## _wp11110 — introduction of news shocks can indeed increase asset price volatility

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### I. Introduction — role of news shocks
- “News shocks” are changes in expectation about the future realization of economic fundamentals (Cochrane (1994); Beaudry and Portier (2004)).
- News shocks may be observable to agents but unobservable to macroeconomists/econometricians.
- DSGE models with news shocks (Beaudry and Portier (2004, 2007); Jaimovich and Rebelo (2009); Schmitt-Grohe and Uribe (2008)) can attribute significant aggregate fluctuations to news shocks.
- West (1988) result: conditioning on information sets that include future fundamentals should reduce the conditional variance of asset prices in present discounted value models (PVM).
- Paper contributions:
  - In a PVM, introduction of news shocks always induces a fall in asset price volatility relative to the same model without news shocks (conditional variance measure).
  - If news shocks are positively correlated with current shocks (“correlated news shocks”), the data generating process for the fundamental is serially correlated and asset price volatility can increase in PVM relative to fundamentals while holding the unconditional variance of the fundamental constant.
  - In general equilibrium, introducing news about future productivity need not decrease asset price volatility; providing more information about the exogenous process may increase the conditional variance of asset prices because the endogenous dividend/cash-flow process is affected by the information set.
  - The paper incorporates news shocks about technology and monetary policy in a canonical, closed-economy DSGE model with sticky prices; the model is parameterized to illustrate potential large increases in asset price conditional variance from news shocks.
  - Practical implication: empirical event studies of monetary policy impacts on equity prices may be biased if they ignore monetary policy news shocks.

### II. Partial-equilibrium example — definitions and key algebra
- Definition of “news” (consistent with Jaimovich and Rebelo (2009); Schmitt-Grohe and Uribe (2008)):
  - z_t is news about f_{t+j} if there exists integer j > 0 such that Var(f_{t+j} | I_t) < Var(f_{t+j} | H_t) with z_t ∈ I_t but z_t ∉ H_t.
- Present discounted value model (PVM):
  - x_t(F_t) = sum_{j=0}^∞ β^j P(f_{t+j} | F_t).
- West (1988) proposition:
  - Given H_t ⊂ I_t, Var(x_t(I_t) | I_{t-1}) ≤ Var(x_t(H_t) | H_{t-1}).
- Example fundamental with current shock ε_t and news z_t:
  - f_t = f_{t-1} + ε_t + z_{t-1}.
  - (ε_t, z_t)' are jointly i.i.d. zero mean with Var(ε_t) = σ_1^2, Var(z_t) = σ_2^2, Cov(ε_t, z_t) = ρ σ_1 σ_2, −1 < ρ < 1.
  - z_t = ρ (σ_2/σ_1) ε_t + η_t, where η_t orthogonal to ε_t and Var(η_t) = (1−ρ^2) σ_2^2.
  - Alternate representation produces apparent persistence: f_t = f_{t-1} + θ_t + ρ_θ θ_{t-1}.
- Conditional variances of the fundamental:
  - Var(f_t | I_{t-1}) = Var(ε_t) = σ_1^2 ≤ Var(f_t | H_{t-1}) = Var(ε_t + η_{t-1}) = σ_1^2 + (1−ρ^2) σ_2^2.
  - Equality holds if |ρ| = 1.
- Asset prices in two information environments:
  - With news: x_t(I_t) = (1/(1−β)) f_t + (β/(1−β)) z_t.
  - Without news: x_t(H_t) = (1/(1−β)) f_t + (β/(1−β)) ρ (σ_2/σ_1) ε_t.
- Conditional variances of asset prices:
  - Var(x_t(I_t) | I_{t-1}) = (1−β)^{−2} (σ_1^2 + 2β ρ σ_1 σ_2 + β^2 σ_2^2).
  - Var(x_t(H_t) | H_{t-1}) = (1−β)^{−2} (σ_1^2 + 2β ρ σ_1 σ_2 + [β^2 + (1−β^2)(1−ρ^2)] σ_2^2).
  - Difference equals −(1−β^2)(1−ρ^2) σ_2^2, consistent with West (1988): news reduces conditional variance for given univariate representation.
- Unconditional variance of the fundamental:
  - Var(Δ f_t) = Var(ε_t + z_{t-1}) = σ_1^2 + σ_2^2 (does not depend on ρ).
- Key insight:
  - Correlated news shocks (higher ρ) can increase asset price conditional volatility relative to the volatility of the cash flow process while leaving unconditional cash-flow variance unchanged (generates apparent persistence/magnification).
  - Econometricians observing only fundamentals cannot distinguish between persistence and correlated news shocks; asset prices contain information that can help identify news.

