## _wp08284

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### Introduction and research question
- Purpose: study the role of news about future productivity and monetary policy for equity price and exchange rate dynamics in a standard DSGE open economy model.
- Focus: volatility and comovement under alternative information assumptions and monetary policy reactions to productivity shocks.
- Novelty:
  - Monetary policy "news" (communication) can guide expectations and move asset prices even without changes in the policy target.
  - Explicit separation of "current shocks" (surprises to current variables) and "news shocks" (new information about future fundamentals).

### Main contributions
- Asset price volatility condition:
  - A positive correlation between current and news shocks increases volatility of exchange rates and equity prices for given stochastic-process volatilities.
  - Mechanism: when current and news shocks are positively correlated, news about the future reinforce current surprises so asset prices move more than fundamentals.
  - Link to persistence: positive current–news correlation creates a link between today’s and tomorrow’s fundamentals analogous to persistence in standard stochastic processes.
- Comovement of exchange rates and equity return differentials:
  - Comovement depends crucially on the currency denomination of equity returns and monetary policy response to the output gap.
  - News shocks affect equity prices and exchange rates similarly but are not the primary driver of their comovement.
  - If there is no systematic monetary policy or no response to the output gap, equity return differentials measured in investor currency are independent of the exchange rate and depend only on current and news productivity shocks.
  - If monetary policy responds to the output gap:
    - The model can generate a positive (counterfactual) correlation between equity return differentials in investor currency and exchange rates.
    - A data-consistent negative correlation requires a small negative monetary response to the output gap (coefficient must be small enough to guarantee uniqueness of equilibrium).
  - Equity return differentials measured in firm local currency typically show positive correlation with exchange rate returns for reasonable parameter ranges and policy specifications.
  - If productivity shocks are more volatile than monetary shocks, equity returns tend to be more volatile than exchange rate returns.

### Model overview (structure and preferences)
- Two-country, symmetric DSGE with production, sticky prices in local currency, and complete nominal international financial markets.
- Firms: monopolistic competition, linear technology in labor, prices set one period in advance in the currency of the final consumer.
- Households: representative Home household maximizing expected lifetime utility with complete nominal contingent claims.
- Period utility:
  - U(Cs(j), Mt(j)/Pt, Ls(j)) = Cs(j)^(1−ρ)/(1−ρ) + κ1/(1−ε) (Ms(j)/Ps)^(1−ε) − κ2/(1+ψ) Ls(j)^(1+ψ), with ρ>0, ε>0, ψ≥0, κ1>0, κ2>0.
- Consumption aggregator:
  - C_t(j) ≡ [ (1/2)^(1/ω) C_h,t(j)^( (ω−1)/ω ) + (1/2)^(1/ω) C_f,t(j)^( (ω−1)/ω ) ]^( ω/(ω−1) ), ω>0.
  - CES sub-aggregates defined with elasticity λ>1.
- Asset pricing under complete markets:
  - S_t P*_t / P_t = U_c(C*_t) / U_c(C_t).  (equation (11))
  - Price of a security: Z_t = Σ_{s=0}^∞ β^s D_{t,t+s} CF_{t+s}.  (equation (12))
- Representative firm pricing (log-linearized expressions preserved):
  - P_h,t(i) and P*_h,t(i) given by equations (17) and (18) in the source.

### Stochastic processes and news specification
- Productivity processes (two-component):
  - Relative productivity: a^R_t ≡ ln(A_t) − ln(A*_t) = θ a^R_{t−1} + ν^R_{1,t} + ν^R_{2,t−1}, with |θ|<1.  (equation (21))
  - World productivity (unit root): a^W_t ≡ 1/2[ln(A_t)+ln(A*_t)] = a^W_{t−1} + ν^W_{1,t} + ν^W_{2,t−1}.  (equation (22))
  - (ν_{1,t}, ν_{2,t}) jointly i.i.d., mean zero, specified variance–covariance matrix (σ^2_{ν1} …).
  - Interpretation: ν_{1,t} = current productivity shock; ν_{2,t} = news shock (information about a_{t+1}).
- Money supply and monetary policy news:
  - ln(M_t) = ln(M_{t−1}) + μ_t.  (equation (23))
  - ln(M*_t) = ln(M*_{t−1}) + μ*_t.  (equation (24))
  - μ^R_t = ν^R_{3,t} + ν^R_{4,t−1} + χ^R_1 ν^R_{1,t} + χ^R_2 ν^R_{2,t−1} + χ^R_3 ν^R_{2,t}.  (equation (25))
  - μ^W_t = ν^W_{3,t} + ν^W_{4,t−1} + χ^W_1 ν^W_{1,t} + χ^W_2 ν^W_{2,t−1} + χ^W_3 ν^W_{2,t}.  (equation (26))
  - ν_{3,t} and ν_{4,t}: current money shocks and money-news shocks, jointly i.i.d., independent from ν_{1,t} and ν_{2,t}.
  - χ coefficients: monetary policy responses to current and future technology shocks; μ_{1,t} are surprise components, μ_{2,t} are news about future money supply.

### Solution approach and key analytic properties
- Linearization around symmetric steady state; lower case denotes log-deviations; full solution reported in appendix.
- Key mechanisms:
  - How asset prices incorporate news about the future.
  - Distinction between announcements (news shocks) and unanticipated policy changes—announcements affect asset prices differently.
  - Event studies focusing only on unanticipated policy changes may bias inference about policy effects on equity prices.

