Asset Prices in Business Cycle Models Driven by News

and Volatility Shocks∗

Aytek Malkhozov† Maral Shamloo‡

August 2011

Abstract

Shocks to expectations (news) and volatility are important drivers of macroeconomic fluc-

tuations. We explore how incorporating these features into the real business cycle framework

improves our ability to explain key asset pricing facts and the co-movement of macroeco-

nomic and financial variables. News create variation in expected consumption growth which,

together with Epstein-Zin-Weil preferences, increases market prices of risk. Furthermore,

news, as identified in macroeconomic literature and in contrast to long-run risks, explain

key cyclical properties of asset returns and valuations. Finally, agents offset the impact of

volatility shocks with precautionary savings, leading to low risk premia.

Keywords : Anticipated Shocks, Long-Run Risk, Stochastic Volatility, Asset Prices and

Aggregate Fluctuations, Perturbation Methods and Recursive Preferences

JEL Classification : G12, E32, E21, C63

1 Introduction

Our understanding of the shocks driving the business cycle became more nuanced over the last

years. First, agents are not simply buffeted by surprise changes in technology, but receive news

∗We thank Kosuke Aoki, Mikhail Chernov, Jesus Fernandez-Villaverde, Francois Gourio, Stephanie Schmitt-Grohe, Wouter den Haan, Christian Julliard, Leonid Kogan, Xiaoji Lin, Lars Lochstoer, Alex Michaelides, StavrosPanageas, Franck Portier, Ricardo Reis, Bryan Routledge, Harald Uhlig, Raman Uppal, Dimitri Vayanos andAmir Yaron for valuable comments. We also appreciate the comments from seminar participants at the LondonSchool of Economics, Central European University, McGill University faculty of management and McGill Universityeconomics department, University of Montreal, and feedback from conference attendees at the SED 2010 meeting,NBER 2010 Summer Institute and AEA 2011 meeting.†McGill, Desautels Faculty of Management, aytek.malkhozov@mcgill.ca‡International Monetary Fund, mshamloo@imf.org

1

about future fundamentals and form their expectations accordingly. Second, the volatility of tech-

nological shocks varies through time. Both the changes in expectations and changes in volatility

have been identified as major causes of fluctuations in aggregate macroeconomic quantities. In

this paper we use a real business cycle model with recursive preferences to explore the implica-

tions of news and volatility shocks for risk premia and the cyclical properties of asset returns and

valuations.

News shocks, in conjunction with Epstein-Zin-Weil preferences, increase market prices of risk.

In the news driven business cycles paradigm most technological changes are anticipated several

quarters ahead. This generates significant variation in expected consumption growth over this

horizon which is a risk factor if agents are not indifferent towards the timing of the resolution of

uncertainty. At the same time, the immediate effects of news shocks are limited as the production

possibilities of the economy are still unaffected and the unconditional volatility of consumption

growth is low.

Furthermore, news allow us to explain the lead of asset prices over the business cycle quantities,

both as a qualitative phenomenon and quantitatively. Asset pricing models with production

typically fail to reproduce the co-movements between financial and macroeconomic variables.

Anticipated shocks are a natural channel to account for the lead properties: asset prices are

forward looking and incorporate news that will affect productivity only in the future. Notably,

the relative importance of productivity surprises and news at different horizons that is needed to

quantitatively match the lead of asset prices over quantities is consistent with the estimates in

macroeconomic literature.

Our final main result pertains to stochastic volatility. In a production economy agents can

offset volatility shocks by changing the level of precautionary savings and investment. Thereby

in equilibrium higher volatility leads to higher future consumption growth. As a result, prices of

risk remain low.

Variation in conditional means and volatilities became a prominent element of consumption

based asset pricing following the work by Bansal and Yaron (2004). In this paper we look at

how these features in consumption growth can arise as an endogenous response to fundamental

impulses identified in macroeconomic literature. We also extend the focus of the analysis to the

co-movement between asset prices and macroeconomic quantities. This allows us to gain a new

perspective on long-run risk models. Large changes in expectations over one to four quarter

horizon induced by news shocks affect risk premia through the same mechanism as small but very

persistent long-run risks. However, and unlike long-range persistence in consumption growth,

they are also consistent and help to account for cyclical properties of macroeconomic quantities,

asset returns and valuations. These findings are interesting in light of the results documented

in Beeler and Campbell (2009), who show that long-run risks typically imply excessive long-run

predictability of consumption growth, and Backus, Chernov, and Zin (2011), who find that long-

run risk models generate additional dispersion in stochastic discount factor at the cost of its

2

excessive persistence.

Shocks to the volatility of productivity translate into stochastic volatility in consumption

growth. Yet they also affect future expected consumption growth. In other terms stochastic

volatility and long-run risk in consumption are tightly related.

They key element of our model is agents’information set. Allowing for news shocks implies

that agents have more information about future fundamentals than contained in the history of

macroeconomic time series. Additional state variables are required to describe the term structure

of agents expectations. For instance, in our framework it is straightforward to construct an

example where productivity growth series are i.i.d. and yet conditional on agent’s information set

productivity over several future periods is predictable. In the same way, news cannot be identified

from just the time series of consumption.

Because agents optimal policies, as well as asset prices, are forward looking we can identify

news from the co-movement of aggregate quantities by imposing some identification restrictions.

In an article that started the line of research on anticipated shocks Beaudry and Portier (2006) use

data on total factor productivity and stock-market returns to document the importance of news

shocks. More recently, Barsky and Sims (2011) use a similar vector auto-regressive approach using

more comprehensive set of observables and milder identification restrictions. Schmitt-Grohe and

Uribe (2008) undertake a more detailed analysis of the structure of news. Using the restrictions

imposed by a structural model and macroeconomic data only, they estimate the volatilities news

shocks, with permanent and transitory effects and at various horizons. We calibrate our model

to their estimates and show that it can quantitatively explain the lead of asset prices over the

business cycle, thereby confirming the importance of news shocks. Alternatively, because our

model provides a good description of macroeconomic and financial variables dynamics, it can be

used as a structural framework to identify news shocks from asset pricing and macroeconomic

data jointly.

It is important to distinguish between news and noise. The former are actual changes in

fundamentals anticipated one or several periods ahead. The latter are temporary errors. Recent

work in macroeconomics (see for instance Blanchard, L’Huillier, and Lorenzoni (2009)) looks at

the respective roles of news and noise shocks for aggregate fluctuations. This paper deals with the

asset pricing implication of news only. Expectations can be revised - good news about productivity

two quarters ahead can potentially be offset by bad one quarter ahead news the following period

- but agents are correct on average.

The relative size of substitution and wealth effects is important to match both for macroeco-

nomic dynamics and asset prices. Empirically news lead to only a moderate change in consump-

tion on impact implying low wealth effects. Jaimovich and Rebelo (2009) generate a positive

co-movement of consumption, investment and hours following a news shock through a combina-

tion of habit formation in consumption, Greenwood, Hercowitz, and Huffman (1988) preferences,

adjustment costs and variable capital utilization. In our baseline model we mitigate wealth effects

3

by choosing a high value for representative agent’s elasticity of intertemporal substitution. In a

separate section we study the asset pricing implications of Jaimovich and Rebelo (2009).

Limited wealth effects are important to obtain significant variation in expected consumption

growth while keeping the unconditional volatility of realized consumption growth low, which is

the key to the main asset pricing results. This point is not specific to a model with news. For

instance, Kaltenbrunner and Lochstoer (2010) show that in a baseline real business cycle model

with i.i.d. shocks to productivity growth, consumption growth is highly persistent if elasticity of

intertemporal substitution is high enough. In addition to news this channel is also at work in

out model. Yet, contrary to news, its implies counterfactual lead and lag correlations between

macroeconomic and financial variables. A related work by Croce (2010) studies how a small

persistent component in productivity growth translates into more long-run consumption risk when

elasticity of intertemporal substitution is as high as 2.