### III. DSGE model — setup and informational assumptions
- Economy:
  - Production economy with unit population, nominal rigidity (goods prices preset one period in advance), monopolistic competition, linear technology (no capital).
- Households maximize:
  - max_{C_t(j), M_t(j), L_t(j)} E_t sum_{s=t}^∞ [ C_s(j)^{1−ρ}/(1−ρ) + κ_1/(1−ε) (M_s(j)/P_s)^{1−ε} − κ_2/(1+ψ) L_s(j)^{1+ψ} ].
- Consumption basket:
  - C_t(j) ≡ [∫_0^1 C_t(j,i)^{(λ−1)/λ} di]^{λ/(λ−1)}.
- First-order conditions:
  - W_t = κ_2 L_t^ψ C_t^{−ρ}/P_t.
  - (M_t/P_t)^ε = κ_1 C_t^ρ (1−E_t β D_{t,t+1}), with D_{t,t+s} ≡ (C_{t+s}^{−ρ}/P_{t+s})/(C_t^{−ρ}/P_t).
- Firms:
  - Y_t(i) = A_t L_t(i); firm i sets price P_t(i) one period in advance.
  - Optimal preset price: P_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].
- Market clearing:
  - Y_t = A_t L_t = C_t.
- Stochastic processes with news:
  - a_t ≡ ln(A_t) = a_{t-1} + ν_{1,t} + ν_{2,t−1}, with (ν_{1,t}, ν_{2,t})' i.i.d., Var(ν_{1,t}) = σ_{ν1}^2, Var(ν_{2,t}) = σ_{ν2}^2, Cov(ν_{1,t}, ν_{2,t}) = ρ_a σ_{ν1} σ_{ν2}. ν_{1,t} is current productivity shock; ν_{2,t} is news about productivity one period ahead.
  - ln(M_t) = ln(M_{t−1}) + μ_t, with μ_t = ν_{3,t} + ν_{4,t−1} + χ_1 ν_{1,t} + χ_2 ν_{2,t−1} + χ_3 ν_{2,t}, where (ν_{3,t}, ν_{4,t})' are i.i.d., Var(ν_{3,t}) = σ_{ν3}^2, Var(ν_{4,t}) = σ_{ν4}^2, Cov(ν_{3,t}, ν_{4,t}) = ρ_m σ_{ν3} σ_{ν4}. ν_{3,t} is current money shock; ν_{4,t} is money news; χ_1, χ_2, χ_3 are monetary policy response coefficients.
  - Auxiliary variables: μ_{1,t} ≡ ν_{3,t} + χ_1 ν_{1,t} + χ_3 ν_{2,t}, μ_{2,t} ≡ ν_{4,t} + χ_2 ν_{2,t} so μ_t = μ_{1,t} + μ_{2,t−1}.
- Information sets:
  - I_t: agents observe news (ν_{2,t} ∈ I_t).
  - H_t: agents do not observe news until t+1 (ν_{2,t} ∉ H_t).
- Solution approach: log-linearize around initial steady state; full solution in Appendix 2.

### IV. General-equilibrium results — equity price and returns
- Equity price definition and linearization:
  - Q_t = Π_t + E_t[β D_{t,t+1} Q_{t+1}], linearized: 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 r_{t+1} − i_t equals q_{t+1} − E_t q_{t+1}.
  - Variance of excess return equals conditional variance of equity price innovation.
- Surprise component of equity price:
  - q_{t+1} − E_t q_{t+1} = (Λ_1 + Λ_2) ν_{1,t+1} + Λ_2 ν_{2,t+1} + Λ_m {[(1−β) ε + β] μ_{1,t+1} + β μ_{2,t+1} }.
- Definitions of coefficients:
  - Λ_1 ≡ (1−β) ζ/(1−ζ) (ψ+1).
  - Λ_2 ≡ { [ 1 + (1−β) ( (1−ρ)/ρ − ζ/(1−ζ) (ρ+ψ)/ρ ) ] ρ (1−1/ε) + (1−ρ) } β/(ψ+1) ρ+ψ.
  - Λ_m ≡ 1 + (1−β) ( (1−ρ)/ρ − ζ/(1−ζ) (ρ+ψ)/ρ ).
  - ζ = (λ−1)/λ is the steady state labor share of revenue.
- Key general-equilibrium insight:
  - Dividends and profits are endogenous and affected by agents’ information sets; therefore alternative information assumptions can change the dividend process and equity-price volatility.

### IV.B Productivity news shocks — when news raises equity volatility
- Set ρ_a = 0 to isolate general equilibrium effect (so surprise component for non-news agents equals Δ a_{t+1} = ν_{1,t+1} + ν_{2,t}).
- With no monetary shocks/reactions:
  - With news: q_{t+1}(I_{t+1}) − E(q_{t+1}(I_{t+1}) | I_t) = (Λ_1 + Λ_2) ν_{1,t+1} + Λ_2 ν_{2,t+1}.
  - Without news: q_{t+1}(H_{t+1}) − E(q_{t+1}(H_{t+1}) | H_t) = (Λ_1 + Λ_2) (ν_{1,t+1} + ν_{2,t}).
- Difference in conditional variances:
  - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = −(Λ_1 + 2 Λ_2) Λ_1 σ_{ν2}^2.
- Implication:
  - Asset price volatility with news is larger than without news when Λ_1 + 2 Λ_2 < 0 (Λ_1 > 0).
  - Example parameter values producing this sign: ε = 1, ψ = 0, ζ = 2/3, β = .95 and inverse intertemporal substitution ρ ≥ (1 − 1−β/(2β) ζ/(1−ζ))^{−1} ≈ 1.06 (as stated).
  - Note: if ρ = 1, expression (38) is positive for ε = .8, ζ = 2/3, β = .95 (as noted).
- Economic mechanism:
  - A productivity news shock changes expectations and thus consumption and pricing decisions today; under certain parameter values the effect on future profits dominates and produces larger price declines when news is observed, increasing volatility.
- Numerical illustration:
  - Figures (not included here) plot conditional variance with/without news as function of ρ for three ε values; log difference ln Var(...) − ln Var(...) can be very large under the chosen parameters (ψ = 0, β = .95, ζ = 2/3, unity variances for productivity shocks, no correlation).
  - When Λ_1 + 2 Λ_2 = 0 the two conditional variances are equal; log difference → ∞ when Λ_1 + Λ_2 = 0 (equity price unresponsive when agents do not observe news).

