## wpiea2020151-print-pdf

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---

### I. Introduction and objective
- Augment a linearized DSGE model (similar to Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2003)) with a tractable endogenous risk mechanism to support joint analysis of monetary and macroprudential policy.
- State dependent conditional heteroskedasticity mechanism:
  - Conditional variances of structural shocks specified as loglinear functions of the predetermined phase and position of the business or financial cycle, measured by the lagged change in and level of the output or financial gap.
  - Motivated by empirical evidence that macrofinancial volatility tends to be low during business cycle expansions and high during contractions, particularly when coinciding with financial cycle downturns.
- Key modeling goals:
  - Preserve the conditional mean function of the nested homoskedastic linearized DSGE model so standard linear perturbation solution algorithms remain applicable.
  - Allow conditional variances to be state dependent, producing location and scale shifts in conditional distributions and differing quantile impulse responses from mean impulse responses.
  - Provide a computationally inexpensive interim solution that can be bolted onto existing linearized DSGE models without simulation methods.

### II. Theoretical structure and mechanisms
- Model structure and features:
  - Nominal and real rigidities: short run nominal price and wage rigidities (monopolistic competition, staggered reoptimization, partial indexation), habit persistence in consumption, adjustment costs in residential and business investment, variable capital utilization, labor market with involuntary unemployment.
  - Financial sector: households access banks, real estate, and capital markets; banks issue risky mortgage and corporate loans at infrequently adjusted predetermined loan rates; banks accumulate capital out of retained earnings given credit losses to satisfy a capital requirement.
  - Borrowing collateralized against housing and capital stock values.
  - Government instruments: monetary and macroprudential policy via instrument rules.
- Analytical contributions:
  - Rational expectations equilibrium of the linearized DSGE model is invariant to the considered form of state dependent conditional heteroskedasticity when driven by stationary predetermined endogenous variables.
  - Derive an analytical extension of the Kalman (1960) filter to handle specified state dependent conditional heteroskedasticity, enabling estimation by Bayesian maximum likelihood without simulation.

### III. Empirical strategy and data
- Focus economies and sample:
  - Japan and the United States; sample period 1980Q1 to 2019Q3 (empirical focus) and effective estimation sample 1987Q1 to 2019Q3 (estimation results).
- Financial vulnerability indicators:
  - Credit: total credit to the nonfinancial private sector.
  - House price: broad residential property price index.
  - Equity price: broad stock price index.
  - All nominal variables deflated by the seasonally adjusted GDP price deflator.
- Cycle extraction:
  - Log real variables filtered via Hodrick and Prescott (1997) with smoothing parameter 16,000.
- Financial gap construction:
  - Aggregate standardized indicators {x_kit} using panel regression with indicator and economy specific fixed effects; two year lag order estimated by pooled quantile regression at the 0.05 quantile.
  - Estimated financial gap weights:
    - Credit gap weight: 0.72 (reported earlier) and 0.75 (calibration entry) — preserve both as they appear in source.
    - House price gap weight: 0.23 (estimated) and 0.20 (calibrated).
    - Equity price gap weight: 0.05.
  - Estimated financial gap peaks prior to combined bursting of credit-fueled property and stock market bubbles in Japan (early 1990s) and United States (late 2000s).

### IV. Stylized facts of growth at risk (reduced form and structural alignment)
- Definition and estimation:
  - Growth at risk defined as the conditional lower quantile of output growth; conditioning variables are lagged change in and level of the financial gap.
  - Output detrended via HP filter with smoothing parameter 16,000; conditional mean from pooled OLS with dummies; conditional quantiles from pooled quantile regression at 0.05 and 0.95 quantiles.
  - Horizons considered: one and three year lag orders (short and medium run).
- Key stylized facts:
  - The conditional lower quantile of output growth is more volatile than its conditional mean, which is more volatile than its conditional upper quantile.
  - Empirical ordering confirmed across Japan and the United States and across horizons: sample variance(lower quantile) > sample variance(mean) > sample variance(upper quantile).
  - Growth at risk estimates closely track business cycle contractions associated with financial cycle downturns (Japan early 1990s; US early 1990s, early 2000s, late 2000s); do not fully track Japan late 2000s contraction because it did not coincide with a sharp domestic financial cycle downturn.
- Additional empirical regularities:
  - Across economies and horizons, sample correlation between the conditional mean and variance of output growth is negative, ranging from -0.71 to -0.94 in one statement and -0.62 to -0.84 in another statement in the source — preserve both occurrences as reported.
  - Intuition: falls in conditional mean are associated with rises in conditional variance (amplifying lower quantiles); rises in conditional mean are associated with falls in conditional variance (mitigating upper quantiles).
  - If output growth is conditionally normally distributed, conditional α quantile qit,α = μit + zα σit (source explanatory note preserved).

### V. Heteroskedastic linearized DSGE: specification, estimation, and diagnostics
- State space and heteroskedastic specification:
  - Linearized DSGE expressed as augmented linear state space with empirically flexible trend components and state dependent conditional heteroskedasticity for structural shocks.
  - Decompose variables into cyclical and trend components; trend components follow AR(1) convergence to long run balanced growth with constant π, g, n.
  - Structural shocks: AR(1) or iid with conditionally normal heteroskedastic innovations.
  - Conditional variances:
    - Loglinear functions of past change in and level of the output gap for macro shocks (equation (102)).
    - Loglinear functions of past change in and level of the financial gap for financial shocks (equation (103)).
    - These ensure positive conditional variances ĥ^Z_t converging to unconditional variances 2ˆ Z σ.
- Estimation approach:
  - Bayesian maximum likelihood treating augmented state space as conditionally Gaussian; extend Kalman filter to handle state dependent conditional heteroskedasticity.
  - Posterior maximized with the differential evolution algorithm (Storn and Price (1997)).
  - Priors: strongly identified parameters conditional on informative priors; weakly identified calibrated; variance parameters estimated with diffuse priors.
  - Data used for estimation: levels of twelve endogenous variables for Japan or United States over 1986Q3 to 2019Q3 (listed variables preserved).
  - Trend smoothing: difference order 2 with smoothing parameter 16,000 for variables with long run trends; difference order 1 with smoothing parameter 400 otherwise.
- Diagnostic performance:
  - Augmenting linearized DSGE with parsimonious state dependent heteroskedasticity substantially improves goodness of fit to conditional distributions for Japan and the United States.
  - Matches key stylized facts from the growth at risk literature.
  - Sequential dynamic forecast performance (Theil's U, holdout 1990Q2 to 2019Q3):
    - Japan: RMS prediction error is 32 percent lower for inflation and 52 percent lower for output growth (model vs random walk).
    - United States: RMS prediction error is 22 percent lower for inflation and 59 percent lower for output growth.
  - Heteroskedastic specification has little effect on predicted means when parameters and initial conditions are identical to homoskedastic model but substantially affects coverage of forecast confidence intervals:
    - Forecast intervals expand considerably at times of macrofinancial stress (e.g., Japan originating 1992Q4; US originating 2008Q4), and contract slightly during normal times.
    - Quantiles and probability density functions estimated with Monte Carlo simulation with 100,000 replications.

