## NKV model, empirical findings, and policy implications (wpiea2020236-print-pdf)

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### Research question and model overview
- Purpose: present an expanded New Keynesian model (NKV for “New Keynesian Vulnerability”) that endogenizes the price of risk and links it to financial conditions to jointly analyze monetary and macroprudential policies.
- Core mechanism:
  - Conditional volatility of the output gap is proportional to the price of risk, creating a “vulnerability channel” of monetary policy and an intertemporal risk-return tradeoff for policymakers.
- Model scope:
  - Parsimonious four-equation setup: IS curve with a financial accelerator and heteroskedastic wedge; law of motion for financial conditions η_t; Phillips curve; Taylor rule.
  - Vulnerability function V(X_t) ≡ max{ν − %′ X_t, 0} (piecewise-affine; ensures nonnegative volatility).

### Key model equations and structure
- IS curve (with endogenous heteroskedastic shock):
  - y_gap_t = E_t y_gap_{t+1} − (1/σ)(i_t − E_t π_{t+1}) − γ_η η_t − V(X_t) ε_{ygap,t}  (Equation (1))
  - Shock: ε_{ygap,t} ~ N.i.d.(0, σ_y^2); realized wedge ≡ V(X_t) ε_{ygap,t}.
- Financial conditions law of motion (semi-structural and reduced form):
  - η_t ≡ λ_η η_{t−1} + λ_{ηη} η_{t−2} − θ_y y_gap_t − θ_η E_t y_gap_{t+1}  (Equation (2))
  - Under equilibrium/policy, reduced-form: η_t = γ_η η_{t−1} + γ_{ηη} η_{t−2} + γ_ε ε_{y,t}  (Equation (6))
  - Example baseline reduced-form AR(2): η_t = 1.97 η_{t−1} − 1.01 η_{t−2}
- Pricing kernel and price of risk:
  - Log-SDF: m_t = ̃m_t − logβ = a_1 η_{t−1} + a_2 η_{t−2} + a_3 y_gap_{t−1} + b_m V(·) ε_{ygap,t}
  - Conditional volatility of m_t: vol(m_t | F_{t−1}) = |b_m| σ_y (ν − ̃%′[η_{t−1}, η_{t−2}, η_{t−3}, _{ygap,t−1}]′)_+
  - Implication: conditional volatility (price of risk) is piecewise-affine in η and the IS wedge.

### Empirical calibration and data
- Data used:
  - output gap = log-difference between real GDP and CBO potential; annual core PCE inflation; National Financial Conditions Index (NFCI); sample 1973–2017.
- Calibration:
  - NK structural parameters set to textbook Chapter 3 of Galí (2008); additional parameters chosen to match first-order autocorrelations of Δy_gap_t and E_t Δy_gap_{t+1}, and correlations with η_t.
- Fit to moments (reported data moments):
  - corr(Δy_gap_t, Δy_gap_{t−1}) = 0.33
  - corr(E_t Δy_gap_{t+1}, E_{t−1} Δy_gap_t) = 0.82
  - corr(Δy_gap_t, η_t) = −0.43
  - corr(E_t Δy_gap_{t+1}, η_t) = −0.82

### Five stylized empirical facts and model performance
- Stylized Fact 1 — Financial variables predict the tail of the output gap distribution:
  - Empirical: NFCI/FCI significantly predicts the conditional 5th, median, and 95th quantiles of one-quarter-ahead output gap growth (example: p-value FCI = 0.0032).
  - NKV: simulated conditional quantiles reproduce skewness where lower quantiles move sharply with financial conditions while upper quantiles are relatively stable.
- Stylized Fact 2 — Conditional output gap median and volatility correlate negatively:
  - Empirical regression shows a strong negative linear relation; NKV reproduces the negative correlation with high R^2 in simulation.
- Stylized Fact 3 — Financial variables do not predict tails of inflation:
  - Empirical: FCI coefficient lacks significance at the 50 percent level for inflation tails (example: p-value FCI = 0.5421); inflation heteroskedasticity largely driven by past inflation.
  - NKV: captures lack of financial-conditions predictability for inflation tails and produces a positive slope between inflation conditional median and volatility.
- Stylized Fact 4 — The “volatility paradox” (future risk builds during good times):
  - When η is low (loose financial conditions), short-term volatility is low but the conditional volatility elasticity to η is negative in the near term and becomes positive at longer horizons; conditional mean elasticity falls and becomes negative over horizon.
- Stylized Fact 5 — Term structures of Growth-at-Risk cross:
  - Empirical: 5th percentile term structures conditional on initial FCI deciles cross over the projection horizon (bottom decile easier FCI shows lower near-term downside risk but higher medium-term downside risk).
  - NKV: simulations replicate the crossing of growth-at-risk term structures.

