## Appendix A . Data Description

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---

### I. High-level findings and research purpose
- Purpose: Augment an approximate linear DSGE model of the world economy with structural shocks exhibiting potentially asymmetric generalized autoregressive conditional heteroskedasticity (GARCH) effects to jointly decompose levels and volatilities of output and financial conditions.
- Key empirical finding: Very strong evidence of asymmetric autoregressive conditional heteroskedasticity (ARCH) effects is found; risk premia shocks are estimated to contribute disproportionately to cyclical output fluctuations and turbulence during abrupt swings in financial conditions across the fifteen largest national economies.
- Scope and sample:
  - Economies: Australia, Brazil, Canada, China, France, Germany, India, Italy, Japan, Korea, Mexico, Russia, Spain, the United Kingdom, and the United States.
  - Sample period: 1999Q1 through 2017Q4.
- Data and dimensionality:
  - Number of endogenous variables observed: 280.
  - Number of structural shocks driving the model: 281.

### II. Model architecture (theoretical framework)
- Model class and features:
  - An approximate linear DSGE model refined from Vitek (2018) with nominal and real rigidities, extensive macrofinancial linkages, bank and capital-market intermediation, and diverse spillover channels.
- Explicit agents and sectors:
  - Households (three types: bank-intermediated, capital-market-intermediated, credit-constrained), developers, firms (industry, intermediate, final), banks (intermediate and final), government (monetary, fiscal, macroprudential authorities).
  - Construction sector, production sector (including nonrenewable commodities), banking sector (mortgage and corporate lending, regulatory capital requirement), and trade sector.
- Financial features:
  - Internationally diversified short term bond, long term bond and stock portfolios; bilateral nominal exchange rate linkages.
  - Collateralized borrowing with regulatory loan to value ratio limits (Kiyotaki and Moore (1997) financial accelerator mechanism).
  - Bank regulation: countercyclical capital buffer rule, loan-to-value limits, bank regulatory costs and capital accumulation.
- Price and wage rigidities:
  - Calvo-style nominal price rigidity extensions for output, export, import and intermediate sectors (partial indexation allowed).
  - Nominal wage rigidity following Erceg, Henderson and Levin (2000) with involuntary unemployment interpretation along Galí (2011) and partial indexation.
- Solution and equilibrium:
  - Agents optimize intertemporally under rational expectations; equilibrium derived by linearizing first-order conditions around a deterministic steady state with zero inflation, zero productivity and labor force growth, and steady public and national financial wealth (unless stated otherwise).

### III. Empirical framework and estimation strategy
- Linearization and state space:
  - Equilibrium conditions are linearized and written as a multivariate linear rational expectations representation driven by potentially heteroskedastic structural shocks.
  - Response coefficients are functions of behavioral parameters restricted to coincide across economies or groups, plus economy-specific steady-state characteristics.
- Shock set and stochastic structure:
  - Structural shocks include productivity, labor supply, consumption demand, investment demand (residential and business consolidated in estimation), export and import demand, price and wage markups, housing, portfolio and duration risk premia, credit risk premium, liquidity risk premium, currency risk premia, monetary, fiscal and macroprudential policy shocks, among others.
  - Many shocks follow stationary first-order autoregressive processes with innovations conditionally normally distributed and conditional variances following symmetric or asymmetric GARCH specifications.
  - Asymmetric conditional variance specification: threshold GARCH (Glosten, Jaganathan and Runkle (1993)) extension for key risk premia and policy shocks.
- Estimation approach and identification:
  - Joint estimation by full information maximum likelihood of subsets of parameters and variables, conditional on calibrated parameters and observed variables.
  - Restricted estimation model consolidates or eliminates weakly identified shocks (e.g., residential and business investment consolidated; several microfinancial shocks treated as homoskedastic or removed).
  - Identification: all innovations assumed independent conditional on the model (conditional multivariate normality); heteroskedastic innovation processes allowed for many shocks.

### IV. Shocks, volatility modeling, and macrofinancial amplification
- Conditional mean dynamics:
  - Many structural shocks follow AR(1) processes with economy-specific persistence.
- Conditional variance dynamics:
  - Symmetric GARCH for a large set of shocks (productivity, labor, demand, many markup shocks, commodity price markups, selected fiscal shocks).
  - Asymmetric GARCH (threshold GARCH) for key risk premia and monetary policy shocks (housing risk premium, credit risk premium, duration and equity risk premia, monetary policy shock).
- Macrofinancial amplification channels:
  - Loan-to-value constraints, bank capital regulation, credit spreads, and asset prices generate amplification where increases in risk premia produce both mean and variance effects on output and financial conditions.
- Empirical amplification finding:
  - Risk premia shocks disproportionately contribute to cyclical output fluctuations and turbulence during abrupt financial-condition swings (consistent with Adrian, Boyarchenko and Giannone (2017) for the United States).

### V. Policy rules embedded in the model
- Monetary policy:
  - Flexible inflation targeting and managed exchange rate regimes: nominal policy interest rate follows a partial-adjustment rule responding to expected future consumption price inflation, output gap, and (if managed) the nominal effective exchange rate change. Parameters and notation preserved (0    1 for partial adjustment, and response coefficients 1

, 0
Y
, 0
j
).
  - Fixed exchange rate regime: nominal policy rate tracks the leader economy’s rate one-for-one and responds to bilateral exchange rate deviations.
  - Leader economy in a monetary union targets are output-weighted across union members.
- Fiscal policy:
  - Countercyclical fiscal expenditure rules for public consumption and investment with partial adjustment dynamics tracking potential output (parameters 01G).
  - Acyclical fiscal revenue rules for corporate and labor tax rates (parameters 01).
  - Non-discretionary lump sum transfer program stabilizes national financial wealth; discretionary transfers stabilize public financial wealth.
- Macroprudential policy:
  - Countercyclical regulatory capital buffer rule with partial adjustment responding to bank credit growth and contemporaneous housing and equity price changes.
  - Loan-to-value limits follow partial-adjustment rules responding to debt growth and asset price changes.
  - Loan default rates evolve with partial adjustment and depend on output gap and asset price changes.