Finally, as shown in Bansal and Yaron (2004), elasticity of intertemporal substitution higher

than the inverse of the coeffi cient of relative risk aversion is required to translate fluctuations in

expected consumption growth into high market prices of risk.

Extensive evidence on the co-movements between a range of variables along the business

cycle is provided in Backus, Routledge, and Zin (2008). The authors also explore how aggregate

quantities and the risk free-rate react to an anticipated productivity shock. Garleanu, Panageas,

and Yu (2011) explain the lead of stock-market returns over consumption/output by the slow

adoption of technological options in a version of an exchange economy. News shocks can be seen

as reduced form modeling, in the spirit of real business cycle literature, of qualitative mechanisms

suggested by the authors.

The importance of volatility shocks for macroeconomic dynamics has been highlighted in

Justiniano and Primiceri (2008), Fernandez-Villaverde, Guerron-Quintana, Rubio-Ramirez, and

Uribe (2009) and Bloom (2009). At the same time, stochastic volatility has been instrumental

to explain both the level of market risk premia and returns predictability in consumption-based

asset pricing literature. See Bansal and Yaron (2004), Bansal, Kiku, and Yaron (2009), Backus,

Routledge, and Zin (2008), Beeler and Campbell (2009) to mention just some contributions. In

this work we incorporate agent’s equilibrium responses to volatility shocks in the study of asset

prices. These cash-flow effects are typically not accounted for in endowment economy analysis,

contrary to the discount rates effect of stochastic volatility. A notable exception is Backus,

Routledge, and Zin (2008) who allow for dynamic interaction between consumption volatility and

growth in an exchange economy. We provide theoretical justification for this type of linkages and

describe their equilibrium properties.

We solve the models using higher order perturbation methods. Our specification of news

contains a large number of state variables that keep track of the whole term structure of repre-

sentatie agent’s expectations. Methods such as value function iteration are infeasible because of

the dimensionality problem. Our approach draws on Schmitt-Grohe and Uribe (2004) method-

4

ology for approximating a general class of dynamic stochastic general equilibrium models to the

second order. We use Dynare++ software for up to fourth order approximation to capture po-

tential variation in risk premia and the dynamic effects of stochastic volatility. Alternatively,

we use a simple risk-adjusted log-linear approximation of models with recursive preferences and

affi ne shocks described in Malkhozov and Shamloo (2011) to derive simple closed-form solutions

and illustrate a particular economic mechanism. Several recent papers, including Rudebusch and

Swanson (2011), Binsbergen van, Fernández-Villaverde, Koijen, and Rubio-Ramírez (2008) and

Caldara, Fernández-Villaverde, Rubio-Ramírez, and Yao (2008) address the specific issue of using

perturbation methods in the context of models with recursive preferences. Our paper illustrates

how the standard perturbation methods and software packages can be applied.

The rest of the paper is organized as follows. In Section 2 we present the baseline model with

news and discuss its extensions. In Section 3 we present the model with volatility shocks. Finally,

Section 4 concludes.

2 News

2.1 Baseline model

Our baseline setup is a standard real business cycle model augmented along two dimensions:

anticipated productivity shocks and Epstein-Zin-Weil (see Epstein and Zin (1989), Epstein and

Zin (1991) and Weil (1989)) preferences. In the sub-section 2.4 we will discuss extensions, such

as capital adjustment costs or Jaimovich and Rebelo (2009) model.

Preferences. The representative consumer maximizes a utility function defined recursively:

MaxCt

Ut

where

Ut =(C1−1/ψt + β(Et(U

1−γt+1 ))

1−1/ψ1−γ

) 11−1/ψ

This preferences specification allows us to calibrate the elasticity of intertemporal substitution ψ

independently from the coeffi cient of relative risk aversion γ (see Epstein and Zin (1989)). This

separation has an important implication for the agent’s preferences towards the early resolution

of uncertainty. If γ > 1/ψ (γ < 1/ψ) the investor prefers early (late) resolution of uncertainty.

Intuitively, with γ > 1/ψ the agent’s propensity to smooth consumption across states is greater

than his propensity to smooth consumption across time.

Technology. The consumption good is produced according to a constant returns to scale

neoclassical production function

Yt = ZtA1−αt Kα

t

5

where Kt is the stock of capital, hours worked are constant and normalized to 1 and Yt is the

output. Zt and At represent the stationary and non-stationary components of the total factor

productivity respectively. The law of motion of capital is given by

Kt+1 = (1− δ)Kt + It (1)

where It = Yt − Ct. As capital stock can be adjusted in a frictionless way, the marginal rate oftransformation between new capital and consumption, or Tobin’s q, is constant and equal to 1.

Shocks. The specification for the technology shocks is the key element of the model. Denote

the growth in non-stationary component and the stationary component of productivity as:

lnAt − lnAt−1 = x1t

lnZt = x2t

We follow Schmitt-Grohe and Uribe (2008) and specify x1t and x2t as autoregressive and subject

to anticipated (ε1, ε2, ε3) and unanticipated (ε0) innovations

x1t = (1− φA)µA + φAx1t−1 + ε0A,t + ε1A,t−1 + ε2A,t−2 + ε3A,t−3 (2)

x2t = (1− φZ)µZ + φZx2t−1 + ε0Z,t + ε1Z,t−1 + ε2Z,t−2 + ε3Z,t−3 (3)

ε0t are date t productivity surprises. Innovations ε1t−1, ε

2t−2, ε

3t−3 are anticipated one, two and three

periods ahead - they affect date-t productivity, but are period t − 1, t − 2, t − 3 information

respectively. In other words, at date t the agent learns about 4× 2 innovations - ε0t , ε1t , ε

2t and ε

3t

- that affect productivity immediately and in one, two and three periods ahead respectively. We

assume all innovations to be independent and normally distributed.

The shock structure can be easily written as a first order vector autoregressive system

xt+1 = H0 +H1xt +H2εt+1

with x1 and x2t being the first two elements of vector xt, εt ∼ N(0, I8×8) and matrices H0, H1 and

H2 given in Appendix B.

Asset prices. Equilibrium conditions and perturbation solution to the model is presented in

Appendices A and C. The equilibrium stochastic discount factor is

Mt+1 = β

Vt+1

Et(V 1−γt+1

) 11−γ

1/ψ−γ (Ct+1Ct

)−1/ψ

where Vt = maxCt

(Ut) . Provided the agent is not indifferent towards the timing of the resolution

of uncertainty (1/ψ 6= γ) expected consumption growth enters the stochastic discount factor

6

alongside realized consumption growth through the value function term.

In equilibrium the return on any asset i, Ri,t+1, satisfies

Et(Mt+1Ri,t+1

)= 1 (4)

One-period risk-free rate is

Rf,t = (EtMt+1)−1

As shown in Backus, Chernov, and Zin (2011) the conditional entropy of the stochastic discount

factor defined as

Lt(Mt+1) = logEtMt+1 − Et logMt+1

verifies

ELt(Mt+1) ≥ E(

logRit+1 − logRf

t

)Mean excess returns place a lower bound on the entropy of the stochastic discount factor. This

requirement is similar, although not equivalent, to Hansen and Jagannathan (1991) result. We

will use entropy as a measure of the ability of a model to generate high prices of risk.

We also consider three particular assets: a claim on aggregate consumption stream, a claim on

aggregate dividend stream and a real consol bond that pays a geometrically decreasing coupon.

Aggregate dividend is defined as output net of investment and wages

Dt = Yt −WagetNt − It = αYt − It

Total returns - the sum of period flow and capital gain - on consumption and dividend claims

(Rc,t and Rd,t) substituted in the asset pricing equation (4) provide us with a recursive definition

of the price-dividend ratios, that can be easily solved for using perturbation methods

Pc,tCt

= Et

(Mt+1

Ct+1Ct

(Pc,t+1Ct+1

+ 1

))Pd,tDt

= Et

(Mt+1

Dt+1

Dt

(Pd,t+1Dt+1

+ 1

))Finally, we look at the real term premium. This provides us with an additional dimension

along which we can evaluate the asset pricing performance of the model. As discussed in Backus,

Chernov, and Zin (2011) the spread between log yields on n periods (yn) and one period (y1) zero

coupon bonds is informative about the properties (in particular the persistence) of the stochastic

discount factor.

n−1ELt(Mt+1 × ...×Mt+n)− ELt(Mt+1) = −E (yn − y1)

We will use a related metric to simplify the perturbation solution. (see Rudebusch and Swanson

7

(2011) who use the same asset). The price of a consol bond with geometrically decreasing coupon

can be easily defined by a first-order recursion

Pb,t = 1 + δbEt (Mt+1Pb,t+1)

where δb is the rate of coupon decay. By varying δb we can change the duration of the bond1.