### Monetary policy implications and instability mechanism
- Benchmark Taylor rule coefficients: φ_π = 1.5; φ_y = 0.125 (baseline).
- Key finding:
  - Making Taylor rule coefficients arbitrarily large can lead to explosive financial conditions in NKV — i.e., excessive output-gap stabilization can destabilize η_t (Minsky-like “too much stability breeds instability”).
- Stability illustrations:
  - Increasing both Taylor coefficients simultaneously, maximum multiplier for stability reported: coefficients remain stable only up to about 2.1×[1.5,0.125] = [3.15,0.2625].
  - Other reported limits when scaling only one coefficient include [3.525,0.125] and [1.5,0.15].
- Mechanism:
  - Aggressive stabilization drives y_gap and E_t y_gap_{t+1} toward zero, causing equilibrium coefficients γ_η, γ_{ηη} to approach semi-structural λ_η, λ_{ηη}; if the semi-structural AR(2) for η_t is unstable, reducing output gap volatility can push η_t toward an unstable law of motion and violate Blanchard-Kahn conditions.

### Macroprudential policy design and results
- Augmented law of motion with cyclical macroprudential instrument μ_t:
  - η_t = μ_t + λ_η η_{t−1} + λ_{ηη} η_{t−2} − θ_y y_gap_t − θ_η E_t y_gap_{t+1}
  - Example stabilizing design: μ_t = ν_η η_{t−1} + ν_{ηη} η_{t−2} ⇒ η_t = (λ_η + ν_η) η_{t−1} + (λ_{ηη} + ν_{ηη}) η_{t−2} − θ_y y_gap_t − θ_η E_t y_gap_{t+1}
- Main result:
  - A macroprudential rule that stabilizes the autoregressive coefficients of η_t (so that (λ_η + ν_η, λ_{ηη} + ν_{ηη}) correspond to a stable AR(2)) removes the instability risk induced by very aggressive monetary policy.
  - Combined monetary and macroprudential policy can approximate full stabilization (output gap, inflation, financial conditions).
  - Even lagged macroprudential effects can achieve stability if the induced process for η_t is stable.

### Alternative monetary policy path: Taylor rule augmented for expected financial conditions
- Expanded Taylor rule:
  - i_t = φ_π π_t + φ_y y_gap_t − φ_η E_t η_{t+1}  (Equation (7)); example φ_η = 0.1.
- Comparative findings:
  - Extended Taylor rule reduces ergodic standard deviation of η relative to standard Taylor rule; extreme realizations of η become much less likely.
  - Extended rule lowers output gap volatility and places more mass near the mean in ergodic output gap distribution, though it does not always dominate a combined monetary+macroprudential strategy.
  - Initialization effects: from the same high η, the extended rule may imply larger immediate rate hikes (example: extra 115bps in one simulation) producing a sharper near-term output contraction and temporarily higher inflation, highlighting tradeoffs and the role of expectations credibility.

### Policy recommendations and tradeoffs
- Monetary policy focused solely on stabilizing inflation and the output gap risks destabilizing financial conditions if financial conditions are endogenous and their semi-structural dynamics are unstable.
- A suitable combination of macroprudential policy (targeting stability of η) with monetary policy can ensure efficiency and approximate full stabilization of inflation, output gap, and financial conditions.
- When macroprudential tools are unavailable or ineffective, augmenting a Taylor rule with expected financial conditions materially lowers volatility and the probability of extreme financial-condition outcomes, though it cannot fully eliminate volatility nor always substitute for a direct macroprudential counterpart.
- Policymakers face an intertemporal tradeoff: easing policy lowers near-term downside risk via risk-taking but can raise medium-term vulnerability if the vulnerability channel is ignored.