### VI. Market clearing, accounting identities, and external sector
- Market clearing:
  - Final output market: exports equal production less domestic demand (equation reference preserved).
  - Final import market: imports equal total domestic demand.
  - Mortgage and corporate loan supply clear against developer and firm demands.
- External sector and national accounting:
  - Net foreign assets equal the sum of financial wealth of households, developers, firms, banks and government.
  - Current account equals net international investment income plus trade balance; national dynamic budget constraint explicit.
  - Multilateral consistency: world output-weighted trade balance sums to zero; global terms of trade shifter adjusts accordingly.

### Appendix A — Estimation of cyclical components
- Filter used: generalized Hodrick-Prescott filter due to Vitek (2014).
- For variables with long run trends (core price level, output price level, consumption price level, quantity of output, quantity of private consumption, quantity of exports, quantity of imports, price of housing, price of equity, nominal bilateral exchange rate, nominal wage, employment, quantity of public consumption, quantity of public investment, prices of nonrenewable energy and nonenergy commodities):
  - difference order: two
  - smoothing parameter: 16,000
- For variables without long run trends (nominal policy interest rate, nominal short term bond yield, nominal long term bond yield, unemployment rate, fiscal balance ratio):
  - difference order: one
  - smoothing parameter: 400

### Appendix A — Estimation procedure (state space, innovations, and volatility)
- Model setup and solution:
  - Vector stochastic process xt of N nonpredetermined endogenous variables, M observed.
  - Exogenous vector νt of dimension K follows stationary first order stochastic linear difference equation with iid innovations εt such that 1/2 1 |~iid (,) ttK − − HεI0∁N (as stated).
  - Unique stationary solution: xt = C1 xt−1 + C2 νt (equation (254)); solution calculated with Klein (2000) procedure.
- Asymmetric multivariate GARCH for Ht (equation (255)):
  - Ht depends on past H, outer products of past innovations ε and asymmetric innovation vectors η, with diagonal matrices D0, D1, D2, D3 (dimension K).
  - ηt = max(εt, 0) (max( , )t t=η ε 0 as stated).
  - Positive definiteness: diagonal elements of D0 positive; all elements of D1, D1+ D2 and D3 nonnegative.
- State space representation (equations (256)–(257)):
  - yt = F zt
  - zt = G1 zt−1 + G2 εt
  - Initial states: 0|0 ~ N(0, Pzz0) and independent of innovations imply conditional multivariate normality.
- Filtering, smoothing and estimation:
  - Kalman filter (Kalman (1960)) and de Jong (1989) smoother extended for asymmetric multivariate GARCH.
  - Prediction (eqs. (258)–(262)), updating (eqs. (263)–(265)), smoothing (eqs. (266)–(270)); recursive forward and backward evaluation yield smoothed state estimates and mean squared error matrix.
  - Full information maximum likelihood: parameter vector θ (J-dimensional) maximizes conditional loglikelihood ΛT(θ) = Σt λt(θ) (equations (271)–(272)); under regularity θ̂T consistent and asymptotically normal (equation (272)). Consistent estimators of asymptotic variance components A0 and B0 given by (273)–(274).

### Appendix A — Estimation results (calibration and estimated variance parameters)
- Partitioning of parameters:
  - Parameters determining conditional means are calibrated.
  - Parameters determining conditional variances are estimated.
- Calibrated parameter values (as stated):
  - subjective discount factor β set to imply an annualized discount rate of 4 percent
  - habit persistence in consumption Cα = 0.80
  - habit persistence in labor supply Lα = 0.80
  - intertemporal consumption elasticity σ = 1.00
  - intratemporal labor supply elasticity η = 1.00
  - adjustment cost parameters Hχ = 1.50 and Kχ = 1.50
  - depreciation parameters Hδ and Kδ set to imply annualized depreciation rates of 10 percent
  - partial indexation parameters Yγ, Lγ, Xγ, Mγ = 0.80
  - nominal rigidity parameters Yω, Lω, Xω, Mω set to imply average reoptimization intervals of 6 quarters
  - credit constrained household share Cφ = 0.50
  - financial friction parameter Cω set to imply an average adjustment interval of 4 quarters
  - intratemporal import demand elasticity Mψ = 1.00
- Regime and calibration choices:
  - Monetary policy regimes: flexible inflation targeting for Australia, Canada, the Euro Area, Japan, Mexico, Russia, the United Kingdom and the United States; managed exchange rate for Brazil, China, India and Korea (consistent with IMF (2016) de facto classification).
  - Within the Euro Area, Germany is the leader economy.
  - High interbank market contagion economies: advanced economies; low interbank contagion: China.
  - High capital market contagion economies: emerging market economies with open capital accounts; low capital market contagion: China.
  - Quotation currency for FX transactions: issued by the United States.
  - Macroeconomic and financial great ratios, bilateral trade, bank lending, nonfinancial corporate borrowing, portfolio debt and equity investment weights, and world weights calibrated to observed 2016 values (normalized as described).
- Numerical optimization and diagnostics:
  - Conditional loglikelihood numerically maximized over effective sample period 1999Q3 through 2017Q4.
  - Calculations performed to quadruple precision where necessary and double precision otherwise.
  - Point estimates satisfy sufficient conditions for positive definiteness of the conditional covariance matrix.
- Key statistical findings on volatility dynamics:
  - Strong evidence of ARCH effects in structural shocks; null of no ARCH effects rejected at conventional significance levels.
  - Point estimate for hα = 0.15.
  - Strong evidence of asymmetric ARCH effects for some conditional variances (financial and monetary policy shocks); null of symmetric ARCH effects rejected at conventional significance levels.
  - Point estimate for hγ = −0.11 (negative value implies positive financial and monetary policy shocks raise conditional variances less than negative shocks of equal magnitude).
  - Little or no evidence of more persistent asymmetric GARCH effects beyond these asymmetric ARCH effects.
- Goodness of fit and time variation:
  - Estimated conditional variances of structural shocks exhibit substantial time variation and volatility clustering.
  - Volatility generally peaked during the Global Financial Crisis.
  - Approximation errors in evaluating historical decompositions of output volatility are negligible, amounting to 0.03 percent of the conditional variance of output in absolute value, on average across economies and over time.