### Analytical results — Exchange rate predictability and volatility (Appendix A highlights)
- Exchange rate linearized solution (preserved form):
  - s_t = E_{t−1} m^R_t + (1−β) ε (m^R_t − E_{t−1} m^R_t)︸︷︷︸ μ^R_{1,t} + β (E_t m^R_{t+1} − E_{t−1} m^R_t)︸︷︷︸ μ^R_{1,t} + μ^R_{2,t}.
- Predictability implication:
  - Without news shocks (μ^R_{2,t} = 0) and with E_t m^R_{t+1} = m^R_t, surprise s_t − E_{t−1} s_t = [(1−β) ε + β] (m^R_t − E_{t−1} m^R_t).
  - With news shocks, because β typically close to one, s_t places large weight on future money supply; knowledge of current fundamentals insufficient to predict s_t when news shocks matter.
- Conditional variance result:
  - Var_{t−1} s_t = [(1−β) ε + β]^2 Var_{t−1} m^R_t + β^2 Var(μ^R_2).
- Exchange rate volatility with correlated news:
  - Conditional variance of exchange rate returns:
    - Var_t(Δ s^R_t) = [(1−β) ε + β]^2 σ^2_{ν^R_3} + β^2 σ^2_{ν^R_4} + 2 [(1−β) ε + β] β %_m σ_{ν^R_3} σ_{ν^R_4}.
  - Numerical examples:
    - Case 1: σ^2_{ν^R_3} = .5, σ^2_{ν^R_4} = .5, β = 0.95, %_m = 0.6 → Var_t(Δ s^R_t) = 1.545.
    - Case 2: σ^2_{ν^R_3} = 1, σ^2_{ν^R_4} = 0 → Var_t(Δ s^R_t) = 1.
- Interpretation:
  - Positive correlation between news shocks and current shocks increases contemporaneous movement of s_t, raising exchange rate volatility without changing unconditional variance of fundamentals.

### Analytical results — Equity returns and relative returns
- Equity price and return identities (preserved):
  - Pre-dividend price: q_t = β E_t q_{t+1} + (1−β) π_t − β i_t.
  - Return: r_{t+1} = i_t + (q_{t+1} − E_t q_{t+1}).
  - Excess return decomposition: r_{t+1} − i_t = (r^W_{t+1} − i^W_t) + 1/2 (r^R_{t+1} − i^R_t).
- World excess return depends on news shocks (unless ε = 1 and ρ = 1).
- Policy-news transmission:
  - Policy news affect world excess returns only if current monetary policy shocks affect it (μ^W_{1,t+1}, μ^W_{2,t+1} share coefficients).
  - Event studies using only contemporaneous policy surprises may omit effects of news about future policy.
- Equity return volatility increases with news shocks correlated with current shocks; this can also increase conditional variance of dividends and consumption.
- Relative returns (investor-currency vs firm local currency):
  - Define investor-currency relative return: r^R\$_{t+1} ≡ r^R_{t+1} − Δ s_{t+1}.
  - Expression (preserved form) for r^R\$_{t+1} shows dependence on ν^R_{1,t+1}, ν^R_{2,t+1}, structural parameters (ψ, ω, ζ, β, θ).
  - Variance comparisons:
    - Var(a^R_t) = 1/(1−θ)^2 (σ^2_{ν^R_1} + σ^2_{ν^R_2} + 2 θ %_a σ_{ν^R_1} σ_{ν^R_2}).
    - When θ = 0, %_a > 0 unambiguously increases volatility of relative equity returns without affecting volatility of relative productivity level.
    - Effect of persistence depends on θ relative to β.
  - Special case ω = 1: r^R\$_{t+1} = (1−β) (ψ + 1) ζ/(1−ζ) ν^R_{1,t+1} → news shocks have no impact.
  - Firm-local-currency relative return: r^R_{t+1} = Δ s_{t+1} + r^R\$_{t+1}.
    - If monetary policy does not respond to productivity shocks, Δ s_{t+1} uncorrelated with r^R\$_{t+1} and Var(r^R_{t+1}) = Var(Δ s_{t+1}) + Var(r^R\$_{t+1}).
    - If policy responds to productivity shocks, Cov(Δ s_{t+1}, r^R\$_{t+1}) may be positive or negative.

### Exchange rate — equity comovement (analytical covariance)
- If monetary policy does not respond to relative productivity shocks (χ^R coefficients = 0), Cov(Δ s_{t+1}, r^R\$_{t+1}) = 0.
- If monetary policy responds to productivity shocks, Cov(Δ s_{t+1}, r^R\$_{t+1}) ≠ 0; the sign depends on policy accommodation and shock variances.
- Exact covariance components (preserved expression):
  - Cov(Δ s_{t+1}, r^R\$_{t+1}) = (1−β) (ψ + 1) ( (ω−1)/ω ψ + 1 β θ/(1−β) θ + ζ/(1−ζ) ) [1 + (ε−1) (1−β)] χ^R_1 Var(ν_1) + (1−β) (ψ + 1) ( (ω−1)/ω ψ + 1 β/(1−β) θ ) [1 + (ε−1) (1−β)] χ^R_3 Var(ν_2) + (1−β) (ψ + 1) ( (ω−1)/ω ψ + 1 β/(1−β) θ ) β χ^R_2 Var(ν_2).
- Typical outcome: an accommodative monetary response to a positive productivity shock tends to induce currency depreciation and higher equity returns → Cov(r^R\$_{t+1}, Δ s_{t+1}) > 0 if policy is accommodative.

### Quantitative analysis — extensions, calibration, and experiment setup
- Extensions:
  - Staggered pricing via Rotemberg quadratic menu costs: Z_P = α_P/2 (P_{.,t}(i) / P_{.,t−1}(i) − 1)^2; linearization yields four Phillips curves (equations (50)–(53)).
  - Endogenous interest-rate setting (Taylor-type with smoothing and exchange-rate term):
    - i_t = γ i_{t−1} + (1−γ) [ φ_p ∆ p_t + φ_y (y_t − y^{Flex}_t) + 1/2 φ_s ∆ s_t + ν_{3,t} + ν_{4,t} ].
- Calibration (quarterly frequency):
  - ρ = 3.
  - ω = 3.
  - ψ = 0.1.
  - β = 0.95 (annual basis).
  - Average duration of price change = one year.
  - ζ = 2/3.
  - Interest rule coefficients: φ_p = 1.5, φ_y = 1.
  - Shock structure: one current shock and one news shock per process (detailed in appendix).