### Appendix A — analytics of the conditional mean-volatility trade-off slope (summary)
- First-order approximation (linear model):
  - y_{t+1} = y_ss + A(y_t − y_ss) + B ε_{t+1}
  - Conditional variance of y_{t+1} equals B Σ_ε and is independent of the state y_t.
- Second-order approximation:
  - y_{t+1} = y_ss + 1/2 g_{σσ} + A(y_t − y_ss) + B ε_{t+1} + C(y_t − y_ss)^2 + D(y_t − y_ss) ε_{t+1} + E ε_{t+1}^2
  - Conditional mean μ_{2nd} and variance (σ^2_{2nd})^2 expressions show:
    - Conditional variance depends on state y_t only if D ≠ 0.
    - Changes in conditional mean are first-order phenomena; changes in conditional volatility require higher-order terms (D ≠ 0).

### Appendix B — analytical derivations of correlation coefficients (summary)
- Reduced form under Equations (1)–(4):
  - η_t = F_2 η_{t−1} + F_3 η_{t−2} + F_1 ε_{ygap,t}
  - y_gap,t = P_2 η_{t−1} + P_3 η_{t−2} + P_1 ε_{ygap,t}
  - Coefficients [F_1,F_2,F_3] and [P_1,P_2,P_3] are non-linear functions of structural parameters.
- Lemmas and closed-form results (selected):
  - Lemma 2: y_gap,t expressed as an ARMA(2,2) with explicit MA terms in ε_{ygap,t−k}.
  - Lemma 4: dy_gap,t follows an ARMA(2,3) with MA terms up to ε_{ygap,t−3}.
  - Lemma 5: E_t y_gap,t+1 is ARMA(2,1) with MA terms up to ε_{ygap,t−1}.
  - Lemma 6: E_t y_gap,t+1 − y_gap,t is ARMA(2,2) with explicit MA coefficients.
  - Lemma 7: explicit expressions linking E_t y_gap,t+1, dy_gap,t, and E_t dy_gap,t+1 to η_{t−i} and ε_{ygap,t−k}.
- Covariance formulas with η_t (using γ(i) as i-th autocovariance of η_t):
  - cov(η_t, y_gap,t) = P_2 γ(1) + P_3 γ(2) + P_1 F_1
  - cov(η_t, E_t y_gap,t+1) = (P_2 F_2 + P_3) γ(1) + P_2 F_3 γ(2) + P_2 F_1
  - cov(η_t, dy_gap,t) = P_2 γ(1) + (P_3 − P_2) γ(2) − P_3 γ(3) + P_1 F_1 − P_1 F_2 F_1
  - cov(η_t, E_t dy_gap,t+1) = (P_2 F_2 + P_3 − P_2) γ(1) + (P_2 F_3 − P_3) γ(2) + (P_2 F_1 − P_1) F_1
- Autocorrelations of η_t (τ(i) = corr(η_t, η_{t−i})):
  - τ(1) = F_2 / (1 − F_3)
  - τ(2) = (F_3 − F_2^2) / (F_3 − 1)
  - τ(3) = (−F_3^2 + F_2 (F_3 − 2) F_3) / (F_3 − 1)

*Source: wpiea2020236-print-pdf — Appendix B: Analytical Derivations of Correlation Coefficients*

### 1.    Introduction1

### 1.    Introduction1

### Major themes and structure
- 2.    The NKV Model3
  - 2.1.    Links between financial conditions and the price of risk7
- 3.    Empirical Evidence8
  - 3.1.    Data8
  - 3.2.    Model calibration8
  - 3.3.    Stylized Fact 1:  Financial variables predict the tail of the output gap distribution9
  - 3.4.    Stylized Fact 2:  Conditional output gap growth median and volatility correlate negatively10
  - 3.5.    Stylized Fact 3:  Financial variables do not predict tails of inflation11
  - 3.6.    Stylized Fact 4:  The volatility paradox12
  - 3.7.    Stylized Fact 5:  Term structures of growth-at-risk cross14
- 4.    Monetary Policy15
- 5.    Macroprudential Policy18
- 6.    Alternative Policy Paths with Endogenous Risk21
- 7.    Related Literature25
- 8.    Conclusion27
- References29
- Appendix A: The Analytics of the Conditional Mean-Volatility Trade-off Slope31
  - A.1.    The linear model31
  - A.2.    Second-order approximation31

*Canonical source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020236-print-pdf.pdf*

### Appendix B: Analytical Derivations of Correlation Coefficients32

### Appendix B: Analytical Derivations of Correlation Coefficients32

### Introduction and research question
- Purpose: present an expanded New Keynesian (NK) model (NKV for “New Keynesian Vulnerability”) that endogenizes the price of risk and links it to financial conditions to jointly analyze monetary and macroprudential policies.
- Core mechanism: conditional volatility of the output gap is proportional to the price of risk, creating a “vulnerability channel” of monetary policy that generates an intertemporal risk-return tradeoff for policymakers.
- Model scope: parsimonious four-equation setup (IS curve with a financial accelerator and heteroskedastic wedge; law of motion for financial conditions η_t; Phillips curve; Taylor rule) used to evaluate output, inflation, and financial vulnerability under simple policy rules.