### Appendix A — Inference: historical decompositions and main empirical findings
- Historical decompositions methodology:
  - Level decomposition of smoothed state vector into deterministic and stochastic components via contributions from contemporaneous and past smoothed innovations (equation (275)).
  - Volatility decomposition of predicted covariance matrix into deterministic and stochastic components via contributions from contemporaneous conditional variances (equation (276)).
- Main empirical attributions and stylized facts:
  - Cyclical output fluctuations primarily attributed to economy-specific combinations of domestic and foreign macroeconomic and financial shocks, generally mitigated by policy shocks.
  - Buildup to the Global Financial Crisis: supportive macroeconomic and financial shocks caused gradual synchronized global expansion.
  - Global Financial Crisis: adverse macroeconomic and financial shocks concentrated in the United States caused abrupt synchronized contraction, mitigated by supportive policy shocks.
  - Post-crisis recovery: supportive macroeconomic and financial shocks contributed to synchronized recovery; later derailed in parts by adverse financial shocks (Italy, Spain during Euro Area Sovereign Debt Crisis; Brazil and Russia after the Taper Tantrum and commodity price collapse).
  - Over time, average relative contribution from domestic versus foreign shocks decreases with trade openness.
- Financial conditions and volatility attributions:
  - No single linear financial conditions index implied by the approximate linear model; inferred as contributions from financial and monetary policy shocks to output.
  - Financial conditions index decomposed into contributions from domestic and foreign housing risk premium, monetary policy and credit risk premium, duration risk premium, and equity risk premium shocks.
  - Abrupt tightenings of financial conditions amplify cyclical output contractions.
  - Domestic housing risk premium dynamics contributed substantially in Spain, the United Kingdom, and the United States around the Global Financial Crisis.
  - Foreign housing risk premium shocks contributed substantially in Canada, Germany and Mexico, transmitted primarily via trade linkages.
  - Domestic equity risk premium shocks in the United States contributed substantially to worldwide cyclical dynamics via financial linkages.
  - Domestic monetary policy and credit risk premium shocks generally mitigated these dynamics.
  - Duration risk premium decompression contributed substantially to severe contractions in Italy, Spain, Brazil and Russia in crisis episodes; safe-haven inflows mitigated contractions in France and Germany.
- Volatility contributions:
  - Output volatility primarily generated by conditional variances of macroeconomic shocks; relative contribution from domestic vs foreign macro shocks decreases with trade openness.
  - For major net commodity exporters (Australia, Canada, Russia), conditional variances of world terms of trade shocks also contribute substantially to output volatility.
  - Over time, contribution from domestic macro shock variances to output volatility is essentially constant; contribution from foreign macro shock variances varies considerably and peaks during trade disruptions coinciding with financial turbulence.
  - Financial conditions volatility primarily generated by conditional variances of domestic monetary policy and credit risk premium shocks, secondarily by duration risk premium shocks (which dominate during financial market turbulence).
  - Across economies, relative contribution from domestic vs foreign duration risk premium shocks increases with financial depth and decreases with financial openness; U.S. duration risk premium variance makes a dominant global contribution.
  - Conditional variances of housing and equity risk premia shocks contribute little to financial conditions volatility and, by implication, to output volatility across economies and over time.

### Appendix A — Data Description (variables, sources, calibration)
- Sample period: 1999Q1 through 2017Q4.
- Economies: Australia, Brazil, Canada, China, France, Germany, India, Italy, Japan, Korea, Mexico, Russia, Spain, the United Kingdom, and the United States.
- Primary data sources: GDS and WEO databases compiled by the IMF; Bloomberg; Bank for International Settlements.
- Secondary sources where unavailable: IFS database (IMF); ILO database (International Labour Organization).
- Calibration sources: IMF databases where available, otherwise BIS, World Bank Group or World Federation of Exchanges.
- Specific database uses:
  - Macroeconomic great ratios: WEO and WDI databases.
  - Financial great ratios: BIS, CPIS and WFE databases.
  - Bilateral trade weights: DOTS database.
  - Bilateral bank lending and nonfinancial corporate borrowing weights (consolidated ultimate risk basis): BIS database.
  - Bilateral portfolio debt and equity investment weights: CPIS and WFE databases.
- Macroeconomic variables and proxies:
  - Core price level: seasonally adjusted core consumer price index.
  - Output price level: seasonally adjusted gross domestic product price deflator.
  - Consumption price level: seasonally adjusted consumer price index.
  - Quantity of output: seasonally adjusted real gross domestic product.
  - Quantity of private consumption: seasonally adjusted real private consumption expenditures.
  - Quantity of exports: seasonally adjusted real export revenues.
  - Quantity of imports: seasonally adjusted real import expenditures.
  - Price of housing: broad residential property price index.
  - Nominal wage: derived from the quadratically interpolated annual labor income share.
  - Unemployment rate: seasonally adjusted share of total unemployment in the total labor force.
  - Employment: seasonally adjusted total employment.
  - Quantity of public consumption and public investment: quadratically interpolated annual real expenditures of the general government.
  - Fiscal balance: quadratically interpolated annual overall fiscal balance of the general government.
  - Prices of energy and nonenergy commodities: broad commodity price indexes denominated in United States dollars.
- Financial market variables and measurement:
  - Nominal policy interest rate: central bank policy rate.
  - Nominal short term bond yield: three month Treasury bill yield.
  - Nominal long term bond yield: ten year government bond yield.
  - Price of equity: broad stock price index denominated in domestic currency units.
  - Nominal bilateral exchange rate: domestic currency price of one United States dollar.
  - All financial market variables are expressed as period average values.
- Calibration and seasonality:
  - Calibration based on annual data from IMF and other international databases.
  - Bilateral weights and financial linkages constructed from DOTS, BIS (consolidated ultimate risk), CPIS and WFE databases.
  - Note on seasonality: The calibration is a function of the seasonal frequency S, evaluated at 4S=.

