## _wp10249 - References

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---

### Introduction: purpose and contribution
- Proposes a simple alternative to higher-order perturbation and value-function methods for solving RBC models with recursive Epstein-Zin preferences and an affine structure for exogenous state variables.
- Key idea: exploit joint log-normality of shocks and use log-linearization to compute log-linearized dynamics and risk adjustments while retaining computational simplicity.
- Claimed advantages:
  - Linear structure eases description of time-series properties and estimation.
  - Analytical form permits explicit inspection of mechanisms behind quantity dynamics and asset prices.
  - Handles stochastic volatility more easily than standard perturbation methods (avoids third- or higher-order expansions).
- Relationship to literature:
  - Builds on Epstein and Zin (1989, 1991), Weil (1989), Schmitt-Grohe and Uribe (2004), and Duffie, Pan, and Singleton (2000).
  - Contrasts with Backus, Routledge, and Zin (2007) and Uhlig (2010) on which equilibrium conditions to linearize and on consistent computation of risk adjustments.

### Setup: model ingredients and equilibrium
- Preferences:
  - Representative consumer with recursive Epstein-Zin utility:
    - Ut = [Ct^(1-1/ψ) + β (Et(Ut+1^(1-γ)))^( (1-1/ψ)/(1-γ) ) ]^(1/(1-1/ψ))
  - Parameters separate risk aversion γ and EIS ψ; power utility is special case γ = 1 = ψ.
- Technology:
  - Cobb-Douglas production: Yt = Zt At^(1-α) Kt^α
  - Capital law of motion: Kt+1 = (1-δ) Kt + Yt - Ct
  - Marginal product of capital Rt = α (At Kt / )^(1-α) + (1-δ) (as shown)
- Shocks and affine structure:
  - Exogenous vector ut with affine discrete-time dynamics:
    - ut+1 = H0 + H1 ut + Σt εt+1
    - εt ~ N(0, Inε)
  - Time-varying covariance evolution:
    - (Σt Σt')ij = (G0)ij + (G1)ij ut
  - Affine structure implies conditional expectation and conditional variance-covariance of ut are affine in ut, enabling conditional normality of variables up to first order.
- Stationary transformation:
  - Scale nonstationary variables: ˜Xt = Xt A_t^-1
  - Stationary equilibrium equations (examples): equations (3)-(6) in the source (value function, capital law, marginal product, Euler equation).

### Log-Normal Risk Adjustment: approximation technique
- Conjecture: log variables yt are conditionally normal; write equilibrium conditions as
  - f(xt+1, xt, yt, Et[exp(Γ yt+1)]) = 0
- Use identity for log-normal variables:
  - Et[exp(Γ yt+1)] = exp(Γ Et[yt+1] + 1/2 Γ (Var_t yt+1) Γ')
- Linearize resulting condition to obtain a log-linear system where yt is linear in state variables and includes explicit constant "risk-adjustment" terms reflecting differences between stochastic and non-stochastic steady states.
- Advantages highlighted:
  - Captures risk adjustment without second- and higher-order Taylor expansions of all terms.
  - Computationally simple, semi-analytical, and tractable for models with stochastic volatility.

### Example 1: RBC with recursive preferences (no stochastic volatility)
- Shock structure used:
  - ut = [at; zt]
  - at+1 = (1-ρa) μ + ρa at + σa ε1,t+1
  - zt+1 = ρz zt + σz ε2,t+1
- Value function manipulation under normality yields (see equation (9)) a log-linear expression with variance corrections:
  - Linearized value function equation example: ζ1 ˆvt = ζ2 ˆct + ζ3 [ˆat + Et ˆvt+1 + 1/2 (1-γ)^2 Var(ˆvt+1)]
- Linearized system (equations (10)-(13)) produces three equations in unknowns ˆv_t, ˆc_t, ˆk_t and can be written as:
  - [Et ˆv_{t+1}; Et ˆc_{t+1}; ˆk_{t+1}] = R + W [ˆv_t; ˆc_t; ˆk_t] + Q Et [ˆa_{t+1}; ˆz_{t+1}]
- Key interpretation:
  - Vector R summarizes constant risk-adjustment terms (difference between stochastic and non-stochastic steady states). These constants affect levels (e.g., precautionary savings lowering consumption relative to no-risk model) and financial variables (unconditional mean of the risk-free rate).
- Quantities and prices:
  - Log consumption growth: gc_{t+1} = ˜c_{t+1} - ˜c_t + ˜a_t
  - Risk-free rate: rf_t ≈ -ln β - 1/2 (1/ψ - γ)(γ - 1) Var_t(˜v_{t+1}) - 1/2 Var_t(m_{t+1}) + (1/ψ) Et[gc_{t+1}]
    - Note: first-order log-linearization would omit the Var_t(˜v_{t+1}) and Var_t(m_{t+1}) terms.
  - Asset risk premium up to first order: Et[ r_{i,t+1} - r_{f,t} ] = -Cov_t( r_{i,t+1}, m_{t+1} ) - 1/2 Var_t( r_{i,t+1} )
- Entropy (market price of risk) for log-normal M_{t+1}:
  - ln Et M_{t+1} - Et ln M_{t+1} = 1/2 Var_t(m_{t+1})