### IV.C Monetary policy news shocks — cases and interactions
- Decomposition:
  - μ_{1,t} ≡ ν_{3,t} + χ_1 ν_{1,t} + χ_3 ν_{2,t}, μ_{2,t} ≡ ν_{4,t} + χ_2 ν_{2,t}.
- Case 1 — monetary news alone (no policy reaction to productivity shocks):
  - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = − Λ_m 2 (1−β) ε [ (1−β) ε + 2β ] σ_{ν4}^2.
  - Since ε > 0, expression is always negative ⇒ monetary policy announcement alone cannot increase conditional variance of equity price in this simple specification.
- Case 2 — monetary policy reacts to productivity news (assume ν_{4,t} = 0, χ_3 = 0):
  - μ_t(I_t) = χ_1 ν_{1,t} + χ_2 ν_{2,t−1} + ν_{3,t}; μ_t(H_t) = χ_1 (ν_{1,t} + ν_{2,t−1}) + ν_{3,t}.
  - Surprise components with news vs without news are given in text.
  - Difference in conditional variances:
    - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = − {Λ_1 + 2Λ_2 + Λ_m χ_1 [(1−β) ε + β] + Λ_m β χ_2} × {Λ_1 + Λ_m χ_1 [(1−β) ε + β] − Λ_m β χ_2} σ_{ν2}^2.
  - Depending on χ_1, χ_2 and other parameters, the product can be negative or positive; thus monetary policy reactions to productivity news can make news raise conditional equity volatility.
- Example:
  - ε = 1, ψ = 0 and “optimal” monetary reaction replicating flexible-price equilibrium (χ_1 = 1, χ_2 = 0) yields:
    - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = −(Λ_1 + 2Λ_2 + Λ_m)(Λ_1 + Λ_m) σ_{ν2}^2
    - = − [ (1 + β) 1/ρ − β ] [ β + (1−β) 1/ρ ] σ_{ν2}^2,
    - implying news shocks can increase equity price volatility for ρ > 1 + 1/β (plausible value slightly larger than 2 for β = .95 under stated parameter choices).
- Overall implication:
  - Monetary policy news by itself reduces conditional variance in this simple model, but realistic monetary policy responses to productivity news can generate an increase in equity price conditional volatility when news is taken into account.

### Main findings on news shocks and equity price volatility
- In general equilibrium, news shocks about future productivity can significantly increase the volatility of equity prices relative to an environment in which agents do not observe news under plausible parameter values.
- This contrasts with a partial value model (PVM) result (West (1988)) where introducing news shocks reduces volatility, because in general equilibrium agents change behavior and thus affect the cash flow stream.
- Correlated news shocks can either increase or decrease conditional variance of asset prices in general equilibrium depending on parameter values; in a simple PVM example correlated news shocks can increase conditional variance holding Var(∆a_{t+1}) constant.

### Conditions for positive comovement and higher volatility
- A positive correlation between the news shock and equity price requires the coefficient on the news shocks in equation (44) to be positive: Λ_2 + Λ_m χ_2 β > 0, which holds iff
  - [ 1 + (1−β) ( (1−ρ)/ρ − ζ/(1−ζ) (ρ+ψ)/ρ ) ] [ ρ ( 1− 1/ε ) ψ+ 1 ρ+ψ + χ_2 ] + (1−ρ) ψ+ 1 ρ+ψ > 0.
- More simply, if Λ_m > 0, then the condition can be satisfied if χ_2 > −(2/β)Λ_m (text: χ_2 > −2/βm in source).
- To generate both higher volatility with the introduction of news shocks and a positive correlation between the news shock and equity price, inequalities (46) and (48) must hold simultaneously; under assumptions this implies
  - χ_1 < −1 +m β χ_2 m [(1−β)ε+β] . (equation (49) in source)
- Numerical example satisfying both conditions: ε = 1, ψ = 0, ρ = 1, ζ = 2/3, β = .95, χ_1 = .4, χ_2 = .6. (Source note: the value of ρ can be as large as 2 in this example; alternative parameter choices discussed in text.)