### Appendix A — Selected calibrated parameter excerpts (Table 2 entries reproduced exactly as presented)
- Cα 0.80000
- Lα 0.80000
- Cϕ 0.50000
- Dϕ 0.70000
- Hχ 1.50000
- Kχ 1.50000
- Bχ 1.50000
- Cχ 1/0.10000
- A Sϕ 0.12500
- δ 0.01000/S
- Mψ 1.00000
- ρA 0.95000
- iρ 0.80000
- κρ 0.80000
- δρ 0.60000
- νρA 0.40000
- νρN 0.40000
- τρ 0.80000
- Cνρ 0.40000
- Xγ 0.80000
- Mγ 0.80000
- Yθ (1 0.15000)/0.15000+
- Cθ (1 0.01000/ )/(0.01000/SS+
- πξ 2.00000
- Yλ 0.01000
- Yξ 0.50000/S
- Xμ 0.25000
- Mμ 0.15000
- kξE 2.00000
- ξE 0.10000
- Bκζ 0.05000S
- H Vκζ 0.02500S
- S Vκζ 0.01250S
- D Bϕ ζ 0.05000S
- D Vϕ ζ 0.02500S
- F Bϕ ζ 0.05000S
- F Vϕ ζ 0.01250S
- M Yδ ζ 0.10000/S

(Note: Table 2 presents many calibrated parameter values in symbolic form; entries reproduced exactly as in the source.)

### Appendix A — Selected estimated parameter excerpts (Table 3 entries reproduced exactly as presented)
- νωA 0 1.71 10 *** + ×
- X Jω 6 1.95 10 *** + ×
- ,GC νω 1 1.95 10 *** − ×
- νωN 0 5.11 10 *** + ×
- M Jω 3 4.69 10 *** + ×
- ,GI νω 0 9.41 10 *** + ×
- C νω 2 1.22 10 *** + ×
- H νω 1 1.94 10 *** − ×
- τ νω 1 2.89 10 *** − ×
- I νω 1 4.47 10 *** + ×
- ,iS νω 2 1.02 10 *** − ×
- Yk Jω 1 3.39 10 *** + ×
- X νω 0 7.27 10 *** + ×
- B νω 1 4.54 10 *** − ×
- hα 1 1.46 10 *** − ×
- M νω 0 9.53 10 *** + ×
- S νω 0 2.25 10 *** + ×
- hγ 1 1.10 10 *** − − ×
- Y Jω 2 2.80 10 *** + ×
- νωE 0 6.98 10 *** + ×
- L Jω 3 1.74 10 *** + ×
- ,iP νω 2 7.37 10 *** − ×

- Note: Statistical significance at the 1, 5 and 10 percent levels is indicated by ***, ** and *, respectively.

### Appendix A — Conclusion and suggested research avenues
- Conclusion: The DSGE model augmented with structural shocks exhibiting asymmetric GARCH effects finds very strong evidence of asymmetric ARCH effects, enabling joint decomposition of levels and volatilities of output and financial conditions into time-varying shock contributions.
- Main empirical implication: Risk premia shocks estimated to contribute disproportionately to cyclical output fluctuations and turbulence during swings in financial conditions across the fifteen largest national economies.
- Suggested future research directions:
  - Test whether asymmetric ARCH effects persist after solving the DSGE model to second order or allowing time-varying parameters.
  - Assess empirical adequacy of asymmetric ARCH versus asymmetric stochastic volatility (SV) effects as representations of volatility clustering in structural shocks.

*Source: wp18238 - Appendix A . Data Description.*

### Appendix A . Data Description ..........................................................................................

### Appendix A . Data Description ..................................................................................................... 69

### Appendix B . Estimation Results .................................................................................................. 71
- Table 2 . Calibrated Parameter Values ......................................................................... 71
- Table 3 . Estimated Parameter Values .......................................................................... 71

*Source: wp18238 - Appendix A . Data Description.*

### References .............................................................................................................

### wp18238 - References

### I. High-level findings and research purpose
- Purpose: Augment an approximate linear DSGE model of the world economy with structural shocks exhibiting potentially asymmetric generalized autoregressive conditional heteroskedasticity (GARCH) effects to jointly decompose levels and volatilities of output and financial conditions.
- Key empirical finding: Very strong evidence of asymmetric autoregressive conditional heteroskedasticity (ARCH) effects is found; risk premia shocks are estimated to contribute disproportionately to cyclical output fluctuations and turbulence during abrupt swings in financial conditions across the fifteen largest national economies.
- Scope: Analysis covers the fifteen largest national economies in the world: Australia, Brazil, Canada, China, France, Germany, India, Italy, Japan, Korea, Mexico, Russia, Spain, the United Kingdom, and the United States.
- Sample period: 1999Q1 through 2017Q4.
- Data and dimensionality:
  - Number of endogenous variables observed: 280.
  - Number of structural shocks driving the model: 281.