### Impulse-response and smoothing effects (selected quantitative findings)
- Interest rate smoothing reduces volatility of the interest rate differential while increasing volatility of the exchange rate and the relative equity return because "more information about the future is being taken into account on impact."
- Experiment assumptions illustrated in Figure 3:
  - Policy parameters: φp = 1.5, φy = 0.2, φs = 0.1 and γ = 0.6.
  - Autoregressive coefficients: θW = 0.95 and θR = 0.7.
- Qualitative impulse-response results:
  - Current productivity shock: positive current productivity → negative output gap (sticky-price output lags) → lower interest rate if φ_y > 0 → exchange rate depreciates (∆ s > 0); Home equity return increases → Cov(∆ s, r_t) tends to be positive.
  - Productivity news shock: on impact domestic good price falls → demand and actual output increase → output gap positive → nominal exchange rate depreciates to induce future real appreciation → relative equity return increases.
  - Policy shocks: current monetary shocks induce large interest differential and exchange rate appreciation; policy news induces appreciation and a small negative interest differential.
- With interest-rate smoothing γ = 0.6 and persistence θ_R = 0.7, current shocks contain information about multiple future periods, amplifying effects.

### Model specifications, benchmarking, and comparative results (Models 1–6)
- Benchmarking details:
  - Countries evaluated relative to the U.S.: UK, Japan (JP), and Germany (BD); main floating currencies from 1973 to date.
  - Interest rate data: 3-month LIBOR, sample 1987Q1-2007Q4.
  - Equity returns: MSCI series and end-of-period exchange rates against the U.S. dollar; sample 1973Q1-2007Q4.
  - Conditional standard deviation of quarterly relative interest rate scaled to about 0.16 percent to match data.
  - Interest rate shocks attached to the interest rate rule set to variance one per cent of the variance of productivity shocks.
- Model variants:
  - Model 1: Base Model — γ = 0, φy = 0.2, φp = 1.5, φs = 0.1, no correlation between news and current shocks.
  - Model 2: Model 1 with correlated shocks (%_a = %_m = 0.9 for both R and W).
  - Model 3: Model 1 with interest-rate smoothing γ = 0.4.
  - Model 4: Model 3 with correlated shocks (combined effects).
  - Model 5: Model 4 with φy = 0.0 (no monetary response to output gap).
  - Model 6: Model 4 with φy = −0.01 (small negative monetary response to output gap).
- Key comparative outcomes:
  - Correlated news and current shocks and interest-rate smoothing both increase asset price volatility; smoothing can have quantitatively larger effects.
  - Model 5 (φy = 0) lowers exchange rate return volatility and reduces positive correlation between exchange rate and investor-currency equity return.
  - Model 6 (φy = −0.01) matches the data sign for the correlation between exchange rate return and relative equity return in investor currency but has empirical caveats about plausibility.
- Regularities and limitations:
  - Volatility rank typically matched: Var(relative equity return in firm currency) > exchange rate volatility > relative interest rate volatility.
  - Shortcoming: the model underpredicts volatility of the relative return on equities in investor currency (rR$ t+1) in the first four specifications.
  - Comovement: with φy > 0 the model implies positive correlation between exchange rate and investor-currency equity return; φy = 0 makes this correlation zero, closer to some data patterns. Empirical data show positive correlation for firm-currency returns for Germany and UK, negative for Japan; large negative correlation between investor-currency relative return and exchange rate across countries is noted.

### Main conceptual findings and policy implications
- News shocks and volatility:
  - Positive correlation between news and current shocks increases asset price volatility without changing unconditional variance of underlying fundamentals.
  - This provides an economic mechanism explaining part of excess volatility of asset prices and why persistent stochastic processes generate higher asset price volatility.
- Monetary policy communication channel:
  - Monetary policy communication (news) can move asset prices even absent actual policy changes; announcements and unanticipated policy moves have distinct asset-price effects.
  - Monetary policy reaction to the output gap strongly influences comovements between exchange rates and equity returns; a small negative φy can flip the sign of correlations to match some data patterns but may be empirically questionable.
- Empirical caution:
  - Event-study inference using only contemporaneous policy surprises (unanticipated policy changes) can misestimate policy effects because it omits policy-news effects and anticipatory components.

*Source: _wp08284 - References . . . (appendix available from the authors).*

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

### _wp08284 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Introduction and research question
- Purpose: study the role of news about future productivity and monetary policy for equity price and exchange rate dynamics in a standard DSGE open economy model.
- Focus: volatility and comovement under alternative information assumptions and monetary policy reactions to productivity shocks.
- Novelty:
  - Consideration of monetary policy "news" as communication guiding expectations (e.g., fund rate futures move on communications without changes in the federal fund target rate).
  - Explicit separation of "current shocks" (surprises to current variables) and "news shocks" (new information about future fundamentals).

### Main contributions (twofold)
- Identification of a condition that increases asset price volatility:
  - A positive correlation between current and news shocks increases volatility of exchange rates and equity prices for given stochastic-process volatilities.
  - Intuition: asset prices respond to both current shocks and news shocks; when positively correlated, news about the future reinforce current surprises so asset prices move more than fundamentals.
  - Links to persistence: positive current–news correlation creates a link between today’s and tomorrow’s fundamentals analogous to persistence in standard stochastic processes; helps explain why persistent processes generate volatile asset prices.
- Analysis of comovement between exchange rates and equity return differentials:
  - Comovement depends crucially on the currency denomination of equity returns and monetary policy.
  - News shocks affect equity prices and exchange rates similarly and thus are not the primary driver of their comovement.
  - If there is no systematic monetary policy or no response to the output gap, equity return differentials measured in investor currency are independent of the exchange rate and depend only on current and news productivity shocks.
  - If monetary policy responds to the output gap, the model can generate:
    - A positive (counterfactual) correlation between equity return differentials in investor currency and exchange rates.
    - A data-consistent negative correlation only under a small negative monetary response to the output gap (requires the coefficient to be small enough to guarantee uniqueness of equilibrium).
  - The model generates a positive correlation between equity return differentials measured in firm local currency and exchange rate returns for reasonable parameter ranges and policy specifications, consistent with data.
  - If productivity shocks are more volatile than monetary shocks, equity returns tend to be more volatile than exchange rate returns (as observed in the data).