### VI. Priors, calibration, and posterior mode findings
- Prior means and calibrations (selected):
  - Mortgage and corporate loan to value ratio parameters Hϕ and Fϕ: 0.80.
  - Financial friction parameter Cω implies average adjustment interval of 4.0 quarters.
  - Partial adjustment parameter in monetary policy rule Siρ: 0.80.
  - Expected inflation response coefficient πξ prior mean: 1.50.
  - Output gap response coefficient GYξ prior mean: 0.50 (annual frequency).
  - Calibrated parameters:
    - β set to imply discount rate of 4.0 percent.
    - Hδ set to imply housing depreciation rate of 3.0 percent (annual).
    - Kδ set to imply capital depreciation rate of 9.0 percent (annual).
    - Pθ set to imply steady state price markup of 15.0 percent.
    - Steady state labor income share: 0.67.
    - Steady state bank lending rate Ci: 6.0 percent (annual).
    - Steady state default rate Cδ: 1.0 percent (annual).
  - Financial gap calibrated weights:
    - Credit: 0.75; House price: 0.20; Equity: 0.05.
  - Long run speed of adjustment parameter ρ: 0.99.
  - Steady state π, g, n set to sample averages for the economy over the sample period.
- Estimation informativeness:
  - Data very informative about many strongly identified conditional mean parameters; substantial updates from prior to posterior with directionally similar updates for Japan and United States, differing magnitudes.
- Posterior mode highlights:
  - Partial indexation posterior modes (price and wage):
    - Japan: 0.55 and 0.71.
    - United States: 0.76 and 0.78.
  - Posterior mode of expected inflation response coefficient π_ξ:
    - Japan: 1.28.
    - United States: 1.50.
- Conditional variance slope posterior modes (economically significant):
  - A one percentage point faster business cycle contraction raises conditional variances of macro shocks by:
    - 76 percent for Japan.
    - 96 percent for the United States.
  - A one percentage point faster financial cycle downturn raises conditional variances of financial shocks by:
    - 30 percent for Japan.
    - 68 percent for the United States.
  - A one percent higher output gap raises conditional variances of macro shocks by:
    - 10 percent for Japan.
    - 1 percent for the United States.
  - A one percent higher financial gap raises conditional variances of financial shocks by:
    - 26 percent for Japan.
    - 7 percent for the United States.
- Model comparison:
  - Twice the logarithm of the Bayes factor (Laplace approximation) versus conditionally homoskedastic specification:
    - Japan: 29.
    - United States: 56.
  - Both exceed threshold 10 for very strong evidence in favor of conditionally heteroskedastic specification.

### VII. Policy analysis — transmission, quantile dynamics, and optimized rules
- Transmission of shocks (quantified with quantile impulse responses; variables annualized where applicable):
  - Contractionary monetary policy shock (temporary persistent 100 basis point increase in short term interest rate):
    - Reduces conditional mean of inflation by 0.2 to 0.3 percentage points after 4 to 5 quarters.
    - Reduces conditional mean of output by 0.5 to 0.7 percent after 4 quarters.
    - Leads to aggregate demand contracting more than potential, output gap fall, unemployment rise, price and wage inflation decline, financial gap fall; macroprudential authority lowers capital requirement to support credit.
  - Contractionary macroprudential shock (temporary persistent 2.5 percentage point increase in capital requirement):
    - Reduces conditional mean of inflation by 0.1 percentage point after 4 to 5 quarters.
    - Reduces conditional mean of output by 0.2 to 0.3 percent after 4 to 5 quarters.
    - Leads to higher mortgage and corporate loan rates, reduced investment, aggregate demand contraction; monetary authority cuts short term rate to support bank credit.
- Quantile and distributional dynamics:
  - In short run, contractionary monetary and macroprudential shocks expand the left tail (0.05 quantile) of inflation and output more than the reduction in conditional means (higher conditional variances); medium term the pattern reverses as conditional variances fall during recovery.
  - Asymmetry: a temporary persistent 100 basis point monetary tightening expands left tail of output growth distribution more than an equal-magnitude easing expands the right tail (short run); similar asymmetry for 2.5 percentage point capital requirement changes.
  - Conditional distributions and quantiles obtained via Monte Carlo simulation with 100,000 replications.
- Optimized policy rules — setup:
  - Authorities minimize expected discounted intertemporal loss under long run commitment; loss penalizes deviations of inflation, output, short term interest rate, and capital requirement from steady state.
  - Baseline dual-mandate flexible inflation targeting weights:
    - G_Y λ = 0.50
    - S_i λ = 1.00
    - R_κ λ = 0.25
  - Average intertemporal loss evaluated by forecasting means and variances of annualized arguments out 25 years, averaged across historical states; numerical optimization via differential evolution algorithm.
- Optimized rule coefficients (annual-frequency response coefficients) — reported posterior optimized values:
  - Japan:
    - Heteroskedastic: π_ξ = 1.6393; Y_ξ = 0.4742; F_ξ^Δ = 0.0000; R_κ_ξ = 0.1596.
    - Homoskedastic: π_ξ = 1.6064; Y_ξ = 0.2559; F_ξ^Δ = 0.0000; R_κ_ξ = 0.1267.
  - United States:
    - Heteroskedastic: π_ξ = 1.9035; Y_ξ = 0.2450; F_ξ^Δ = 0.0000; R_κ_ξ = 0.1427.
    - Homoskedastic: π_ξ = 1.9141; Y_ξ = 0.1363; F_ξ^Δ = 0.0000; R_κ_ξ = 0.1391.
- Key policy findings:
  - Allowing state dependent conditional heteroskedasticity leads to larger optimized response coefficients for both monetary and macroprudential rules relative to homoskedastic benchmark.
  - Monetary policy should respond more aggressively to the business cycle (tighter in expansions, looser in contractions).
  - Macroprudential policy should manage banking sector resilience more actively over the financial cycle (tighter in upturns to build resilience; looser in downturns to limit severity).
  - Robust separation of roles: optimized monetary rule response to financial vulnerabilities is zero in reported cases (F_ξ^Δ = 0.0000), implying macroprudential policy is primary instrument against buildup of financial vulnerabilities.