### II. Model architecture (theoretical framework)
- Model class:
  - An approximate linear dynamic stochastic general equilibrium (DSGE) model refined from Vitek (2018).
  - Features: range of nominal and real rigidities, extensive macrofinancial linkages, bank and capital-market intermediation, and diverse spillover channels.
- Economic agents and sectors modeled explicitly:
  - Households (three types: bank-intermediated, capital-market-intermediated, credit-constrained), developers, firms (industry, intermediate, final), banks (intermediate and final), government (monetary, fiscal, macroprudential authorities).
  - Construction sector (developers, housing demand and supply, residential investment, mortgage collateral constraints).
  - Production sector (industries, nonrenewable commodities for energy and nonenergy, business investment, capital utilization).
  - Banking sector (mortgage and corporate lending, regulatory capital requirement, bank capital accumulation, loan supply frictions).
  - Trade sector (exports, imports, export and import price setting, real and nominal effective exchange rates).
- Financial features:
  - Portfolio structures include internationally diversified short term bond, long term bond and stock portfolios; linkages through bilateral nominal exchange rates.
  - Collateralized borrowing: developer and firm borrowing constrained by regulatory loan to value ratio limits, following Kiyotaki and Moore (1997) financial accelerator mechanism.
  - Bank regulation: countercyclical capital buffer rule, loan-to-value limits for mortgages and corporate loans; bank regulatory costs and capital accumulation modeled.
- Price and wage rigidities:
  - Calvo-style nominal price rigidity extensions for output, export, import and intermediate sectors (partial indexation allowed).
  - Nominal wage rigidity following Erceg, Henderson and Levin (2000) with involuntary unemployment interpretation along Galí (2011) and partial indexation.
- Optimality and equilibrium:
  - Agents optimize intertemporally under rational expectations; equilibrium derived by linearizing first-order conditions around a deterministic steady state with zero inflation, zero productivity and labor force growth, and steady public and national financial wealth (unless stated otherwise).

### III. Empirical framework and estimation strategy
- Linearization and state space:
  - The DSGE equilibrium conditions are analytically linearized and consolidated into a multivariate linear rational expectations representation driven by potentially heteroskedastic structural shocks.
  - Response coefficients are functions of behavioral parameters restricted to coincide across economies or groups, plus economy-specific steady-state characteristics.
- Shocks and stochastic structure:
  - Structural shocks include productivity, labor supply, consumption demand, investment demand (residential and business consolidated in estimation), export and import demand, price and wage markups, housing, portfolio and duration risk premia, credit risk premium, liquidity risk premium, currency risk premia, monetary, fiscal and macroprudential policy shocks, among others.
  - Many shocks follow stationary first-order autoregressive processes; innovations are conditionally normally distributed with conditional variances following symmetric or asymmetric GARCH specifications.
  - Asymmetric conditional variance specification: threshold GARCH (Glosten, Jaganathan and Runkle (1993)) extension of GARCH for key risk premia and policy shocks.
- Estimation approach:
  - Joint estimation by full information maximum likelihood of subsets of parameters and variables, conditional on calibrated parameters and observed variables.
  - A restricted version of the model is used for estimation where weakly identified shocks are consolidated or eliminated:
    - Residential and business investment demand shocks consolidated into a single investment demand shock.
    - Corporate and labor income tax rate shocks consolidated into a tax rate shock.
    - Liquidity risk premium, mortgage and corporate loan rate markup, mortgage and corporate loan default, transfer payment, capital requirement, and mortgage and corporate loan to value limit shocks are eliminated (treated as homoskedastic or removed).
  - Impulse response analysis can be based on the unrestricted version of the model.
- Identification assumptions:
  - As an identifying restriction, all innovations are assumed independent conditional on the model (conditional multivariate normality).
  - Many innovation processes are allowed to be heteroskedastic; some shocks modeled as homoskedastic where identification is weak.
- Data transformations and observables:
  - Estimation conditions on cyclical components of 280 endogenous variables across the fifteen economies for 1999Q1–2017Q4.
  - Observed macro and financial variables include: core price level, output price level, consumption price level, output quantity, private consumption quantity, exports, imports, nominal policy interest rate (shadow rate substituted during ELB for systemic advanced economies), nominal short and long term bond yields, housing and equity prices, nominal bilateral exchange rates, nominal wage, unemployment rate, employment, public consumption and investment quantities, fiscal balance ratio, prices of nonrenewable energy and nonenergy commodities.

### IV. Shocks, volatility modeling, and macrofinancial amplification
- Conditional mean dynamics: many structural shocks follow AR(1) processes (productivity, labor supply, consumption demand, investment and trade demand shocks, risk premia), with economy-specific persistence parameters.
- Conditional variance dynamics:
  - Symmetric GARCH processes for a large set of shocks (productivity, labor supply, demand shocks, many markup shocks, commodity price markups, selected fiscal shocks).
  - Asymmetric GARCH (threshold GARCH) processes for key risk premia and monetary policy shocks, capturing that positive and negative innovations can have different impacts on conditional variance (e.g., housing risk premium, credit risk premium, duration and equity risk premia, monetary policy shock).
- Macrofinancial amplification:
  - The model’s macrofinancial linkages (loan-to-value constraints, bank capital regulation, credit spreads, asset prices) generate amplification channels where increases in risk premia produce both mean and variance effects on output and financial conditions.
  - Empirical evidence: risk premia shocks are estimated to disproportionately contribute to cyclical output fluctuations and turbulence during abrupt swings in financial conditions (consistent with Adrian, Boyarchenko and Giannone (2017) for the United States).