### Model overview
- Structure:
  - Two-country, symmetric DSGE with production, sticky prices in local currency, and complete nominal international financial markets.
  - Firms: monopolistic competition, linear technology in labor, goods traded but markets segmented, prices set one period in advance in the currency of the final consumer.
  - Households: representative Home household maximizes expected lifetime utility subject to a budget constraint with complete nominal contingent claims.
- Preferences and consumption aggregation:
  - Period utility: U(Cs(j), Mt(j)/Pt, Ls(j)) = Cs(j)^(1−ρ)/(1−ρ) + κ1/(1−ε) (Ms(j)/Ps)^(1−ε) − κ2/(1+ψ) Ls(j)^(1+ψ), with ρ>0, ε>0, ψ≥0, κ1>0, κ2>0.
  - Consumption basket C_t(j) ≡ [ (1/2)^(1/ω) C_h,t(j)^( (ω−1)/ω ) + (1/2)^(1/ω) C_f,t(j)^( (ω−1)/ω ) ]^( ω/(ω−1) ), ω>0.
  - CES sub-aggregates: C_h,t(j) and C_f,t(j) defined with elasticity λ>1.
- Asset pricing under complete markets:
  - S_t P*_t / P_t = U_c(C*_t) / U_c(C_t).  (equation (11))
  - Price Z_t of a security with payoff CF_{t+s} (in Home currency): Z_t = Σ_{s=0}^∞ β^s D_{t,t+s} CF_{t+s}, where D_{t,t+s} is the stochastic discount factor.  (equation (12))
- Firms’ optimal pricing (representative expressions):
  - P_h,t(i) = λ/(λ−1) E_{t−1} [ D_{t−1,t} W_t A_t C_t ] / E_{t−1} [ D_{t−1,t} C_t ].  (equation (17))
  - Analogous expression for foreign-market price P*_h,t(i) with S_t adjustment.  (equation (18))

### Stochastic processes and information assumptions
- Productivity processes (two-component structure: current shock ν1,t and news shock ν2,t):
  - Relative productivity: a^R_t ≡ ln(A_t) − ln(A*_t) = θ a^R_{t−1} + ν^R_{1,t} + ν^R_{2,t−1}, with |θ|<1.  (equation (21))
  - World productivity (unit root): a^W_t ≡ 1/2[ln(A_t)+ln(A*_t)] = a^W_{t−1} + ν^W_{1,t} + ν^W_{2,t−1}.  (equation (22))
  - (ν_{1,t}, ν_{2,t}) are jointly i.i.d. over time with mean zero and specified variance–covariance matrix (σ^2_{ν1} …).
  - Interpretation: ν_{1,t} = current productivity shock; ν_{2,t} = news shock providing information about productivity one period in advance (on a_{t+1}).
  - Two-component formulation permits analysis of both mean-reverting relative dynamics and unit-root world trend.
- Money supply processes and monetary policy news:
  - ln(M_t) = ln(M_{t−1}) + μ_t.  (equation (23))
  - ln(M*_t) = ln(M*_{t−1}) + μ*_t.  (equation (24))
  - μ^R_t = ν^R_{3,t} + ν^R_{4,t−1} + χ^R_1 ν^R_{1,t} + χ^R_2 ν^R_{2,t−1} + χ^R_3 ν^R_{2,t}.  (equation (25))
  - μ^W_t = ν^W_{3,t} + ν^W_{4,t−1} + χ^W_1 ν^W_{1,t} + χ^W_2 ν^W_{2,t−1} + χ^W_3 ν^W_{2,t}.  (equation (26))
  - ν_{3,t} and ν_{4,t} are current money shocks and money-news shocks, jointly i.i.d., independent from ν_{1,t} and ν_{2,t}.
  - χ coefficients denote monetary policy responses to current and future technology shocks; χ_2 ν_{2,t−1} is the delayed monetary response to productivity news.
  - Composite definitions for interpretation:
    - μ^R_{1,t} ≡ ν^R_{3,t} + χ^R_1 ν_{1,t} + χ^R_3 ν^R_{2,t};
    - μ^R_{2,t} ≡ ν^R_{4,t} + χ^R_2 ν^R_{2,t};
    - μ^W_{1,t} ≡ ν^W_{3,t} + χ^W_1 ν_{1,t} + χ^W_3 ν^W_{2,t};
    - μ^W_{2,t} ≡ ν^W_{4,t} + χ^W_2 ν^W_{2,t};
    - so μ^R_t = μ^R_{1,t} + μ^R_{2,t−1}, μ^W_t = μ^W_{1,t} + μ^W_{2,t−1}.
  - Interpretation: μ_{1,t} are surprise components (unanticipated policy changes); μ_{2,t} are news about future money supply including delayed policy responses to productivity news.

### Solution approach and properties
- Solution: log-linearization around an initial fully symmetric steady state; lower case denotes log-deviations. Full solution reported in the appendix.
- Key mechanisms highlighted analytically:
  - How asset prices (exchange rates and equity prices) incorporate news about the future.
  - Distinction between an “announcement” (news shock) and an unanticipated actual policy change—announcements affect asset prices differently from unanticipated policy moves.
  - Event studies focusing only on unanticipated policy changes may bias inference about policy effects on equity prices.

### Model assumptions and calibration-related remarks
- Initial steady state: A_0 = A*_0 = 1 and P_0 = P*_0 = S_0 = 1, determining M_0 = M*_0; no news at time 0.
- Complete nominal asset markets: allows pricing equities as present discounted sums of future profits and isolates asset pricing from portfolio allocation effects.
- Extensions:
  - Staggered pricing and rule-based interest rate setting (section 4) evaluated numerically and contrasted to data for the United States, Japan, Germany, and the UK.
  - An extension assuming serially correlated error terms (without explicit news shocks) can generate even higher asset price volatility than the news-shock model; authors prefer separating news and current shocks to clarify mechanisms.
  - In an extension, imposing quadratic Rotemberg adjustment costs and standard Taylor-rule monetary policy yields monetary responses to relative productivity consistent with the model specification.