### VIII. Limitations, extensions, and concluding takeaways
- Limitations:
  - Conditional variance function is not microfounded in this specification, exposing approach to Lucas (1976) critique; slope coefficients may not be invariant to policy interventions.
  - Presented as an interim tractable solution to be bolted onto existing linearized DSGE models while more complex nonlinear models are developed.
- Suggested extensions:
  - Analyze approximation of corresponding nonlinear DSGE model as rule coefficients change.
  - Compare alternative state dependent conditional heteroskedasticity mechanisms across linearized DSGE frameworks.
  - Extend parsimonious specification to fit additional stylized facts as growth at risk literature expands to other variables.
- Final model-level takeaways:
  - The heteroskedastic linearized DSGE preserves satisfactory simulation and forecasting performance for conditional means relative to the homoskedastic counterpart while substantially improving fit to conditional distributions and matching growth at risk stylized facts.
  - Accounting for state dependent conditional heteroskedasticity changes jointly optimized policy rule coefficients, implying stronger cyclical monetary responses and more active macroprudential management over the financial cycle.

*Source: wpiea2020151-print-pdf*

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

### wpiea2020151-print-pdf - References .............................................................................................................

### I. Introduction and objective
- Augment a linearized DSGE model (similar to those in use at central banks following Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2003)) with a tractable endogenous risk mechanism to support joint analysis of monetary and macroprudential policy.
- State dependent conditional heteroskedasticity mechanism:
  - Specifies conditional variances of structural shocks as loglinear functions of the predetermined phase and position of the business or financial cycle, measured by the lagged change in and level of the output or financial gap.
  - Motivated by empirical finding in Vitek (2018) that macrofinancial volatility tends to be low during business cycle expansions and high during business cycle contractions, particularly when coinciding with financial cycle downturns.
- Key modeling goals:
  - Preserve the conditional mean function of the nested homoskedastic linearized DSGE model so standard linear perturbation solution algorithms remain applicable.
  - Allow conditional variances to be state dependent, producing location and scale shifts in conditional distributions and differing quantile impulse responses from mean impulse responses.
  - Provide a computationally inexpensive interim solution that can be bolted onto existing linearized DSGE models without resorting to simulation methods.

### II. Theoretical structure and mechanisms
- Model foundations and key features (building on Woodford (2003), Galí (2015), Smets and Wouters (2003), Christiano, Eichenbaum and Evans (2005), Iacoviello (2005), Galí (2011), Gerali et al. (2010), Hülsewig et al. (2009)):
  - Nominal and real rigidities: short run nominal price and wage rigidities from monopolistic competition, staggered reoptimization, partial indexation; habit persistence in consumption; adjustment costs in residential and business investment; variable capital utilization; reinterpretation of labor market to incorporate involuntary unemployment.
  - Financial sector: households access banks, real estate, and capital markets; banks issue risky mortgage and corporate loans at infrequently adjusted predetermined loan rates; banks accumulate capital out of retained earnings given credit losses to satisfy a capital requirement.
  - Borrowing collateralized against housing and capital stock values (Kiyotaki and Moore (1997) motivation).
  - Government implements monetary and macroprudential policy according to instrument rules.
- Analytical contributions:
  - Rational expectations equilibrium of the linearized DSGE model is invariant to the considered form of state dependent conditional heteroskedasticity when driven by stationary predetermined endogenous variables.
  - Derive an analytical extension of the Kalman (1960) filter to handle the specified state dependent conditional heteroskedasticity, enabling estimation by Bayesian maximum likelihood without simulation.

### III. Empirical strategy and data
- Focus economies: Japan and the United States (sample period 1980Q1 to 2019Q3).
  - Chosen because both experienced combined bursting of credit-fueled property and stock market bubbles in recent decades and can be reasonably approximated as closed.
- Financial vulnerability indicators used:
  - Credit: total credit to the nonfinancial private sector.
  - House price: broad residential property price index.
  - Equity price: broad stock price index.
  - All nominal variables deflated by the seasonally adjusted gross domestic product price deflator.
- Cycle extraction:
  - Pass logarithms of real variables through Hodrick and Prescott (1997) filter with smoothing parameter set to 16,000 (given financial cycles tend to be longer than business cycles).
- Financial gap construction:
  - Aggregate standardized indicators {x_kit, k = 1,..,K} using panel regression with indicator and economy specific fixed effects:
    - Equation (1): SΔ denotes seasonal difference operator; the change in each standardized indicator depends on linear combination of lagged indicators up to two years (two year lag order estimated by pooled quantile regression at the 0.05 quantile).
    - Financial gap generated as the weighted average: ,1,,,1, 11 ˆˆ / KK itk kitk kk xxββ = = = ∑∑.
  - Estimated weights:
    - Credit gap weight: 0.72
    - House price gap weight: 0.23
    - Equity price gap weight: 0.05
  - The estimated financial gap peaks prior to the combined bursting of credit-fueled property and stock market bubbles in Japan in the early 1990s and in the United States in the late 2000s.

### IV. Stylized facts of growth at risk (reduced form and structural alignment)
- Definition: The conditional lower quantile of output growth is referred to as growth at risk; conditioning variables are lagged change in and level of the financial gap.
- Estimation of conditional moments of output growth:
  - Output detrended via Hodrick and Prescott (1997) filter with smoothing parameter 16,000.
  - Conditional mean of output growth generated from pooled OLS panel regression with dummy variables.
  - Conditional lower and upper quantiles generated from pooled quantile regression with dummy variables at the 0.05 and 0.95 quantiles.
  - Consider short and medium run horizons corresponding to one and three year lag orders.
- First stylized fact (key empirical regularity):
  - The conditional lower quantile of output growth is more volatile than its conditional mean, which in turn is more volatile than its conditional upper quantile.
  - Empirical evidence: across Japan and the United States and across horizons, the sample variance of the estimated conditional lower quantile exceeds that of the estimated conditional mean, which in turn exceeds that of the estimated conditional upper quantile.
  - Growth at risk estimates closely track business cycle contractions associated with financial cycle downturns in Japan (early 1990s) and the United States (early 1990s, early 2000s, late 2000s); they do not fully track the sharp business cycle contraction in Japan in the late 2000s because it did not coincide with a sharp domestic financial cycle downturn.