### V. Policy rules embedded in the model
- Monetary policy:
  - Flexible inflation targeting and managed exchange rate regimes: nominal policy interest rate follows a partial-adjustment rule responding to expected future consumption price inflation, output gap, and (if managed) the nominal effective exchange rate change. Parameters: 0    1 for partial adjustment, and response coefficients 1

, 0
Y
, 0
j
 (text notation preserved).
  - Fixed exchange rate regime: nominal policy rate tracks the leader economy’s rate one-for-one and responds to bilateral exchange rate deviations.
  - Leader economy in a monetary union targets are output-weighted across union members.
- Fiscal policy:
  - Countercyclical fiscal expenditure rules for public consumption and investment with partial adjustment dynamics tracking potential output (parameters 01G).
  - Acyclical fiscal revenue rules for corporate and labor tax rates (parameters 01).
  - Non-discretionary lump sum transfer program that stabilizes national financial wealth; discretionary transfers stabilize public financial wealth.
  - Fiscal reaction functions and government budget constraint explicitly modeled; long-term yields and short-term yields linked through money and interbank market relationships with credit and liquidity risk premia.
- Macroprudential policy:
  - Countercyclical regulatory capital buffer rule with partial adjustment dynamics responding to bank credit growth and contemporaneous changes in housing and equity prices (parameters and notation preserved).
  - Loan-to-value ratio limits for mortgages and corporate loans follow partial-adjustment rules, responding to mortgage or corporate debt growth and changes in housing or equity prices.
  - Loan default rates evolve with partial adjustment and depend on output gap and asset price changes.
  - These macroprudential tools interact with lending spreads, loan supply, and bank capital dynamics in the model.

### VI. Market clearing, accounting identities, and equilibrium conditions
- Final goods, import, and bank loan market clearing conditions are explicit:
  - Final output market: exports equal production less domestic demand (equation reference preserved in text).
  - Final import market: imports equal total domestic demand.
  - Mortgage and corporate loan supply clear against developer and firm demands.
- External sector and net foreign asset dynamics:
  - Net foreign assets equal the sum of financial wealth of households, developers, firms, banks and government.
  - Current account equals net international investment income plus trade balance; national dynamic budget constraint explicit.
  - Multilateral consistency: world output-weighted trade balance sums to zero; global terms of trade shifter adjusts accordingly.

*Italicized source: Content drawn exclusively from wp18238 - References (wp18238 - References .............................................................................................................; canonical PDF https://www.imf.org/-/media/files/publications/wp/2018/wp18238.pdf).*

### Appendix A.

### Appendix A.

### Estimation of cyclical components
- Cyclical components estimated with the generalized Hodrick-Prescott filter due to Vitek (2014).
- For variables with long run trends (core price level, output price level, consumption price level, quantity of output, quantity of private consumption, quantity of exports, quantity of imports, price of housing, price of equity, nominal bilateral exchange rate, nominal wage, employment, quantity of public consumption, quantity of public investment, prices of nonrenewable energy and nonenergy commodities):
  - difference order: two
  - smoothing parameter: 16,000
- For variables without long run trends (nominal policy interest rate, nominal short term bond yield, nominal long term bond yield, unemployment rate, fiscal balance ratio):
  - difference order: one
  - smoothing parameter: 400

### Estimation procedure (state space, innovations, and volatility)
- Model setup:
  - Vector stochastic process xt of N nonpredetermined endogenous variables, M observed.
  - Exogenous vector νt of dimension K follows stationary first order stochastic linear difference equation with iid innovations εt such that 1/2 1 |~iid (,) ttK − − HεI0∁N (as stated).
  - Unique stationary solution expressed as xt = C1 xt−1 + C2 νt (equation (254)).
  - Unique stationary solution calculated with the procedure due to Klein (2000).
- Asymmetric multivariate GARCH specification for the conditional covariance matrix Ht (equation (255)):
  - Ht depends on past H, outer products of past innovations ε and asymmetric innovation vectors η, with diagonal matrices D0, D1, D2, D3 (dimension K).
  - ηt = max(εt, 0) (max( , )t t=η ε 0 as stated).
  - Positive definiteness conditions: diagonal elements of D0 positive; all elements of D1, D1+ D2 and D3 nonnegative.
- State space representation for observed vector yt and full state vector zt (equations (256)–(257)):
  - yt = F zt
  - zt = G1 zt−1 + G2 εt
  - Initial states: 0|0 ~ N(0, Pzz0) and independent of innovations imply conditional multivariate normality.
- Filtering and smoothing:
  - Kalman filter (Kalman (1960)) and de Jong (1989) smoother, extended for asymmetric multivariate GARCH.
  - Prediction equations (equations (258)–(262)) and updating equations (equations (263)–(265)) specified; smoothing equations (equations (266)–(270)) provided.
  - Recursive forward evaluation (eqs. (258)–(265)) and backward evaluation (eqs. (266)–(269)) yield smoothed state estimates and mean squared error matrix.
- Full information maximum likelihood:
  - Parameter vector θ (J-dimensional) maximizes conditional loglikelihood ΛT(θ) = Σt λt(θ) (equations (271)–(272)).
  - Under regularity, estimator θ̂T is consistent and asymptotically normal as stated (equation (272)).
  - Consistent estimators of the asymptotic variance components A0 and B0 given by (273)–(274) with definitions of at and bt as in the text.
  - If distributional assumptions hold, information matrix equality holds and A0 = B0.