The term premium is defined as the difference between the yield on the bond and its yield under

risk neutrality

TPt = log

(δbPb,tPb,t − 1

)− log

(δbPb,t

Pb,t − 1

)where

Pb,t = 1 + δbR−1f,tEt

(Pb,t+1

)2.2 Calibration

Calibration of shocks is a key element of the model. We follow Schmitt-Grohe and Uribe (2008)

to calibrate the relative importance of surprises and news at various horizons for permanent

and stationary components of productivity. See Table 1 and Figure 1. Permanent changes in

productivity are largely anticipated several quarters ahead, with volatilities of one, two and three

quarter news each significantly larger than the volatility of unanticipated permanent shocks. The

effect of technological surprises is mostly transitory. Yet anticipated transitory changes are as

important.

[TABLE I HERE]

[FIGURE 1 HERE]

We calibrate technology-related parameters of the model to values that are standard in business

cycle literature (see respective results tables). Consistently with long-run risk literature (see

Bansal and Yaron (2004), Kaltenbrunner and Lochstoer (2010)) and the intuition that empirically

identified responses to news shocks imply moderate wealth effects our baseline value for the

elasticity of intertemporal substitution is set to 1.5. Baseline relative risk aversion coeffi cient is

10. In each calibration we scale the shocks to match the volatility of consumption growth in the

1947-2010 sample2 (1.47%). This is approximately half of the value targeted by Bansal and Yaron

(2004) and Kaltenbrunner and Lochstoer (2010). Trivially, return volatilities and risk premia

increase significantly if the model is calibrated to deliver annual consumption volatility of ∼ 3%

(sample starting in the pre-World War II period). The same holds for Sharpe ratios when higher

relative risk aversion is assumed (as it is the case, for instance, in Croce (2010)). Finally, we do not

1Explicitly, the duration of this bond is 1− δbe−y + δbe−y/(1− δbe−y), where y is the bond yield.2This allows us to be consistent with the sample used in Schmitt-Grohe and Uribe (2008) to identify news

shocks.

8

assume any operating leverage for the consumption and dividend claim which would mechanically

increase returns volatility and risk premia.

2.3 Results

News help explain, in a simple framework, salient financial and macroeconomic facts, and, most

importantly, the cyclical properties of asset prices. Extending our analysis to macroeconomic

dynamics and macro-finance co-movements allows us to gain a new perspective on asset pricing

mechanisms.

Asset prices. Our benchmark calibration that features permanent news shocks and elasticity of

intertemporal substitution equal to 1.5 generates high entropy (variance) of the stochastic discount

factor and closely matches the mean and volatility of the risk free rate and the stock-market

Sharpe ratios observed in the data. We take the claim on consumption stream (or equivalently

levered claim on consumption stream when we focus on Sharpe ratios) as the model counterpart

to aggregate stock-market. These results are obtained consistently with the low volatility of

realized consumption growth. News have little effect on asset prices if elasticity of intertemporal

substitution is low. With elasticity equal to 1.5 the entropy of the stochastic discount factor

increases dramatically and news account for 22% of the increase. See Table 2 for the comparison of

benchmark calibration with no-news and low elasticity of intertemporal substitution calibrations.

[TABLE II HERE]

Two mechanisms are at work, both are controlled by the elasticity of intertemporal substitution

parameter. If wealth effects are small the initial response of consumption to productivity shocks

is moderate. This accounts for low unconditional volatility of realized consumption growth. The

agent chooses to increase consumption progressively and, in response to news shocks, to delay

part of the increase until productivity effectively improves. See impulse responses of consumption

growth in Figure 2 The former effect (described in Kaltenbrunner and Lochstoer (2010)) creates

variation in expected consumption growth over the long run, the latter (driven by news) - variation

over one to three quarters horizon. With Epstein-Zin-Weil preferences this variation in expected

consumption growth increases the volatility of the stochastic discount factor.3

[FIGURE 2 HERE]

News naturally increase autocorrelations of quarterly consumption growth at 1 to 3 period

lag, creating a hump-shape in autocorrelation function which is consistent with the data. This

accounts for only part of the additional predictability in consumption growth produced by news.

The other part is driven by the state varibales in the agent’s information set. At the same time

3Specifically, if the elasticity of intertemporal substitution is different from the inverse of relative risk aversioncoeffi cient, which is the case if both parameters are calibrated to be greater than one.

9

news do not decrease the rate of autocorrelations decay and thus do not add to excessive long-run

persistence. See Figure 3.

[FIGURE 3 HERE]

A higher value for the elasticity of intertemporal substitution parameter (ψ = 2) further

mitigates wealth effects and increases risk premia for a given volatility of realized consumption

growth. Yet in our baseline setting it implies counterfactually high persistence in consumption

growth. See Table 3 and Figure 3. Relative risk aversion parameter has virtually no effect on

quantities dynamics.4 Higher relative risk aversion coeffi cient (γ = 20) naturally increases the

volatility of the stochastic discount factor and the Sharpe ratios. Yet it also implies a large

negative real term premium.5 See Table 3. While data on real rates are not easily available and

complete, Rudebusch and Swanson (2011) argue that real term premium is either slightly positive

(US) or slightly negative.(UK). Both possibilities are inconsistent with the magnitude of premium

obtained when γ = 20.6

[TABLE III HERE]

Macroeconomic quantities. Our benchmark calibration matches the volatilities and, impor-

tantly, correlations of consumption, output and investment growth without assuming any ad-

ditional frictions. See Table 2. Real business cycle models without news imply near perfect

co-movement of these three aggregates. Correlations, in particular between consumption and in-

vestment, are lower in the data. News allow to de-couple the movements in consumption, output

and investment. Future productivity prospects do not relax the current resource constraint of the

economy and agents have to choose between increasing consumption and increasing investment.

In our benchmark calibration consumption increases on good news and investment falls.7 Output,

consumption and investment rise when productivity gains become effective. See Figure 2.

The responses of main macroeconomic aggregates described above are consistent with the

identification in Barsky and Sims (2011). It is important that the elasticity of intertemporal

substitution is not low to avoid a counterfactual collapse in investment that would reduce future

output. On the other hand if elasticity of intertemporal substitution becomes as high as ∼ 2.5

consumption falls and investment goes up on good news.

4This is exactly true in the second order approximation.5In the setup we consider interest rates fall in recessions, long term real bonds are a hedge and always command

a negative premium as seen in all calibrations.6Notice that real term premium is directly related to the properties of the stochastic discount factor. There

exists a tension in the models with recursive preferences between increasing Sharpe ratios for stocks and keepingthe real term premium moderate.

7Because agent faces the choice between consumption today and investment in capital stock that will remainproductive for several periods in the future, the threshold for elasticity of intertemporal substitution at whichsubstitution effect dominates the income effect is higher than 1.

10

In contrast to Barsky and Sims (2011), Beaudry and Portier (2006) argue that news create a

joint boom in consumption, output and investment (as well as hours worked). This is (trivially)

not possible in our baseline model - agents don’t have any additional adjustment margin to

increase output on good news and either consumption or investment has to fall. As a matter

of fact, macroeconomic literature points out the diffi culty to generate such a "news boom" in a

real business cycle framework, even in presence of flexible labor and various frictions. See Barro

and King (1984) and Beaudry and Portier (2007). Jaimovich and Rebelo (2009) propose a model

that is able to reproduce this pattern. We will look at its asset pricing implication in the next

sub-section.