### Comparison with perturbation methods (theoretical and quantitative)
- The log-normal risk method is a truncated version of second-order perturbation:
  - Second-order perturbation solution approximation:
    - Yt - Ȳ ≈ g_s(Ȳ)(St - Ṡ) + 1/2 g_{ss}(Ȳ)(St - Ṡ)^2
  - Log-normal risk keeps second-order terms involving variances of state variables (risk-adjustment) but omits quadratic terms in endogenous state variables (second-order dynamic terms).
- Numerical calibration (monthly):
  - Table 1 calibration highlights (monthly frequency):
    - β = 0:9983
    - α = 0:34
    - μ = 0:0017
    - δ = 0:0087
    - γ = 5
    - ψ = 1:5
    - ρA = 0:14
    - ρZ = 0:0
    - σ^2_A = 0:059
    - σ_Z^2 = 0:01
- Table 2: first-order term coefficients match exactly between perturbation and log-normal risk; second-order dynamic terms are zero in log-normal risk; second-order variance (risk-adjustment) terms are very close but not identical.
- Impulse response comparison (Figures 1 and 2):
  - Dynamic responses of capital, consumption and risk-free rates to one-standard-deviation shocks in a(t) and z(t) are nearly identical between second-order perturbation and log-normal risk (responses measured as percentage deviations from stochastic steady state).
- Stochastic steady-state comparison (Table 3):
  - K_ss: 0:8666 (Perturbation) vs 0:8732 (Log-Normal Risk) — % Diff 0:01
  - C_ss: 0:0379 vs 0:0382 — % Diff 0:01
  - r_fss: -0:0068 vs -0:0069 — % Diff 0:01
  - V_ss: -8:2019 vs -8:2513 — % Diff 0:01
  - Conclusion: stochastic steady states from both methods differ by less than 1 percent for variables shown.

### Example 2: stochastic volatility in a simple growth model
- Setup (CRRA preferences with 1/ψ = γ and no stationary Z shocks):
  - â_{t+1} = μ + σ_t ε_{a,t+1}
  - σ_{t+1}^2 = (1-φ) θ + φ σ_t^2 + ω ε_{σ,t+1}
  - Innovations: E_t[ ε_{a,t+1} ε_{σ,t+1} ] = 0
  - Affine representation:
    - H0 = [ μ ; (1-φ) θ ], H1 = [ 0 0 ; 0 φ ], Σ_t = [ σ_t 0 ; 0 ω ]
- Solution form conjectured:
  - ˆc_t = R_c + α ˆa_t + β σ_t^2
  - Var_t(ˆc_{t+1}) = [ α^2 + β^2 φ ] σ_t^2 + β^2 ω^2
  - Conditional variance of consumption growth inherits productivity variance process leading to an AR(1)-type variance dynamics:
    - (σ_{c,t+1})^2 = (1-φ) ( c_a^2 θ + c_σ^2 ω^2 ) + φ (σ_{c,t})^2 + c_a^2 ω ε_{t+1}
  - Coefficients β and R are solved by equating coefficients on σ_t^2 and constants.
- Qualitative comparison with perturbation:
  - Capturing stochastic volatility fully with perturbation requires fourth-order expansion; dynare++ can perform fourth-order approximations but is computationally expensive and a "black box".
  - The log-normal risk method yields algebraic (quadratic) equations for coefficients and gives a computationally simple alternative that is argued to be close to dynare++ fourth-order results.
- Calibration notes:
  - Table 4 standard parameters (Example 2):
    - α = 0:34
    - δ = 1 - 0:9
    - μ = 1/12 0:02
    - β = 0:98^{1/12}
    - γ = 1 = ψ = 0:2 noted in text (CRRA case)
  - Table 5 (stochastic volatility parameters and implied consumption volatility) described but numerical entries for Table 5 beyond the source excerpt are not provided here.
- Advantages emphasized:
  - Directly obtains first-order effects of stochastic volatility on quantities and asset prices via log-linearization and explicit risk adjustments, avoiding higher-order perturbation complications.
  - Matches or closely approximates higher-order perturbation/dynare++ results with far lower computational cost.