### Estimation results
- Partitioning of parameters:
  - Parameters determining conditional means are calibrated.
  - Parameters determining conditional variances are estimated.
- Calibrated parameter values (as stated):
  - subjective discount factor β set to imply an annualized discount rate of 4 percent
  - habit persistence in consumption Cα = 0.80
  - habit persistence in labor supply Lα = 0.80
  - intertemporal consumption elasticity σ = 1.00
  - intratemporal labor supply elasticity η = 1.00
  - adjustment cost parameters Hχ = 1.50 and Kχ = 1.50
  - depreciation parameters Hδ and Kδ set to imply annualized depreciation rates of 10 percent
  - partial indexation parameters Yγ, Lγ, Xγ, Mγ = 0.80
  - nominal rigidity parameters Yω, Lω, Xω, Mω set to imply average reoptimization intervals of 6 quarters
  - credit constrained household share Cφ = 0.50
  - financial friction parameter Cω set to imply an average adjustment interval of 4 quarters
  - intratemporal import demand elasticity Mψ = 1.00
- Regime and calibration choices:
  - Monetary policy: flexible inflation targeting for Australia, Canada, the Euro Area, Japan, Mexico, Russia, the United Kingdom and the United States; managed exchange rate for Brazil, China, India and Korea (consistent with IMF (2016) de facto classification).
  - Within the Euro Area, Germany is the leader economy.
  - High interbank market contagion economies: advanced economies; low interbank contagion: China.
  - High capital market contagion economies: emerging market economies with open capital accounts; low capital market contagion: China.
  - Quotation currency for FX transactions: issued by the United States.
  - Macroeconomic and financial great ratios, bilateral trade, bank lending, nonfinancial corporate borrowing, portfolio debt and equity investment weights, and world weights calibrated to observed 2016 values (normalized as described).
- Estimated variance parameters and numerical optimization:
  - Conditional loglikelihood numerically maximized over effective sample period 1999Q3 through 2017Q4.
  - Calculations performed to quadruple precision where necessary and double precision otherwise.
  - Point estimates satisfy sufficient conditions for positive definiteness of the conditional covariance matrix.
- Key statistical findings on volatility dynamics:
  - Strong evidence of ARCH effects in structural shocks; null of no ARCH effects rejected at conventional significance levels.
  - Point estimate for hα = 0.15.
  - Strong evidence of asymmetric ARCH effects for some conditional variances (financial and monetary policy shocks); null of symmetric ARCH effects rejected at conventional significance levels.
  - Point estimate for hγ = −0.11 (negative value implies positive financial and monetary policy shocks raise conditional variances less than negative shocks of equal magnitude).
  - Little or no evidence of more persistent asymmetric GARCH effects beyond these asymmetric ARCH effects.
- Goodness of fit and time variation:
  - Estimated conditional variances of structural shocks exhibit substantial time variation and volatility clustering.
  - Volatility generally peaked during the Global Financial Crisis.
  - Approximation errors in evaluating historical decompositions of output volatility are negligible, amounting to 0.03 percent of the conditional variance of output in absolute value, on average across economies and over time.

### Inference: historical decompositions and main empirical findings
- Historical decompositions methodology:
  - Level decomposition of smoothed state vector into deterministic and stochastic components via contributions from contemporaneous and past smoothed innovations (equation (275)).
  - Volatility decomposition of predicted covariance matrix into deterministic and stochastic components via contributions from contemporaneous conditional variances (equation (276)).
- Main empirical attributions and stylized facts:
  - Cyclical output fluctuations primarily attributed to economy-specific combinations of domestic and foreign macroeconomic and financial shocks, generally mitigated by policy shocks.
  - Buildup to the Global Financial Crisis: supportive macroeconomic and financial shocks caused gradual synchronized global expansion.
  - Global Financial Crisis: adverse macroeconomic and financial shocks concentrated in the United States caused abrupt synchronized contraction, mitigated by supportive policy shocks.
  - Post-crisis recovery: supportive macroeconomic and financial shocks contributed to synchronized recovery; later derailed in parts by adverse financial shocks (Italy, Spain during Euro Area Sovereign Debt Crisis; Brazil and Russia after the Taper Tantrum and commodity price collapse).
  - Over time, average relative contribution from domestic versus foreign shocks decreases with trade openness.
- Financial conditions index (inferred measure):
  - No single linear financial conditions index implied by the approximate linear model; instead inferred as the contribution from financial and monetary policy shocks to output.
  - Financial conditions index decomposed into contributions from domestic and foreign housing risk premium, monetary policy and credit risk premium, duration risk premium, and equity risk premium shocks.
  - Abrupt tightenings of financial conditions amplify cyclical output contractions.
  - Domestic housing risk premium dynamics contributed substantially to cyclical dynamics in Spain, the United Kingdom, and the United States around the Global Financial Crisis.
  - Foreign housing risk premium shocks contributed substantially to cyclical dynamics in Canada, Germany and Mexico, transmitted primarily via trade linkages.
  - Domestic equity risk premium shocks in the United States contributed substantially to worldwide cyclical dynamics via financial linkages.
  - Domestic monetary policy and credit risk premium shocks generally mitigated these dynamics.
  - Duration risk premium decompression contributed substantially to severe contractions in Italy, Spain, Brazil and Russia in crisis episodes; safe-haven inflows mitigated contractions in France and Germany.
- Volatility contributions:
  - Output volatility primarily generated by conditional variances of macroeconomic shocks; relative contribution from domestic vs foreign macro shocks decreases with trade openness.
  - For major net commodity exporters (Australia, Canada, Russia), conditional variances of world terms of trade shocks also contribute substantially to output volatility.
  - Over time, contribution from domestic macro shock variances to output volatility is essentially constant; contribution from foreign macro shock variances varies considerably and peaks during trade disruptions coinciding with financial turbulence.
  - Financial conditions volatility primarily generated by conditional variances of domestic monetary policy and credit risk premium shocks, secondarily by duration risk premium shocks (which dominate during financial market turbulence).
  - Across economies, relative contribution from domestic vs foreign duration risk premium shocks increases with financial depth and decreases with financial openness; U.S. duration risk premium variance makes a dominant global contribution.
  - Conditional variances of housing and equity risk premia shocks contribute little to financial conditions volatility and, by implication, to output volatility across economies and over time.

### Conclusion and research avenues
- The DSGE model augmented with structural shocks exhibiting asymmetric GARCH effects finds very strong evidence of asymmetric ARCH effects, enabling joint decomposition of levels and volatilities of output and financial conditions into time-varying shock contributions.
- Risk premia shocks estimated to contribute disproportionately to cyclical output fluctuations and turbulence during swings in financial conditions across the fifteen largest national economies.
- Suggested future research directions (computationally demanding):
  - Test whether asymmetric ARCH effects persist after solving the DSGE model to second order or allowing time-varying parameters.
  - Assess empirical adequacy of asymmetric ARCH versus asymmetric stochastic volatility (SV) effects as representations of volatility clustering in structural shocks.