### Role of the stochastic discount factor (SDF) and monetary policy
- Surprise component of the equity return (equation (50)):
  - r_{t+1} − E_t r_{t+1} = q_{t+1} − E_t q_{t+1} = ∑_{s=0}^{∞} β^s [(1−β)(E_{t+1}−E_t) π_{t+s+1} + β (E_{t+1}−E_t) d_{t+s+1,t+2+s}] = ∑_{s=0}^{∞} β^s [(1−β)(E_{t+1}−E_t) π_{t+s+1} − β i_{t,t+1}].
- First part: present discounted value of expectation changes in future dividends. Second part: present discounted value of expectation changes in future discount factors, which reduces to the nominal interest rate at time t+1 because E_t d_{t+s+1,t+2+s} = 0 for s>0 and E_{t+1} d_{t+1,t+2} = −i_{t+1}.
- Introducing productivity news shocks does not affect equity price volatility through the SDF in the model when monetary policy does not react to them because coefficients on ν_{2,t} and ν_{2,t+1} are identical in i_{t+1}(I_{t+1}) and i_{t+1}(H_{t+1}) (equations (51) and (52)):
  - i_{t+1}(I_{t+1}) =(1−β){[(1−ε)μ_{1,t+1}+μ_{2,t+1}] + (ψ+ 1)/(ρ+ψ) ρ (1−1/ε) (ν_{1,t+1}+ν_{2,t+1})},
  - i_{t+1}(H_{t+1}) =(1−β){(1−ε)(μ_{1,t+1}+μ_{2,t}) + (ψ+ 1)/(ρ+ψ) ρ (1−1/ε) (ν_{1,t+1}+ν_{2,t})}.
- The SDF can affect equity price volatility only if monetary policy reacts to productivity news shocks. Example: if ε = 1, then
  - i_{t+1}(I_{t+1}) = (1−β) μ_{2,t+1},  i_{t+1}(H_{t+1}) = 0,
  - hence monetary reaction to news changes SDF volatility and thus equity price volatility.
- Conclusion: modeling monetary policy response to productivity news shocks is key for matching empirical asset price features in a DSGE with news shocks.

### Correlated news shocks and conditional variance
- With Var(∆a_{t+1}) = Var(ν_{1,t+1}+ν_{2,t}) = σ^2_{ν1} + σ^2_{ν2} independent of the correlation ρ_a between ν_{1,t+1} and ν_{2,t+1}, the conditional variance of q_{t+1} depends on ρ_a:
  - Var_t q_{t+1} = (Λ_1 + Λ_2)^2 σ^2_{ν1} + 2 Λ_1 Λ_2 ρ_a σ_{ν1} σ_{ν1} + Λ_2^2 σ^2_{ν2}. (equation (53))
- Therefore increasing ρ_a can increase equity price volatility if Λ_2 > 0 or decrease it if Λ_2 < 0 (given Λ_1 > 0). Correlated news shocks can mimic effects of persistent current shocks because information about the future conveyed by a persistent current shock is similar to that carried by correlated news shocks.

### Impact of monetary policy shocks on equity prices and implications for empirical work
- Monetary policy news shocks affect excess equity return if and only if current monetary policy shocks also affect the excess return, because μ_{1,t+1} and μ_{2,t+1} share the same coefficient in equation (35) and their impact requires
  - [ 1 + (1−β) ( (1−ρ)/ρ − ζ/(1−ζ) (ρ+ψ)/ρ ) ] ≠ 0.
- Implication: event-study estimates of the effect of current policy shocks on equity returns may be biased because policy announcements often contain information about future interest rates (μ_{2,t+1}) as well as current policy (μ_{1,t+1}). Regressing equity returns on changes in federal funds futures at FOMC announcements may omit the effect of news about the future and produce biased estimates (Rigobon and Sack (2004) noted event-study bias).
- Example evidence: Okina and Shiratsuka (2004) find a Bank of Japan policy announcement affecting the long end of the yield curve, illustrating quantitatively significant effects of news shocks on asset prices.

### Conclusions and implications for future research
- Introducing news shocks in general equilibrium can raise equity price volatility because agents alter behavior and endogenous cash flows respond to information; therefore additional information does not necessarily reduce volatility of endogenous variables.
- Correlated news shocks can either increase or decrease volatility in general equilibrium depending on parameters, differing from PVM intuition.
- Econometric specifications that omit news shocks may be biased when estimating fundamental shock impacts on asset prices; news shocks can be observationally equivalent to serially correlated fundamentals from the econometrician’s viewpoint, but models with persistent fundamentals cannot replicate certain empirical features that news shocks can.
- The paper’s DSGE model is deliberately simple and not suited for full quantitative evaluation against data, but the general equilibrium effects identified can be numerically sizable.
- Recommended areas for future research: quantitative analysis of asset price volatility in DSGE models with news shocks and development of empirical identification strategies to disentangle news and current shocks effects on asset prices.

*Source: _wp11110 - introduction of news shocks can indeed increase asset price volatility.*

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### _wp11110 - References ........................................................................................................ 32