### Calibration of stochastic volatility (consumption and productivity)
- Calibration rows (as displayed in the source table):
  - BKY1 0:00002916    0:980    0:0000068
  - BY 0:00006084    0:987    0:0000023
  - BKY2 0:00005184    0:999    0:0000028
  - θφω
  - BKY1 0:00006601    0:980    0:0000155
  - BY 0:00013870    0:987    0:0000052
  - BKY2 0:00011787    0:999    0:0000064
- The top panel (rows 1 to 3) reports calibration of consumption stochastic volatility according to Bansal, Kiku, and Yaron (2006, 2007) and Bansal and Yaron (2004).
- The lower panel reports the calibration of stochastic volatility in productivity that implies the consumption stochastic volatilities shown in the top panel.

### Simulation results and approximation-order comparisons
- Simulations:
  - Results are obtained using dynare++ averaged over 30,000 simulations.
  - The model is approximated up to fourth-order; impulse responses from first-, second-, third-, and fourth-order approximations are superimposed (Figure 3).
- Key findings on approximation orders:
  - Consumption and risk-free rates do not respond to a shock to the volatility of productivity to the first-order.
  - The second-order approximation is characterized as "purely noise" for these responses.
  - The third- and fourth-order approximations capture the dynamics of consumption and the risk-free rate responses to a volatility shock; the first- and second-order approximations do not qualitatively match these results.
  - Expected qualitative dynamics following a positive shock to volatility:
    - Consumption initially drops as consumers save more when risk rises (immediate effect).
    - As volatility returns to steady state, consumption increases and can overshoot steady state because agents accumulate capital to counter higher volatility and production capabilities increase.
    - The risk-free rate decreases as agents save more, then increases slowly as consumption returns to its unconditional mean.
- Comparison with log-normal risk method:
  - Figure 4 compares dynare++ fourth-order results with the log-normal risk adjustment method.
  - Responses using the log-normal risk method are very close to the fourth-order approximation responses.
  - The log-normal risk results are much smoother than the dynare++ average-over-30,000-simulations output.

### Implications for asset prices and risk premia
- Literature context:
  - A strand of asset pricing literature (Bansal and Yaron (2004); Bansal, Kiku, and Yaron (2006, 2007)) argues that consumption stochastic volatility is an extra risk factor that increases premia for a given consumption volatility, assuming shocks to consumption growth and consumption volatility are uncorrelated; increased consumption volatility reduces consumption via precautionary savings, implying a negative correlation between consumption growth and consumption growth volatility.
- Results when stochastic volatility in consumption arises from productivity volatility:
  - If stochastic volatility in consumption is generated by stochastic volatility in productivity, the mechanism suggested in the endowment-asset-pricing literature is unlikely to emerge in equilibrium.
  - Even when innovations to productivity growth and productivity growth variance are uncorrelated, the analysis shows that innovations to consumption growth and consumption growth variance have a positive correlation:
    - Cov_t(g_c t+1 ; σ_c t+1 
 2 ) =c_σ c 2 x ω 2 :
  - Interpretation:
    - Stochastic volatility in consumption, when driven by productivity volatility, becomes a hedge for expected consumption growth and thereby reduces risk premia—opposite to the mechanism assumed in the endowment economy asset pricing literature.
- Parameter dependence:
  - Figure 5 reports the correlation between consumption growth and variance of consumption growth for different calibrations and values of ψ.
  - The source notes: "Higher elasticity of intertemporal substitution implies a stronger negative correlation." (juxtaposed with the model’s positive-correlation result when volatility is productivity-driven)

### Methodological conclusions and recommendations
- Computational methods:
  - The paper suggests solving real business cycle models using approximation techniques common in the asset pricing literature; log-linearization as presented is a convenient tool to capture dynamics of quantities, asset pricing, and welfare implications.
  - The log-linear (log-normal risk-adjustment) method is explicitly related to standard higher-order perturbation approaches; the paper shows that the log-normal risk adjustment closely matches fourth-order perturbation results in the examples studied.
  - The approach offers a computationally efficient way to solve and estimate models with stochastic volatility.
- Practical implication for modelers:
  - For dynamics and asset pricing implications driven by stochastic volatility, higher-order approximations (third- and fourth-order) or log-normal risk adjustments are necessary to capture qualitatively correct responses to volatility shocks; first- and second-order approximations can be misleading (noise or no response).

