
- Policy responses and spillovers:
  - Euro depreciates initially → temporary boost to European exporters and widening of the output gap.
  - Medium-term foreign demand decline dampens activity in Europe and China.
  - ECB and PBC implement expansionary monetary policies to mitigate losses.

### Temporary Decrease in US Domestic Demand (Section 4.2)
- Shock: one-quarter 1 percent reduction in the US output gap due to lower domestic demand.
- Immediate effects:
  - Headline inflation decreases by 0.2 percent.
  - US Fed cuts interest rates by about 25 basis points.
- Global spillovers:
  - EA and China output gaps contract by approximately 0.1 to 0.15 percent.
  - Inflation decelerates by about 0.05 to 0.1 percent in EA and China.
  - EA and China central banks reduce policy rates, but to a lesser extent than the US.

### Temporary Decrease in China’s Domestic Demand (Section 4.3)
- Shock: one-quarter 1 percent reduction in China’s output gap.
- Effects and policy response:
  - Inflation in China declines by 0.1 percent.
  - PBC eases monetary policy by approximately 3 basis points.
  - China’s real effective exchange rate weakens, enabling PBC intervention to strengthen the currency without sacrificing price competitiveness.
- Model calibration note:
  - Trade linkages calibrated via a trade matrix among four economic blocks (including rest of world); spillovers from China’s slowdown are significantly smaller than US-originated shocks despite China’s growing global role.

### Temporary Increase in Global Oil Price (Section 4.4)
- Shock: 10 percent increase in the real price of oil, lasting eight quarters.
- Global impact:
  - Higher production costs and reduced consumer spending → inflation and output effects across all regions.
  - Central banks raise policy rates to curb second-round effects on core inflation.
- Quantitative regional effects:
  - US headline inflation rises by 0.4 percent.
  - US Fed increases interest rates by over 25 basis points.
  - US output gap narrows by approximately 0.05 percent within eight quarters.
  - ECB raises policy rate by around 3 basis points.
- Transmission asymmetry:
  - Inflationary impact and monetary policy response are particularly pronounced in the US due to high energy consumption and a low fuel tax wedge.

### Shift in Euro Area Risk Appetite (Section 4.5)
- Shock: one-quarter negative shock to EA risk premium of 1 percentage point.
- Immediate dynamics:
  - Euro depreciates against US dollar and Renminbi → initial rise in EA inflation.
  - In model without explicit macro-financial banking linkages, temporary risk premium increase appears to stimulate EA activity via improved price competitiveness.
- Policy and medium-term effects:
  - ECB raises policy rate by approximately 50 basis points and maintains elevated rates → euro appreciation and decline in EA inflation.
  - Medium-term euro appreciation mildly increases US inflation, prompting marginal Fed rate increase; US economic impact remains modest.
  - China mirrors US adjustments via renminbi depreciation, then marginal renminbi appreciation as euro appreciation restores competitiveness.

### Comparison with Other Global Models (Section 4.6)
- Models compared: ECB-G, IMF’s annual FSGM, NiGEM; reproduced comparative framework from Dieppe et al. (2017).
- Shock set for comparison:
  - Temporary 25 basis points increase in US monetary policy rates.
  - Temporary 1 percent reduction in US GDP (domestic demand).
  - Temporary 1 percent reduction in China’s GDP (domestic demand).
  - Temporary 10 percent rise in real oil prices.
- Key comparative findings:
  - US monetary policy shock (25 basis points): US GDP declines between 0.1 and 0.6 percent across models. G3MOD responses for EA, China, and global economy closely align with FSGM and NiGEM; ECB-G shows more pronounced responses.
  - Temporary 1 percent negative US demand shock: reduces real global GDP by between 0.3 and 0.4 percent; impacts are relatively consistent across models with some deviation for China in FSGM.
  - Temporary 1 percent negative Chinese demand shock: slightly smaller negative impact on world GDP than US demand shock; US and EA experience minor negative deviations. Cross-model impacts show remarkable consistency.
  - Temporary 10 percent oil price rise: world GDP contracts by 0.02 to 0.15 percent; individual country impacts are more varied. G3MOD’s elasticities are smaller than other models, reflecting calibration to a 20-year macroeconomic dataset to capture average historical behaviors.
- Overall conclusion: reasonable consistency among different global models despite structural and calibration differences; ECB-G possibly captures stronger financial spillovers.