### I. INTRODUCTION — role of news shocks
- “News shocks” are changes in expectation about the future realization of economic fundamentals (Cochrane (1994); Beaudry and Portier (2004)).
- News shocks are attractive because they are observable to agents but may be unobservable to macroeconomists/econometricians.
- DSGE models with news shocks (Beaudry and Portier (2004, 2007); Jaimovich and Rebelo (2009); Schmitt-Grohe and Uribe (2008)) can attribute significant aggregate fluctuations to news shocks.
- West (1988) result: conditioning on information sets that include future fundamentals should reduce the conditional variance of asset prices in present discounted value models (PVM).
- Paper contributions:
  - In a PVM, introduction of news shocks always induces a fall in asset price volatility relative to the same model without news shocks (conditional variance measure).
  - If news shocks are positively correlated with current shocks (“correlated news shocks”), the data generating process for the fundamental is serially correlated and asset price volatility can increase in PVM relative to fundamentals while holding the unconditional variance of the fundamental constant.
  - In general equilibrium, introducing news about future productivity need not decrease asset price volatility; providing more information about the exogenous process may increase the conditional variance of asset prices because the endogenous dividend/cash-flow process is affected by the information set.
- The paper incorporates news shocks about technology and monetary policy in a canonical, closed-economy DSGE model with sticky prices; the model is parameterized to illustrate potential large increases in asset price conditional variance from news shocks.
- Practical implication: empirical event studies of monetary policy impacts on equity prices may be biased if they ignore monetary policy news shocks.

### II. PARTIAL-EQUILIBRIUM EXAMPLE — definitions and key algebra
- Definition of “news” (consistent with Jaimovich and Rebelo (2009); Schmitt-Grohe and Uribe (2008)):
  - z_t is news about f_{t+j} if there exists integer j > 0 such that Var(f_{t+j} | I_t) < Var(f_{t+j} | H_t) with z_t ∈ I_t but z_t ∉ H_t.
- Present discounted value model (PVM):
  - x_t(F_t) = sum_{j=0}^∞ β^j P(f_{t+j} | F_t).
- West (1988) proposition in this context:
  - Given H_t ⊂ I_t, Var(x_t(I_t) | I_{t-1}) ≤ Var(x_t(H_t) | H_{t-1}).
- Example fundamental process with current shock ε_t and news z_t:
  - f_t = f_{t-1} + ε_t + z_{t-1}.
  - (ε_t, z_t)' are jointly i.i.d. zero mean with Var(ε_t) = σ_1^2, Var(z_t) = σ_2^2, Cov(ε_t, z_t) = ρ σ_1 σ_2, −1 < ρ < 1.
  - z_t = ρ (σ_2/σ_1) ε_t + η_t, where η_t orthogonal to ε_t and Var(η_t) = (1−ρ^2) σ_2^2.
  - Alternate representation produces apparent persistence: f_t = f_{t-1} + θ_t + ρ_θ θ_{t-1}.
- Conditional variances of the fundamental:
  - Var(f_t | I_{t-1}) = Var(ε_t) = σ_1^2 ≤ Var(f_t | H_{t-1}) = Var(ε_t + η_{t-1}) = σ_1^2 + (1−ρ^2) σ_2^2.
  - Equality holds if |ρ| = 1.
- Asset prices in two information environments:
  - With news (agents observe z_t at t): x_t(I_t) = (1/(1−β)) f_t + (β/(1−β)) z_t.
  - Without news (agents observe z_t only with one-period delay): x_t(H_t) = (1/(1−β)) f_t + (β/(1−β)) ρ (σ_2/σ_1) ε_t.
- Conditional variances of asset prices:
  - Var(x_t(I_t) | I_{t-1}) = (1−β)^{−2} (σ_1^2 + 2β ρ σ_1 σ_2 + β^2 σ_2^2).
  - Var(x_t(H_t) | H_{t-1}) = (1−β)^{−2} (σ_1^2 + 2β ρ σ_1 σ_2 + [β^2 + (1−β^2)(1−ρ^2)] σ_2^2).
  - Difference equals −(1−β^2)(1−ρ^2) σ_2^2, consistent with West (1988): news reduces conditional variance for given univariate representation.
- Unconditional variance of the fundamental:
  - Var(Δ f_t) = Var(ε_t + z_{t-1}) = σ_1^2 + σ_2^2 (does not depend on ρ).
- Key insights:
  - Correlated news shocks (higher ρ) can increase asset price conditional volatility relative to the volatility of the cash flow process while leaving unconditional cash-flow variance unchanged (generates apparent persistence/magnification).
  - Econometricians observing only fundamentals cannot distinguish between persistence and correlated news shocks; asset prices contain information that can help identify news.