### The NKV model — structure and key equations
- IS curve (with endogenous heteroskedastic shock):
  - y_gap_t = E_t y_gap_{t+1} − (1/σ)(i_t − E_t π_{t+1}) − γ_η η_t − V(X_t) ε_{ygap,t}  (Equation (1))
  - Vulnerability function: V(X_t) ≡ max{ν − %′ X_t, 0} (piecewise-affine; ensures nonnegative volatility)
  - Shock: ε_{ygap,t} ~ N.i.d.(0, σ_y^2) and the realized wedge is _{ygap,t} ≡ V(X_t) ε_{ygap,t}
- Financial conditions law of motion (semi-structural):
  - η_t ≡ λ_η η_{t−1} + λ_{ηη} η_{t−2} − θ_y y_gap_t − θ_η E_t y_gap_{t+1}  (Equation (2))
  - Under equilibrium and policy, reduced-form may be written η_t = γ_η η_{t−1} + γ_{ηη} η_{t−2} + γ_ε ε_{y,t}  (Equation (6))
  - Example baseline reduced-form AR(2) in paper: η_t = 1.97 η_{t−1} − 1.01 η_{t−2}
- Pricing kernel and price of risk link:
  - Log-SDF: m_t = ̃m_t − logβ = a_1 η_{t−1} + a_2 η_{t−2} + a_3 y_gap_{t−1} + b_m V(·) ε_{ygap,t}
  - Conditional volatility of m_t: vol(m_t | F_{t−1}) = |b_m| σ_y (ν − ̃%′[η_{t−1}, η_{t−2}, η_{t−3}, _{ygap,t−1}]′)_+
  - Implication: conditional volatility (price of risk) is piecewise-affine in η and the IS wedge.

### Empirical calibration and fit
- Data: output gap = log-difference between real GDP and CBO potential; annual core PCE inflation; National Financial Conditions Index (NFCI); sample 1973–2017.
- Calibration approach: structural NK parameter values held to textbook Chapter 3 of Galí (2008); additional parameters chosen to match first-order autocorrelations of Δy_gap_t and E_t Δy_gap_{t+1}, and correlations with η_t.
- Fit to moments (Table 3 reported):
  - Data moments: corr(Δy_gap_t, Δy_gap_{t−1}) = 0.33; corr(E_t Δy_gap_{t+1}, E_{t−1} Δy_gap_t) = 0.82; corr(Δy_gap_t, η_t) = −0.43; corr(E_t Δy_gap_{t+1}, η_t) = −0.82
  - VAR and NKV reported to provide comparable fits to targeted moments (see Table 3 in source).

### Five stylized empirical facts and model performance
- Stylized Fact 1 — Financial variables predict the tail of the output gap distribution:
  - Empirical: NFCI/FCI significantly predicts the conditional 5th quantile, median, and 95th quantile of one-quarter-ahead output gap growth (p-value example: p-value FCI = 0.0032).
  - NKV: simulated conditional quantiles reproduce skewness pattern where lower quantiles move sharply with financial conditions while upper quantiles are relatively stable.
- Stylized Fact 2 — Conditional output gap median and volatility correlate negatively:
  - Empirical regression (example reported): strong negative linear relation; model reproduces negative correlation with high R^2 in simulation (figures in source).
- Stylized Fact 3 — Financial variables do not predict tails of inflation:
  - Empirical: FCI coefficient lacks significance at the 50 percent level for inflation tails (example p-value FCI = 0.5421); inflation’s conditional heteroskedasticity is largely driven by past inflation.
  - NKV: qualitatively captures lack of financial-conditions predictability for inflation tails and positive slope between inflation conditional median and volatility.
- Stylized Fact 4 — The “volatility paradox” (future risk builds during good times):
  - Empirical and model: when η is low (loose financial conditions), short-term volatility is low but conditional volatility elasticity to η is negative in near term and becomes positive at longer horizons; conditional mean elasticity falls and becomes negative over horizon.
- Stylized Fact 5 — Term structures of Growth-at-Risk cross:
  - Empirical: 5th percentile of conditional Δy_gap term structures conditional on initial FCI deciles cross over projection horizon (bottom decile easier FCI shows lower near-term downside risk but higher medium-term downside risk).
  - NKV: simulations replicate crossing of growth-at-risk term structures.