### Historical Interpretation (Section 5) — Methodology and Purpose
- Approach: use Kalman filter in the QPM suite to estimate unobserved variables (gaps and trends) and shocks from observed time-series data to build coherent narratives and evaluate calibration/performance.
- Sample and focus: decomposition performed for 2006Q1 to 2023Q4, emphasizing the global financial crisis (GFC) and the COVID-19 pandemic.
- Uses: interpret historical developments, estimate initial conditions for forecasts, and decompose external forecasts by estimating cyclical positions based on observed inflationary pressure and monetary stance.

### Historical Interpretation of the US Economy (Section 5.1) — Findings
- Growth and potential:
  - Outside GFC and COVID-19 periods, US GDP growth fluctuated between 2 and 3 percent; model estimates stable potential growth.
  - COVID-19 caused downturn in both actual level and potential growth (potential turned negative), with subsequent rebound.
- Output gap decomposition (Figure 8): 
  - Overheated cycle before GFC driven by positive domestic and spillover demand shocks.
  - Deep recession during GFC exacerbated by stringent lending conditions (negative BLT contribution); recovery aided by easing of lending conditions.
  - Post-GFC: neutral output gap 2014–2017, followed by above-trend phase pre-COVID-19 despite restrictive monetary policy.
- COVID-19 dynamics:
  - Large negative domestic demand shocks in early 2020 from imposed restrictions.
  - Recovery driven by unprecedented fiscal stimulus (reflected as positive domestic demand shocks) and monetary easing.
  - Subsequent overheated state persisted despite policy tightening, driven by accumulated savings and delayed consumption.
- Inflation dynamics:
  - With some exceptions, US inflation hovered around target; core inflation stable while non-core (food and energy) more volatile.
  - Commodity price shocks (including Russia-Ukraine war) pushed headline inflation above medium-term target; commodity price collapses after GFC and in 2015–2016 produced decelerations.
  - Core inflation moved with real marginal cost changes driven by domestic demand pressures, exchange rate volatility, imported inflation, and second-round commodity effects.
  - Post-pandemic acceleration in core inflation linked to supply bottlenecks; inflation began to decelerate more rapidly in 2023 as the Fed tightened monetary stance.
- Non-core inflation:
  - Non-core Phillips curve shows higher volatility than core due to fluctuations in real marginal costs and global oil and food price gaps.
  - Non-core inflation less persistent than core inflation, decelerating quickly once global oil and food inflation normalized.

*IMF Working Paper — G3MOD: A Multi-Country Global Forecasting Model (chapter/section 4).*

### 5.2    Historical Interpretation of the Euro Area Economy

### 5.2    Historical Interpretation of the Euro Area Economy

### Output and growth trajectory
- EA output gap: trajectory suggested an overheated cycle before the GFC, primarily driven by positive domestic and foreign demand shocks, followed by a deep recession during and after the GFC, exacerbated by economic spillovers from the European sovereign debt crisis during 2012-2014.
- Recovery path: gradual recovery with loose bank lending conditions and a rebound in foreign demand; pre-pandemic period during 2018– 2019 saw a slightly positive output gap driven by further rebounds in domestic and foreign demand.
- Pandemic impact: sharp decline in economic activity during the pandemic due to stringent quarantine measures and travel restrictions; substantial negative domestic demand shocks in early-2020 compounded by a decline in foreign demand.
- Post-pandemic recovery: slower compared to the US, primarily driven by loosening of bank lending conditions; positive effects of fiscal policy indirectly captured through increased domestic demand shocks.
- From 2023: unwinding of adverse supply shocks, moderating commodity price increases, and the launch of the ECB’s tightening cycle turned monetary conditions contractionary, alleviating overheated demand pressures, anchoring inflation expectations, and contributing to disinflation over the forecast horizon.