### Model implications summarized as bullets
- Volatility:
  - Positive current–news correlation → higher asset price volatility (exchange rates and equities) for given shock variances.
  - Persistence in fundamentals (or equivalently current–news correlation) provides an economic explanation for volatile asset prices.
- Comovement:
  - News shocks affect exchange rates and equities similarly; therefore, news is not the dominant driver of their comovement.
  - Monetary policy responses to the output gap strongly influence the sign and magnitude of the correlation between equity return differentials (investor-currency) and exchange rates.
  - Equity return differentials measured in firm local currency typically show positive correlation with exchange rate returns for reasonable parameter ranges.
- Relative volatilities:
  - If productivity shocks are more volatile than monetary shocks, equity returns > exchange rate returns in volatility, matching empirical patterns.
- Policy-news channel:
  - Monetary policy communication (news) can move asset prices even absent actual policy changes; this communication channel has distinct effects from unanticipated actual policy changes.

*Source: _wp08284 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . .*

### appendix available from the authors, we report some of the results without assumingχ

### _wp08284 - appendix available from the authors, we report some of the results without assumingχ

### A. News Shock and Exchange Rate Predictability
- Key equation for the exchange rate (linearized solution):
  - s_t = E_{t−1} m^R_t + (1−β) ε (m^R_t − E_{t−1} m^R_t)︸︷︷︸ μ^R_{1,t} + β (E_t m^R_{t+1} − E_{t−1} m^R_t)︸︷︷︸ μ^R_{1,t} + μ^R_{2,t}.
- Interpretation:
  - The exchange rate depends only on the relative money supply, which itself is affected by current shocks (μ^R_{1,t}) and news shocks (μ^R_{2,t}).
  - Without news shocks (μ^R_{2,t} = 0) and with E_t m^R_{t+1} = m^R_t:
    - s_t = E_{t−1} m^R_t + [(1−β) ε + β] (m^R_t − E_{t−1} m^R_t).
    - The surprise to the exchange rate, s_t − E_{t−1} s_t, is [(1−β) ε + β] times the surprise to the relative money supply.
    - If ε is close to 1, knowing fundamentals should allow predicting s_t — yet Meese and Rogoff (1983) show this fails empirically.
  - With news shocks:
    - If agents know future monetary policy one period in advance (and ν_{3,t} = 0, χ_1 = 0, χ_3 = 0), m^R_t = E_{t−1} m^R_t and
      - s_t = (1−β) m^R_t + β E_t m^R_{t+1}.
    - The surprise becomes s_t − E_{t−1} s_t = (1−β)(m^R_t − E_{t−1} m^R_t) + β (E_t m^R_{t+1} − E_{t−1} m^R_{t+1}), where E_t m^R_{t+1} − E_{t−1} m^R_{t+1} = μ^R_{2,t} ≠ 0.
    - Because s_t places large weight on future money supply (β typically close to one), knowledge of current fundamentals is insufficient to predict exchange rates when news shocks matter.
- Conclusion:
  - News shocks explain why exchange rates are hard to predict using contemporaneous fundamentals: asset prices heavily depend on future information.

### B. Exchange Rate Volatility
- Conditional variance result:
  - Var_{t−1} s_t = [(1−β) ε + β]^2 Var_{t−1} m^R_t + β^2 Var(μ^R_2).
- Implications:
  - If Var(μ^R_2) > 0 (news shocks about future money supply or monetary reaction to future productivity), conditional variance of s_t can exceed that of the relative money supply for any ε ≥ 1.
  - Generating realistic excess volatility relative to unconditional variance of fundamentals (e.g., relative money supply) requires additional structure.
- News shocks correlated with current shocks can generate excess volatility relative to unconditional variance of fundamentals:
  - With no monetary response to productivity shocks, unconditional variance of relative money supply growth:
    - Var(Δ m^R_t) = Var(ν^R_{3,t} + ν^R_{4,t−1}) = σ^2_{ν3} + σ^2_{ν4}.
  - Conditional variance of exchange rate returns:
    - Var_t(Δ s^R_t) = Var([(1−β) ε + β] ν^R_{3,t} + β ν^R_{4,t})
    = [(1−β) ε + β]^2 σ^2_{ν^R_3} + β^2 σ^2_{ν^R_4} + 2 [(1−β) ε + β] β %_m σ_{ν^R_3} σ_{ν^R_4}.
  - For ε = 1, to have exchange rate returns more volatile than money supply growth one needs:
    - %_m > 1−β 2 / 2β σ_{ν^R_4} / σ_{ν^R_3} > 0 (preserves exact expression as in source).
  - Numerical examples:
    - Case 1: σ^2_{ν^R_3} = .5, σ^2_{ν^R_4} = .5, β = 0.95, %_m = 0.6 → Var_t(Δ s^R_t) = 1.545.
    - Case 2: σ^2_{ν^R_3} = 1, σ^2_{ν^R_4} = 0 → Var_t(Δ s^R_t) = 1.
- Forces at work:
  - ν^R_4 provides accurate news about the future, which reduces asset volatility (West (1988) proposition).
  - Positive correlation between news shocks and current shocks increases contemporaneous movement of s_t (in same direction), raising exchange rate volatility without changing unconditional variance of fundamentals (current and lagged shocks enter fundamentals at different times).
- Economic interpretation:
  - Correlated news shocks capture effects of persistent underlying processes (e.g., interest rate smoothing) that create “noisy” news about the future similar to current surprises, increasing asset price volatility.
- Generality:
  - Any asset price that is a discounted sum of future fundamentals exhibits similar sensitivity: a uni-directional update to the future path of cash flows, {cf_{t+s}}_{s≥0}, moves asset price z_t substantially even if Δ cf_t does not become more volatile.