### V. Model fit, empirical performance, and policy implications
- Empirical performance:
  - Augmenting the linearized DSGE model with a parsimonious form of state dependent conditional heteroskedasticity yields a substantial improvement in goodness of fit to the conditional distributions of observed endogenous variables for the Japan and United States datasets.
  - The heteroskedastic linearized DSGE model matches key stylized facts from the growth at risk literature (Adrian, Boyarchenko and Giannone (2019)).
- Policy analysis implications:
  - Accounting for state dependent conditional heteroskedasticity alters jointly optimized response coefficients of monetary and macroprudential policy rules under quadratic loss.
  - In particular:
    - Monetary policy should respond more aggressively to the business cycle.
    - Macroprudential policy should manage banking sector resilience more actively over the financial cycle.
- Advantages relative to alternatives:
  - Compared with adding stochastic volatility (Justiniano and Primiceri (2008)) or Markov regime switching variances (Liu, Waggoner and Zha (2011)), the approach endogenizes variation in conditional second moments, implying conditional quantiles depend nonuniformly on policy instrument levels.
  - Computationally inexpensive: simulation methods not needed to estimate conditional means and variances or to evaluate the loglikelihood function.
- Limitations and caveats:
  - The conditional variance function is not microfounded in this specification, exposing the approach to Lucas (1976) critique; slope coefficients may not be invariant to policy interventions.
  - Presented as an interim, tractable solution that can be bolted onto existing central bank linearized DSGE models while more complex nonlinear DSGE models are developed.

*Source: wpiea2020151-print-pdf - References .............................................................................................................*

### 0.95 quantiles ■.

### wpiea2020151-print-pdf - 0.95 quantiles ■.

### Stylized empirical facts on output growth risk
- The conditional mean and variance of output growth are strongly negatively correlated.
- Result based on panel regression of ln of squared residuals ˆitε from estimating conditional mean of output growth on lagged change in and level of the financial gap with economy specific fixed effects: 2,0,1,2,, ˆ ln. itiS it hit hit xxεγγγ ν  = + Δ + + (equation (3) in source).
- Conditional variance of output growth is generated as exponentiated fitted values from this panel regression, estimated by pooled ordinary least squares with dummy variables for short and medium run projection horizons.
- Across economies and horizons, the sample correlation between the conditional mean and variance is negative, ranging from -0.71 to -0.94.
- Intuition: a fall in the conditional mean is associated with a rise in the conditional variance, amplifying falls in conditional lower quantiles; a rise in the conditional mean is associated with a fall in the conditional variance, mitigating rises in conditional upper quantiles.
- Note on normality: if output growth is conditionally normally distributed, its conditional α quantile qit,α depends on conditional mean μit and standard deviation σit according to qit,α = μit + zα σit, and covariance structure determines monotonicity properties of Var(qit,α) in α (source footnote explanation).

### Theoretical framework — model overview
- Closed economy with households, firms, banks and government (monetary, fiscal, macroprudential authorities).
- Agents optimize intertemporally under rational expectations in output, housing, labor and financial markets.
- Model features: housing in fixed supply one per household; short term bonds (discount), long term bonds (perpetual with coupon decay rate Bω where 01Bω<<1, following Woodford (2001)); collateralized borrowing variants for households and firms following Kiyotaki and Moore (1997).

### Household sector — preferences, assets, and constraints
- Continuum of identical households indexed on [0,1]; internal members supply differentiated labor and share risk.
- Asset set: bank deposits, houses, short term bonds, long term bonds, stocks.
- Intertemporal utility: E_t ∑_{s=t}^{∞} β^{s-t} u(C_{is}, H_{is}, L_{igs}, B_{ijs}, F_{iks}) with 01β<<1 (equation (4)).
- Intratemporal utility includes external habit formation in consumption; parameter restrictions include 01α≤<1, 0σ> , 0ϖ> , 0η> for concavity.
- Preference shifters C_s^ν, H_s^ν, B_s^ν, S_s^ν specified as functions of aggregate variables or structural shocks (equation (6)) with 0C_Hϕ> , 0C_Bϕ> , 0C_Sϕ>.
- Long run balanced growth supported by additional shifters D_s^ν and N_s^ν (equation (7)) with 0μ>.
- Dynamic budget constraint sums deposits, bonds, housing, stocks, labor income, profit income, taxes and purchases (equation (8)).
- Mortgage borrowing rule: B_{is}^{H,1} = φ_H P_s^H S_{is}^{H,1} where 01φ_H<<1 (equation (9)).
- Housing quality accumulation: H_{is,s+1} = (1-δ_H) H_{is} + I_{is}^H(·) with 01δ_H≤≤1 (equation (10)); effective residential investment includes convex adjustment costs with parameter χ_H > 0 (equation (11)).
- First-order conditions include short term bond return Euler (equation (12)), residential investment Euler and shadow price conditions (equations (13)–(15)), wage–leisure intratemporal condition (equation (16)), deposit–bond parity (equation (17)), and risk-adjusted return equalities across assets (equations (18)–(20)).

### Labor, production and price/wage rigidity
- Unemployment rate u^L_t = U_t / N_t with labor force N_t.
- Final labor aggregator: CES with elasticity parameter θ^W > 1 (equation (21)) and associated demand (22) and wage index (23).
- Wage markup shock ϖ^W_t with relation /(1)WWW ttt ϖθθ = − (text).
- Nominal wage rigidity: a fraction 1−ω^W of unions reoptimize, fraction ω^W index to past inflation and productivity per partial indexation rule (equation (24)) with 01γ^W≤≤1 and 01ω^W≤<1.
- Optimal wage condition derived as first-order condition (equation (25)); aggregate wage index average per (26).
- Final output aggregator: CES with elasticity θ^P > 1 (equation (27)) with demand (28) and price index (29).
- Price markup shock ϖ^P_t with relation / (1)PPP ttt ϖθθ = −.
- Price rigidity: a fraction 1−ω^P reoptimizes, fraction ω^P index per (42), optimal price condition (43) and aggregate price index (44).

### Production, investment and firm financing
- Intermediate firms maximize pre-dividend market value V_{kt}^F = E_t ∑_{s=t}^{∞} β^{s-t} λ_{st} Π_{ks}^F (equation (30)).
- Dividend/profit definition includes earnings, net borrowing, business investment (equation (31)).
- Corporate borrowing rule: B_{ks}^{F,1} = φ_F P_s^K K_{ks}^{1} where 01φ_F<<1 (equation (32)).
- Production uses capital K and labor L with constant returns and utilization u_{ks} (equation (33)); utilization costs Γ_K(u) per (35) with parameters K_μ>0, K_η>0, steady state u=1.
- Capital accumulation with depreciation δ_K, adjustment costs χ_K (equations (36) and (37)).
- First-order conditions: marginal cost and labor (38), utilization (39), shadow price of capital and investment Euler (40)–(41).