### Representative equations and solution-approach highlights (selected)
- Simulation-average and approximation statement:
  - "Figure 3 shows the dynare++ results from the 30,000 simulations. We approximate the model up to fourth-order..."
- Correlation result (as presented in the source text):
  - Cov_t(g_c t+1 ; σ_c t+1 
 2 ) =c_σ c 2 x ω 2 :
- Euler equation and value-function structure (selected forms provided in appendices):
  - Value function (scaled variables): V(K_t;A_t) = A_t^{-1} V( K_t A_t^{-1} ; A_t A_t^{-1} )
  - Euler-like condition (scaled form in source): βE_t[ A_t^{-1=ψ} E_t(V^{1-γ}_{t+1})^{(γ-1)/ (1-γ)} ... ] = 1
- Solution methodology:
  - Appendices detail solutions via linearization, method of undetermined coefficients, forward/backward solution of eigenvalue decompositions, and explicit coefficient solutions for Example 1 and Example 2.
  - The log-normal risk-adjustment approach is shown to be closely related to perturbation methods: the log-linear approximation drops quadratic state terms (g_xx; h_xx) and computes g_{σσ}, h_{σσ} analogously but ignoring g_xx terms.

*Source: _wp10249 - References (PDF chapter/section)*

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

### _wp10249 - References

### Introduction: purpose and contribution
- Proposes a simple alternative to higher-order perturbation and value-function methods for solving RBC models with recursive Epstein-Zin preferences and an affine structure for exogenous state variables.
- Key idea: exploit joint log-normality of shocks and use log-linearization to compute log-linearized dynamics and risk adjustments while retaining computational simplicity.
- Claimed advantages:
  - Linear structure eases description of time-series properties and estimation.
  - Analytical form permits explicit inspection of mechanisms behind quantity dynamics and asset prices.
  - Handles stochastic volatility more easily than standard perturbation methods (avoids third- or higher-order expansions).
- Relationship to literature:
  - Builds on Epstein and Zin (1989, 1991), Weil (1989), Schmitt-Grohe and Uribe (2004), and Duffie, Pan, and Singleton (2000).
  - Contrasts with Backus, Routledge, and Zin (2007) and Uhlig (2010) on which equilibrium conditions to linearize and on consistent computation of risk adjustments.

### Setup: model ingredients and equilibrium
- Preferences:
  - Representative consumer with recursive Epstein-Zin utility:
    - Ut = [Ct^(1-1/ψ) + β (Et(Ut+1^(1-γ)))^( (1-1/ψ)/(1-γ) ) ]^(1/(1-1/ψ))
  - Parameters separate risk aversion γ and EIS ψ; power utility is special case γ = 1 = ψ.
- Technology:
  - Cobb-Douglas production: Yt = Zt At^(1-α) Kt^α
  - Capital law of motion: Kt+1 = (1-δ) Kt + Yt - Ct
  - Marginal product of capital Rt = α (At Kt / )^(1-α) + (1-δ) (as shown)
- Shocks and affine structure:
  - Exogenous vector ut with affine discrete-time dynamics:
    - ut+1 = H0 + H1 ut + Σt εt+1
    - εt ~ N(0, Inε)
  - Time-varying covariance evolution:
    - (Σt Σt')ij = (G0)ij + (G1)ij ut
  - Affine structure implies conditional expectation and conditional variance-covariance of ut are affine in ut, enabling conditional normality of variables up to first order.
- Stationary transformation:
  - Scale nonstationary variables: ˜Xt = Xt A_t^-1
  - Stationary equilibrium equations (examples): equations (3)-(6) in the source (value function, capital law, marginal product, Euler equation).