### C. Equity Return Volatility
- Equity pricing and return expressions:
  - Pre-dividend price: q_t = β E_t q_{t+1} + (1−β) π_t − β i_t.
  - Return on equity: r_{t+1} = i_t + (q_{t+1} − E_t q_{t+1}).
  - Excess return decomposition (home vs world): r_{t+1} − i_t = (r^W_{t+1} − i^W_t) + 1/2 (r^R_{t+1} − i^R_t).
- World excess return (compact expression in source):
  - r^W_{t+1} − i^W_t = {ρ (1 − 1/ε) + (1−ρ) [1 + (1−β) (1 − 1/ε)]} β ψ + 1/(ρ + ψ) (ν^W_{1,t+1} + ν^W_{2,t+1}) + (1−β) ζ/(1−ζ) (ψ + 1) [ν^W_{1,t+1} − β (1 − 1/ε) (ν^W_{1,t+1} + ν^W_{2,t+1})] + [1 + (1−β) (1−ρ)/ρ − (1−β) ζ/(1−ζ) (ρ + ψ)/(ρ)] [ (1−β) ε (μ^W_{1,t+1}) + β (μ^W_{1,t+1} + μ^W_{2,t+1}) ].
- Key points:
  - News shocks about future productivity affect world excess returns unless ε = 1 and ρ = 1 (unlikely).
  - Variance of world excess return relative to variance of world productivity growth depends on correlation %_a between ν^W_{1,t+1} and ν^W_{2,t+1}.
  - Policy-news transmission: policy news affect world excess returns only if current monetary policy shocks affect it — μ^W_{1,t+1} and μ^W_{2,t+1} share same coefficient. Hence event studies using only contemporaneous policy surprises (e.g., changes in federal funds futures) may omit effects of news about future policy and misestimate impacts.
  - Equity return volatility increases with news shocks correlated with current shocks, but this can also increase conditional variance of other fundamentals (dividends, consumption).
- Dividends and consumption:
  - Surprise to dividends: π^W_t − E_{t−1} π^W_t = c^W_t − E_{t−1} c^W_t + ζ/(1−ζ) [ (ψ + 1) (a^W_t − E_{t−1} a^W_t) − (ρ + ψ) (c^W_t − E_{t−1} c^W_t) ].
  - Consumption depends on news shocks and has variance increasing in correlation between current and news shocks:
    - c^W_t = 1/ρ { (1−β) ε (μ^W_{1,t}) + β (μ^W_{1,t} + μ^W_{2,t}) } + (ψ + 1)/(ρ + ψ) { E_{t−1} a^W_t + β [1 − 1/ε] (ν^W_{1,t} + ν^W_{2,t}) }.
- Relative equity returns (investor currency vs firm local currency):
  - Define relative return in investor currency: r^R\$_{t+1} ≡ r^R_{t+1} − Δ s_{t+1}.
  - For investor-currency relative return:
    - r^R\$_{t+1} = (1−β) (ψ + 1) [ (ω−1)/ω ψ + 1 β θ/(1−β) θ + ζ/(1−ζ) ] ν^R_{1,t+1} + (1−β) (ψ + 1) [ (ω−1)/ω ψ + 1 β/(1−β) θ ] ν^R_{2,t+1}.
  - Unconditional variance expression (exact form preserved from source) shows dependence on σ^2_{ν^R_1}, σ^2_{ν^R_2}, and %_a.
  - Comparison with volatility of relative productivity:
    - Var(a^R_t) = 1/(1−θ)^2 (σ^2_{ν^R_1} + σ^2_{ν^R_2} + 2 θ %_a σ_{ν^R_1} σ_{ν^R_2}).
    - When θ = 0, %_a > 0 unambiguously increases volatility of relative equity returns without affecting volatility of relative productivity level.
    - When θ ≠ 0, variance ratio Var(r^R\$_{t+1}) / Var(a^R_t) depends crucially on (β θ/(1−β) θ)^2 / (1/(1−θ)^2); effect of persistence increases volatility for θ < β, but effect declines when θ > β.
  - Special case ω = 1 (elasticity of substitution unity):
    - r^R\$_{t+1} = (1−β) (ψ + 1) ζ/(1−ζ) ν^R_{1,t+1} → news shocks have no impact (terms of trade provide full risk sharing).
  - Relative equity return in firm local currency:
    - r^R_{t+1} = Δ s_{t+1} + r^R\$_{t+1}.
    - If monetary policy does not respond to productivity shocks, Δ s_{t+1} is uncorrelated with r^R\$_{t+1} and Var(r^R_{t+1}) = Var(Δ s_{t+1}) + Var(r^R\$_{t+1}).
    - When monetary policy responds to productivity shocks, Cov(Δ s_{t+1}, r^R\$_{t+1}) may be positive or negative; thus the model can generate either sign for comovement and either larger or smaller Var(r^R_{t+1}) relative to Var(Δ s_{t+1}).

### D. Exchange Rate and Equity Price Comovement
- Currency denomination and policy response matter:
  - If monetary policy does not respond to relative productivity shocks (χ^R coefficients = 0), exchange rate does not depend on relative productivity shocks while r^R\$_{t+1} does not depend on relative money supply shocks → Cov(Δ s_{t+1}, r^R\$_{t+1}) = 0.
  - If monetary policy responds to productivity shocks, Cov(Δ s_{t+1}, r^R\$_{t+1}) ≠ 0; sign depends on whether policy accommodates productivity shocks and on shock variances.
- Exact covariance components (preserved):
  - Cov(Δ s_{t+1}, r^R\$_{t+1}) = (1−β) (ψ + 1) ( (ω−1)/ω ψ + 1 β θ/(1−β) θ + ζ/(1−ζ) ) [1 + (ε−1) (1−β)] χ^R_1 Var(ν_1) + (1−β) (ψ + 1) ( (ω−1)/ω ψ + 1 β/(1−β) θ ) [1 + (ε−1) (1−β)] χ^R_3 Var(ν_2) + (1−β) (ψ + 1) ( (ω−1)/ω ψ + 1 β/(1−β) θ ) β χ^R_2 Var(ν_2).
- Typical outcome:
  - A positive productivity shock in a sticky-price model typically induces monetary easing and currency depreciation → Cov(r^R\$_{t+1}, Δ s_{t+1}) > 0 if policy is accommodative.
  - If monetary response is not accommodative, covariance can be negative.
- Relation for relative return in firm currency:
  - If Cov(r^R\$_{t+1}, Δ s_{t+1}) > 0 then Cov(r^R_{t+1}, Δ s_{t+1}) = Var(Δ s_{t+1}) + Cov(r^R\$_{t+1}, Δ s_{t+1}) > 0.
  - In the no-policy-response case, Var_t(r^R_{t+1}) = Var_t(Δ s_{t+1}) + Var(r^R\$_{t+1}) and Corr(r^R_{t+1}, Δ s_{t+1}) = Var(Δ s_{t+1}) / sqrt{ Var(Δ s_{t+1}) [ Var(Δ s_{t+1}) + Var(r^R\$_{t+1}) ] } ≤ 1.
  - Contrast with Hau and Rey (2006): their model predicts Corr(r^R_{t+1}, Δ s_{t+1}) = 1 (perfect correlation) because of a single relative shock; our model allows much smaller correlation consistent with data.