### Inflation dynamics and decomposition
- EA inflation historically exhibited more deviation from its medium-term target than US inflation.
- Major historical episodes:
  - GFC and European Debt Crisis: subdued domestic inflationary pressures, notably affecting core prices.
  - Commodity price swings: collapse after the GFC and in 2015-2016 led to significant deceleration in overall inflation; commodity shocks like those following the Russia-Ukraine war pushed headline inflation above the medium-term target.
- Core inflation:
  - Before the post-COVID-19 recovery, EA core inflation fluctuated below the ECB’s 2 percent inflation target.
  - Supply chain disruptions during the pandemic caused significant adverse supply shocks in 2021.
  - Core inflation accelerated due to EUR depreciation and cost spillovers from non-core prices.
  - As real marginal costs decreased, core inflation began to decelerate in 2023 with the fading of adverse supply shocks and moderating commodity prices.
  - The subdued cyclical position of real economic activity also contributed to lower real marginal costs; inflation expectations adjusted to the ECB tightening cycle, further contributing to core deceleration.
- Non-core inflation:
  - Mostly mirrored global commodity cycles, particularly through movements in real marginal costs and impacts of global oil and food price gaps.
  - Influenced by the cyclical position of real economic activity and the real exchange rate gap.
  - Post-pandemic acceleration exacerbated by the rebound in global commodity prices and the euro’s depreciation.
  - Russia halting natural gas exports to the EU in response to economic sanctions after its invasion of Ukraine in 2022 exacerbated acceleration.
  - By end-2023, tighter monetary conditions and a slightly negative cyclical position of real economic activity emerged; real exchange rate appreciation significantly contributed to decreasing real marginal costs and deceleration of non-core inflation.

### Energy and commodity specifics
- Natural gas versus oil:
  - Natural gas price increased fifteenfold in European Union between January 2022 and its peak level in September 2022.
  - Oil price increase was about 50 percent in the period between January 2022 and its peak in June 2022.
  - Consequently, the global oil price gap does not fully capture the impact of the European energy crisis on non-core inflation dynamics in 2022.
- Policy and market responses in 2023:
  - European countries lowered non-core inflation by re-routing energy imports.
  - Deceleration reinforced by normalization of commodity prices and strengthening of the EUR following ECB tightening efforts.
- EU re-routing and demand reduction measures:
  - To compensate for reductions in Russian imports, the EU increased imports from other sources, including Norway, the US, and the Middle East, and reduced gas demand across the bloc, including voluntary 15 percent gas use reduction targets.

### Model-based interpretation (G3MOD) and link to EA experience
- G3MOD captures EA dynamics: overheating pre-GFC, deep recession during and after the GFC, COVID-19 demand collapse, and diverse post-pandemic inflationary pressures.
- Real marginal costs and inflation linkages:
  - Decreases in real marginal costs contributed to core inflation deceleration in 2023.
  - Real exchange rate appreciation contributed to decreasing real marginal costs and non-core inflation deceleration.
- Calibration notes relevant to EA:
  - Larger inflation expectations persistence in the EA compared to the US required a more prolonged economic acceleration to generate noticeable inflationary pressures.
  - Adjusting EA Phillips curves to be more backward-looking (increasing values of 훼1 in equation 8 and 훼5 in equation 10) could improve fit during post-GFC recovery periods but would worsen fit during the troughs of the GFC and COVID-19.

*Source: IMF Working Papers — G3MOD: A Multi-Country Global Forecasting Model (section 5.2).*

### 7.   Translating External Forecast into QPM language

### 7.   Translating External Forecast into QPM language

### Purpose and scope
- G3MOD is a tool to manipulate and interpret projections of key external variables—GDP growth, inflation, exchange rates, and commodity prices—within a QPM-consistent framework.
- The procedures reproduce headline values of an external forecast while decomposing those forecasts into gap and trend components, and produce quarterly versions when the original forecast is annual.
- The example application uses the IMF’s published WEO forecasts for the US, Euro Area, and China (CPI, GDP, exchange rates), and commodity prices (food, oil).