### Log-Normal Risk Adjustment: approximation technique
- Conjecture: log variables yt are conditionally normal; write equilibrium conditions as
  - f(xt+1, xt, yt, Et[exp(Γ yt+1)]) = 0
- Use identity for log-normal variables:
  - Et[exp(Γ yt+1)] = exp(Γ Et[yt+1] + 1/2 Γ (Var_t yt+1) Γ')
- Linearize resulting condition to obtain a log-linear system where yt is linear in state variables and includes explicit constant "risk-adjustment" terms reflecting differences between stochastic and non-stochastic steady states.
- Advantages highlighted:
  - Captures risk adjustment without second- and higher-order Taylor expansions of all terms.
  - Computationally simple, semi-analytical, and tractable for models with stochastic volatility.

### Example 1: RBC with recursive preferences (no stochastic volatility)
- Shock structure used:
  - ut = [at; zt]
  - at+1 = (1-ρa) μ + ρa at + σa ε1,t+1
  - zt+1 = ρz zt + σz ε2,t+1
- Value function manipulation under normality yields (see equation (9)) a log-linear expression with variance corrections:
  - Linearized value function equation example: ζ1 ˆvt = ζ2 ˆct + ζ3 [ˆat + Et ˆvt+1 + 1/2 (1-γ)^2 Var(ˆvt+1)]
- Linearized system (equations (10)-(13)) produces three equations in unknowns ˆv_t, ˆc_t, ˆk_t and can be written as:
  - [Et ˆv_{t+1}; Et ˆc_{t+1}; ˆk_{t+1}] = R + W [ˆv_t; ˆc_t; ˆk_t] + Q Et [ˆa_{t+1}; ˆz_{t+1}]
- Key interpretation:
  - Vector R summarizes constant risk-adjustment terms (difference between stochastic and non-stochastic steady states). These constants affect levels (e.g., precautionary savings lowering consumption relative to no-risk model) and financial variables (unconditional mean of the risk-free rate).
- Quantities and prices:
  - Log consumption growth: gc_{t+1} = ˜c_{t+1} - ˜c_t + ˜a_t
  - Risk-free rate: rf_t ≈ -ln β - 1/2 (1/ψ - γ)(γ - 1) Var_t(˜v_{t+1}) - 1/2 Var_t(m_{t+1}) + (1/ψ) Et[gc_{t+1}]
    - Note: first-order log-linearization would omit the Var_t(˜v_{t+1}) and Var_t(m_{t+1}) terms.
  - Asset risk premium up to first order: Et[ r_{i,t+1} - r_{f,t} ] = -Cov_t( r_{i,t+1}, m_{t+1} ) - 1/2 Var_t( r_{i,t+1} )
- Entropy (market price of risk) for log-normal M_{t+1}:
  - ln Et M_{t+1} - Et ln M_{t+1} = 1/2 Var_t(m_{t+1})

### Comparison with perturbation methods (theoretical and quantitative)
- The log-normal risk method is a truncated version of second-order perturbation:
  - Second-order perturbation solution approximation:
    - Yt - Ȳ ≈ g_s(Ȳ)(St - Ṡ) + 1/2 g_{ss}(Ȳ)(St - Ṡ)^2
  - Log-normal risk keeps second-order terms involving variances of state variables (risk-adjustment) but omits quadratic terms in endogenous state variables (second-order dynamic terms).
- Numerical calibration (monthly):
  - Table 1 calibration highlights (monthly frequency):
    - β = 0:9983
    - α = 0:34
    - μ = 0:0017
    - δ = 0:0087
    - γ = 5
    - ψ = 1:5
    - ρA = 0:14
    - ρZ = 0:0
    - σ^2_A = 0:059
    - σ_Z^2 = 0:01
- Table 2: first-order term coefficients match exactly between perturbation and log-normal risk; second-order dynamic terms are zero in log-normal risk; second-order variance (risk-adjustment) terms are very close but not identical.
- Impulse response comparison (Figures 1 and 2):
  - Dynamic responses of capital, consumption and risk-free rates to one-standard-deviation shocks in a(t) and z(t) are nearly identical between second-order perturbation and log-normal risk (responses measured as percentage deviations from stochastic steady state).
- Stochastic steady-state comparison (Table 3):
  - K_ss: 0:8666 (Perturbation) vs 0:8732 (Log-Normal Risk) — % Diff 0:01
  - C_ss: 0:0379 vs 0:0382 — % Diff 0:01
  - r_fss: -0:0068 vs -0:0069 — % Diff 0:01
  - V_ss: -8:2019 vs -8:2513 — % Diff 0:01
  - Conclusion: stochastic steady states from both methods differ by less than 1 percent for variables shown.