### IV. Quantitative Analysis

#### A. Model Extension and Calibration
- Two key extensions:
  - Staggered pricing (Rotemberg quadratic menu costs):
    - Adjustment costs: Z_P = α_P/2 (P_{.,t}(i) / P_{.,t−1}(i) − 1)^2.
    - Firms price to market; linearization yields four Phillips curves (Home/Foreign, market-specific) in equations (50)–(53):
      - ∆ p_{h,t} = β E_t ∆ p_{h,t+1} + λ^{−1} α_p (w_t − p_t − a_t + p_t − p_{h,t}).
      - ∆ p_{f,t} = β E_t ∆ p_{f,t+1} + λ^{−1} α_p (w^*_t − p^*_t − a^*_t + p^*_t − (p_{f,t} − s_t)).
      - ∆ p^*_{h,t} = β E_t ∆ p^*_{h,t+1} + λ^{−1} α_p (w_t − p_t − a_t + p_t − (p^*_{h,t} + s_t)).
      - ∆ p^*_{f,t} = β E_t ∆ p^*_{f,t+1} + λ^{−1} α_p (w^*_t − p^*_t − a^*_t + p^*_t − p^*_{f,t}).
  - Endogenous interest rate setting (Taylor-type rule with smoothing and exchange rate term):
    - i_t = γ i_{t−1} + (1−γ) [ φ_p ∆ p_t + φ_y (y_t − y^{Flex}_t) + 1/2 φ_s ∆ s_t + ν_{3,t} + ν_{4,t} ].
    - Output gap (y_t − y^{Flex}_t) is log difference between sticky and flexible-price output; under symmetry, interest rate responds to relative productivity shocks when φ_y ≠ 0 because output gap depends on productivity via terms of trade.
- Calibration (quarterly frequency):
  - ρ = 3.
  - ω = 3.
  - ψ = 0.1.
  - β = 0.95 (annual basis).
  - Average duration of price change = one year.
  - ζ = 2/3.
  - Interest rule coefficients: φ_p = 1.5, φ_y = 1.
  - Shock structure: one current shock and one news shock per process (detailed in appendix).

#### B. Impulse Responses (selected qualitative results)
- Figure summaries (as described in source):
  - Figure 1 (no interest rate smoothing γ = 0, no exchange rate smoothing φ_s = 0, no persistence in productivity):
    - Current productivity shock:
      - Positive current productivity → negative output gap (sticky-price output lags) → lower interest rate if φ_y > 0 → exchange rate depreciates (∆ s > 0).
      - Home firms’ revenue increases → relative return on Home equity increases → Cov(∆ s, r_t) tends to be positive.
    - Productivity news shock:
      - On impact, domestic good price falls (anticipation of higher future productivity) → demand and actual output increase → output gap positive → nominal exchange rate depreciates to induce future real appreciation consistent with future productivity improvement → relative equity return increases due to higher future profit.
    - With both current and news productivity shocks, r^R_t reacts more than ∆ s_t and is positively correlated with ∆ s_t because r^R_{t+1} = r^R\$_{t+1} + ∆ s_t.
    - Policy shocks:
      - Current monetary shocks induce large interest differential and exchange rate appreciation; policy news induces appreciation and a small negative interest differential (anticipatory lower inflation and subsequent easing).
  - Figure 2 (interest rate smoothing γ = 0.6, productivity persistence θ_R = 0.7):
    - Current shocks contain information about multiple future periods (not only one period ahead), amplifying the effects described above (qualitative description in source).
- Dependence on monetary policy response:
  - If φ_y = 0 (no monetary response to productivity), exchange rate does not react to current relative productivity shocks; relative equity return in firm currency is more volatile than exchange rate (Var(r^R_{t+1}) = Var(r^R\$_{t+1}) + Var(∆ s_{R,t})).
  - If φ_y > 0 and monetary policy is accommodative, Cov(r^R\$_{t+1}, ∆ s_{t+1}) > 0 ⇒ Var(r^R_{t+1}) > Var(∆ s_{R,t}).
  - If φ_y < 0, sign ambiguous; relative volatility depends on productivity vs interest rate variances.
- General quantitative implication:
  - News shocks correlated with current shocks, persistence, and interest rate smoothing can jointly generate larger equity and exchange rate volatility relative to fundamentals, with comovement and magnitudes sensitive to monetary policy reaction coefficients and shock variances.

*IMF Working Paper _wp08284 (appendix available from the authors) — content unit as provided*

### section 3. As we can see, the interest rate differential becomes much less volatile with smoothing. On

### _wp08284 - section 3. As we can see, the interest rate differential becomes much less volatile with smoothing. On

### Impulse responses and smoothing effects
- Interest rate smoothing reduces volatility of the interest rate differential while increasing volatility of the exchange rate and the relative equity return because "more information about the future is being taken into account on impact."
- Experiment assumptions illustrated in Figure 3:
  - Policy parameters: φp = 1.5, φy = 0.2, φs = 0.1 and γ = 0.6.
  - Exchange rate smoothing: small positive coefficient (implicitly present).
  - Autoregressive coefficients: θW = 0.95 and θR = 0.7.
- Under these assumptions:
  - Equity return tends to be more volatile than the exchange rate because the exchange rate does not depend on world shocks while Home variables do.
  - The Home interest rate, affected by world shocks, remains less volatile than the exchange rate.