### Monetary policy implications and results
- Benchmark Taylor rule coefficients: φ_π = 1.5; φ_y = 0.125 (baseline).
- Key finding: unlike standard NK, making Taylor rule coefficients arbitrarily large (more “activist”) can lead to explosive financial conditions in the NKV model — i.e., too much output-gap stabilization can destabilize η_t and the equilibrium (Minsky-like “too much stability breeds instability”).
- Stability illustrations:
  - When increasing both Taylor coefficients simultaneously, maximum multiplier for stability reported: coefficients remain stable only up to about 2.1×[1.5,0.125] = [3.15,0.2625]. Other limits when scaling only one coefficient are reported in source (e.g., [3.525,0.125] or [1.5,0.15]).
- Mechanism: aggressive stabilization drives y_gap and E_t y_gap_{t+1} toward zero, which causes equilibrium coefficients γ_η, γ_{ηη} to approach semi-structural λ_η, λ_{ηη}; if the semi-structural AR(2) for η_t is unstable, reducing output gap volatility can push η_t toward an unstable law of motion and violate Blanchard-Kahn conditions.

### Macroprudential policy implications
- Augment model with a cyclical macroprudential instrument μ_t that directly affects η_t contemporaneously:
  - η_t = μ_t + λ_η η_{t−1} + λ_{ηη} η_{t−2} − θ_y y_gap_t − θ_η E_t y_gap_{t+1}
  - Example stabilizing design: μ_t = ν_η η_{t−1} + ν_{ηη} η_{t−2} yields semi-structural η_t = (λ_η + ν_η) η_{t−1} + (λ_{ηη} + ν_{ηη}) η_{t−2} − θ_y y_gap_t − θ_η E_t y_gap_{t+1}
- Main result: a macroprudential rule that stabilizes the autoregressive coefficients of η_t (i.e., makes (λ_η + ν_η, λ_{ηη} + ν_{ηη}) correspond to a stable AR(2)) removes the instability risk induced by very aggressive monetary policy; combined monetary and macroprudential policy can approximate full stabilization (output gap, inflation, financial conditions).
- Even lagged macroprudential effects can achieve stability if the induced process for η_t is stable.

### Alternative monetary policy path: Taylor rule augmented for expected financial conditions
- Proposed expanded Taylor rule:
  - i_t = φ_π π_t + φ_y y_gap_t − φ_η E_t η_{t+1}  (Equation (7)) with example φ_η = 0.1
- Comparative results:
  - Extended Taylor rule reduces ergodic standard deviation of η relative to standard Taylor rule; extreme realizations of η become much less likely (histogram comparisons in source).
  - Extended rule yields lower output gap volatility and more mass near the mean in ergodic output gap distribution, though it does not fully eliminate volatility or always dominate a combined monetary+macroprudential strategy.
  - Initialization considerations: when comparing impulse responses starting from the same high η, the extended rule may imply larger immediate rate hikes (example reported: extra 115bps in one simulation) that produce a sharper near-term output contraction and temporarily higher inflation — illustrating tradeoffs and the importance of credible expectations.

### Summary of policy recommendations and tradeoffs
- Monetary policy that fully focuses on stabilizing inflation and the output gap (as in standard NK prescriptions) risks destabilizing financial conditions if financial conditions are endogenous and their semi-structural dynamics are unstable.
- A suitable combination of macroprudential policy (targeting stability of η) with monetary policy can ensure efficiency and can approximate full stabilization of inflation, output gap, and financial conditions.
- When macroprudential tools are not available or effective, augmenting a Taylor rule with a term for expected financial conditions can materially lower volatility and the probability of extreme financial-condition outcomes, though it cannot fully eliminate volatility or always substitute for a direct macroprudential counterpart.
- Policymakers face an intertemporal tradeoff: easing policy lowers near-term downside risk via risk-taking but can raise medium-term vulnerability if the vulnerability channel is ignored.