Macro-finance co-movements. Without news stock-market returns in the model are near-

perfectly correlated with contemporaneous changes in macroeconomic aggregates. This is coun-

terfactual - the lead of asset prices over macroeconomic quantities is a salient feature of the data.

See Figures 4 and 5. Backus, Routledge, and Zin (2007) and Backus, Routledge, and Zin (2008)

provide aditional evidence on this point. Our model with news quantitatively reproduces the lead

of the stock-market returns over the GDP growth. See Figure 4. Anticipated changes in pro-

ductivity are the most natural channel to account for this phenomenon - asset prices are forward

looking and incorporate expected changes in the production possibilities of the economy. The rel-

ative importance of wealth and susbstitution effects again plays a critical role - initial responses

of quantities have to be moderate and most of the adjustment to the new steady states has to

happen when productivity gains are effectively realized. In other terms if wealth effects are strong

agents immediately "consume away" future anticipated gains. See Figure 4, panel 2 in particular.

[FIGURE 4 HERE]

The lead of prices over quantities is observed across a range of financial8 and macroeconomic

variables and these patterns are matched by the benchmark model. See Figure 5.

[FIGURE 5 HERE]

Several comments are in order. First, the structure of anticipated shocks implied by the asset

prices is quantitatively consistent with the estimates carried out in macroeconomic literature.

This provides additional evidence for the importance of news. In addition, our results suggest

that shocks anticipated one extra quarter ahead should also be incorporated.

Second, even in the calibration with news, consumption growth has a high positive contem-

poraneous correlation (0.92) and investment - a negative contamporaneous correlation (−0.22)

with excess returns, suggesting that the initial responses of consumption and investment to news

8In our model risk premia (including the term premium) are constant. This is exactly true in the second-orderapproximation of the model, but is also verified with a third-order solution robustness check. As a result at thisstage we do not consider risk premia in the analysis of macro-finance co-movements. We will address the questionof time-varying risk premia in the section dedicated to volatility shocks.

11

in our benchmark calibration is too strong and wealth effects should be even lower. As pointed

out above, this increases the effect of news on the market prices of risk, but implies excessive

long-run persistence in consumption growth. Third, while news shock produce macro-finance

co-movement patterns consistent with the data, high elasticity of intertemporal substitution also

generates excessive long-run predictability of consumption growth by price-dividend ratios. In

other words, in the simple framework where elasticity of intertemporal substitution parameter

controls all the shock propagation channels there is a tension between increasing short-run and

decreasing long-run predictability.

2.4 Extensions

The model discussed above provides a useful framework to understand the effects of news on asset

prices and their co-movement with macroeconomic aggregates. In this section we augment this

baseline framework.

Capital adjustment costs and pricing of dividend claim. Following a standard approach in

finance literature the previous section analyzed the claim on consumption stream (or equivalently

levered claim on consumption stream when we focus on Sharpe ratios) as the model counterpart

to aggregate stock-market. None of the calibrations discussed above can account for risk premia

and cyclical properties of returns if the capital claim is considered instead. This is the case despite

the fact that the stochastic discount factor is volatile enough as seen from its entropy, the Sharpe

ratio of consumption claim and the term premium. The reason is that the properties of capital

claim payouts (Dt = αYt − It) do not match the properties of public equity market dividends

in the data. Without news, a positive technological surprise increases investments more than

the output times capital share. As a result, consumption and dividend growth are negatively

correlated. See also Kaltenbrunner and Lochstoer (2010). In the presence of news, investment

falls initially on a positive shock to expectations but then rebounds strongly, leading to positive

correlation of dividend and consumption growth but negative autocorrelation in dividend growth.

The latter matters in the context of long-run risk pricing mechanism. See Table 4. In the data, as

reported for instance in Bansal and Yaron (2004), dividend growth is both positively correlated

with consumption growth and positively auto-correlated.

[TABLE IV HERE]

To improve on the pricing of the dividend claim by introducing variation in Tobin’s marginal q

we examine two types of capital adjustment costs used respectively in Jermann (1998) and Croce

(2010). Equation (1) of the benchmark model becomes respectively

Kt+1 = (1− δ + ϕ (It/Kt))Kt

12

where ϕ (It/Kt) = κ1 (It/Kt)1−1/τ + κ2 and at the balanced growth path ϕ (I/K) = I/K and

ϕ′ (I/K) = 1 and

Kt+1 = (1− δ − υ (It/Yt))Kt + It

υ (I/Y ) = 0, υ′ (I/Y ) = 0, υ′′ (I/Y ) = ζ.9

Imposing capital adjustment of the form and magnitude assumed in Jermann (1998) increases

the volatility of excess returns on dividend claim to 1.77% and its risk premium to approximately

0.3%. These numbers can be mechanically increased if we calibrated our model to higher volatility

of consumption growth, higher risk aversion or assumed operating leverage. See Table 5.

[TABLE V HERE]

As adjusting capital is costly investment volatility is too low relative to consumption, which

itself becomes more volatile. The latter effect decreases the entropy of the stochastic discount fac-

tor for a fixed level of consumption volatility. Unlike in the benchmark model where consumption

increases and investment drops on good news, in a model with adjustment costs consumption falls

on good news to allow for a progressive increase in investment.10 Because this growth is gradual,

dividends, which are the difference between capital share α times output and investment, are more

aligned with consumption and business cycle in general. As a result, consistently with the data,

returns on dividend claim lead the output and consumption growth. However, they have a high

contemporaneous correlation with investment which now increases immediately after a positive

change in expectations occurs. See Figure 6.

[FIGURE 6 HERE]

In a calibration with adjustment costs of form and magnitude suggested in Croce (2010) the

volatility of excess returns on dividend claim and its risk premium are only 0.25% and 0.05%

respectively. As discussed by the author this calibration imposes smaller overall adjusmtment

loss than in Jermann (1998). Yet they are suffi cient to match the key co-movements of dividend

claim returns and all business cycle quantities very closely. See Figure 6. Consumption rises

and investment falls on good news as in the benchmark case, but those initial responses are very

small - consumption smoothing and investment smoothing motives compensate each other. As a

result all quantities effectively start adjusting to their steady state levels only when productivity

changes. Investment volatility is not too low relative to consumption. Finally, the model does

not generate excessive long-run predictability in consumption growth (corr(∆c,∆c−1

)= 14.2%

declining to corr(∆c,∆c−4

)= 5.10%) because the mechanism at work is not lower wealth effect

embedded in preferences but costly capital adjustment

9Marginal rate of transformation between new capital and consumption (q) with the two forms of adjustment

costs can be written respectively as qt = 1/ϕ′(ItKt

)= (I/K)

−1/τ(It/Kt)

1/τ and qt = 11−υ′(It/Yt)(Kt/Yt)

.10Consumption is therefore negatively auto-correlated.

13

Prices of risk in Jaimovich and Rebelo (2009). Jaimovich and Rebelo (2009) show that an

appropriately calibrated combination of investment adjustment costs, variable capital utilisation,

and Greenwood, Hercowitz, and Huffman (1988) preferences creates an increase in consumption,

investment and hours worked in response to good news about productivity - thereby solving a

long standing problem in macroeconomic literature. The model is primarily designed to produce

the qualitative result of a joint boom and does not necessarily provide a quantitative description

of all the aspects of the economy. For instance, we can show that returns on dividend claim

negatively lead the business cycle. See the discussion of representative firm’s value in Jaimovich

and Rebelo (2009)

We will focus on two dimensions of the model - market prices of risk and cyclical properties of

excess returns - to illustrate the full potential of news shocks. We consider the same model and

calibrate it in the same way as the authors but define preferences recursively to be able to choose

the risk aversion parameter independently. We set it to 10 in order to create a wedge between

the relative risk aversion and the elasticity of intertemporal substitution (equal to 1) and allow

expected consumption growth risk to be priced11. We refer the reader to the original Jaimovich

and Rebelo (2009) paper and Appendix D for the detailed description of the model.