*Source: wp18238 - Appendix A.*

### Appendix A. Data Description

### Appendix A. Data Description

### Sample and economies
- Sample period: 1999Q1 through 2017Q4.
- Economies: Australia, Brazil, Canada, China, France, Germany, India, Italy, Japan, Korea, Mexico, Russia, Spain, the United Kingdom, and the United States.

### Data sources
- Primary sources: GDS and WEO databases compiled by the IMF; Bloomberg; Bank for International Settlements.
- Secondary sources where unavailable: IFS database compiled by the IMF; ILO database produced by the International Labour Organization.
- Calibration sources: IMF databases where available, and otherwise the Bank for International Settlements, the World Bank Group or the World Federation of Exchanges.
- Specific database uses:
  - Macroeconomic great ratios: WEO and WDI databases.
  - Financial great ratios: BIS, CPIS and WFE databases.
  - Bilateral trade weights (goods, CIF basis): DOTS database.
  - Bilateral bank lending and nonfinancial corporate borrowing weights (consolidated ultimate risk basis): BIS database.
  - Bilateral portfolio debt and equity investment weights: CPIS and WFE databases.

### Macroeconomic variables and measurement/proxies
- Core price level: seasonally adjusted core consumer price index.
- Output price level: seasonally adjusted gross domestic product price deflator.
- Consumption price level: seasonally adjusted consumer price index.
- Quantity of output: seasonally adjusted real gross domestic product.
- Quantity of private consumption: seasonally adjusted real private consumption expenditures.
- Quantity of exports: seasonally adjusted real export revenues.
- Quantity of imports: seasonally adjusted real import expenditures.
- Price of housing: broad residential property price index.
- Nominal wage: derived from the quadratically interpolated annual labor income share.
- Unemployment rate: seasonally adjusted share of total unemployment in the total labor force.
- Employment: seasonally adjusted total employment.
- Quantity of public consumption: quadratically interpolated annual real consumption expenditures of the general government.
- Quantity of public investment: quadratically interpolated annual real investment expenditures of the general government.
- Fiscal balance: quadratically interpolated annual overall fiscal balance of the general government.
- Prices of energy and nonenergy commodities: broad commodity price indexes denominated in United States dollars.

### Financial market variables and measurement
- Nominal policy interest rate: central bank policy rate.
- Nominal short term bond yield: three month Treasury bill yield.
- Nominal long term bond yield: ten year government bond yield.
- Price of equity: broad stock price index denominated in domestic currency units.
- Nominal bilateral exchange rate: domestic currency price of one United States dollar.
- All financial market variables are expressed as period average values.

### Calibration and estimation notes
- Calibration is based on annual data from IMF and other international databases as noted above.
- Bilateral weights and financial linkages are constructed from DOTS, BIS (consolidated ultimate risk), CPIS and WFE databases.
- Note on seasonality: The calibration is a function of the seasonal frequency S, evaluated at 4S=.

### Table 2. Calibrated Parameter Values (excerpted notation as presented)
- Cα 0.80000
- Lα 0.80000
- Cϕ 0.50000
- Dϕ 0.70000
- Hχ 1.50000
- Kχ 1.50000
- Bχ 1.50000
- Cχ 1/0.10000
- A Sϕ 0.12500
- δ 0.01000/S
- Mψ 1.00000
- ρA 0.95000
- iρ 0.80000
- κρ 0.80000
- δρ 0.60000
- νρA 0.40000
- νρN 0.40000
- τρ 0.80000
- Cνρ 0.40000
- Xγ 0.80000
- Mγ 0.80000
- Yθ (1 0.15000)/0.15000+
- Cθ (1 0.01000/ )/(0.01000/SS+
- πξ 2.00000
- Yλ 0.01000
- Yξ 0.50000/S
- Xμ 0.25000
- Mμ 0.15000
- kξE 2.00000
- ξE 0.10000
- Bκζ 0.05000S
- H Vκζ 0.02500S
- S Vκζ 0.01250S
- D Bϕ ζ 0.05000S
- D Vϕ ζ 0.02500S
- F Bϕ ζ 0.05000S
- F Vϕ ζ 0.01250S
- M Yδ ζ 0.10000/S

(Note: Table 2 presents many calibrated parameter values in symbolic form; the above reproduces entries exactly as they appear in the source.)

### Table 3. Estimated Parameter Values (excerpted notation as presented)
- νωA 0 1.71 10 *** + ×
- X Jω 6 1.95 10 *** + ×
- ,GC νω 1 1.95 10 *** − ×
- νωN 0 5.11 10 *** + ×
- M Jω 3 4.69 10 *** + ×
- ,GI νω 0 9.41 10 *** + ×
- C νω 2 1.22 10 *** + ×
- H νω 1 1.94 10 *** − ×
- τ νω 1 2.89 10 *** − ×
- I νω 1 4.47 10 *** + ×
- ,iS νω 2 1.02 10 *** − ×
- Yk Jω 1 3.39 10 *** + ×
- X νω 0 7.27 10 *** + ×
- B νω 1 4.54 10 *** − ×
- hα 1 1.46 10 *** − ×
- M νω 0 9.53 10 *** + ×
- S νω 0 2.25 10 *** + ×
- hγ 1 1.10 10 *** − − ×
- Y Jω 2 2.80 10 *** + ×
- νωE 0 6.98 10 *** + ×
- L Jω 3 1.74 10 *** + ×
- ,iP νω 2 7.37 10 *** − ×

Note: Statistical significance at the 1, 5 and 10 percent levels is indicated by ***, ** and *, respectively.

*Source: IMF Working Paper — Appendix A. Data Description (wp18238 - Appendix A. Data Description).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18238.pdf_