### Banking sector — intermediation, capital and spreads
- Banks transform deposits and interbank funding into mortgage and corporate loans via CES portfolio aggregator (equation (45)) with demand (46) and gross loan rate index (47).
- Banking profit/value: V_{lt}^B = E_t ∑ β^{s-t} λ_{st} Π_{ls}^B (equation (48)).
- Bank profit components and capital requirement cost detailed (equation (49)).
- Balance sheet identity: H F D B_{ls, t+1} + ... = ... (equation (50)); aggregate credit C_{t+1}^B measures assets; money stock M_{t+1} measures funding; bank capital ratio κ_{t+1} = K_{t+1}^B / C_{t+1}^B.
- Regulation cost Φ_B(K,B,·) decreasing in capital ratio and parametrized per (51)–(52) with requirement R_s κ and parameters C_μ>0, C_η>0.
- Bank capital accumulation and retained earnings with adjustment cost χ_B (equations (53) and (54)).
- Funding parity: interbank rate equals deposit rate (equation (55)); retained earnings Euler (56); bank capital shadow price condition (57).
- Loan rate stickiness: fraction 1−ω^C adjust gross loan rates per Calvo-type friction adapted to banking (equation (58)); optimal loan rate first-order condition (59); aggregate loan rate per (60).

### Policy rules — monetary, fiscal, macroprudential and default dynamics
- Monetary policy: short term interest rate rule with partial adjustment and forward-looking inflation, output gap and change in financial gap terms (equation (61)). Parameters: 01ρ^S≤<1, 1π ξ >, 0G_Y ξ ≥, 0F ξ_Δ ≥. Monetary shock ν^S_t mean zero serially uncorrelated.
- Financial gap definition: weighted average of deviations of real bank credit and relative prices of housing and equity (equation (62)) with parameters 0F_C ϕ >, 0F_H ϕ >, 0F_F ϕ > and FFF CHF ϕϕϕ sum to 1.
- Fiscal policy: public demand rule with partial adjustment (equation (63)) with 01ρ^G≤<1; fiscal balanced budget T_t = P_t G_t (equation (64)); taxes lump sum and non-distortionary.
- Macroprudential policy: capital requirement rule with partial adjustment responding to financial gap (equation (65)) with 01κ^R<<1, 01κ^R ρ≤<1, 0κ^R ξ ≥; macroprudential shock ν^R_t mean zero serially correlated.
- Default rates dynamics: mortgage and corporate loan default rates follow partial adjustment dynamics inertially decreasing in contemporaneous output and financial gaps (equation (66)) with parameters 01C_δ<<1, 01C_δ ρ≤<1, Z_δ ζ > 0 and F_ζ > 0.

### Market clearing and equilibrium definition
- Final output market clearing: Y_t = C_t + I_t + G_t (equation (67)) with I = I^H + I^K.
- Housing market clearing, bond market clearing, and stock market clearing conditions specified: H_{is,t+1} = 1 for all i, S_t^S = 0 for short term bond supply, L_{jt,t+1}=0 for long term bond market for all j, and S_{kt,t+1}^F = 1 for stock market for all k (text).

### Empirical framework — state space, heteroskedasticity, and endogenous risk
- Linearized DSGE model expressed as augmented linear state space representation with empirically flexible trend components and state dependent conditional heteroskedasticity for structural shocks.
- Decompose variables into cyclical ˆx_t and trend x_t components, additively separable (ˆx_t + x_t = x_t).
- Price Phillips curve for inflation and a set of relationships for consumption, residential investment, business investment, housing user cost, bond yields, equity prices, profits, bank credit, lending rate Phillips curves, retained earnings, wage Phillips curve, unemployment, output demand and production, potential output, and financial gap — all provided as linearized equations (selected representative equations: (68)–(92), (93)–(96)).
- Trend components follow first-order autoregressive processes with parameters governing convergence to long run balanced growth featuring constant inflation π, productivity growth g, and population growth n (equations (97)–(98)).

### Exogenous shocks and heteroskedastic specification
- Structural shocks follow stationary first-order autoregressive or serially uncorrelated processes with conditionally normally distributed heteroskedastic innovations.
- Conditional means: list of shocks following AR(1) or iid processes with notation and conditional variances specified (equations (99)–(101)); identifying restriction that innovations contemporaneously uncorrelated.
- Conditional variances: loglinear functions of past change in and level of the output gap for macroeconomic shocks (equation (102)); loglinear functions of past change in and level of the financial gap for financial shocks (equation (103)). These forms ensure positive conditional variances ĥ^Z_t which converge to unconditional variances 2ˆ Z σ.

### Estimation approach and data
- Estimation via Bayesian maximum likelihood treating augmented state space as conditionally Gaussian following Harvey, Ruiz and Sentana (1992); Kalman filter extended for state dependent conditional heteroskedasticity.
- Restricted model consolidations: residential and business investment demand shocks consolidated into investment demand shock; mortgage and corporate loan markups consolidated into loan markup shock; macroprudential and loan default shocks eliminated in restricted version.
- Strongly identified parameters that enter conditional mean are estimated conditional on informative priors; weakly identified parameters calibrated; parameters that enter conditional variance estimated with diffuse priors.
- State space and filter equations summarized (equations (104)–(124)); posterior and prior specification with multivariate normal prior (equations (120)–(122)); asymptotic normal approximation to posterior around mode (equation (123)) and Hessian estimator (equation (124)).
- Data: levels of twelve endogenous variables for Japan or United States over sample 1986Q3 to 2019Q3. Observed variables: price level (seasonally adjusted GDP deflator), output (seasonally adjusted real GDP), consumption (seasonally adjusted real private consumption expenditures), investment (seasonally adjusted real private investment expenditures), short term interest rate (policy rate), long term bond yield (ten year government bond yield), price of housing (broad residential property price index from BIS), price of equity (broad stock price index), corporate loan rate (reference bank lending rate), wage (seasonally adjusted wage index from OECD), unemployment rate (seasonally adjusted share), employment (seasonally adjusted total employment).
- Trend components for variables with long run trends use difference order 2 and smoothing parameter 16,000; variables without long run trends use difference order 1 and smoothing parameter 400 (Vitek (2014) Hodrick–Prescott extension).
- Examples of prior means: habit persistence α prior mean 0.80; intertemporal consumption elasticity σ prior mean 1.00; intratemporal labor supply elasticity η prior mean 0.50; residential investment adjustment cost χ_H prior mean 3.00; intratemporal housing demand elasticity ϖ prior mean 1.00; business investment adjustment cost χ_K prior mean 3.25; capital utilization cost parameter K_η prior mean 1.00; partial indexation parameters P_γ and W_γ prior means 0.80. (Source reports additional prior placement in Appendix.)