### III. DSGE MODEL — setup and informational assumptions
- Economy: production economy with unit population, nominal rigidity (goods prices preset one period in advance), monopolistic competition, linear technology (no capital).
- Households maximize:
  - max_{C_t(j), M_t(j), L_t(j)} E_t sum_{s=t}^∞ [ C_s(j)^{1−ρ}/(1−ρ) + κ_1/(1−ε) (M_s(j)/P_s)^{1−ε} − κ_2/(1+ψ) L_s(j)^{1+ψ} ].
- Consumption basket:
  - C_t(j) ≡ [∫_0^1 C_t(j,i)^{(λ−1)/λ} di]^{λ/(λ−1)}.
- First-order conditions highlighted:
  - W_t = κ_2 L_t^ψ C_t^{−ρ}/P_t (labor supply).
  - (M_t/P_t)^ε = κ_1 C_t^ρ (1−E_t β D_{t,t+1}) (money demand), with D_{t,t+s} ≡ (C_{t+s}^{−ρ}/P_{t+s})/(C_t^{−ρ}/P_t) the pricing kernel.
- Firms:
  - Y_t(i) = A_t L_t(i); firm i sets price P_t(i) one period in advance to maximize discounted profits.
  - Optimal preset price: P_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].
- Market clearing:
  - Y_t = A_t L_t = C_t.
- Stochastic processes with news:
  - a_t ≡ ln(A_t) = a_{t−1} + ν_{1,t} + ν_{2,t−1}, where (ν_{1,t}, ν_{2,t})' are i.i.d., Var(ν_{1,t}) = σ_{ν1}^2, Var(ν_{2,t}) = σ_{ν2}^2, Cov(ν_{1,t}, ν_{2,t}) = ρ_a σ_{ν1} σ_{ν2}. ν_{1,t} is current productivity shock; ν_{2,t} is news about productivity one period ahead.
  - ln(M_t) = ln(M_{t−1}) + μ_t, with μ_t = ν_{3,t} + ν_{4,t−1} + χ_1 ν_{1,t} + χ_2 ν_{2,t−1} + χ_3 ν_{2,t}, where (ν_{3,t}, ν_{4,t})' are i.i.d., Var(ν_{3,t}) = σ_{ν3}^2, Var(ν_{4,t}) = σ_{ν4}^2, Cov(ν_{3,t}, ν_{4,t}) = ρ_m σ_{ν3} σ_{ν4}. ν_{3,t} is current money shock; ν_{4,t} is money news; χ_1, χ_2, χ_3 are monetary policy response coefficients.
  - Auxiliary variables: μ_{1,t} ≡ ν_{3,t} + χ_1 ν_{1,t} + χ_3 ν_{2,t}, μ_{2,t} ≡ ν_{4,t} + χ_2 ν_{2,t} so μ_t = μ_{1,t} + μ_{2,t−1}.
- Information sets:
  - I_t denotes agents’ information when they observe news (ν_{2,t} ∈ I_t).
  - H_t denotes agents’ information when they do not observe news until t+1 (ν_{2,t} ∉ H_t).
- Solution approach: log-linearize around initial steady state (lowercase for log deviations); full solution in Appendix 2.

### IV. GENERAL-EQUILIBRIUM RESULTS — equity price and returns
- Equity price definition in general equilibrium:
  - Q_t = Π_t + E_t[β D_{t,t+1} Q_{t+1}], linearized: 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}), so excess return r_{t+1} − i_t equals q_{t+1} − E_t q_{t+1}.
  - Variance of excess return equals conditional variance of equity price innovation (focus of analysis).
- Surprise component of equity price (equation with shocks):
  - q_{t+1} − E_t q_{t+1} = (Λ_1 + Λ_2) ν_{1,t+1} + Λ_2 ν_{2,t+1} + Λ_m {[(1−β) ε + β] μ_{1,t+1} + β μ_{2,t+1} },
  - where Λ_1 ≡ (1−β) ζ/(1−ζ) (ψ+1),
    Λ_2 ≡ { [ 1 + (1−β) ( (1−ρ)/ρ − ζ/(1−ζ) (ρ+ψ)/ρ ) ] ρ (1−1/ε) + (1−ρ) } β/(ψ+1) ρ+ψ,
    Λ_m ≡ 1 + (1−β) ( (1−ρ)/ρ − ζ/(1−ζ) (ρ+ψ)/ρ ),
    and ζ = (λ−1)/λ is the steady state labor share of revenue.
- Key general-equilibrium insight: dividends and profits are endogenous and affected by agents’ information sets; therefore alternative information assumptions can change the dividend process and equity-price volatility.

### IV.B Productivity news shocks — when news raises equity volatility
- Set ρ_a = 0 to isolate general equilibrium effect (so surprise component for non-news agents equals Δ a_{t+1} = ν_{1,t+1} + ν_{2,t}).
- With no monetary shocks/reactions for simplicity:
  - With news: q_{t+1}(I_{t+1}) − E(q_{t+1}(I_{t+1}) | I_t) = (Λ_1 + Λ_2) ν_{1,t+1} + Λ_2 ν_{2,t+1}.
  - Without news: q_{t+1}(H_{t+1}) − E(q_{t+1}(H_{t+1}) | H_t) = (Λ_1 + Λ_2) (ν_{1,t+1} + ν_{2,t}).
- Difference in conditional variances:
  - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = −(Λ_1 + 2 Λ_2) Λ_1 σ_{ν2}^2.
- Implication:
  - Asset price volatility with news is larger than without news when Λ_1 + 2 Λ_2 < 0 (Λ_1 > 0).
  - Example parameter values producing this sign: ε = 1, ψ = 0, ζ = 2/3, β = .95 and inverse intertemporal substitution ρ ≥ (1 − 1−β/(2β) ζ/(1−ζ))^{−1} ≈ 1.06 (as stated).
  - Note: if ρ = 1, expression (38) is positive for ε = .8, ζ = 2/3, β = .95 (as noted).
- Economic mechanism:
  - In general equilibrium a productivity news shock affects expectations and thus consumption and pricing decisions today, changing current and future profits differently across information sets. Under certain parameter values, the effect on future profits dominates and produces larger price declines when news is observed, increasing volatility.
- Numerical illustration (described):
  - Figures (not included here) plot conditional variance with/without news as function of ρ for three ε values; log difference ln Var(...) − ln Var(...) can be very large under the chosen parameters (ψ = 0, β = .95, ζ = 2/3, unity variances for productivity shocks, no correlation).
  - When Λ_1 + 2 Λ_2 = 0 the two conditional variances are equal; log difference → ∞ when Λ_1 + Λ_2 = 0 (equity price unresponsive when agents do not observe news).