Expected consumption growth varies a lot at short horizon , but there is no long-run pre-

dictability - while the initial response of consumption to news is small, it goes to its new steady

state almost immediately when productivity changes. See Figure 7. As a result when the economy

is buffeted by 3 quarter anticipated shocks, the entropy of the stochastic discount factor (6.52%)

is more than six times higher than in the economy subject to technological surprises only (1.08%)

keeping the realized consumption volatility constant (1.47%). When calibrated to have 1,2,3-

quater productivity news as well as surprises the model produces extremely accurate description

of the cyclical properties of excess returns. See Figure 6.

3 Volatility shocks

In this section we look at the asset pricing implications of changes in macroeconomic volatility.

Exchnage economy literature typically makes the assumption that volatility acts on discount rates

only. Yet in a production setting it also has real effects, it is in fact an important driver of business

cycle fluctuations, and can influence expected cash flows.

3.1 Model and calibration

We consider a real business cycle model where the variance of productivity shocks is itself stochas-

tic. The baseline model of Section 2 (see sub-section 2.1 of Section 2) remains the same except

for productivity dynamics that become:

11As noted before risk aversion has no impact on the quantities dynamics.

14

lnAt+1 − lnAt = µA + σt

(√1− ρ2εA,t+1 + ρεσ,t+1

)

σ2t+1 = (1− λ) θ + λσ2t + ωεσ,t+1. (5)

where Et (εA,t+1εσ,t+1) = 0. In addition we focus on permanent shocks only as those matter for

asset prices and set

lnZt = 1

As reported in Fernandez-Villaverde, Guerron-Quintana, Rubio-Ramirez, and Uribe (2009)

and Malkhozov and Shamloo (2011), we need a third order expansion to capture the dynamic

effects of stochastic volatility on the other variables of the model with standard perturbation

methods. We use a fourth order-expansion for robustness. We also solve the model with a log-

linearization procedure outlined in Malkhozov and Shamloo (2011). The latter approach delivers

a tractable closed -form approximation and allows us to undertake our calibration exercise.

We calibrate the parameters of the productivity variance process (5) in such a way that in

equilibrium the long termmean, mean reversion and volatility of conditional consumption variance

are the same as assumed in Bansal, Kiku, and Yaron (2009). We choose this benchmark following

Beeler and Campbell (2009) who show that in Bansal, Kiku, and Yaron (2009) stochastic volatility

plays a crucial role for asset pricing results.12 See Malkhozov and Shamloo (2011) and Appendix

E for the details of the procedure and Table 6 for the resulting parameter values. In order to be

consistent with Bansal, Kiku, and Yaron (2009) we calibrate to monthly frequency. The other

parameters of the model are standard in business cycle literature and presented in Table 7 together

with the results.

[TABLE VI HERE]

Finally, we examine two cases of the interaction between volatility and productivity, namely

ρ = 0 and ρ = −0.1. The first calibration isolates the effects of volatility shocks. The second is

motivated by Bloom (2009). The author suggests that macroeconomic uncertainty shocks have an

effect on aggregate productivity as higher uncertainty hinders the effi cient reallocation of capital

between firms. This happens because firms subject to non-convex adjustment costs postpone

investment and hiring decisions until resulting ineffi ciencies outweigh the effect of adjustment

penalties. We incorporate this feature in reduced form by choosing ρ < 0.13

12This is in contrast with Bansal and Yaron (2004) where volatility is important for returns predictability butnot the size of risk premia. See Beeler and Campbell (2009).13In Bloom (2009) the effect of volatility shocks on productivity is transitory. In our framework transitory

changes in productivity increase the volatility of macroeconomic variables but have close to no effect on riskpremia. We therefore assume that the effect of volatility shocks is permanent in order to investigate the possibleimportance of this channel.

15

3.2 Results

By construction volatility shocks lead to the variation in conditional variance of consumption

growth and variation in risk premia identical to Bansal, Kiku, and Yaron (2009). Therefore we

focus on the interaction of volatility with other variables of the model.

Volatility shocks increase the unconditional volatility of macroeconomic quantities - annualized

volatility of consumption growth goes up from 2.57% to 3.59% - but have little effect on the risk

premium of the consumption claim - it decreases slightly from 0.95% to 0.92%. See Table 7. The

result is due to endogenous consumption choice. In a production economy higher volatility can

be offset by higher precautionary savings. As seen on Figure 8 a positive variance shock leads

to an immediate drop in consumption and an increase in investment. This in turn means higher

future output and higher future consumption. At the same time, responses of consumption and

investment generate more variability in macroeconomic aggregates.

[TABLE VII HERE]

[FIGURE 8 HERE]

Another way to understand the low risk premia is to consider the consumption dynamics

implied by the model and compare them to the assumptions made in long-run risk literature.

First, the correlation between innovations to consumption volatility and innovations to realized

consumption growth is negative. In our calibration it is equal to −0.63 when volatility is at the

steady state. Second and more important, stochastic volatility and long run risk in consumption

growth are related. With Epstein-Zin-Weil preferences these two factors are the main drivers of the

level of risk premia. And because higher volatility implies higher expected consumption growth,

they tend to offset each other. Appendix E derives these results analytically in a log-linearized

setting.

In contrast consumption based asset pricing literature typically assumes away the two addi-

tional equilibrium channels by setting the correlation between volatility and consumption growth

shocks to 0 and allowing for no link between stochastic volatility and long run risk, which explains

the difference in results14

Consider the case when higher volatility causes a permanent drop in productivity. As before,

on volatility shock the agent would like to increase her buffer stock savings. Yet this is precisely

the time when productivity and output drop and the agent has less resources to offset the effects

of higher uncertainty. The risk premium of the consumption claim is 1.27%. However, the model

generates high volatility in consumption growth and returns. As a result the Sharpe ratio is low.

14Backus, Routledge, and Zin (2008) consider an endowment economy where volatility drives expected consump-tion growth in a way similar to our model but has no immediate effect on realized consumption growth. As aresult, in their model stock-market goes up when volatility increases.

16

4 Conclusion

In this paper we show that the paradigms of news driven business cycle fluctuations and volatility

shock driven business cycle fluctuations recently advanced in macroeconomic literature help our

understanding of the asset pricing mechanisms.

We find that news help to explain the magnitude of market prices of risk by creating variation in

expected, as opposed to realized, consumption growth and, crucially, to match cyclical properties

of asset returns and valuations. This is true both in a simple real business cycle framework and

models designed to replicate additional features of the data.

Our baseline model provides a good description of salient macro-finance facts. Its implication

for the co-movement of aggregate quantities and asset returns can be used to identify news shocks

using both macroeconomic and financial data The calibration in this paper is based on estimation

by Schmitt-Grohe and Uribe (2008) that used macroeconomic data only. Yet it provides a close

quantitative description of the lead of asset returns over the business cycle. This consistency is

evidence of the importance of news shocks.

Both the model with news and the model with stochastic volatility lead us to re-examine

the assumptions about consumption growth made in the analysis of endowment economies. We

suggest predictability in consumption growth conditional on the additional variables of agent’s

information set, in conjunction with Epstein-Zin-Weil preferences, as a channel to increase the

market prices of risk while avoiding excessive long-range predictability, that is typically the result

of models with highly persistent consumption growth. Furhtermore, the real effects of macroeco-

nomic volatility shocks have to be taken into account. Agents build up precautionary savings

in order to counter their effect. As a result, in equilibrium, innovations to stochastic volatility

should exhibit negative correlation with innovations to realized consumption growth and positive

correlations with innovations to expected consumption growth or long-run risk.

17

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19

Appendix

A Perturbation solution

An approximate analytical solution to the model can be easily found using perturbation methods.

Following the notations laid out in Schmitt-Grohe and Uribe (2004) we write the system of

equilibrium conditions that recursively define the model as

Etf (yt+1, yt, xt+1, xt) = 0

where yt is the vector of control variables and xt is a vector containing state variables (notice that

the notation in this sub-section are independent of the rest of the paper). In general we don’t

need to specify the exact distribution of the vector of shocks. For each order of approximation

we need to specify the corresponding order of the moments of shocks distribution.