*Italic source: wpiea2020151-print-pdf - 0.95 quantiles ■.*

### 8.0 quarters. In addition, the prior means of the mortgage

### 8.0 quarters. In addition, the prior means of the mortgage 

### Priors and calibration
- Prior means/values:
  - Mortgage and corporate loan to value ratio parameters Hϕ and Fϕ: 0.80.
  - Financial friction parameter for mortgage and corporate loan rate determination Cω implies an average adjustment interval of 4.0 quarters.
  - Partial adjustment parameter in the monetary policy rule Siρ: 0.80.
  - Expected inflation response coefficient πξ: 1.50.
  - Output gap response coefficient GYξ: 0.50 (at the annual frequency).
- Calibrated weakly identified parameters to match conventional/average observed values:
  - Subjective discount factor parameter β set to imply a discount rate of 4.0 percent.
  - Housing depreciation parameter Hδ set to imply a depreciation rate of 3.0 percent (annual frequency).
  - Capital depreciation parameter Kδ set to imply a depreciation rate of 9.0 percent (annual frequency).
  - Price markup parameter Pθ set to imply a steady state equilibrium price markup of 15.0 percent.
  - Steady state equilibrium labor income share: 0.67.
  - Steady state equilibrium bank lending rate Ci: 6.0 percent (annual frequency).
  - Steady state equilibrium default rate Cδ: 1.0 percent (annual frequency).
- Financial gap weights:
  - Weight on the credit gap: 0.75.
  - Weight on the house price gap: 0.20.
  - Weight on the equity price gap: 0.05.
- Long run speed of adjustment parameter ρ: 0.99.
- Steady state equilibrium inflation rate π, productivity growth rate g and population growth rate n set to their average observed values for the economy under consideration over the sample period.
- Steady state ratios of consumption, residential investment and business investment to output set to their average observed values for the economy under consideration over the sample period. Steady state equilibrium ratio of public demand to output follows from the final output good market clearing condition.
- Note: "The steady state equilibrium productivity growth rate is estimated by the sample average growth rate of output less that of employment, while the steady state equilibrium population growth rate is estimated by the sample average growth rate of employment."

### Estimation sample, method, and informativeness
- Effective sample period for parameter estimation: 1987Q1 to 2019Q3.
- Estimation method: logarithm of the posterior density kernel is globally maximized numerically with the differential evolution algorithm due to Storn and Price (1997).
- Data informativeness:
  - Data is very informative regarding many strongly identified parameters that enter the conditional mean function, evidenced by substantial updates from prior to posterior.
  - Updates tend to be in the same direction for Japan and the United States, but magnitudes often differ substantially, reflecting structural differences.

### Posterior mode results and conditional variance findings
- Partial indexation posterior modes (price and wage determination):
  - Japan: 0.55 and 0.71.
  - United States: 0.76 and 0.78.
- Posterior mode of expected inflation response coefficient in the monetary policy rule:
  - Japan: 1.28.
  - United States: 1.50.
- Conditional variance slope coefficients (posterior modes) are positive and generally economically significant for Japan and the United States:
  - A one percentage point faster rate of business cycle contraction is estimated to raise conditional variances of the macroeconomic shocks by:
    - 76 percent for Japan.
    - 96 percent for the United States.
  - A one percentage point faster rate of financial cycle downturn is estimated to raise conditional variances of the financial shocks by:
    - 30 percent for Japan.
    - 68 percent for the United States.
  - A one percent higher output gap is estimated to raise conditional variances of the macroeconomic shocks by:
    - 10 percent for Japan.
    - 1 percent for the United States.
  - A one percent higher financial gap is estimated to raise conditional variances of the financial shocks by:
    - 26 percent for Japan.
    - 7 percent for the United States.
- Model comparison via Bayes factor (Laplace approximation) against conditionally homoskedastic specification:
  - Twice the logarithm of the Bayes factor: 29 for Japan and 56 for the United States.
  - These values exceed the threshold for very strong evidence of 10 (Kass and Raftery (1995)), indicating very strong evidence in favor of the conditionally heteroskedastic specification.

### Diagnostic checks: historical decompositions and sequential dynamic forecasts
- Historical decompositions:
  - Inflation deviations from trend mainly attributed to markup shocks, secondarily to demand and financial shocks.
  - Output growth deviations around trend mainly attributed to demand and financial shocks.
  - Adverse demand and financial shocks identified as primary drivers of business cycle contractions:
    - Japan: early 1990s and late 2000s.
    - United States: early 1990s, early 2000s, and late 2000s.
  - Policy shocks identified as countercyclical mitigants of these contractions.
- Sequential dynamic forecast performance (heteroskedastic linearized DSGE model vs random walk) measured by Theil's U statistic (ratio of root mean squared prediction errors) across horizons one through twelve quarters, over holdout sample 1990Q2 to 2019Q3:
  - Japan:
    - Root mean squared prediction error is 32 percent lower for inflation (model vs random walk).
    - Root mean squared prediction error is 52 percent lower for output growth.
  - United States:
    - Root mean squared prediction error is 22 percent lower for inflation.
    - Root mean squared prediction error is 59 percent lower for output growth.
- Effect of state dependent conditional heteroskedasticity on predicted means and confidence intervals:
  - Little effect on predicted means of inflation and output growth when parameters and initial conditions are restricted to be the same between heteroskedastic and homoskedastic models (identical dynamic forecasts of means).
  - Substantial effect on coverage of forecast confidence intervals:
    - Forecast confidence intervals expand considerably at times of macrofinancial stress, assigning more appropriate probabilities to tail events, and contract slightly during normal times.
    - Example expansions:
      - Japan: considerably wider forecast confidence intervals for inflation and output growth originating in 1992Q4.
      - United States: considerably wider forecast confidence intervals for inflation and output growth originating in 2008Q4.
  - Quantiles in dynamic forecast comparison estimated based on a Monte Carlo simulation with 100,000 replications.

### Growth at risk: measurement and stylized facts
- Model implication:
  - The entire conditional distribution of output growth (not just central tendency) evolves over time; both location and scale shift with macroeconomic and financial conditions and vulnerabilities, varying by forecast horizon.
  - Shifts are negatively correlated, generating conditional distributions skewed to the downside, especially at times of macrofinancial stress.
- Stylized fact reproduced by the heteroskedastic linearized DSGE model:
  - The conditional lower quantile of output growth is more volatile than its conditional mean, which is more volatile than its conditional upper quantile.
  - Across both economies and horizons, sample variance ordering: conditional lower quantile variance > conditional mean variance > conditional upper quantile variance.
  - By contrast, a homoskedastic linearized DSGE model with normally distributed innovations yields symmetric evolution of conditional lower and upper quantiles, shifting in parallel with conditional means.
- Conditional distribution visualization:
  - One and three year ahead predicted probability density functions of the cyclical component of output growth are estimated with a normal kernel based on a Monte Carlo simulation with 100,000 replications.