### Key procedural steps
- Data collection and transformation:
  - Users collect data for the US, Euro Area, China, and for the 10 other economies combined into the “rest of the world” block.
  - An automated data transformation script applies frequency conversion, log-linearization, and constructs a single rest of the world block as a weighted average based on nominal GDP sizes.
- Estimation of initial conditions:
  - The Kalman filter estimates initial conditions of the economy based on observed data and the model’s structural assumptions, independently of future information.
  - Users may apply expert judgment to reflect assessments or overwrite the filter estimates (example: level shift in the GDP trend during the COVID-19 period).
- Forecasting and imposing external forecasts:
  - The model imposes annual-frequency projections for key GDP growth, inflation, and commodity prices from the WEO and interpolates them to quarterly frequency using the Kalman filter.
  - Simultaneously, the filter estimates unobserved variables (gaps and trends) consistently with the forecast story, including inflationary pressures and projected monetary policy measures.
  - Expert judgment can be used to fine-tune decompositions, for example aligning the projected US policy rate with the Federal Reserve’s expectations via the Fed dot plot.

### Technical approach and interpretation
- Interpolation and decomposition:
  - Annual published WEO forecast numbers are interpolated to quarterly frequency by the Kalman filter.
  - The filter estimates gap and trend components of real variables in accordance with projected monetary policy and inflation outlooks for the G3 economies.
- Information usage:
  - The procedure uses the inflation outlook, commodity price movements, and exchange rate fluctuations to help determine the strength of demand-side inflationary pressures and the output gaps underpinning GDP projections.
- Role of expert judgment:
  - Expert judgment supplements the parsimonious G3MOD where needed—e.g., when the unconditional G3MOD projection of the US policy rate is not aligned with Federal Reserve expectations, or when a WEO forecast is outdated or the forecast story has significantly changed.

### Forecast comparison, validation, and adjustments
- Unconditional versus external forecast:
  - Before imposing an external forecast, the unconditional model forecast (derived from estimated initial conditions and cyclical positions without imposed assumptions) is compared with the external forecast (WEO).
  - Significant, inexplicable discrepancies require revisiting the estimation of initial conditions to avoid assuming low-probability shocks when both forecasts share similar narratives.
- Use cases for adjustments:
  - Under extraordinary circumstances, expert judgments can reflect updated monetary policy decisions (example: revision of Fed dot plot pointing to a prolonged restrictive monetary policy).
  - Expert judgments were used historically to inform the model about a level shift in the GDP trend during COVID-19 to avoid implausible large negative output gaps.

### Outputs, applications, and flexibility
- Primary output:
  - G3MOD decomposes a given external forecast into quarterly gap-trend projections consistent with QPM frameworks; it does not inherently produce its own external forecast in this example.
- Alternative uses:
  - The model can generate a pure model ‘unconditional’ forecast to demonstrate implications of estimated initial conditions and the model’s underlying narrative.
- Institutional applications:
  - G3MOD codes can produce in-house QPM-consistent external assumptions for IMF staff and other users.
  - G3MOD output aligned with GAS assumptions can be used directly by Fund staff using QPM models in their external assumptions.
  - IMF TA recipients can independently generate external assumptions for their regular forecasting activities after training and capacity building; example trade-weighted guidance: Bank of Mauritius foreign demand gap weights could be 30 percent US and 70 percent Euro Area.
- Practical benefits:
  - Maps annual institutional forecasts into quarterly gap-trend projections while maintaining the underlying narrative of the original external forecast.
  - Supports central banks and other policy institutions in creating baseline and alternative scenarios for the external environment with enhanced flexibility and timeliness.

### Implications for policy analysis and forecasting practice
- G3MOD enables users to:
  - Produce quarterly, gap-trend consistent external narratives from annual institutional forecasts.
  - Incorporate projected monetary policy measures and inflationary pressures into the estimation of unobserved variables.
  - Use expert judgment to align model-implied policy rates or other paths with institutional expectations (e.g., Fed dot plot).
- Institutional relevance:
  - Suited for central banks and policy institutions with semi-structural gap-trend frameworks and limited resources for in-house multi-country projections.
  - Useful for regular forecasting rounds, policy-contingent forecasts, risk assessments, and capacity building for IMF TA recipients.

*Italic: Content derived from "7.   Translating External Forecast into QPM language" of the G3MOD chapter in the source PDF.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024254-print-pdf.pdf_