### Example 2: stochastic volatility in a simple growth model
- Setup (CRRA preferences with 1/ψ = γ and no stationary Z shocks):
  - â_{t+1} = μ + σ_t ε_{a,t+1}
  - σ_{t+1}^2 = (1-φ) θ + φ σ_t^2 + ω ε_{σ,t+1}
  - Innovations: E_t[ ε_{a,t+1} ε_{σ,t+1} ] = 0
  - Affine representation:
    - H0 = [ μ ; (1-φ) θ ], H1 = [ 0 0 ; 0 φ ], Σ_t = [ σ_t 0 ; 0 ω ]
- Solution form conjectured:
  - ˆc_t = R_c + α ˆa_t + β σ_t^2
  - Var_t(ˆc_{t+1}) = [ α^2 + β^2 φ ] σ_t^2 + β^2 ω^2
  - Conditional variance of consumption growth inherits productivity variance process leading to an AR(1)-type variance dynamics:
    - (σ_{c,t+1})^2 = (1-φ) ( c_a^2 θ + c_σ^2 ω^2 ) + φ (σ_{c,t})^2 + c_a^2 ω ε_{t+1}
  - Coefficients β and R are solved by equating coefficients on σ_t^2 and constants.
- Qualitative comparison with perturbation:
  - Capturing stochastic volatility fully with perturbation requires fourth-order expansion; dynare++ can perform fourth-order approximations but is computationally expensive and a "black box".
  - The log-normal risk method yields algebraic (quadratic) equations for coefficients and gives a computationally simple alternative that is argued to be close to dynare++ fourth-order results.
- Calibration notes:
  - Table 4 standard parameters (Example 2):
    - α = 0:34
    - δ = 1 - 0:9
    - μ = 1/12 0:02
    - β = 0:98^{1/12} (presented as 0:98^{1/12} and 1/12 formatting in source)
    - γ = 1 = ψ = 0:2 noted in text (CRRA case)
  - Table 5 (stochastic volatility parameters and implied consumption volatility) described but numerical entries for Table 5 beyond the source excerpt are not provided here.
- Advantages emphasized:
  - Directly obtains first-order effects of stochastic volatility on quantities and asset prices via log-linearization and explicit risk adjustments, avoiding higher-order perturbation complications.
  - Matches or closely approximates higher-order perturbation/dynare++ results with far lower computational cost.

_Italic: Source: _wp10249 - References (PDF chapter/section)

### 2. The top panel shows the calibration of consumption stochastic volatility according to Bansal,

### _wp10249 - 2. The top panel shows the calibration of consumption stochastic volatility according to Bansal,

### Calibration of stochastic volatility (consumption and productivity)
- Calibration rows (as displayed in the source table):
  - BKY1 0:00002916    0:980    0:0000068
  - BY 0:00006084    0:987    0:0000023
  - BKY2 0:00005184    0:999    0:0000028
  - θφω
  - BKY1 0:00006601    0:980    0:0000155
  - BY 0:00013870    0:987    0:0000052
  - BKY2 0:00011787    0:999    0:0000064
- The top panel (rows 1 to 3) reports calibration of consumption stochastic volatility according to Bansal, Kiku, and Yaron (2006, 2007) and Bansal and Yaron (2004).
- The lower panel reports the calibration of stochastic volatility in productivity that implies the consumption stochastic volatilities shown in the top panel.

### Simulation results and approximation-order comparisons
- Simulations:
  - Results are obtained using dynare++ averaged over 30,000 simulations.
  - The model is approximated up to fourth-order; impulse responses from first-, second-, third-, and fourth-order approximations are superimposed (Figure 3).
- Key findings on approximation orders:
  - Consumption and risk-free rates do not respond to a shock to the volatility of productivity to the first-order.
  - The second-order approximation is characterized as "purely noise" for these responses.
  - The third- and fourth-order approximations capture the dynamics of consumption and the risk-free rate responses to a volatility shock; the first- and second-order approximations do not qualitatively match these results.
  - The expected qualitative dynamics described:
    - Following a positive shock to volatility, consumption initially drops (consumers save more as risk in the economy rises — immediate effect).
    - As volatility returns to steady state, consumption increases and can overshoot the steady state because agents accumulate capital to counter higher volatility and production capabilities increase.
    - The risk-free rate decreases as agents save more relative to pre-shock levels, then increases slowly as consumption returns to its unconditional mean.
- Comparison with log-normal risk method:
  - Figure 4 compares dynare++ fourth-order results with the log-normal risk adjustment method.
  - Responses using the log-normal risk method are very close to the fourth-order approximation responses.
  - The log-normal risk results are much smoother than the dynare++ average-over-30,000-simulations output.