### IV.C Monetary policy news shocks — cases and interactions
- Decomposition: μ_{1,t} ≡ ν_{3,t} + χ_1 ν_{1,t} + χ_3 ν_{2,t}, μ_{2,t} ≡ ν_{4,t} + χ_2 ν_{2,t}.
- Case 1 — monetary news alone (no policy reaction to productivity shocks):
  - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = − Λ_m 2 (1−β) ε [ (1−β) ε + 2β ] σ_{ν4}^2.
  - Since ε > 0, expression is always negative ⇒ monetary policy announcement alone cannot increase conditional variance of equity price in this simple specification.
- Case 2 — monetary policy reacts to productivity news (assume ν_{4,t} = 0, χ_3 = 0 for simplicity):
  - μ_t(I_t) = χ_1 ν_{1,t} + χ_2 ν_{2,t−1} + ν_{3,t};
    μ_t(H_t) = χ_1 (ν_{1,t} + ν_{2,t−1}) + ν_{3,t}.
  - Surprise components:
    - With news: q_{t+1}(I_{t+1}) − E(...) = {Λ_1 + Λ_2 + Λ_m χ_1 [(1−β) ε + β]} ν_{1,t+1} + (Λ_2 + Λ_m χ_2 β) ν_{2,t+1} + Λ_m [(1−β) ε + β] ν_{3,t}.
    - Without news: q_{t+1}(H_{t+1}) − E(...) = {Λ_1 + Λ_2 + Λ_m χ_1 [(1−β) ε + β]} (ν_{1,t+1} + ν_{2,t}) + Λ_m [(1−β) ε + β] ν_{3,t}.
  - Difference in conditional variances:
    - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = − {Λ_1 + 2Λ_2 + Λ_m χ_1 [(1−β) ε + β] + Λ_m β χ_2} × {Λ_1 + Λ_m χ_1 [(1−β) ε + β] − Λ_m β χ_2} σ_{ν2}^2.
  - Depending on χ_1, χ_2 and other parameters, the product can be negative or positive; thus monetary policy reactions to productivity news can make news raise conditional equity volatility.
- Example: ε = 1, ψ = 0 and “optimal” monetary reaction replicating flexible-price equilibrium (χ_1 = 1, χ_2 = 0) yields:
  - Var(q_{t+1}(I_{t+1}) | I_t) − Var(q_{t+1}(H_{t+1}) | H_t) = −(Λ_1 + 2Λ_2 + Λ_m)(Λ_1 + Λ_m) σ_{ν2}^2
  - = − [ (1 + β) 1/ρ − β ] [ β + (1−β) 1/ρ ] σ_{ν2}^2,
  - implying news shocks can increase equity price volatility for ρ > 1 + 1/β (plausible value slightly larger than 2 for β = .95 under stated parameter choices).
- Overall implication: monetary policy news by itself reduces conditional variance in this simple model, but realistic monetary policy responses to productivity news can generate an increase in equity price conditional volatility when news is taken into account.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11110.pdf*

### introduction of news shocks can indeed increase asset price volatility.

### _wp11110 - introduction of news shocks can indeed increase asset price volatility.

### Main findings on news shocks and equity price volatility
- In general equilibrium, news shocks about future productivity can significantly increase the volatility of equity prices relative to an environment in which agents do not observe news under plausible parameter values.
- This contrasts with a partial value model (PVM) result (West (1988)) where introducing news shocks reduces volatility, because in general equilibrium agents change behavior and thus affect the cash flow stream.
- Correlated news shocks can either increase or decrease conditional variance of asset prices in general equilibrium depending on parameter values; in a simple PVM example correlated news shocks can increase conditional variance holding Var(∆a_{t+1}) constant.

### Conditions for positive comovement between equity price and productivity news shock, and for higher volatility
- A positive correlation between the news shock and equity price requires the coefficient on the news shocks in equation (44) to be positive. Specifically, Λ_2 + Λ_m χ_2 β > 0, which holds iff
  [
  1 + (1−β)
  (
  1−ρ
  ρ
  −
  ζ
  1−ζ
  ρ+ψ
  ρ
  )
  ][
  ρ
  (
  1−
  1
  ε
  )
  ψ+
  1
  ρ+ψ
  +χ_2
  ]+(1−ρ)
  ψ+
  1
  ρ+ψ
  >0.
- More simply, if Λ_m > 0, then the condition can be satisfied if χ_2 > −(2/β)Λ_m (text: χ_2 > −2/βm in source).
- To generate both higher volatility with the introduction of news shocks and a positive correlation between the news shock and equity price, inequalities (46) and (48) must hold simultaneously. Assuming the first term in (46) is positive, the second term must be negative, implying
  χ_1 < −1 +m β χ_2 m [(1−β)ε+β] . (equation (49) in source)
- Numerical example satisfying both conditions: ε = 1, ψ = 0, ρ = 1, ζ = 2/3, β = .95, χ_1 = .4, χ_2 = .6. (Source note: the value of ρ can be as large as 2 in this example; alternative parameter choices discussed in text.)