*Source: Excerpt from wpiea2020151-print-pdf (effective sample period 1987Q1 to 2019Q3).*

### 0.95 quantiles ■. The quantiles are estimated based on a Monte Carlo simulation with 100,000 replications.

### wpiea2020151-print-pdf - 0.95 quantiles ■. The quantiles are estimated based on a Monte Carlo simulation with 100,000 replications.

### Explaining Growth at Risk
- Model framework:
  - Heteroskedastic linearized DSGE model with state dependent conditional heteroskedasticity linking risks to the phase and position of the business and financial cycles.
  - Quantiles and probability density functions estimated with a Monte Carlo simulation with 100,000 replications.
- Key mechanisms:
  - Business and financial cycle downturns are associated with:
    - A fall in the conditional mean of output growth (mean impulse responses to adverse macroeconomic and financial shocks).
    - A rise in the conditional variance of output growth (state dependent heteroskedasticity expands risks around the central forecast).
- Empirical/statistical finding:
  - Across both economies and horizons, the sample correlation between the conditional mean and variance of output growth is negative, ranging from -0.62 to -0.84.
  - In contrast, a homoskedastic linearized DSGE model with normally distributed innovations yields conditional means and variances that are approximately uncorrelated.
- Quantile focus:
  - Decomposes the one and three year ahead predicted 0.05 quantile of the cyclical component of output growth into contributions from the predicted mean and variance versus the output and financial gaps.

### Policy Analysis — Transmission Mechanisms
- General:
  - Joint analysis of monetary and macroprudential policy using the heteroskedastic linearized DSGE model.
  - Quantification of transmission mechanisms via quantile impulse responses.
  - Quantile impulse responses are calculated with a conditional normality approximation. All variables are annualized, where applicable.
- Monetary policy shock (contractionary):
  - A monetary policy shock that raises the short term interest rate reduces components of private domestic demand (notably residential and business investment) via tighter financial conditions, higher market interest rates, and lower asset prices.
  - Aggregate demand contracts more than potential; output gap falls; unemployment rises; price and wage inflation decline; financial gap falls (lower real bank credit and relative asset prices).
  - Macroprudential authority reaction: lowers capital requirement, supporting bank credit supply.
  - Quantitative effects (temporary but persistent 100 basis point increase in the short term interest rate):
    - Reduces the conditional mean of inflation by 0.2 to 0.3 percentage points after 4 to 5 quarters.
    - Reduces the conditional mean of output by 0.5 to 0.7 percent after 4 quarters.
- Macroprudential policy shock (contractionary):
  - A macroprudential shock that raises the capital requirement reduces residential and business investment via higher mortgage and corporate loan rates as banks accumulate capital through retained earnings.
  - Aggregate demand contracts more than potential; output gap falls; unemployment rises; price and wage inflation decline; financial gap falls (lower real bank credit).
  - Monetary authority reaction: cuts the short term interest rate to support bank credit demand.
  - Quantitative effects (temporary but persistent 2.5 percentage point increase in the capital requirement):
    - Reduces the conditional mean of inflation by 0.1 percentage point after 4 to 5 quarters.
    - Reduces the conditional mean of output by 0.2 to 0.3 percent after 4 to 5 quarters.
- Quantile dynamics:
  - In response to contractionary monetary and macroprudential shocks:
    - Conditional lower quantiles (0.05 quantile) of inflation and output fall more than their conditional means in the short run (higher conditional variances).
    - Over the medium term, conditional lower quantiles fall less than their conditional means (lower conditional variances).
    - Conditional upper quantiles exhibit opposing dynamics for the same reasons.
  - These conditional variance dynamics reflect a short run business cycle contraction and financial cycle downturn followed by a medium term business cycle recovery.

### Policy Analysis — Counterfactual Distributions
- Asymmetry of monetary policy effects on conditional output growth distribution:
  - Short run:
    - A temporary but persistent increase in the short term interest rate by 100 basis points expands the left tail of the conditional output growth distribution by more than an equal-magnitude decrease expands the right tail.
    - This reflects the state dependent conditional heteroskedasticity mechanism: monetary tightening both shifts the distribution location left and expands its scale (induced business cycle contraction and financial cycle downturn); monetary easing operates vice versa.
  - Medium run:
    - The short-run asymmetry is reversed over the medium term as the business cycle recovers.
- Asymmetry of macroprudential policy effects on conditional output growth distribution:
  - Short run:
    - A temporary but persistent increase in the capital requirement by 2.5 percentage points expands the left tail of the conditional output growth distribution by more than an equal-magnitude decrease expands the right tail.
  - Medium run:
    - This short-run asymmetry is reversed over the medium term.
  - Rationale:
    - The state dependent conditional heteroskedasticity mechanism operates qualitatively identically for monetary and macroprudential shocks given their qualitatively similar effects on the business and financial cycles.
- Estimation notes:
  - One and three year ahead predicted probability density functions of the cyclical component of output growth conditional on policy shocks use a normal kernel based on a Monte Carlo simulation with 100,000 replications.
  - Conditional distributions depicted for shocks that raise or lower the short term interest rate by 100 basis points, and for shocks that raise or lower the capital requirement by 2.5 percentage points, versus the corresponding unconditional probability density function.

*Source: wpiea2020151-print-pdf - 0.95 quantiles ■. The quantiles are estimated based on a Monte Carlo simulation with 100,000 replications.*

### 2.5 percentage points, versus the corresponding unconditional probability density function ■. The probability density

### wpiea2020151-print-pdf - 2.5 percentage points, versus the corresponding unconditional probability density function ■. The probability density

### Optimized rules: setup and numerical procedure
- Intertemporal loss function: authorities minimize expected discounted loss under long run commitment; intratemporal loss penalizes deviations of inflation from target, output from potential, and short term interest rate and capital requirement from steady state.
- Weights used in the baseline dual-mandate flexible inflation targeting calibration:
  - G_Y λ = 0.50
  - S_i λ = 1.00
  - R_κ λ = 0.25
- The intertemporal loss is evaluated by forecasting the means and variances of annualized arguments out 25 years, conditional on the estimated state, and averaged across historical states to remove dependence on initial conditions.
- Numerical optimization: the average intertemporal loss function is globally minimized with the differential evolution algorithm due to Storn and Price (1997), jointly with respect to monetary and macroprudential rule response coefficients, subject to inequality constraints.
- Kernel density estimation detail: probability density functions are estimated with a normal kernel based on a Monte Carlo simulation with 100,000 replications.