### Implications for asset prices and risk premia
- Literature context:
  - A strand of asset pricing literature (Bansal and Yaron (2004); Bansal, Kiku, and Yaron (2006, 2007)) argues that consumption stochastic volatility is an extra risk factor that increases premia for a given consumption volatility, assuming shocks to consumption growth and consumption volatility are uncorrelated; increased consumption volatility reduces consumption via precautionary savings, implying a negative correlation between consumption growth and consumption growth volatility.
- Results when stochastic volatility in consumption arises from productivity volatility:
  - If stochastic volatility in consumption is generated by stochastic volatility in productivity, the mechanism suggested in the endowment-asset-pricing literature is unlikely to emerge in equilibrium.
  - Even when innovations to productivity growth and productivity growth variance are uncorrelated, the analysis shows that innovations to consumption growth and consumption growth variance have a positive correlation:
    - Cov_t(g_c t+1 ; σ_c t+1 
 2 ) =c_σ c 2 x ω 2 :
    - Note in the source text this correlation is reported as positive, implying consumption growth increases when there are shocks to consumption growth volatility.
  - Interpretation:
    - Stochastic volatility in consumption, when driven by productivity volatility, becomes a hedge for expected consumption growth and thereby reduces risk premia—opposite to the mechanism assumed in the endowment economy asset pricing literature.
- Parameter dependence:
  - Figure 5 reports the correlation between consumption growth and variance of consumption growth for different calibrations and values of ψ.
  - The source notes: "Higher elasticity of intertemporal substitution implies a stronger negative correlation." (The text juxtaposes that empirical/analytic observation with the model’s positive-correlation result when volatility is productivity-driven.)

### Methodological conclusions and recommendations
- Computational methods:
  - The paper suggests solving real business cycle models using approximation techniques common in the asset pricing literature; log-linearization as presented is a convenient tool to capture dynamics of quantities, asset pricing, and welfare implications.
  - The log-linear (log-normal risk-adjustment) method is explicitly related to standard higher-order perturbation approaches; the paper shows that the log-normal risk adjustment closely matches fourth-order perturbation results in the examples studied.
  - The approach offers a computationally efficient way to solve and estimate models with stochastic volatility.
- Practical implication for modelers:
  - For dynamics and asset pricing implications driven by stochastic volatility, higher-order approximations (third- and fourth-order) or log-normal risk adjustments are necessary to capture qualitatively correct responses to volatility shocks; first- and second-order approximations can be misleading (noise or no response).

### Representative equations and solution-approach highlights (selected from appendices)
- Simulation-average and approximation statement:
  - "Figure 3 shows the dynare++ results from the 30,000 simulations. We approximate the model up to fourth-order..."
- Correlation result (as presented in the source text):
  - Cov_t(g_c t+1 ; σ_c t+1 
 2 ) =c_σ c 2 x ω 2 :
- Euler equation and value-function structure (selected forms provided in appendices):
  - Value function (scaled variables): V(K_t;A_t) = A_t^{-1} V( K_t A_t^{-1} ; A_t A_t^{-1} )
  - Euler-like condition (scaled form in source): βE_t[ A_t^{-1=ψ} E_t(V^{1-γ}_{t+1})^{(γ-1)/ (1-γ)} ... ] = 1  (see source for full expressions)
- Solution methodology:
  - The appendices detail solutions via linearization, method of undetermined coefficients, forward/backward solution of eigenvalue decompositions, and explicit coefficient solutions for Example 1 (no stochastic volatility) and Example 2 (with stochastic volatility).
  - The log-normal risk-adjustment approach is shown to be closely related to perturbation methods: the log-linear approximation drops quadratic state terms (g_xx; h_xx) and computes g_{σσ}, h_{σσ} analogously but ignoring g_xx terms.

*Source: _wp10249 - 2. The top panel shows the calibration of consumption stochastic volatility according to Bansal, (PDF): https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10249.pdf*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10249.pdf_