The solution is given by the equilibrium policy function for yt and the laws of motion for xt:

yt = g(xt, σ)

xt+1 = h(xt, σ) + σηεt+1

where σ is a parameter scaling the size of uncertainty.

At the non-stochastic steady state σ = 0 (we set it equal to 1 otherwise). We define the

non-stochastic steady state as vectors (x; y) such that

f (y, y, x, x) = 0

We expand the functions g and h around the σ = 0 and xt = x point. Our benchmark is a

second order approximation that can be written as

[g(xt, σ)]i = [g(x, 0)]i + [gx(x, 0)]i xt+1

2xTt [gxx(x, 0)]i xt+

1

2[gσσ(x, 0)]i [σ] [σ]

[h(xt, σ)]i = [h(x, 0)]i + [hx(x, 0)]i xt+1

2xT [hxx(x, 0)]i xt+

1

2[hσσ(x, 0)]i [σ] [σ]

where xt = xt − x.Schmitt-Grohe and Uribe (2004) show that the remaining terms of the Taylor expansion are

zero i.e. gσ = hσ = gσx = hσx = 0.

gσ = hσ = 0 implies that the first order approximation is not affected by the volatility of

the shock and produces no adjustment for risk. gσx = hσx = 0 implies that a second-order

20

approximation can only produce constant risk premia. We check for time variation in risk premia

with a third-order approximation. We use Dynare++ software to this effect.

Furthermore, the risk aversion parameter γ does not affect the non-stochastic steady state

values. Neither does it enter the expressions for hx or hxx evaluated at the non-stochastic steady-

state. In other terms, to the second order, risk aversion matters for the difference between the

stochastic steady state and the non-stochastic steady state but not the dynamics of the variables.

Finally, we can apply the solution method to any transformation of the variables. As it

is standard in macroeconomics, we find the second order approximation in the logarithms of

variables.

For the model with stochastic volatility we use the fourth order perturbation around the

non-stochastic steady state.

B VAR(1) representation of information structure

(2) and (3) can be written as a first order vector autoregressive system. For parsimony reason we

omit the transitory shocks part and present only 2. Define

xt =

x1t

ε1A,tε2A,tε3A,tε2A,t−1ε3A,t−1ε3A,t−2

and

H0 =

(1− φA)µA

0

0

0

0

0

0

H1 =

ρA 1 0 0 1 0 1

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 0 1 0

H2 =

σ0A 0 0 0

0 σ0µ 0 0

0 0 σ1µ 0

0 0 0 σ2µ

0 0 0 0

0 0 0 0

0 0 0 0

Then

xt+1 = H0 +H1xt +H2εt+1

is an equivalent representation of (2).

21

C Solving the baseline model

Problem and equilibrium conditions

The social planner’s problem is

V (Kt, xt) = maxCt

(Ut)

where Vt is the problem’s continuation value or simply the value function. The maximization is

subject to the resource constraint, the law of motion of capital and exogenous variables dynamics

Ct + It = ZtA1−αt Kα

t

Kt+1 = (1− δ)Kt + It

xt+1 = H0 +H1xt +H2εt+1

The Bellman equation

Vt = maxCt

(C1−1/ψt + β(Et(V

1−γt+1 ))

1−1/ψ1−γ

) 11−1/ψ

The first order conditions with respect to consumption

C−1/ψt = −βEt

(V 1−γt+1

) γ−1/ψ1−γ Et

(V −γt+1VKt+1

∂Kt+1

∂Ct

)The envelope condition with respect to capital

VKt = βV1ψ

t Et(V 1−γt+1

) γ−1/ψ1−γ Et

(V −γt+1VKt+1

∂Kt+1

∂Kt

)Combining the two

Et

[Mt+1

(∂Kt+2

∂Ct+1

)−1∂Kt+1

∂Ct

∂Kt+2

∂Kt+1

]= 1

where Mt+1 is the stochastic discount factor

Mt+1 = β

Vt+1

Et(V 1−γt+1

) 11−γ

1/ψ−γ (Ct+1Ct

)−1/ψ

Without capital adjustment costs ∂Kt+1∂Ct

= −1 and ∂Kt+1∂Kt

= 1− δ + αZtA1−αt Kα−1

t .

22

Stationarized model and steady state

Since labor augmenting productivity process At has a unit root our economy is growing. For all

the variables Xt inheriting the unit root define Xt as Xt =Xt

At−1and Vt = V (Kt, At, x

(−1)t ). Since

the value function is homogeneous of degree one in Kt and At (simply observe that all equations

in the model are)

V (Kt, At, x(−1)t ) = At−1V (

Kt

At−1,AtAt−1

, x(−1)t )

VtAt−1

= maxCt

((CtAt−1

)1− 1ψ

+

(1

At−1

)1− 1ψ

β(EtV1−γt+1 )

1− 1ψ

1−γ

) 1

1− 1ψ

Vt = maxCt

((Ct

)1− 1ψ

+ A1− 1

ψ

t β(EtV1−γt+1 )

1− 1ψ

1−γ

) 1

1− 1ψ

(6)

As compared to models with time-additive preferences the value function Vt (and the expression

certainty equivalent(EtV

1−γt+1

) 11−γ) appear in the Euler equation and therefore has to be approxi-

mated. We add (6) to the set of conditions that define equilibrium. The other conditions in their

stationary form are

Et

βA−1/ψt

Vt+1

Et

(V 1−γt+1

) 11−γ

1/ψ−γ (

Ct+1

Ct

)−1/ψ︸ ︷︷ ︸

Mt+1

(1− δ + αZt+1A

1−αt+1 K

α−1t+1

) = 1

Ct + It = ZtA1−αt Kα

t

Kt+1At = (1− δ)Kt + It

xt+1 = H0 +H1xt +H2εt+1

where x1t = ln A.

23

At the non-stochastic steady state

x = (I −H1)−1H0

A = exp(x1)

Z = exp(x2)

K = A

(β−1A1/ψ − (1− δ)

αZ

) 1

α− 1

I = (A− (1− δ))K

V =

(C1−

1ψ

1− βA1−1ψ

) 1

1− 1ψ

Models with capital adjustment costs

Compared to the previous section, a few modification are required to accommodate adjustment

costs. First the (stationarized) laws of motion of capital become respectively

Kt+1At = (1− δ + ϕ(It/Kt

))Kt

and

Kt+1At = (1− δ − υ(It/Yt

))Kt + It

Next∂Kt+1

∂Ct= − 1

qt

and∂Kt+1

∂Kt

= 1− δ + qtαZtA1−αt Kα−1

t

where qt = 1/ϕ′(ItKt

)and qt = 1

1−υ′(It/Yt)(Kt/Yt)for the two respective cases. The Euler equation

becomes

Et

βA−1/ψt

Vt+1

Et

(V 1−γt+1

) 11−γ

1/ψ−γ (

Ct+1

Ct

)−1/ψ(1− δ) qt+1 + αZt+1A

1−αt+1 K

α−1t+1

qt

= 1

We calibrated the adjustment costs not to have any effects at the non-stochastic steady state and

it remains the same as before.