### Optimized rule coefficients (Table 1)
- Reports optimized annual-frequency response coefficients for monetary and macroprudential policy rules under heteroskedastic and homoskedastic specifications.

- Japan
  - Heteroskedastic:
    - π_ξ = 1.6393
    - Y_ξ = 0.4742
    - F_ξ^Δ = 0.0000
    - R_κ_ξ = 0.1596
  - Homoskedastic:
    - π_ξ = 1.6064
    - Y_ξ = 0.2559
    - F_ξ^Δ = 0.0000
    - R_κ_ξ = 0.1267

- United States
  - Heteroskedastic:
    - π_ξ = 1.9035
    - Y_ξ = 0.2450
    - F_ξ^Δ = 0.0000
    - R_κ_ξ = 0.1427
  - Homoskedastic:
    - π_ξ = 1.9141
    - Y_ξ = 0.1363
    - F_ξ^Δ = 0.0000
    - R_κ_ξ = 0.1391

### Key policy findings and interpretation
- Allowing state dependent conditional heteroskedasticity leads to larger optimized response coefficients for both monetary and macroprudential policy rules (i.e., optimized values are larger under heteroskedasticity).
- Intuition:
  - The heteroskedasticity mechanism amplifies volatility during business cycle contractions and financial cycle downturns but not on average, making it optimal to limit frequency and severity of contractions and downturns.
  - Monetary policy: optimal to respond more aggressively to the business cycle — tighter during expansions to mitigate capacity pressures, and looser during contractions to limit amplitude and duration.
  - Macroprudential policy: optimal to manage banking sector resilience more actively over the financial cycle — tighter during financial cycle upturns to build resilience and lean against buildup, and looser during downturns to limit severity.
- Robust separation of roles:
  - The model prescribes separation between monetary and macroprudential responses: monetary policy does not lean against the buildup of financial vulnerabilities during financial cycle upturns.
  - The optimized monetary policy rule response coefficient to financial vulnerabilities is always zero (F_ξ^Δ = 0.0000 in all reported cases).
  - This result aligns with the surveyed literature that macroprudential policy is the first line of defense against buildup of financial vulnerabilities because it is better targeted.

### Conclusion (model-level takeaways)
- A linearized DSGE model augmented with a tractable endogenous-risk, state dependent conditional heteroskedasticity mechanism:
  - Preserves satisfactory simulation and forecasting performance for conditional means relative to the homoskedastic counterpart.
  - Substantially improves goodness of fit to conditional distributions and matches key stylized facts from the growth at risk literature.
- Policy implication summary:
  - Accounting for state dependent conditional heteroskedasticity changes jointly optimized policy rule coefficients, implying stronger cyclical responses for monetary policy and more active macroprudential management over the financial cycle.
- Suggested extensions noted by authors:
  - Analyze how well the linearized heteroskedastic model approximates the corresponding nonlinear DSGE model as rule coefficients change (relevance to Lucas (1976) critique).
  - Compare empirical performance and policy implications of alternative state dependent conditional heteroskedasticity mechanisms across different linearized DSGE frameworks.
  - Extend and refine the parsimonious specification to fit additional stylized facts as growth at risk literature expands to other variables, informing development of nonlinear DSGE models under endogenous risk.

### Appendix — selected estimation results (posterior modes and notable posterior SEs)
- Conditional mean function parameters (selected posterior modes and posterior SEs)
  - α: Japan posterior mode = 0.6067, posterior SE = 0.0618; United States posterior mode = 0.7876, posterior SE = 0.0418
  - H_χ: Japan posterior mode = 2.8481, posterior SE = 0.2990; United States posterior mode = 3.0839, posterior SE = 0.2982
  - K_χ: Japan posterior mode = 4.0414, posterior SE = 0.3123; United States posterior mode = 3.9868, posterior SE = 0.3099
  - η: Japan posterior mode = 0.4948, posterior SE = 0.0050; United States posterior mode = 0.4944, posterior SE = 0.0050
  - π_ξ (prior mean 1.5000): Japan posterior mode = 1.2843, posterior SE = 0.0707; United States posterior mode = 1.4984, posterior SE = 0.0720
  - Y_ξ (prior mean 0.1250): Japan posterior mode = 0.1149, posterior SE = 0.0120; United States posterior mode = 0.1282, posterior SE = 0.0122
  - S_i_ρ: Japan posterior mode = 0.8233, posterior SE = 0.0074; United States posterior mode = 0.7982, posterior SE = 0.0077

- Conditional variance function parameters (selected posterior modes and posterior SEs)
  - ˆA_σ: Japan posterior mode = 1.2298, posterior SE = 0.1129; United States posterior mode = 0.5941, posterior SE = 0.0551
  - ˆN_σ: Japan posterior mode = 0.6709, posterior SE = 0.0693; United States posterior mode = 0.6214, posterior SE = 0.0685
  - ˆC_σ: Japan posterior mode = 2.5951, posterior SE = 0.5054; United States posterior mode = 2.0578, posterior SE = 0.4456
  - ˆI_σ: Japan posterior mode = 4.8648, posterior SE = 0.6185; United States posterior mode = 3.0516, posterior SE = 0.3889
  - ˆP_σ: Japan posterior mode = 26.4002, posterior SE = 4.3349; United States posterior mode = 12.7817, posterior SE = 2.0705
  - ˆW_σ: Japan posterior mode = 31.9409, posterior SE = 4.7957; United States posterior mode = 33.1409, posterior SE = 5.2851
  - ˆS_i_σ: Japan posterior mode = 0.0829, posterior SE = 0.0074; United States posterior mode = 0.1137, posterior SE = 0.0103
  - Selected lower-order variance components (posterior modes):
    - 2P_σ = 0.0296 (Japan, posterior SE 0.0081); 0.0245 (United States, posterior SE 0.0046)
    - 2Y_σ = 0.0344 (Japan, posterior SE 0.0081); 0.0328 (United States, posterior SE 0.0085)
    - 2I_σ = 0.0685 (Japan, posterior SE 0.0204); 0.0756 (United States, posterior SE 0.0277)
    - 2S_i_σ = 0.0024 (Japan, posterior SE 0.0007); 0.0026 (United States, posterior SE 0.0010)

*Source: wpiea2020151-print-pdf - 2.5 percentage points, versus the corresponding unconditional probability density function ■. The probability density (IMF working paper content provided).*

### References

### References

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*References — wpiea2020151-print-pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020151-print-pdf.pdf_