### Role of the stochastic discount factor (SDF) and monetary policy
- The surprise component of the equity return is split as (equation (50)):
  r_{t+1} − E_t r_{t+1} = q_{t+1} − E_t q_{t+1} = ∑_{s=0}^{∞} β^s [(1−β)(E_{t+1}−E_t) π_{t+s+1} + β (E_{t+1}−E_t) d_{t+s+1,t+2+s}] = ∑_{s=0}^{∞} β^s [(1−β)(E_{t+1}−E_t) π_{t+s+1} − β i_{t,t+1}].
- The first part: present discounted value of expectation changes in future dividends. The second part: present discounted value of expectation changes in future discount factors, which reduces to the nominal interest rate at time t+1 because E_t d_{t+s+1,t+2+s} = 0 for s>0 and E_{t+1} d_{t+1,t+2} = −i_{t+1}.
- Introducing productivity news shocks does not affect equity price volatility through the SDF in the model when monetary policy does not react to them because coefficients on ν_{2,t} and ν_{2,t+1} are identical in i_{t+1}(I_{t+1}) and i_{t+1}(H_{t+1}) (equations (51) and (52)):
  i_{t+1}(I_{t+1}) =(1−β){[(1−ε)μ_{1,t+1}+μ_{2,t+1}] + (ψ+ 1)/(ρ+ψ) ρ (1−1/ε) (ν_{1,t+1}+ν_{2,t+1})},
  i_{t+1}(H_{t+1}) =(1−β){(1−ε)(μ_{1,t+1}+μ_{2,t}) + (ψ+ 1)/(ρ+ψ) ρ (1−1/ε) (ν_{1,t+1}+ν_{2,t})}.
- The SDF can affect equity price volatility only if monetary policy reacts to productivity news shocks. Example: if ε = 1, then
  i_{t+1}(I_{t+1}) = (1−β) μ_{2,t+1},  i_{t+1}(H_{t+1}) = 0,
  hence monetary reaction to news changes SDF volatility and thus equity price volatility.
- Conclusion: modeling monetary policy response to productivity news shocks is key for matching empirical asset price features in a DSGE with news shocks.

### Correlated news shocks and conditional variance
- With Var(∆a_{t+1}) = Var(ν_{1,t+1}+ν_{2,t}) = σ^2_{ν1} + σ^2_{ν2} independent of the correlation ρ_a between ν_{1,t+1} and ν_{2,t+1}, the conditional variance of q_{t+1} depends on ρ_a:
  Var_t q_{t+1} = (Λ_1 + Λ_2)^2 σ^2_{ν1} + 2 Λ_1 Λ_2 ρ_a σ_{ν1} σ_{ν1} + Λ_2^2 σ^2_{ν2}. (equation (53))
- Therefore increasing ρ_a can increase equity price volatility if Λ_2 > 0 or decrease it if Λ_2 < 0 (given Λ_1 > 0). Correlated news shocks can mimic effects of persistent current shocks because information about the future conveyed by a persistent current shock is similar to that carried by correlated news shocks.

### Impact of monetary policy shocks on equity prices and implications for empirical work
- Monetary policy news shocks affect excess equity return if and only if current monetary policy shocks also affect the excess return, because μ_{1,t+1} and μ_{2,t+1} share the same coefficient in equation (35) and their impact requires
  [
  1 + (1−β)
  (
  1−ρ
  ρ
  −
  ζ
  1−ζ
  ρ+ψ
  ρ
  )
  ]
  ≠ 0.
- Implication: event-study estimates of the effect of current policy shocks on equity returns may be biased because policy announcements often contain information about future interest rates (μ_{2,t+1}) as well as current policy (μ_{1,t+1}). Regressing equity returns on changes in federal funds futures at FOMC announcements may omit the effect of news about the future and produce biased estimates (Rigobon and Sack (2004) noted event-study bias).
- Example evidence: Okina and Shiratsuka (2004) find a Bank of Japan policy announcement affecting the long end of the yield curve, illustrating quantitatively significant effects of news shocks on asset prices.

### Conclusions and implications for future research
- Introducing news shocks in general equilibrium can raise equity price volatility because agents alter behavior and endogenous cash flows respond to information; therefore additional information does not necessarily reduce volatility of endogenous variables.
- Correlated news shocks can either increase or decrease volatility in general equilibrium depending on parameters, differing from PVM intuition.
- Econometric specifications that omit news shocks may be biased when estimating fundamental shock impacts on asset prices; news shocks can be observationally equivalent to serially correlated fundamentals from the econometrician’s viewpoint, but models with persistent fundamentals cannot replicate certain empirical features that news shocks can.
- The paper’s DSGE model is deliberately simple and not suited for full quantitative evaluation against data, but the general equilibrium effects identified can be numerically sizable.
- Recommended areas for future research: quantitative analysis of asset price volatility in DSGE models with news shocks and development of empirical identification strategies to disentangle news and current shocks effects on asset prices.

*Source: _wp11110 - introduction of news shocks can indeed increase asset price volatility.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11110.pdf_