### Conclusion
- The NKV model is a parsimonious semi-structural framework that endogenizes volatility via a vulnerability function and links financial conditions, the price of risk, and macroeconomic outcomes.
- Empirically, the model matches key stylized facts: negative contemporaneous correlation between conditional mean and volatility of output gap growth; crossing term structures of Growth-at-Risk; financial-conditions predictability of output tails but not inflation tails; and the volatility paradox.
- Policy implication: accounting for endogenous risk is essential — coordinated macroprudential policies that stabilize financial conditions improve the feasibility and welfare properties of monetary policy; alternatively, vulnerability-aware monetary rules can reduce volatility when macroprudential tools are limited.

*Source: Appendix B: Analytical Derivations of Correlation Coefficients32, wpiea2020236-print-pdf.*

### References

### wpiea2020236-print-pdf - References

### References
- The section provides the bibliography cited in the chapter, listing works on financial cycles, leverage, monetary policy, liquidity, credit spreads, and related macro-financial modeling.

### Appendix A: The Analytics of the Conditional Mean-Volatility Trade-off Slope
- Research question:
  - What is the lowest approximation order for which a DSGE model can generate a non-trivial relationship between the conditional mean and conditional variance of its variable?
  - Model focus: two variables y_gap and π jointly denoted y ≡ (y_gap, π), approximated around y_ss and driven by an N.i.d. shock vector ε_t. Conditional distribution P(y_{t+1} | F_t) is analyzed.

- A.1. The linear model (first-order):
  - First-order policy function:
    - y_{t+1} = y_ss + A(y_t − y_ss) + B ε_{t+1}
  - Conditional distribution:
    - P(y_{t+1} | F_t) = N(y_ss + A(y_t − y_ss), B Σ_ε)
  - Key implication:
    - The conditional variance of y_{t+1} equals B Σ_ε and is independent of the state y_t.
  - Remark 1:
    - Σ_ε denotes the standard deviation of the exogenous disturbance.
    - E ε_{t+1}^p = { 0 if p is odd; Σ_ε,p p!! if p is even } where p!! = 1×3×...×(p−1).

- A.2. Second-order approximation:
  - Second-order policy function:
    - y_{t+1} = y_ss + 1/2 g_{σσ} + A(y_t − y_ss) + B ε_{t+1} + C(y_t − y_ss)^2 + D(y_t − y_ss) ε_{t+1} + E ε_{t+1}^2
  - Conditional moments:
    - μ_{2nd} = y_ss + 1/2 g_{σσ} + A(y_t − y_ss) + C(y_t − y_ss)^2 + E Σ_{ε,2}
    - (σ^2_{2nd})^2 = (B + D(y_t − y_ss))^2 Σ_{ε,2} + E^2 Σ_{ε,4} (3 − 1)
      - Simplified form: (B + D(y_t − y_ss))^2 Σ_{ε,2} + 2 E^2 Σ_{ε,4}
  - Key implications:
    - The conditional distribution is non-normal (E ε_{t+1}^2 is χ^2(1)).
    - Whether conditional variance is time/state-varying hinges on D:
      - If D = 0, conditional variance is constant:
        - D = 0 ⇒ (σ^2_{2nd})^2 = B^2 Σ_{ε,2} + 2 E^2 Σ_{ε,4}
      - If D ≠ 0, conditional variance depends on the state y_t via D(y_t − y_ss).
    - Skew involves a mixture of normal and χ^2-distributed components:
      - skew = E[( (B + D(y_t − y_ss)) ε_{t+1} + E(ε_{t+1}^2 − Σ_{ε,2}) )^3]
    - Higher moments remain time/state invariant unless D ≠ 0.
  - Interpretation:
    - Changes in the conditional mean are first-order phenomena; changes in conditional volatility require higher-order terms (D ≠ 0).

### Appendix B: Analytical Derivations of Correlation Coefficients
- Model setup and reduced form:
  - Under Equations (1)–(4), η_{t−1} and η_{t−2} are the only state variables.
  - Reduced form:
    - η_t = F_2 η_{t−1} + F_3 η_{t−2} + F_1 ε_{ygap,t}
    - y_gap,t = P_2 η_{t−1} + P_3 η_{t−2} + P_1 ε_{ygap,t}
  - Coefficients [F_1,F_2,F_3] and [P_1,P_2,P_3] are non-linear functions of structural parameters.
  - The appendix derives laws of motion for y_gap,t, E_t y_gap,t+1, dy_gap,t, and E_t dy_gap,t+1 as functions of F and P coefficients.