### Model evaluation: data, moments, and benchmarking
- Countries evaluated relative to the U.S.: UK, Japan (JP), and Germany (BD); main floating currencies from 1973 to date; several model specifications reported in Table 1.
- Data sources and sample periods:
  - Interest rate data: 3-month LIBOR, sample period 1987Q1-2007Q4.
  - Equity returns: MSCI series (total equity return indices) and end-of-period exchange rates against the U.S. dollar for UK, Japan, Germany; sample period 1973Q1-2007Q4.
- Benchmarking:
  - The conditional standard deviation of the (quarterly) relative interest rate is set to about 0.16 percent to match the data by scaling the variance-covariance matrix of the underlying stochastic processes.
  - Interest rate shocks attached to the interest rate rule are assumed minimal: their variance is one per cent of the variance of productivity shocks.
- Reporting conventions:
  - For interest rate data: both conditional (σt(x)) and unconditional (σ(x)) standard deviations are reported; conditional standard deviations computed by running simple univariate autoregressive regressions.
  - For equity and exchange rate returns: only unconditional standard deviations and correlations reported in the data because equity and exchange rate returns have very small predictable components in the data; model reports both conditional and unconditional to show little difference between them.

### Model specifications and comparative effects (Models 1–6)
- Model 1: Base Model — bare-bone monetary policy rule with γ = 0, φy = 0.2, φp = 1.5, φs = 0.1, and no correlation between news and current shocks.
- Model 2: Model 1 with correlated shocks (%
a =%m = 0.9 for both R and W). Correlated news and current shocks increase asset price volatility relative to the volatility of the real relative interest rate.
- Model 3: Model 1 with interest rate smoothing (γ = 0.4). Interest rate smoothing has an effect similar to correlated news shocks but quantitatively larger because smoothing provides information about the entire future path of interest rates.
- Model 4: Model 3 with correlated shocks (combines interest rate smoothing and correlated shocks), generating higher asset price volatility.
- Model 5: Model 4 without monetary policy reaction to the output gap (φy = 0.0). Shutting down policy response to output gap:
  - Lowers exchange rate return volatility.
  - Makes volatility of equity return in firm currency comparable to relative return in investor currency.
  - Removes the positive correlation between exchange rate and relative equity return in investor currency that arises when φy > 0.
- Model 6: Model 4 with negative monetary policy reaction to output gap (φy = −0.01). Model 6:
  - Matches the data in terms of the sign of the correlation between exchange rate return and relative equity return in investor currency.
  - Has empirical caveats: empirical estimates of φy are typically positive; a small negative φy may reflect different definitions of the output gap (flexible-price output used in this model).
  - Uniqueness of a stable equilibrium is guaranteed as long as the negative value is sufficiently small.

### Key quantitative regularities and model successes/limitations
- Volatility ranking captured by the model (consistent with the data in most cases):
  - Variance of the relative return on equities in firm local currency (rR t+1) > exchange rate volatility (∆s t+1) > relative interest rate volatility (iR t).
- Model shortcoming:
  - The model misses the relative size of the volatility of the relative return on equities in investor currency (rR$ t+1): in the first four specifications (where monetary policy responds to the output gap), rR$ t+1 is much less volatile in the model than in the data.
  - When φy = 0 (Model 5) the model reduces the exchange rate return volatility, bringing firm-currency equity return volatility closer to investor-currency return volatility — a possible but not necessarily realistic adjustment.
- Comovement findings:
  - With φy > 0 (monetary policy accommodates productivity shocks) the model implies a positive correlation between exchange rate and the relative equity return in investor currency; setting φy = 0 makes this correlation zero, closer to some data patterns.
  - Empirically: positive correlation between exchange rate return and relative return in firm local currency for Germany, UK, and many other countries; negative correlation for Japan and a few other countries (consistent with Hau and Rey (2006)).
  - Data also show a large negative correlation between relative return in investor currency and exchange rate for all countries considered — surprising but not inconsistent with the model.
  - Model 6 (φy = −0.01) matches the data sign for the correlation between exchange rate and equity returns in investor currency, but at the cost of somewhat lower exchange rate volatility.

### Main conceptual findings and mechanisms
- Role of news shocks:
  - If news shocks are positively correlated with current shocks, asset prices become more volatile without affecting volatility of underlying fundamental processes.
  - Intuition: asset prices respond to both current shocks and today’s news shocks; fundamentals evolve according to current shocks and yesterday’s news shocks. If current and news shocks are positively correlated, news about future fundamentals carries information similar to today’s surprise, so asset prices can move more in the same direction than fundamentals.
  - This mechanism explains part of the excess volatility of asset prices relative to fundamentals and why persistent stochastic processes generate higher asset price volatility.
- Monetary policy reaction to output gap:
  - Has a significant impact on comovements between exchange rate and equity return.
  - A small negative φy can flip the sign of certain correlations to match data, but empirical plausibility is questionable.

### Model solution and equilibrium conditions (high-level)
- Notation: log deviations from the initial symmetric steady state denoted by lower-case letters; * denotes foreign variables; superscript R denotes relative variables; superscript W denotes world average.
- Benchmark equilibrium and extended model equations include:
  - Money, consumption, bond-pricing, wage, price-level, labor, resource constraints, and equity pricing equations (equations A55–A66 and subsequent).
  - Relative-variable solutions (equations A67–A75) express s t, pR t, ρcR t, τ t, lR t, wR t, qR t, πR t, rR t in terms of expectations, money, productivity and news terms, and structural parameters (ψ, ω, ζ, β, ε, θ, ν, etc.).
  - World-variable solutions (equations A76–A82) provide pW t, cW t, lW t, wW t, πW t, qW t, rW t+1 in closed form given the shock structure.
  - Extended model equilibrium (staggered prices and rule-based interest rate setting) summarized in equations A83–A101, including the interest rate rules:
    - iR t = γ iR t−1 + (1−γ)[φp ∆pR t + φy (lR t − lFlex,R t) + φs ∆s t + νR 3,t + νR 4,t−1] (A98)
    - iW t = γ iW t−1 + (1−γ)[φp ∆pW t + φy (lW t − lFlex,W t) + νW 3,t + νW 4,t−1] (A99)
  - Productivity laws of motion:
    - aR t = θR aR t−1 + (1−θR)(νR 1,t + νR 2,t−1) (A100)
    - aW t = θW aW t−1 + (1−θW)(νW 1,t + νW 2,t−1) (A101)

*Source: _wp08284 - section 3. As we can see, the interest rate differential becomes much less volatile with smoothing. On*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08284.pdf_