24

Asset prices

Asset pricing equations can be easily added to the equilibrium conditions

Et (Mt+1Rf,t) = 1

Pc,t

Ct= Et

(AtMt+1

Ct+1

Ct

(Pc,t+1Ct+1

+ 1

))

Pd,t

Dt

= Et

(AtMt+1

Dt+1

Dt

(Pd,t+1

Dt+1

+ 1

))

Pb,t = 1 + δbEt

(Mt+1Pb,t+1

)At the steady state

Rf = β−1A1/ψ

Pc

C=Pd

D=

βA1−1/ψ

1− βA1−1/ψ

Pb =1

1− δbβA−1/ψ

D Solving Jaimovich and Rebelo model

The elements of Jaimovich and Rebelo (2009) model:

Ut =(Ct + η0N

η1t H

νt C

1−νt

)1− 1ψ + βEt

(U

1−γ1− 1

ψ

t+1

) 1− 1ψ

1−γ

Kt+1 = Kt (1− δ (ut)) + It

(1− υ

(ItIt−1

))

Ht+1 = Hνt C

1−νt

Ct + It = (utKt)α (NtAt)

1−α

Equilibrium conditions

Et[Kernt

(1 + (1− ν) η0N

η1t H

νt C−νt

)−Disctt+1

(V3,t+1 + V2,t+1 (1− ν)Hν

t C−νt

)]= 0

Et

[Kerntη0η1N

η1−1t Hν

t C1−νt +Disctt+1V3,t+1 (1− α) (utKt)

αN−αt A1−αt

]= 0

25

Et

[Disctt+1

(−δ′ (ut)V1,t+1 + V3,t+1αu

α−1t Kα−1

t (NtAt)1−α)]

= 0

Et[V1,t −Disctt+1

(V1,t+1 (1− δ (ut)) + V3,t+1αu

αtK

α−1t (NtAt)

1−α)] = 0

Et

[V2,t −Kerntη0νN

η1t H

ν−1t C1−νt −Disctt+1V2,t+1νHν−1

t C1−νt

]= 0

Et

V3,t − V1,t(

1− υ(It−1It−2

)− It−1

It−2υ′(It−1It−2

))−Disctt+1V1,t+1

(ItIt−1

)2υ′′(

ItIt−1

) = 0

where

Kernt = Ct + η0Nη1t H

νt C

1−νt

Disctt+1 = βA− 1ψ

t Et

(V

1−γ1− 1

ψ

t+1

) γ− 1ψ

1−γ

V

1ψ−γ

1− 1ψ

t+1

The stochastic discount factor can be found in the following way

Mt+1 =∂Vt/∂Ct+1∂Vt/∂Ct

= β

V

1

1− 1ψ

t+1

Et

(V

1−γ1− 1

ψ

t+1

) 11−γ

1ψ−γ

Kernt+1 + V2,t+1Ht+1C−1t+1

(1−ν)ν

Kernt + V2,tHtC−1t

(1−ν)ν

And the Tobin’s marginal q is given by

qt =1

1− υ(

ItIt−1

)− It

It−1υ′(

ItIt−1

)We use Jaimovich and Rebelo (2009) calibration β = 0.9985, η1 = 1.4, ν = 0.999, α = 0.36,

υ(

II−1

)= υ′

(II−1

)= 0, υ′′

(II−1

)= 1.3, δ (u) = 0.025, δ

′′(u)uδ′(u) = 0.15 and set ψ arbitrary close to

1 and γ = 10.

The model can be solved, as previously, by approximating it around the non-stochastic steady

state.

26

E Inspecting the stochastic volatility model

Malkhozov and Shamloo (2011) show how a log-linear solution to the model described in section

(assuming ρ = 0) can be obtained. For any Xt, denote xt = log XtAt−1

. Consumption growth can

be written (see Malkhozov and Shamloo (2011))

∆ct+1 = const+ ∆caat + ∆ckkt + (λ− 1 + ckkc) cσ︸ ︷︷ ︸∆cσ

σ2t + caσtεat+1 + cσωε

σt+1

Now the conditional variance of consumption growth is

V art (∆ct+1) = c2aσ2t + c2σω

2

After substituting (5)

V art (∆ct+1) = (1− ϕ)(c2aθ + c2σω

2)︸ ︷︷ ︸

θc

+ λ︸︷︷︸λc

V art−1 (∆ct) + c2aω︸︷︷︸ωc

εσt .

Conditional variance of consumption growth follows the same process as assumed in Bansal and

Yaron (2004) and Bansal, Kiku, and Yaron (2009). All parameters can be solved for in closed form

(see Malkhozov and Shamloo (2011)). It is easy to find productivity growth variance dynamics

that map into given θc, λc, ωc.

Furthermore we see that consumption variance drives expected consumption growth

∆ct+1 = const′ + ∆caat + ∆ckkt +(λ− 1 + ckkc) cσ

c2a︸ ︷︷ ︸∆cσc

V art (∆ct+1) + caσtεat+1 + cσωε

σt+1

If consumption falls when volatility increases cσ < 0

Covt (∆ct+1, V art+1 (∆ct+2)) = cσc2aω

2 < 0

and (ckkc is negligible)

∆cσc =(λ− 1 + ckkc) cσ

c2a> 0

27

Table (1): Schmitt-Grohe and Uribe (2008) calibration of permanent and transitory components ofproductivity

Permanent

Persistence φA 0.14Un-anticipated σ0A (%) 0.591-quarter anticipated σ1A (%) 2.302-quarter anticipated σ2A (%) 1.303-quarter anticipated σ3A (%) 1.10

Transitory

Persistence φZ 0.89Un-anticipated σ0Z (%) 2.701-quarter anticipated σ1Z (%) 0.562-quarter anticipated σ2Z (%) 0.563-quarter anticipated σ3Z (%) 3.00

28

Table (2): Macro and asset pricing results with news - main

Data Calibrations

Benchmark No newsNo news,low EIS

News,low EIS

Technology α = 0.34, δ = 0.025, µA= 0.4%, φA= 0Preferences γ = 10, β = 0.9985

Shocks/News a, news a, no news a, no news a, newsEIS (ψ) 1.5 1.5 0.1 0.1

E (∆c) % 1.60 1.60 1.60 1.60 1.60σ (∆c) % 1.47 1.47 1.47 1.47 1.47

σ (∆c) /σ (∆y) 0.61 0.52 0.57 1.05 1.04σ (∆i) /σ (∆y) 3.09 2.32 1.97 0.73 5.40corr (∆c,∆y) 0.83 0.73 0.93 1.00 0.66corr (∆c,∆i) 0.54 0.49 0.84 1.00 −0.29corr (∆y,∆i) 0.92 0.95 0.98 1.00 0.53

E (rf ) % 0.90 1.12 1.18 15.5 15.5σ (rf ) % 1.10 0.57 0.49 1.10 1.30L(M)% 28.6 23.4 4.31 4.29

E (y − y) % −0.65 −0.51 −0.21 −0.26E (rc − rf ) % 0.82 0.72 −0.01 −0.10σ (rc − rf ) % 2.09 2.00 0.08 0.50

E (rc − rf ) /σ (rc − rf ) 0.33 0.39 0.36 −0.15 −13.6E (rd − rf ) % 0.00 0.03 0.01 0.00σ (rd − rf ) % 0.02 0.07 0.08 0.02

E (rd − rf ) /σ (rd − rf ) 0.33 0.08 0.36 0.15 0.03

This table reports parameters and key annualised macroeconomic and asset pricing moments of thebenchmark calibration (permanent news and high elasticity of intertemporal substitution) compared tono-news and low elasticity of intertemporal substitution calibations. Data are 1947Q1-2010Q4.

29

Table (3): Macro and asset pricing results with news - additional

Data Calibrations

High RRA High EIS PersistenceAll shocks,persistence

Technology α = 0.34, δ = 0.025, µA= 0.4%

Shocks/News a, news a, news a, news a, z news

Persistence φA= 0 φA= 0 φA= 0.14φA= 0.14φZ= 0.89

EIS (ψ) 1.5 2 1.5 1.5RRA (γ) 20 10 10 10Discount (β) 0.9985 0.9975 0.9984 0.9984

E (∆c) % 1.60 1.60 1.60 1.60 1.60σ (∆c) % 1.46 1.47 1.47 1.47 1.47

σ (∆c) /σ (∆y) 0.61 0.52 0.46 0.54 0.33σ (∆i) /σ (∆y) 3.09 2.32 2.45 2.30 2.63corr (∆c,∆y) 0.83 0.73 0.69 0.73 0.57corr (∆c,∆i) 0.54 0.49 0.47 0.48 0.40corr (∆y,∆i) 0.92 0.95 0.96 0.95 0.98

E (rf ) % 0.90 0.55 1.14 1.33 1.52σ (rf ) % 1.10 0.57 0.59 0.58 0.64L(M)% 125 37.9 29.1 18.9

E (y − y) % −1.30 −0.70 −0.70 −0.40E (rc − rf ) % 1.69 1.18 0.83 0.54σ (rc − rf ) % 2.09 2.66 2.10 1.72

E (rc − rf ) /σ (rc − rf ) 0.33 0.81 0.44 0.40 0.32E (rd − rf ) % 0.00 0.00 0.00 0.00σ (rd − rf ) % 0.02 0.02 0.01 0.03

E (rd − rf ) /σ (rd − rf ) 0.33 0.16 0.09 0.08 0.03

This table reports parameters and key annualised macroeconomic and asset pricing moments of additionalcalibrations of the model with news, namely the high elasticity of intertemporal substitution and highrelative risk aversion cases and calibrations with persistence and transitory shocks in productivity growth.Data are 1947Q1-2010Q4.