- Lemma 2: level of the output gap (ARMA(2,2))
  - y_gap,t = F_2 y_gap,t−1 + F_3 y_gap,t−2 + P_1 ε_{ygap,t} + (F_1 P_2 − F_2 P_1) ε_{ygap,t−1} + (F_1 P_3 − F_3 P_1) ε_{ygap,t−2}

- Remark 3:
  - If ε_{ygap,t} ∼ N(0,1), one can define ̃ε_{ygap,t} = P_1 ε_{ygap,t} ∼ N(0, P_1^2) and write y_gap,t as a standard ARMA(2,2) with noise variance P_1^2.

- Lemma 4: change in output gap (ARMA(2,3))
  - dy_gap,t = F_2 dy_gap,t−1 + F_3 dy_gap,t−2 + P_1 ε_{ygap,t} + (F_1 P_2 − (F_2 + 1) P_1) ε_{ygap,t−1} + (F_1 (P_3 − P_2) − (F_3 − F_2) P_1) ε_{ygap,t−2} − (F_1 P_3 − F_3 P_1) ε_{ygap,t−3}

- Lemma 5: conditional mean of the output gap (ARMA(2,1))
  - E_t y_gap,t+1 = F_2 E_{t−1} y_gap,t + F_3 E_{t−2} y_gap,t−1 + P_2 F_1 ε_{ygap,t} + P_3 F_1 ε_{ygap,t−1}

- Lemma 6: conditional mean of the change in output gap (ARMA(2,2))
  - E_t y_gap,t+1 − y_gap,t = F_2 E_{t−1} dy_gap,t + F_3 E_{t−2} dy_gap,t−1 + (P_1 − F_1) ε_{ygap,t} + (F_1 P_2 − F_2 P_1 − F_1) ε_{ygap,t−1} + (F_1 P_3 − F_3 P_1) ε_{ygap,t−2}

- Lemma 7: expressions linking variables to η
  - If η_t = F_2 η_{t−1} + F_3 η_{t−2} + F_1 ε_{ygap,t} and y_gap,t = P_2 η_{t−1} + P_3 η_{t−2} + P_1 ε_{ygap,t}, then:
    - E_t y_gap,t+1 = (P_2 F_2 + P_3) η_{t−1} + P_2 F_3 η_{t−2} + P_2 F_1 ε_{ygap,t}
    - dy_gap,t = P_2 η_{t−1} + (P_3 − P_2) η_{t−2} − P_3 η_{t−3} + P_1 ε_{ygap,t} − P_1 ε_{ygap,t−1}
    - E_t dy_gap,t+1 = (P_2 F_2 + (P_3 − P_2)) η_{t−1} + (P_2 F_3 − P_3) η_{t−2} + (P_2 F_1 − P_1) ε_{ygap,t}

- Remark 8: covariances with η_t
  - Using γ(i) as the i-th order autocovariance of η_t, the appendix gives closed-form expressions:
    - cov(η_t, y_gap,t) = P_2 γ(1) + P_3 γ(2) + P_1 F_1
    - cov(η_t, E_t y_gap,t+1) = (P_2 F_2 + P_3) γ(1) + P_2 F_3 γ(2) + P_2 F_1
    - cov(η_t, dy_gap,t) = P_2 γ(1) + (P_3 − P_2) γ(2) − P_3 γ(3) + P_1 F_1 − P_1 F_2 F_1
    - cov(η_t, E_t dy_gap,t+1) = (P_2 F_2 + P_3 − P_2) γ(1) + (P_2 F_3 − P_3) γ(2) + (P_2 F_1 − P_1) F_1

- Remark 9: autocovariances and autocorrelations of η_t
  - γ(1), γ(2), and γ(3) are computable from η_t = F_2 η_{t−1} + F_3 η_{t−2} + F_1 ε_{ygap,t}.
  - First three autocorrelation coefficients τ(i) = corr(η_t, η_{t−i}) are:
    - τ(1) = F_2 / (1 − F_3)
    - τ(2) = (F_3 − F_2^2) / (F_3 − 1)
    - τ(3) = (−F_3^2 + F_2 (F_3 − 2) F_3) / (F_3 − 1)

- Computational note:
  - The companion Mathematica files contain all underlying derivations.

*Source: wpiea2020236-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020236-print-pdf.pdf_