30

Table (4): Properties of aggregate dividends

Calibrations

Benchmark No news Jermann costs I/Y costs

corr (∆d,∆c) 0.25 −0.90 0.96 −0.89corr (∆d,∆y) −0.91 −0.98 0.91 −0.95corr (∆d,∆i) −0.99 −1.00 0.16 −0.98

corr(∆d,∆d−1

)−0.24 −0.02 −0.24 −0.18

corr(∆d,∆d−2

)−0.08 −0.02 −0.08 −0.06

corr(∆d,∆d−3

)−0.05 −0.02 −0.05 −0.04

corr(∆d,∆d−4

)−0.00 −0.02 −0.00 −0.00

corr(∆d,∆d−5

)−0.00 −0.02 −0.00 −0.00

corr(∆d,∆d−6

)−0.00 −0.02 −0.00 −0.00

corr(∆d,∆d−7

)−0.00 −0.02 −0.00 −0.00

This table reports quarterly correlations of dividend growth with aggregate macroeconomic quantitiesand its own lags for a range of calibrations.

31

Table (5): Macro and asset pricing results with adjustment costs

Data Calibrations

Jermann costs I/Y costs

Technology α = 0.34, δ = 0.025, µA= 0.4%, φA= 0Preferences β = 0.9985, ψ = 1.5, γ = 10Shocks/News a, news

Adjustment costs τ = 0.23 ζ = 0.15

E (∆c) % 1.60 1.60 1.60σ (∆c) % 1.46 1.47 1.47

σ (∆c) /σ (∆y) 0.61 1.28 0.75σ (∆i) /σ (∆y) 3.09 0.54 1.53corr (∆c,∆y) 0.83 0.99 0.99corr (∆c,∆i) 0.54 0.79 0.96corr (∆y,∆i) 0.92 0.86 0.99

E (rf ) % 0.90 1.64 1.49σ (rf ) % 1.10 1.04 0.90L(M)% 4.74 13.9

E (rc − rf ) % 0.08 0.31σ (rc − rf ) % 0.52 1.14

E (rc − rf ) /σ (rc − rf ) 0.33 0.15 0.27E (rd − rf ) % 0.28 0.05σ (rd − rf ) % 1.77 0.25

E (rd − rf ) /σ (rd − rf ) 0.33 0.16 0.21

This table reports parameters and key annualised macroeconomic and asset pricing moments of modelswith adjustment costs. Data are 1947Q1-2010Q4.

32

Table (6): Calibratrion of conditional variance process

Conditional variance of consumption growth

Long term mean θc 5.18× 10−5

Mean reversion λc 0.999Volatility of variance ωc 2.80× 10−6

Corresponding conditional variance of productivity growth

Long term mean θ 3.81× 10−4

Mean reversion λ 0.999Volatility of variance ω 3.39× 10−5

This table reports Bansal, Kiku and Yaron (200) calibration of conditional variance of consumptiongrowth and parameters of conditional variance of productivity growth process that endogenously giverise to those dynamics in our stochastic volatility model.

33

Table (7): Macro and asset pricing moments in models with volatility shocks

Calibrations

Constantvolatility

Stochasticvolatility

Bloom (2009)effect

Technology α = 0.34, δ = 0.025, µA= 0.15%Preferences β = 0.9989, ψ = 1.5, γ = 5Variance θ = 3.81× 10−4, λ = 0.999

Volatility of variance (ω) 0 3.39× 10−5 3.39× 10−5

Correlation (ρ) 0 0 −0.1

E (∆c) % 1.80 1.80 1.80σ (∆c) % 2.57 3.59 3.57

σ (∆c) /σ (∆y) 0.55 0.60 0.61σ (∆i) /σ (∆y) 2.00 2.09 2.13corr (∆c,∆y) 0.93 0.85 0.86corr (∆c,∆i) 0.84 0.60 0.60corr (∆y,∆i) 0.98 0.92 0.92

E (rc − rf ) % 0.95 0.92 1.27σ (rc − rf ) % 3.44 7.20 7.36

E (rc − rf ) /σ (rc − rf ) 0.27 0.13 0.17

1 year R2 − 0.06 0.062 years R2 − 0.11 0.115 years R2 − 0.24 0.25

This table reports parameters and selected annualised macroeconomic and asset pricing moments for themodels with stochastic volatility. The last three rows report R2 of predictive regressions of stock-marketexcess returns at various horizons on price-dividend ratios. The claim on consumption stream is takenas model counterpart of the aggregate stock-market.

34

Figure (1): Productivity news and surprises in Schmitt-Grohe and Uribe (2008)

This figure illustrates Schmitt-Grohe and Uribe (2008) estimates of the volatilities of un-anticipated and1,2,3-quarter anticipated shocks to permanent and transitory components of productivity

35

Figure (2): Impulse responses to productivity news and surprises

This figure reports the responses (in percentage deviations from the steady state) of macroeconomicand financial variables (in logs) to one standard deviation un-anticipated and 3-quarter anticipatedproductivity shocks in the benchmark calibration of the model and the calibration without news.

36

Figure (3): Autocorrelation of consumption growth in the baseline model

This figure reports quarterly autocorrelations of consumption growth in the benchmark, no-news, highand low elasticity of intertemporal susbstitution calibrations. Data are 1947Q1-2010Q4.

37

Figure (4): Stock-market lead over output growth in the baseline model

This figure reports lead and lag correlations of excess stock-market returns and output growthCorr(rc,t+i − rf,t−1+i,∆ct) comparing a range of calibrations. Data are 1947Q1-2010Q4.

38

Figure (5): Lead and lag macro-finance correlations in the baseline model

This figure reports lead and lag correlations of excess returns, risk-free rate and price-dividend ratioswith output, investment and consumption growth. Claim on consumption stream is taken as modelcounterpart of the aggregate stock-market. Data are 1947Q1-2010Q4.

39

Figure (6): Cyclical properties of stock-market returns in extended models

This figure reports lead and lag correlations of excess stock-market returns with output, investment andconsumption growth in the models with the two types of adjustment costs and Jaimovich and Rebelo(2009) model. Claim on representative firm’s dividend stream is taken as the model counterpart ofthe aggregate stock-market in the first two cases and claim on aggregate consumption stream in theJaimovich and Rebelo (2009) case. Data are 1947Q1-2010Q4.

40

Figure (7): Consumption response to news in Jaimovich and Rebelo (2009).

This figure reports the impulse responses (in percentage deviations from the steady state) of consumptionand (for illustrative purpose) consumption growth (both in logs) to 3-quarter anticipated and surprisechanges in productivity in Jaimovich and Rebelo (2009) model.

41

Figure (8): Impulse responses to volatility shocks

This figure reports the impulse responses (in percentage deviations from the steady state) of macro-economic and financial variables (in logs) to one standard deviation shock to conditional variance ofproductivity growth in a simple model with stochastic volatility (crossed line) and the model wherevolatility has an effect on productivity (solid line). Claim on consumption stream is taken as modelcounterpart of the aggregate stock-market.

42